Assessing Game Balance with AlphaZero: Exploring Alternative Rule Sets in Chess
Nenad Tomašev, Ulrich Paquet, Demis Hassabis, Vladimir Kramnik
AAssessing Game Balance with AlphaZero:Exploring Alternative Rule Sets in Chess
Nenad Tomašev * DeepMind
Ulrich Paquet * DeepMind
Demis Hassabis
DeepMind
Vladimir Kramnik
World Chess Champion2000–2007 § Abstract
It is non-trivial to design engaging and balancedsets of game rules. Modern chess has evolved overcenturies, but without a similar recourse to history,the consequences of rule changes to game dynam-ics are difficult to predict. AlphaZero provides analternative in silico means of game balance assess-ment. It is a system that can learn near-optimalstrategies for any rule set from scratch, withoutany human supervision, by continually learningfrom its own experience. In this study we useAlphaZero to creatively explore and design newchess variants. There is growing interest in chessvariants like Fischer Random Chess, because ofclassical chess’s voluminous opening theory, thehigh percentage of draws in professional play,and the non-negligible number of games that endwhile both players are still in their home prepara-tion. We compare nine other variants that involveatomic changes to the rules of chess. The changesallow for novel strategic and tactical patterns toemerge, while keeping the games close to theoriginal. By learning near-optimal strategies foreach variant with AlphaZero, we determine whatgames between strong human players might looklike if these variants were adopted. Qualitatively,several variants are very dynamic. An analyticcomparison show that pieces are valued differ-ently between variants, and that some variants aremore decisive than classical chess. Our findingsdemonstrate the rich possibilities that lie beyondthe rules of modern chess. * Equal contribution § Classical (2000–2006); FIDE and Undisputed (2006–2007)
1. Introduction
Rule design is a critical part of game development, andsmall alterations to game rules can have a large effect on agame’s overall playability and the resulting game dynam-ics. Fine-tuning and balancing rule sets in games is oftena laborious and time-consuming process. Automating thebalancing process is an open area of research (Jaffe et al.,2012; de Mesentier Silva et al., 2017), and machine learn-ing and evolutionary methods have recently been used tohelp game designers balance games more efficiently (An-drade et al., 2005; Leigh et al., 2008; Halim et al., 2014;Grau-Moya et al., 2018). Here we examine the potential ofAlphaZero (Silver et al., 2018) to be used as an explorationtool for investigating game balance and game dynamics un-der different rule sets in board games, taking chess as anexample use case.Popular games often evolve over time and modern-day chessis no exception. The original game of chess is thought tohave been conceived in India in the 6th century, from whereit initially spread to Persia, then the Muslim world and laterto Europe and globally. In medieval times, European chesswas still largely based on Shatranj, an early variant orig-inating from the Sasanian Empire that was based on theIndian Chatura˙nga (Murray, 1913). Notably, the queen andthe bishop (alfin) moves were much more restricted, andthe pieces were not as powerful as those in modern chess.Castling did not exist, but the king’s leap and the queen’sleap existed instead as special first king and queen moves.Apart from checkmate, it was also possible to win by baringthe opposite king, leaving the piece isolated with the entiretyof its army having been captured. In Shatranj, stalemate wasconsidered a win, whereas these days it is considered a draw.The evolution of chess variants over the centuries can beviewed through the lens of changes in search space complex-ity and the expected final outcome uncertainty throughoutthe game, the latter being emphasized by modern rules andseen as important for the overall entertainment value (Cin-cotti et al., 2007). Modern chess was introduced in the15th century, and is one of the most popular games to date, a r X i v : . [ c s . A I] S e p ssessing Game Balance with AlphaZero captivating the imagination of players around the world.The interest in further development of chess has not sub-sided, especially considering a decreasing number of de-cisive games in professional chess and an increasing re-liance on theory and home preparation with chess engines.This trend, coupled with curiosity and desire to tinker withsuch an inspiring game, has given rise to many variants ofchess that have been proposed over the years (Gollon, 1968;Pritchard, 1994; Wikipedia, 2019). These variants involvealterations to the board, the piece placement, or the rules,to offer players “something subtle, sparkling, or amusingwhich cannot be done in ordinary chess” (Beasly, 1998).Probably the most well-known and popular chess variant isthe so-called Chess960 or Fischer Random Chess, wherepieces on the first rank are placed in one of 960 randompermutations, making theoretical preparation infeasible.Chess and artificial intelligence are inextricably linked. Tur-ing (1953) asked, “Could one make a machine to play chess,and to improve its play, game by game, profiting from itsexperience?” While computer chess has progressed steadilysince the 1950s, the second part of Alan Turing’s questionwas realised in full only recently. AlphaZero (Silver et al.,2018) demonstrated state-of-the-art results in playing Go,chess, and shogi. It achieved its skill without any humansupervision by continuously improving its play by learn-ing from self-play games. In doing so, it showed a uniqueplaying style, later analysed in Game Changer (Sadler &Regan, 2019). This in turn gave rise to new projects likeLeela Chess Zero (Lc0, 2018) and improvements in exist-ing chess engines. CrazyAra (Czech et al., 2019) employsa related approach for playing the Crazyhouse chess vari-ant, although it involved pre-training from existing humangames. A model-based extension of the original AlphaZerosystem was shown to generalise to domains like Atari, whilemaintaining its performance on chess even without an exactenvironment simulator (Schrittwieser et al., 2019). Alp-haZero has also shown promise beyond game environments,as a recent application of the model to global optimisationof quantum dynamics suggests (Dalgaard et al., 2020).AlphaZero lends itself naturally to the problem of findingappealing and well-balanced rule sets, as no prior gameknowledge is needed when training AlphaZero on any par-ticular game. Therefore, we can rapidly explore differentrule sets and characterise the arising style of play throughquantitative and qualitative comparisons. Here we examineseveral hypothetical alterations to the rules of chess throughthe lens of AlphaZero, highlighting variants of the game thatcould be of potential interest for the chess community. Onesuch variant that we have examined with AlphaZero, No-castling chess, has been publicly championed by VladimirKramnik (Kramnik, 2019), and has already had its momentin professional play on 19 December 2019, when Luke Mc- Shane and Gawain Jones played the first-ever grandmasterNo-castling match during the London Chess Classic. Thiswas followed up by the very first No-castling chess tourna-ment in Chennai in January 2020, which resulted in 89%decisive games (Shah, 2020).
2. Methods
In this section we motivate nine alterations to the modernchess rules, describe the key components of AlphaZerothat are used in the analysis in Section 3, and outline howAlphaZero was trained for Classical chess and each of thenine variants.
There are many ways in which the rules of chess could bealtered and in this work we limit ourselves to consideringatomic changes that keep the game as close as possible toclassical chess. In some cases, secondary changes neededto be made to the 50-move rule to avoid potentially infinitegames. The idea was to try to preserve the symmetry andthe aesthetic appeal of the original game, while hoping touncover dynamic variants with new opening, middlegame orendgame patterns and a novel body of opening theory. Withthat in mind, we did not consider any alterations involvingchanges to the board itself, the number of pieces, or theirarrangement. Such changes were outside of the scope ofthis initial exploration. Rule alterations that we examine arelisted in Table 1. The variants in Table 1 are by no meansnew to this paper, and many are guised under other names:Self-capture is sometimes referred to as “Reform Chess” or“Free Capture Chess”, while Pawn-back is called “Wren’sGame” by Pritchard (1994). None have yet come underintense scrutiny, and the impact of counting stalemate as awin is a lingering open question in the chess community.Each of the hypothetical rule alterations listed in Table 1could potentially affect the game either in desired or unde-sired ways. As an example, consider No-castling chess. Onepossible outcome of disallowing castling is that it wouldresult in an aggressive playing style and attacking games,given that the kings are more exposed during the game andit takes time to get them to safety. Yet, the inability to easilysafeguard one’s own king might make attacking itself a poorchoice, due to the counterattacking opportunities that openup for the defending side. In Classical chess, players usuallycastle prior to launching an attack. Therefore, such a changecould alternatively be seen as leading to unenterprising playand a much more restrained approach to the game.Historically, the only way to assess such ideas would havebeen for a large number of human players to play the gameover a long period of time, until enough experience andunderstanding has been accumulated. Not only is this a long ssessing Game Balance with AlphaZero Variant Primary rule change Secondary rule changeNo-castling Castling is disallowedthroughout the game -No-castling (10) Castling is disallowedfor the first 10 moves (20 plies) -Pawn one square Pawns can only move by one square -Stalemate=win Forcing stalemate is a winrather than a draw -Torpedo Pawns can move by 1 or 2 squaresanywhere on the board. En passant canconsequently happen anywhere on the board. -Semi-torpedo Pawns can move by two squareboth from the 2nd and the 3rd rank -Pawn-back Pawns can move backwardsby one square, but only back to the2nd/7th rank for White/Black Pawn moves do not counttowards the 50 move rulePawn-sideways Pawns can also move laterallyby one square. Captures areunchanged, diagonally upwards Sideway pawn moves do notcount towards the 50 move ruleSelf-capture It is possible to captureone’s own pieces -
Table 1.
A list of considered alterations to the rules of chess. process, but it also requires the support of a large numberof players to begin with. With AlphaZero, we can automatethis process and simulate the equivalent of decades of humanplay within a day, allowing us to test these hypotheses insilico and observe the emerging patterns and theory for eachof the considered variations of the game.Figure 1 illustrates each of the variants with an exampleposition.
AlphaZero is an adaptive learning system that improvesthrough many rounds of self-play (Silver et al., 2018). Itconsists of a deep neural network f θ with weights θ thatcompute ( p , v ) = f θ ( s ) (1)for a given position or state s . The network outputs a vec-tor of move probabilities p with elements p ( s (cid:48) | s ) as priorprobabilities for considering each move and hence each nextstate s (cid:48) . If we denote game outcome numerically by +1 ,for a win, 0 for a draw and − for a loss, the network addi- We’ve suppressed notation somewhat; the probabilities aretechnically over actions or moves a in state s , but as each action a deterministically leads to a separate next position s (cid:48) , we use theconcise p ( s (cid:48) | s ) in this paper. tionally outputs a scalar value v ∈ ( − , which estimatesthe expected outcome of the game from position s .The two predictions in (1) are used in Monte Carlo treesearch (MCTS) to refine the assessment of a board position.The prior network p assigns weights to candidate moves at a“first glance” of the board, yielding an order in which movesare searched with MCTS. The output v can be viewed asa neural network evaluation function for position s . Thestatistical estimates of the game outcomes after each moveare refined through MCTS, which runs repeated simulationsof how the game might unfold up to a certain ply depth.In each MCTS simulation, f θ is recursively applied to asequence of positions (or nodes) up to a certain ply depthif they have not been processed in an earlier simulation. Atmaximum ply depth, the position is evaluated with (1), andthat evaluation is “backed up” to the root, for each nodeadjusting its “action selection rule” to alter which moveswill be selected and expanded in the next MCTS simulation.After a number of such MCTS simulations, the root movethat was visited (or expanded) most is played. We trained AlphaZero from scratch for each of the rulealterations in Table 1, with the same set of model hyperpa- ssessing Game Balance with AlphaZero o0Z0Zpo0 ZpopZ0Z0 Z0M0ANO0 PO0LPOBZ S0Z0ZKZR a b c d e f g h (a) An example from No-castling chess: This is a typical po-sition where both kings haven’t found immediate safety andremain exposed into the middlegame. rZ0ZkZ0s apo0lpop pZnobm0Z O0Z0o0Z0 Z0OPZNZ0 S0A0J0ZR a b c d e f g h (b) An example from No-castling(10) chess: The play tends tobe slower and more strategic, to allow for later castling. Here,on the 11th move, Black castles at the very first opportunityand White castles immediately after as well. rZ0lkZ0s obo0Zpa0 m0m0o0Z0 O0ZPANO0 S0Z0J0ZR a b c d e f g h (c) An example from Pawn-one-square chess: Black just movedthe knight to a5. In Classical chess this would seem counter-intuitive due to the potential of playing the pawn to b4, forkingthe knights. Here, however, the pawn cannot move to thatsquare in a single move, justifying the manoeuvre. Z0Z0Z0Z0 Z0Z0ZKZk Z0Z0ZNZ0 Z0Z0Z0Z0 a b c d e f g h (d) An example from Stalemate=win chess: An endgame posi-tion that would have been a draw in Classical chess is now awin instead.
Figure 1.
Examples of new strategic and tactical themes that arise in the explored chess variants. Figure 1e continues on the followingpage. 4 ssessing Game Balance with AlphaZero Z0ZkZ0Zr Z0O0Z0Z0 Z0A0ZpZ0 pZ0Z0JpZ Z0Z0S0Z0 a b c d e f g h (e) An example from Torpedo chess: White needs to generaterapid counterplay, and does so with a torpedo move: b4-b6.Black responds with Rh1, to which White promotes to a queenwith yet another torpedo move, b6-b8=Q. Z0ZnZ0a0 o0Z0oPZn PZ0ZNZ0Z ZQZBA0OP Z0ZRJ0ZR a b c d e f g h (f) An example from Semi-torpedo chess: The ability to rapidlyadvance pawns from the 3rd/6th rank enables Black the fol-lowing energetic option: d6-d4, resulting in a forced tacticalsequence. See Game AZ-19 in Appendix B.6 for details. rZ0lka0s ZbZnZpop pZnZpZ0Z ZpopO0Z0 O0O0ANZ0 S0ZQJBZR a b c d e f g h (g) An example from Pawn-back chess: Here, Black uses thispossibility to challenge White’s central pawns, while openingup the diagonal for the b7 bishop, by a pawn-back move d5-d6. rZbl0skZ ZpZ0Zpap ZNZpA0Z0 Z0Z0Z0ZP PO0Z0JPZ S0ZQSBZ0 a b c d e f g h (h) An example from Pawn-sideways chess: After sacrificingthe knight on f2 the previous move, Black utilises a sidewayspawn move f7-e7 for tactical purposes, opening the f-file to-wards the White king, while attacking the knight on d6. rZbZ0akZ opZ0ZpZ0 l0Zpo0A0 O0Z0O0Z0 Z0JRZBZR a b c d e f g h (i) An example from Self-capture chess: a self-capture moveRxh4 generates threats against the Black king.
Figure 1. (Continued from previous page.)
Examples of new strategic and tactical themes that arise in the explored chess variants.5 ssessing Game Balance with AlphaZero rameters. The models were trained for 1 million trainingsteps, with a batch size of 4096 and allowing for an average0.12 samples per position from self-play games. In orderto encourage exploration during training, a small amountof noise was injected in the prior move probabilities (1) be-fore search, sampled from a Dirichlet
Dir(0 . distribution,followed by a renormalization step (Silver et al., 2018). Fur-ther diversity was promoted by stochastic move selection inthe first 30 plies of each of the training self-play games, byselecting the final moves proportionally to the softmax ofthe MCTS visit counts. The remaining game moves fromply 31 onwards were selected as top moves based on MCTS.Training self-play games were generated using 800 MCTSsimulations per move.The absence of baselines makes it hard to formally assessthe strength of each model, which is why it was importantto couple the quantitative analysis and metrics observedat training and test time with a qualitative assessment incollaboration with Vladimir Kramnik, a renowned chessgrandmaster and former world chess champion. As therule changes that are considered in this study are mostlyminor in practical terms, it is reasonable to assume that thetrained models are of similar strength, although it is equallyreasonable to expect that some of them could be further fine-tuned to account for the differences in game length and theaverage number of legal moves that need to be consideredat each position. Given the nature of the study, the highlevel of observed play in trained models, and the number ofrule alterations considered, we decided not to pursue sucha potentially laborious process, as it would not alter any ofthe high-level conclusions that we present and discuss.
3. Quantitative assessment
There are marked differences between the styles of chessthat arises from each of the rule alterations Aesthetically,each variant has its own appeal, and we highlight them fur-ther in Section 4. Here we provide a quantitative comparisonbetween variants, to complement the qualitative observa-tions. Using a large quantity of self-play games, we inferthe expected draw rate and first-move advantage for eachvariant, expressed as the expected score for White (Section3.2). We then illustrate how the same opening can lead tovastly different outcomes under different chess variants inSection 3.3, and that these opening-specific differences candiffer from the aggregate differences across all openings.An analysis of the utilisation of the newly introduced op-tions made possible by the new rule alterations in Section3.4 shows that the non-classical moves are used in a largepercentage of games, often multiple times per game, in eachof the variants. This suggests that the new options are in-deed useful, and contribute to the game. We estimate thediversity of opening play by looking at the opening trees which we construct from AlphaZero’s network priors (1) forthe first couple of moves and show that the breadth of open-ing possibilities in each of these chess variants seems to beinversely related to their relative decisiveness (Section 3.5).Sections 3.6 and 3.7 highlight the difference in opening playaccording to the prior distributions of the variants. Rule ad-justments, especially those affecting piece mobility, are alsoexpected to affect the relative material value of the pieces.Finally, Section 3.8 provides approximations for piece val-ues in each of the variants, computed from a sample of10,000 fast-play AlphaZero games.
For each chess variant, we generated a diverse set of N = 10 , AlphaZero self-play games at 1 second permove, and N = 1 , games at 1 minute per move. Theoutcomes of the fast self-play games are presented in Figure2a; the longer games follow in Figure 2b. As AlphaZero isapproximately deterministic given the same MCTS depthand number of rollouts, we promote diversity in games bysampling the first 20 plies in each game proportional to thesoftmax of the MCTS visit counts, followed by playing thetop moves for the rest of the game.In addition to that, we generated a set of N = 1 , fast-play games from fixed starting positions arising from theDutch Defence, Chigorin Defence, Alekhine Defence andKing’s Gambit for each of the variants, as further discussedin Section 3.3.The two sets of diverse self-play games are used in Section3.2 to compare the decisiveness of each variant, in Section3.4 to analyse how many special moves are used, and inSection 3.8 to estimate piece values across variants.A selection of these games is presented in Appendix B. It is widely hypothesised that classical chess is theoreticallydrawn; that the odds π = ( π win , π draw , π lose ) of white win-ning, drawing and losing are (0 , , at optimal play. Wedetermine how favourable for white or how “drawish” differ-ent variants are by estimating the expected scores and drawrates at non-optimal play under the same conditions. Wekeep the conditions that chess variants are played againstthemselves with AlphaZero fixed, like the move selectioncriteria or Monte Carlo Tree Search (MCTS) evaluationtime.The overall decisiveness in the generated game sets dependson the time controls involved. We see in Figures 2a and2b that across all variations the percentage of drawn gamesincreases with longer thinking times, and longer thinkingtimes also affect the expected score for White, as shown inTable 2. This suggests that the starting position might be ssessing Game Balance with AlphaZero ClassicalNo-castlingNo-castling (10)Pawn one squareStalemate=winTorpedoSemi-torpedoPawn-backPawn-sidewaysSelf-capture 7721110604709100020861306532872871 8820844190028891860671918103916088158783 409449394400394723591308313346White wins Draw Black wins (a) The game outcomes of 10,000 AlphaZero games played at1 second per move for each different chess variant.
ClassicalNo-castlingNo-castling (10)Pawn one squareStalemate=winTorpedoSemi-torpedoPawn-backPawn-sidewaysSelf-capture 182911925932761517 979968985988971894964990980981 33434136452White wins Draw Black wins (b) The game outcomes of 1,000 AlphaZero games played at 1minute per move for each different chess variant.
Figure 2.
AlphaZero self-play game outcomes under different time controls. As moves are determined in a deterministic fashion given thesame conditions, diversity was enforced by sampling the first 20 plies in each game proportional to their MCTS visit counts. Across allvariations the percentage of drawn games increases with longer thinking times. This seems to suggest that the starting position might betheoretically drawn in these chess variants, like in Classical chess, and that some of the variants are simply harder to play, involving morecalculation and richer patterns.
Variant Training 1sec 1minClassical .
1% 51 .
8% 50 . No castling .
7% 53 .
3% 51 . No castling (10) .
5% 51 .
0% 50 . Pawn one square .
5% 51 .
6% 50 . Stalemate=win .
9% 53 .
0% 51 . Torpedo .
0% 56 .
8% 54 . Semi-torpedo .
7% 53 .
6% 50 . Pawn-back .
0% 51 .
1% 50 . Pawn-sideways .
8% 52 .
8% 50 . Self-capture .
2% 52 .
6% 50 . Table 2.
Empirical score for White under different game conditions,for each chess variant: self-play games at the end of model training,1 second per move games, and 1 minute per move games. Diversityin 1 second per move games and 1 minute per move games wasenforced by sampling the first 20 plies in each game proportionalto their MCTS visit counts. theoretically drawn in these chess variants, like in Classicalchess, and that some of the variants are simply harder toplay, involving more calculation and richer patterns. Wehypothesise that the relative differences in AlphaZero’s winrates might translate to differences in human play, althoughthis hypothesis would need to be practically validated in thefuture. Yet, in absence of any existing human games, wecan use these results as a preliminary guess of what thoseresults might be, assuming that what is difficult to calculatefor AlphaZero may be difficult for human players as well. 3.2.1. I
NFERENCE FOR GAME ODDS
To compare variants, we first infer the odds of their out-comes under set playing conditions. For a given variant,let the game outcomes G be n win wins and n lose losses forwhite, and n draw = N − n win − n lose draws. If we assumea uniform Dirichlet prior on π and multinomial likelihoodfor winning, drawing or losing, the posterior distribution isDirichlet, p ( π |G ) = Dir( n win + 1 , n draw + 1 , n lose + 1) . (2)3.2.2. D RAW RATES
To compare the decisiveness of chess variants, we inferthe probability that variant A has a lower draw rate thanvariant B, given the games played G A and G B under thesame conditions: p ( π Adraw < π
Bdraw ) = (cid:90) (cid:90) I (cid:2) π Adraw < π
Bdraw (cid:3) p ( π A |G A ) p ( π B |G B ) d π A d π B . (3)The integral is not available in closed form; we evaluate itwith a Monte Carlo estimate by drawing pairs of samplesfrom p ( π A |G A ) and p ( π B |G B ) – using (2) – and computingthe fraction of times that samples satisfy π Adraw < π
Bdraw .Figure 3a provides an indication of the relative decisiveness of variants, when played by AlphaZero at approximately 1second per move, and Figure 3b provides the comparison at This approach follows MacKay (2003, Chapter 37.1).7 ssessing Game Balance with AlphaZero T o r p e d o S e m i - t o r p e d o N o - c a s t li n g S t a l e m a t e = w i n S e l f - c a p t u r e P a w n - s i d e w a y s C l a ss i c a l P a w n o n e s qu a r e N o - c a s t li n g ( ) P a w n - b a c k TorpedoSemi-torpedoNo-castlingStalemate=winSelf-capturePawn-sidewaysClassicalPawn one squareNo-castling (10)Pawn-back (a) A draw rate comparison p ( π rowdraw < π columndraw ) at approxi-mately 1 seconds per move, on 10,000 AlphaZero games pervariation. T o r p e d o S e m i - t o r p e d o N o - c a s t li n g S t a l e m a t e = w i n C l a ss i c a l P a w n - s i d e w a y s S e l f - c a p t u r e N o - c a s t li n g ( ) P a w n o n e s qu a r e P a w n - b a c k TorpedoSemi-torpedoNo-castlingStalemate=winClassicalPawn-sidewaysSelf-captureNo-castling (10)Pawn one squarePawn-back (b) A draw rate comparison p ( π rowdraw < π columndraw ) at approx-imately 1 minute per move, on 1,000 AlphaZero games pervariation. T o r p e d o S e m i - t o r p e d o N o - c a s t li n g S t a l e m a t e = w i n S e l f - c a p t u r e P a w n - s i d e w a y s C l a ss i c a l P a w n o n e s qu a r e N o - c a s t li n g ( ) P a w n - b a c k TorpedoSemi-torpedoNo-castlingStalemate=winSelf-capturePawn-sidewaysClassicalPawn one squareNo-castling (10)Pawn-back (c) A comparison of expected scores p ( e row > e column ) at 1second per move, on 10,000 games per variation. T o r p e d o N o - c a s t li n g S e m i - t o r p e d o S t a l e m a t e = w i n C l a ss i c a l S e l f - c a p t u r e P a w n - s i d e w a y s N o - c a s t li n g ( ) P a w n o n e s qu a r e P a w n - b a c k TorpedoNo-castlingSemi-torpedoStalemate=winClassicalSelf-capturePawn-sidewaysNo-castling (10)Pawn one squarePawn-back (d) A comparison of expected scores p ( e row > e column ) at 1minute per move, on 1,000 games per variation. Figure 3.
A comparison of draw rates. The most decisive chess variants under both time controls are Torpedo, Semi-torpedo, No-castlingand Stalemate=win. These four variants also give White the largest first-move advantage.
XPECTED SCORES
The decisiveness of a chess variant under imperfect playdoes not necessarily have to correspond to the first-moveadvantage. In classical chess, White scores higher on aver-age. Top-level chess players tend to press for an advantagewith the White pieces and defend with the Black pieces,looking for opportunities to counter-attack. The reason isthe first-move advantage; it is an initiative that, with goodplay, persists throughout the opening phase of the game.This not a universal property that would hold in any game , ssessing Game Balance with AlphaZero as playing the first move might also disadvantage a playerin some types of games. It is therefore important to estimatethe effect of the rule changes on the first-move advantagein each chess variant, expressed as the expected score forWhite.The expected score for White is defined as: e = π win + π draw (4)for a particular set of conditions like time controls, themove selection criteria and the AlphaZero model playingthe game. Given the game outcomes G A and G B of variantsA and B, the probability of white having a higher first-moveadvantage in variant A is p ( e A > e B ) = (cid:90) (cid:90) I (cid:2) π Awin + π Adraw > π
Bwin + π Bdraw (cid:3) p ( π A |G A ) p ( π B |G B ) d π A d π B , (5)which we again evaluate with a Monte Carlo estimate.White’s first-move advantage with approximately 1 secondand 1 minute per move in AlphaZero games is comparedin Figures 3c and 3d respectively. The relative orderingof variations follows the ranking in general decisiveness,suggesting that the new chess variants that are more decisivein AlphaZero games are also more advantageous for White,possibly due to an increase in dynamic attacking options. To further illustrate how different alterations of the rule setwould require players to adjust their opening repertoires, weprovide a comparison of how favourable specific openingpositions are for the first player, for each of the variants pre-viously introduced in Table 1. Figure 4 shows the win, draw,and loss percentages for White under 1 second per move, forthe Dutch Defence, Chigorin Defence, Alekhine Defenceand King’s Gambit, on a sample of 1000 self-play games.The only variant we did not include in these comparisonsis Pawn one square, as the lines used in the comparisonsinvolve the double-pawn-moves which are not legal in thatvariant.These four opening systems are not considered to be themost principled ways of playing Classical chess. They aretherefore particularly interesting for establishing if a certainrule change pushes the evaluation of each of these openingsfrom “slightly inferior” to “unsound” or “unplayable”.In case of Dutch Defence in Figure 4a, we see that it ismore favourable for White in Torpedo and Stalemate=winchess than in Classical chess. This is in line with the over-all increase in decisiveness in those variations, but is notmore favourable in case of No-castling chess, despite No-castling chess otherwise being more decisive than Classical chess. We can already see in this one example that theoverall differences in decisiveness between variants are notequally distributed across all possible opening lines, andthat the evaluation of the difference in the expected scorewill depend on the style of opening play.In case of Chigorin Defence in Figure 4b, Pawn-sidewayschess seems to be refuting the variation, based on our initialfindings. In a smaller sample of games played at 1 minuteper move, we have seen a 100% score being achieved byAlphaZero in this line of Pawn-sideways chess, though theseare still preliminary conclusions. To the human eye theline does not appear to be very forcing; it is not a shorttactical refutation, but results in a fairly long-term strategicadvantage, which AlphaZero converts into a win. This linealso seems to be harder to defend in No-castling chess andTorpedo, but not in Stalemate=win chess, unlike the DutchDefence.The Alekhine Defence in Figure 4c seems to be less sound inall of the variations considered, compared to Classical chess,with a major increase in decisiveness in Pawn-sidewayschess, No-castling chess and Torpedo chess.Finally, King’s Gambit in Figure 4d seems to give a substan-tial advantage to Black across all chess variants considered,although in No-castling chess and Torpedo chess, White hassomewhat better winning chances than in Classical chess.Pawn-sideways chess, again, seems to be the worst of thevariants to consider playing this line in. Still, in our prelimi-nary experiments with games at longer thinking times, mostgames would still ultimately end in a draw. This suggeststhat it is still likely a playable opening, when played at avery high level with deep calculation.
Several of the variants that are explored in this study involveadditional move options that are not permitted under therules of Classical chess, like additional pawn moves andself-captures. It is not clear from the outset how often thesenewly introduced moves would be utilised in each of thevariants. Will they make a difference? We use the set of10,000 games at 1 second per move from Section 3.1 toquantify how often the additional moves are played.3.4.1. T
ORPEDO MOVES
In Semi-torpedo chess, of all games have at least onetorpedo move, and . of all moves played in the gameare torpedo moves. In Torpedo chess, these percentagesare even higher: of games utilise torpedo moves andthese represent . of all moves played in the game.Furthermore, . of games featured pawn promotionswith a torpedo move, highlighting the speed at which apassed pawn can be promoted to a queen. ssessing Game Balance with AlphaZero ClassicalPawn backSemi-torpedoTorpedoStalemate = winNo-castlingNo-castling (10)Self-capturePawn sideways 218123220339282221229224139 743855719595670749752750835 392261664830192626Dutch DefenceWhite wins Draw Black wins (a) Dutch Defence (1. d4 f5)
ClassicalPawn backSemi-torpedoTorpedoStalemate = winNo-castlingNo-castling (10)Self-capturePawn sideways 197231245 393193 359254220 720 782756716 569780 619728769 273 21133938272218117Chigorin DefenceWhite wins Draw Black wins (b) Chigorin Defence (1. d4 d5 2. c4 Nc6)
ClassicalNo-castlingNo-castling (10)Pawn backSemi-torpedoTorpedoStalemate=winSelf-capturePawn-sideways 204 444388261296 432228262 434 778 534602721673 536735717 552 182210183132372114Alekhine’s DefenceWhite wins Draw Black wins (c) Alekhine Defence (1. e4 Nf6)
ClassicalPawn backSemi-torpedoTorpedoStalemate = winNo-castlingNo-castling (10)Self-capturePawn sideways 6326669954118296445 703776679630614 669674717574 234198255271332213297219381King’s GambitWhite wins Draw Black wins (d) King’s Gambit (1. e4 e5 2. f4)
Figure 4.
The same opening position can give vastly different degrees of advantage to either play, depending on the variant underconsideration, as shown here by the number of games won, drawn and lost for AlphaZero as White when playing at approximately 1second per move, for a sample of 1000 games, while always playing the best move without any additional noise being added for playdiversity. The stochasticity captured in the results stems from the asynchronous execution of MCTS threads during search. Therefore,these results indicate how favorable the ’main line’ continuation is, for each of the following openings: the Dutch Defence, the ChigorinDefence, Alekhine Defence and the King’s Gambit.
ACKWARDS AND LATERAL PAWN MOVES
In Pawn-back chess, . of the games involved a back-wards pawn move. In Pawn-sideways chess, . ofgames features lateral pawn moves, and a total of . ofall moves in the game were lateral pawn moves, as the recon-figuring of pawn formations was common in AlphaZero’splaying style in this chess variant.3.4.3. S ELF - CAPTURES
In Self-capture chess, . of games featured self-capturemoves, which represented . of all moves played. Themost common self-captures involved sacrificing a pawn( . ), although sacrificing a bishop ( . ) or a knight( . ) was not uncommon. Rook self-capture sacrificeswere rare ( . ) and occasionally AlphaZero would self- capture a queen ( ), though these were mostly unnecessarycaptures in winning positions, given that AlphaZero was notincentivised to win in the fastest possible way.3.4.4. W INNING THROUGH STALEMATE
In Stalemate=win chess the percentage of all decisive gamesthat were won by stalemate rather than mate in AlphaZerogames was . , though this number is inflated due tothe fact that AlphaZero would often stylistically stalematerather than mate the opponent in positions where both arepossible.The percentages listed above suggest that the rule changesfeatured in these chess variants did indeed leave a traceon how the game is being played, and that they are usefuladditional options that can potentially change the game dy- ssessing Game Balance with AlphaZero namics. Yet, it is important to note that the resulting gamesare still of approximately similar length, as shown in Figure8 in Appendix A, with some changes in the empirical dura-tion of decisive games. This means that playing a game inone of these chess variants is unlikely to prolong or shortenthe game by a large amount, meaning that classical time con-trols should still be appropriate. Note that the numbers inFigure 8 that correspond to the number of plies in AlphaZerogames are an upper bound on game length, since AlphaZerowas trained without discounting, and would therefore notplay the fastest winning sequence in its decisive games. For a game to be appealing, it has to be rich enough inoptions that these options do not get quickly exhausted, asplay would then become repetitive. We use the averageinformation content (entropy) of the first T = 20 pliesof play from each variant’s prior as a surrogate diversitymeasure. The trained AlphaZero policy priors model themove probabilities of the positions in self-play training data,and reflects the statistics at which opening lines appear there.An entropy of zero corresponds to there being one and onlyone forcing sequence of moves to be playable for Whiteand Black, all other moves leading to substantially worsepositions for each side. A higher entropy implies a widerand more balanced opening tree of variations, leading to amore diverse set of middlegame positions. The intuition thatthere would be many more plausible opening lines in slowervariants like Pawn one square, holds true experimentally.In simulation, more decisive variants like Torpedo chesstypically have fewer plausibly playable opening lines.The decomposition of the entropy as a statistical expectationcan help identify whether there exist defensive lines thatequalise the game in an almost forcing way. In Classicalchess, one such defensive resource is the Berlin Defencein the Ruy Lopez, taking the sting out of 1. e4. We showin Section 3.5.2 that AlphaZero, when trained on Classicalchess, expresses a strong preference for the Berlin Defence,similarly to the human consensus on the solidity of theBerlin endgame. Without the option to castle, this particularline disappears in No-castling chess.3.5.1. A VERAGE INFORMATION CONTENT
The prior network from (1) defines the probability of apriori considering move a t in state s t , but as move a t leadsto state s t +1 deterministically, we shall abbreviate the priorwith p ( s t +1 | s t ) .The prior is a weighted list of possible moves for state s t thatare utilised in AlphaZero’s MCTS search. The weights spec-ify how plausible each move is before MCTS calculation;they specify candidates for consideration. In information- Variant Entropy Equivalent 20-ply gamesNo-castling 27.65 . × Torpedo 27.89 . × Self-capture 27.94 . × No-castling (10) 27.97 . × Classical 28.58 . × Stalemate=win 29.01 . × Semi-torpedo 31.63 . × Pawn-back 32.30 . × Pawn-sideways 34.16 . × Pawn one square 38.95 . × Uniform random 64.96 . × Table 3.
The average information content in nats in the first 20plies of the AlphaZero prior for each chess variant. The uniformrandom baseline assumes an equal probability for each move inClassical chess, and provides rough indication of the ratio between“plausible” and “possible” games according to the AlphaZero prior.The uniform random baseline depends on the number of legalmoves per position, and is marginally different but of the samemagnitude for other variations. theoretic terms, the entropy H ( s t ) = − (cid:88) s t +1 p ( s t +1 | s t ) log p ( s t +1 | s t ) (6)is a function of state s t and represents the number of nats (orbits, if log is used) that are needed to encode the weightedmoves in position s t .If there are M ( s t ) legal moves in state s t , then the num-ber of candidate moves m ( s t ) – the number that a topplayer would realistically consider – is much smaller than M ( s t ) . In de Groot (1946)’s original framing, M ( s t ) is aplayer’s legal freedom of choice, while m ( s t ) is their ob-jective freedom of choice. Iida et al. (2003) hypothesisethat m ( s t ) ≈ (cid:112) M ( s t ) on average. Because p ( s t +1 | s t ) isa distribution on all legal moves, we define the number ofcandidate moves m ( s t ) by m ( s t ) = exp( H ( s t )) ; (7)it is the number of uniformly weighted moves that could beencoded in the same number of nats as p ( s t +1 | s t ) . We provide insight into the diversity of the prior opening treethrough two quantities, the move sequence entropy H ( t ) atdepth t from the opening position, and the average numberof candidate moves at ply t , M ( t ) . As an illustrative example, if the number of candidate movesis m ( s t ) = 3 for some p ( s t +1 | s t ) that might put non-zero masson all of its moves, then m ( s t ) is also equal to the number ofcandidate moves of a probability vector p = [ , , , , . . . , that puts equal non-zero mass on only three moves.11 ssessing Game Balance with AlphaZero Move sequence entropy
Let s = s t = [ s , s , . . . s t ] be the sequence of states after t plies, starting at s , theinitial position. The prior probability – without search – ofmove sequence s t is p ( s t | s ) = (cid:81) tτ =1 p ( s τ | s τ − ) . Theentropy of the move sequence is H ( t ) = − (cid:88) s t p ( s t ) log p ( s t )= E s t ∼ p ( s t ) (cid:104) − log p ( s t ) (cid:105) , (8)where the starting position s is dropped from notation forbrevity. An entropy H ( t ) = 0 implies that, according tothe prior, one and only one reasonable opening line couldbe considered by White and Black up to depth t , with alldeviations form that line leading to substantially worse posi-tions for the deviating side. A higher H ( t ) implies that wewould a priori expect a wider opening tree of variations, andconsequently a more diverse set of middlegame positions. Average number of candidate moves
The entropy of achess variant’s prior opening tree is an unwieldy number thatdoesn’t immediately inform us how many move options wehave in each chess variant. A more naturally interpretablenumber is the expected number of (good) candidate movesat each ply as the game unfolds. The average number ofcandidate moves at ply t is M ( t ) = (cid:88) s t p ( s t ) m ( s t ) = E s t ∼ p ( s t ) (cid:104) m ( s t ) (cid:105) . (9)Both the sums in (8) and (9) are over an exponential numberof move sequences. We compute Monte Carlo estimates of H ( t ) and M ( t ) by sampling sequences from p ( s ) andaveraging the negative log probabilities of those sequencesto obtain H ( t ) , or averaging m ( s t ) over all samples at depth t to obtain M ( t ) . We defer a presentation of the breakdownof the average number of candidate moves per variant toFigure 11 in Appendix A, and will encounter M ( t ) next inFigure 6 when Classical and No-castling chess are comparedside by side.The entropy of the AlphaZero prior opening tree is givenin Table 3 for each variation. Similar to the calculation in(7) we give an estimate of the equivalent number of 20-plysequences as exp( H ( t )) . As a baseline comparison, wetake a prior distribution for Classical chess where all legalmoves are equally playable, and estimate the entropy ofthe “Uniform random” move selection criteria. It affordsus a crude estimate of the number of possible classicalopenings, as opposed to the number of plausibly playable orcandidate openings. The estimates in Table 3 for Classicalchess and "Uniform random Classical chess” corroboratethe claim that the number of playable opening lines – aplayer’s objective freedom of choice – is roughly the squareroot of the number of legal opening lines (Iida et al., 2003). − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . . D e n s i t y ( h i s t og r a m ) No-castlingTorpedoSelf-captureNo-castling (10)ClassicalStalemate=winSemi-torpedoPawn-backPawn-sidewaysPawn one square
Figure 5.
Histograms of − log p ( s ) when s ∼ p ( s ) for each vari-ant. Following (8), the means of these distributions give the en-tropies in Table 3. The individual histograms are separately pre-sented in Figure 9 in Appendix A. The two variants that have the largest entropy and hencelargest opening tree in Table 3, Pawn-sideways and Pawnone square, also happen to be among the most drawish,according to Figures 3a and 3b. The two variants that havethe smallest opening trees under our analysis, No-castlingand Torpedo, are also the most decisive and give Whitesome of the largest advantages, according to Figures 3a to3d. Importantly, we estimate the size of the opening trees ofthese more decisive versions to still be of the same order ofmagnitude as that of Classical chess.Figure 5 (a separate figure for each variant appears in Fig-ure 9 in Appendix A) visualises the density of − log p ( s ) when state sequences s are drawn from p ( s ) . The meanof each density is the entropy of (8), and an overlap in thehistograms of two variants implies that their opening treescontain a similar number of lines that are considered ascandidates with similar odds. In Figure 5, a histogram thatis shifted to the left means that fewer move sequences areconsidered a priori , and each has higher probability. A his-togram that is shifted to the right implies that a larger varietyof move sequences are a priori considered, and each has tobe considered with a smaller probability. “Uniform random”is shown in Figure 9j, and would appear as a tall narrowspike centred around 64 in this figure. In the followingsection, we shall use log probability histograms as a tool tohighlight the differences between Classical and No-castlingchess.3.5.2. C LASSICAL VS . N O - CASTLING CHESS
In Classical chess AlphaZero has a strong preference forplaying the Berlin Defence 1. . . e5 2. Nf3 Nc6 3. Bb5 Nf6in response to 1. e4, and here 4. O-O is White’s main reply, ssessing Game Balance with AlphaZero Variant Entropy Equiv. 21-ply gamesClassical (e4) 23.72 . × Classical (Nf3) 29.54 . × No-castling (e4) 27.42 . × No-castling (Nf3) 28.40 . × Table 4.
The average information content in nats of the AlphaZeroprior for Classical and No-castling chess, estimated on the 20 pliesfollowing 1. e4 and 1. Nf3. which is not an option in no-castling chess. Yet, castling isalso an integral part of most other lines in the Ruy Lopez, af-fecting each move when considering relative preferences. Inthe absence of castling, AlphaZero does not have as stronga preference for a particular line for Black after 1. e4, sug-gesting either that it is not as easy to fully neutralise White’sinitiative, or alternatively that there is a larger number ofpromising defensive options.To indicate the difference between Classical and No-castlingchess, we compare the prior’s opening trees after 1. e4and 1. Nf3 in Figure 6. If we examine the density of − log p ( s | s ) under p ( s | s ) , where s is the boardposition after either 1. e4 or 1. Nf3, we see a marked shift inthe characteristics of the AlphaZero prior opening trees (seeFigures 6a and 6b). Statistically, the AlphaZero prior after1. e4 is much more forcing than after 1. Nf3 in Classicalchess. This is also evident from the average informationcontent of the 20 plies after 1. e4 and 1. Nf3 in Table 4. InNo-castling chess, 1. e4 seems as flexible as 1. Nf3, with amuch wider variety of emerging preferential lines of play inthe AlphaZero model.Figure 6 additionally shows the average number of candi-date moves at each ply. In Classical chess, White has moreoptions than Black in both lines, the difference slowly di-minishing over time as the first-move advantage decreases.1. Nf3 offers more options, as it is less forcing. In No-castling chess, there seems to be a higher number of effec-tive available moves for both sides after 1. e4 in the firstcouple of plies, based on the AlphaZero model.The Berlin Defence is a contributing factor to the narroweropening tree footprint we see in Figure 6a. As defensivetool for Black, Vladimir Kramnik successfully used theBerlin Defence in his World Championship Match withGarry Kasparov in 2000. He describes his choice as follows: “ Back in the 90s, the engines of the time seemedto think that White had the advantage in theBerlin endgame, giving evaluations around +1in White’s favour. I thought that things weren’tas simple, given that Black’s only real problemwas the loss of castling rights, and the difficultyof connecting rooks. The first time that I had a deeper look at it was when I was preparing for thematch with Kasparov, and I thought that the open-ing was a good choice against Kasparov’s playingstyle. Pursuing it required a belief in instinct andthe human assessment of the position. Nowadays,it is considered to be a very solid opening, andmodern engines assess most arising positions asbeing equal. ” We compare how similar opening trees are by consideringhow likely a given sequence of moves is under two variants.To compare, we define one variant p as the reference variant,and generate a move sequence s according to its prior. TheKullback-Leibler divergence is a measure of how likely suchsequences of moves are under the opening book of variant q compared to that of p . Given two distributions p ( s ) and q ( s ) , the Kullback-Leibler divergence from q to p is therelative entropy of variant p with respect to q , D KL [ p (cid:107) q ] = (cid:88) s p ( s ) log p ( s ) q ( s )= E s ∼ p ( s ) (cid:104) log p ( s ) − log q ( s ) (cid:105) . (10)It is the expected number of extra nats (or bits if log isused) that is required to compress move sequences fromvariant p using variant q ’s opening book distribution. Thecalculation in (10) involves a sum that is exponential in thelength of s , and we estimate it with a Monte Carlo averageof log p ( s ) /q ( s ) over sampled sequences from p ( s ) .A legal move in variant p may be illegal in variant q , inwhich case there is no way in which sequences in p can beencoded in q . The Kullback-Leibler divergence in (10) isthen infinite. More formally, this happens when q ( s t +1 | s t ) puts zero mass on state transitions which are possible in p .We therefore need to ensure that the reference variant p ischosen so that its legal moves are a subset of those of q . InTable 5 we show all divergences with respect to Classicalchess, and distinguish between two kinds of variants:1. variants that add moves to Classical chess, and whoselegal moves are supersets of Classical chess;2. variants that remove legal moves from Classical chess,and whose moves are subsets of Classical chess.The legal moves of Stalemate=win correspond to that ofClassical chess, and it is included as both a superset and asubset in Table 5. The density of samples from (10) is givenin Figure 10 in Appendix A. The divergence is largest forvariants that introduce the largest number of additional pawnmoves or the most restrictions. Self-capture chess, despite ssessing Game Balance with AlphaZero − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . D e n s i t y ( h i s t og r a m ) Classical (e4)Classical (Nf3) (a) The density of (negative) log likelihoods for opening linesin Classical chess after 1. e4 and 1. Nf3 when move sequencesare sampled from the AlphaZero prior. There is a markeddifference in overlap between the histograms, suggesting thatAlphaZero a priori considers “narrower” opening lines after1. e4 than after 1. Nf3. We identify the samples s at the highlikelihood spike with a particular line in the Berlin Defence. − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . D e n s i t y ( h i s t og r a m ) No-castling (e4)No-castling (Nf3) (b) The density of (negative) log likelihoods for opening lines inNo-castling chess after 1. e4 and 1. Nf3 when move sequencesare sampled from the AlphaZero prior. Without the option ofcastling a king to safety, the prior opening trees after 1. e4 and1. Nf3 have more similar “distributional footprints” comparedto Classical chess in Figure 6a.
Ply t . . . . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Classical (e4) (w)Classical (e4) (b)Classical (Nf3) (w)Classical (Nf3) (b) (c) The average number of candidate moves M ( t ) , as computedwith (9), for Classical chess. Ply t . . . . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s No-castling (e4) (w)No-castling (e4) (b)No-castling (Nf3) (w)No-castling (Nf3) (b) (d) The average number of candidate moves M ( t ) , as computedwith (9), for No-castling chess. Figure 6.
The diversity of responses to 1. e4 and 1. Nf3 in Classical and No-castling chess, as well as the average number of candidatemoves available for White and Black at each ply. The spike is in the classical chess 1. e4 response distribution is at 1. . . e5 2. Nf3 Nc63. Bb5 Nf6 4. O-O Nxe4 5. Re1 Nd6 6. Nxe5 Nxe5 7. Bf1 Be7 8. Rxe5 O-O 9. d4 Bf6 10. Re1 Re8 11. c3, a known equalising line in theBerlin Defence, leading to drawish positions. the plethora of additional opportunities for self-capture, isstatistically closer to Classical chess because of the lowfrequency at which the extra moves are played.
Although the relative entropy expresses how many morenats are required to encode prior moves of one variant givenanother, it does not tell us whether one variant’s player isconsidering the right candidate moves when playing another variant. How many more candidate moves should a playerQ, who was trained on one variant of chess, take into consid-eration when wanting to play at player P’s level in anothervariation? Let q ( s ) be the candidate prior for the variationthat player Q was trained on, and p ( s ) the prior for variantP, variant that Q wants to play. We define the combination ssessing Game Balance with AlphaZero Ply t . . . . . . . . A v e r a g e a dd i t i o n a l c a nd i d a t e m o v e s Pawn-sidewaysSelf-capturePawn one squarePawn-backNo-castlingSemi-torpedoTorpedoNo-castling (10)Stalemate=win
Figure 7.
The average number of additional candidate moves A q ( t ) that a Classical player Q with prior q ( s t +1 | s t ) should consider inorder to match player P’s candidate moves from prior p ( s ) for each of the evaluated variants; see (15). (The order of the variants in thelegend matches their ordering at ply t = 20 .) Variant p Variant q D KL [ p (cid:107) q ] S up e r s e t s Classical Stalemate=win 2.59Classical Self-capture 5.24Classical Semi-torpedo 10.35Classical Pawn-back 11.70Classical Torpedo 11.89Classical Pawn-sideways 24.23 S ub s e t s Stalemate=win Classical 2.50No-castling (10) Classical 7.17No-castling Classical 13.19Pawn one square Classical 20.28
Table 5.
Differences in the opening tree of the new chess variantsand Classical chess. These are expressed as Kullback-Leibler(KL) divergences, the direction depending on whether a particularvariant is a superset or a subset of Classical chess, based on the rulechange. In all cases but Stalemate=win the reverse KL divergencesare infinite as when there are legal opening lines s in variant p thatdon’t exist in q , and hence for which q ( s ) = 0 when p ( s ) is not(contributing − log 0 to the divergence). of the two priors as the normalized supremum r ( s t +1 | s t ) = max (cid:8) p ( s t +1 | s t ) , q ( s t +1 | s t ) (cid:9)(cid:80) s (cid:48) t +1 max (cid:8) p ( s (cid:48) t +1 | s t ) , q ( s (cid:48) t +1 | s t ) (cid:9) . (11)There is a particular reason behind our choice of definitionfor the combined prior in (11): The number of candidatemoves that the combination of players P and Q would con-sider, is always smaller than the sum of candidate movesthat P and Q would consider individually.Put more formally, define the number of candidate moves forthe combined player as the number of uniformly weighedmoves that could be encoded in the same number of nats as r ( s t +1 | s t ) , m r ( s t ) = exp − (cid:88) s t +1 r ( s t +1 | s t ) log r ( s t +1 | s t ) . (12)For any choice of priors p and q the number of candidatemoves that are considered by the combined player in state s t is lower bounded by m r ( s t ) ≤ m p ( s t ) + m q ( s t ) , (13)which we prove in Appendix A.1.We now define the difference additional( s t ) = m r ( s t ) − m q ( s t ) (14)to represent the number of additional candidate moves thatplayer Q should consider, to play at the level of P in position The perceptive reader would recognise equation (12) as equa-tion (7). We restate it here with a subscript to indicate the explicitdependence on the distribution.15 ssessing Game Balance with AlphaZero s t . The additional number of candidates additional( s t ) iszero when the priors match, q = p , and intuitively Q doesn’tneed to consider any further candidate moves. The numberof additional moves may be negative; intuitively, Q putsenough weight on all candidates that P deems important,and doesn’t need to consider any further candidate moves.The number of additional candidate moves and is upperbounded by additional( s t ) ≤ m p ( s t ) according to (13); atthe very worst, Q would additionally have to consider all ofP’s candidates.We consider positions up to ply t plies sampled from priorfor P, and at ply t evaluate how many additional candidatemoves Q should consider on average: A q ( t ) = E s t ∼ p ( s t ) (cid:104) additional( s t ) (cid:105) . (15)The expectation is estimated with a Monte Carlo averageover samples from p ( s t ) .Figure 7 shows the average additional number of candidatemoves if Q is taken as the Classical chess prior, with P iterat-ing over all other variants. From the outset, Pawn one squareplaces of its prior mass on 1. d3, 1. e3, 1. c3 and 1. h3,which together only account for of Classical’s priormass. As pawns are moved from the starting rank and piecesare developed, A q ( t ) slowly decreases for Pawn one square.As the opening progresses, Stalemate=win slowly driftsfrom zero, presumably because some board configurationsthat would lead to drawn endgames under Classical rulesmight have a different outcome. Torpedo puts of itsprior mass on one move, 1. d4, whereas the Classical prior isbroader (its top move, 1. d4, occupies of its prior mass).The truncated plot value for Torpedo is A q (1) = − . , sig-nifying that the first Classical candidate moves effectivelyalready include those of Torpedo chess. There is a slowupward drift in the average number of additional candidatesthat a Classical player has to consider under Self-capturechess as a game progresses. We hypothesise that it can, inpart, be ascribed to the number of reasonable self-capturingoptions increasing toward the middle game. Material plays an important role in chess, and is often usedto assess whether a particular sequence of piece exchangesand captures is favourable. Material sacrifices in chess aremade either for concrete tactical reasons, e.g. mating attacks,or to be traded off for long-term positional strengthening ofthe position. Understanding the material value of pieces inchess helps players master the game and is one of the veryfirst pieces of chess knowledge taught to beginners. Changesto the rules of chess affect piece mobility, and hence also therelative value of pieces. Without a basic estimate of what therelative piece values in each variant are, it would be harderfor human players to start playing these chess variants. As a guide, we provide an experimental approximation to piecevalues based on outcomes of AlphaZero games under 1second per move.We approximate piece values from the weights of a linearmodel that predicts the game outcome from the differencein numbers of each piece only. As background, the realAlphaZero evaluation v in ( p , v ) = f θ ( s ) is the output ofa deep neural network with weights θ . The expected gameoutcome v is the result of a final tanh activation to ensurean output in ( − , . If z ∈ {− , , } indicates the playingside’s game outcome, AlphaZero’s loss function includes themean squared error ( z − v ) (Silver et al., 2018). We createa simplified evaluation function g w ( s ) that only takes piececounts on the board into consideration. For a position s weconstruct a feature vector d def = [1 , d p , d N , d B , d R , d Q ] thatcontains the integer differences between the playing sideand their opponent’s number of pawns, knights, bishops,rooks and queens. We define g w with weights w ∈ R as g w ( s ) = tanh( w T d ) . (16)When trained on the 10,000 AlphaZero self-play board po-sitions from Section 3.1 for each variant, the piece weights w provide an indication of their relative importance. Let ( s, z ) ∼ games represent a sample of a position and finalgame outcome from a variant’s self-play games. We min-imise (cid:96) ( w ) = E ( s,z ) ∼ games (cid:104)(cid:0) z − g w ( s ) (cid:1) (cid:105) (17)empirically over w , and normalise weights w by w p to yieldthe relative piece values. The recovered piece values foreach of the chess variants are given in Table 6.Variant p N B R Q Classical 1 3.05 3.33 5.63 9.5No castling 1 2.97 3.13 5.02 9.49No castling (10) 1 3.14 3.40 5.37 9.85Pawn one square 1 2.95 3.14 5.36 9.62Stalemate=win 1 2.95 3.13 4.76 8.96Self-capture 1 3.10 3.22 5.34 9.42Pawn-back 1 2.65 2.85 4.67 9.39Semi-torpedo 1 2.72 2.95 4.69 8.3Torpedo 1 2.25 2.46 3.58 7.12Pawn-sideways 1 1.8 1.98 2.99 5.92
Table 6.
Estimated piece values from AlphaZero self-play gamesfor each variant.
In Classical chess, piece values vary based on positionalconsiderations and game stage. The piece values in Table6 should not be taken as a gold standard, as the sample ofAlphaZero games that they were estimated on does not fullycapture the diversity of human play, and the game lengthsdo not correspond to that of human games, which tend to be ssessing Game Balance with AlphaZero shorter. For comparison, we have included the piece valueestimates that we obtain by applying the same method toClassical chess, showing that the estimates do not deviatemuch from the known material values. Over the years,many material systems have been proposed in chess. Themost commonly used one (Capablanca & de Firmian, 2006)gives 3–3–5–9 for values of knights, bishops, rooks andqueens. Another system (Kaufman, 1999) gives 3.25–3.25–5–9.75. Yet, bishops are typically considered to be morevaluable than the knights, and there is usually an additiveadjustment while in possession of a bishop pair. The rookvalue varies between 4.5 and 5.5 depending on the systemand the queen values span from 8.5 to 10. The relativepiece values estimated on the AlphaZero game sample forClassical chess, 3.05–3.33–5.63–9.5, do not deviate muchfrom the existing systems. This suggests that the estimatesfor the new chess variants are likely to be approximatelycorrect as well.We can see similar piece values estimated for No-castling,No-castling(10), Pawn-one-square chess, Self-capture andStalemate=win. This is not surprising, given that thesevariants do not involve a major change in piece mobility.Estimated piece values look quite different in the remain-ing variations, where pawn mobility has been increased:Pawn-back, Semi-torpedo, Torpedo and Pawn-sideways.In Pawn-sideways chess, minor pieces seem to be worthapproximately two pawns, which is in line with our anec-dotal observations when analysing AlphaZero games, assuch exchanges are frequently made. Like Torpedo chess,pawns become much stronger and more valuable than be-fore. Changes in Pawn-back and Semi-torpedo are not aspronounced.
4. Qualitative assessment
To evaluate the differences in play between the set of chessvariations considered in this study, we couple the quantita-tive assessment of the variations with expert analysis basedon a large set of representative games. While the overalldecisiveness and opening diversity add to the appeal of anychess variation, the subjective questions of aesthetic valueand the types of positions, moves and patterns that arise arenot possible to fully capture quantitatively. For providinga deep qualitative assessment of the appeal of these chessvariations, we rely on the experience of chess grandmasterVladimir Kramnik, an ex-world chess champion and an au-thority on the game. By characterising typical patterns, wehope to provide players with insights to help them judge forthemselves if they would find some of these chess variantsinteresting enough to try out in practice. What we providehere are preliminary findings.The detailed qualitative assessment of the chess variantspresented in this article, along with typical motifs and illus- trative games, is provided in the Appendix (Section B). Forthis analysis, we use the 1,000 1-minute per move games ofSection 3.1 as well as 200 1-minute per move games from adiverse set of early opening positions that all of the majoropening systems. By looking at the former, we were ableto assess AlphaZero’s preferred style of play in each chessvariant, and by looking at the latter, we could assess how thetreatment of different opening lines changes and which ofthose become more or less promising under each of the rulechanges. Figure 1 shows an illustrative example position foreach of the considered chess variants.What follows is a short summary of the main takeawaysfrom the qualitative analysis for each of the variants, pro-vided by GM Vladimir Kramnik.
No-castling chess is a potentially exciting variant, giventhat king safety is often compromised for both players, al-lowing for simultaneous attacking and counter-attacking andthe equality, when reached, tends to be dynamic in naturerather than “dry”. The multitude of approaches to evacuatethe king, and their timing, adds complexity to the openingplay.
No-castling (10) , where castling is not permitted forthe first 10 moves (20 plies) is a partial restriction, ratherthan an absolute one – which does not change the gameto the same extent. Due to castling being such a powerfuloption, the lines preferred by AlphaZero all tend to involvecastling, only delayed – resulting in a preference for slower,closed positions, and a less attractive style of play. Suchpartial castling restrictions can be considered if the desire isto sidestep opening theory and preparation, but this may notbe of interest for the wider chess audience.
Pawn one square chess variant may appeal to players whoenjoy slower, strategic play – as well as a training tool forunderstanding pawn structures, due to the transpositionalpossibilities when setting up the pawns. The reduced pawnmobility makes it harder to launch fast attacks, making thegame overall less decisive.
Stalemate=win chess has little effect on the opening andmiddlegame play, mostly affecting the evaluation of certainendgames. As such, it does not increase decisiveness of thegame by much, as it seems to almost always be possible todefend without relying on stalemate as a drawing resource.Therefore, this chess variant is not likely to be useful forsidestepping known theory or for making the game substan-tially more decisive at the high level. The overall effect ofthe change seems to be minor.
Torpedo and Semi-torpedo chess both make the gamemore dynamic and more decisive, and Torpedo chess inparticular leads to new motifs and changes in all stagesof the game. Creating passed pawns becomes very impor-tant, as they are hard to stop. The attacking possibilitiesmake Torpedo chess quite appealing, and it is likely to be of ssessing Game Balance with AlphaZero interest for players that enjoy tactical play. Pawn-back chess makes it possible to regain control of theweakened squares in the position and remove some squareweaknesses. It also introduces additional possibilities foropening up diagonals and making squares available for thepieces. Counter-intuitively, even though moving the piecesbackwards is usually a defensive manoeuvre, this can makemore aggressive options possible, given that pawns cannow be pushed further earlier on, as there is always anoption of moving them back to cover the weakened squares.AlphaZero has a strong preference for playing the Frenchdefence with Black, which is particularly interesting.
Pawn-sideways chess is incredibly complex, resulting inpatterns that are at times quite “alien” when one is usedto classical chess. The pawn structures become very fluidand it is impossible to create permanent pawn weaknesses.Given how important this concept is in classical chess, thischess variant requires us to rethink how we approach anygiven position, making it very concrete and relying on deepcalculation. Restructuring the pawn formation takes time,and players need to use that time for creating other types ofadvantages. Many of AlphaZero games in this variant havebeen quite tactical, some involving novel tactics that are notpossible under classical rules.
Self-capture chess is quite entertaining, as it introduces ad-ditional options for sacrificing material – and material sacri-fices have a certain aesthetic appeal. Self-capture moves canfeature in all stages of the game. Not every game involvesself-captures, as giving away material is not always required,but they do feature in a substantial percentage of the games,and in some games they occur multiple times. Self-capturemoves can be used to open files and squares for the piecesin the attack; opening up a blockade by sacrificing a pawnin the pawn chain; or in defence, while escaping the matingnet.
5. Conclusions
We have demonstrated how AlphaZero can be used for pro-totyping board games and assessing the consequences ofrule changes in the game design process, as demonstrated onchess, where we have trained AlphaZero models to evaluate9 different chess variants, representing atomic changes to therules of classical chess. Training an AlphaZero model underthese rule changes helped us effectively simulate decades ofhuman play in a matter of hours, and answer the “what if”question: what the play would potentially look like underdeveloped theory in each chess variant. We believe that asimilar approach could be used for auto-balancing game me-chanics in other types of games, including computer games,in cases when a sufficiently performant reinforcement learn-ing system is available. To assess the consequences of the rule changes, we coupledthe quantitative analysis of the trained model and self-playgames with a deep qualitative analysis where we identifiedmany new patterns and ideas that are not possible underthe rules of classical chess. We showed that there severalchess variants among those considered in this study thatare even more decisive than classical chess: Torpedo chess,Semi-torpedo chess, No-castling chess and Stalemate=winchess.We additionally quantified the arising diversity of openingplay and the intersection of opening trees between chessvariations, showing how different the opening theory is foreach of the rule changes. There is a negative correlationbetween the overall opening diversity and decisiveness, asthe decisive variants likely require more precise play, withfewer plausible choices per move. For each of the chessvariants, we estimated the material value of each of thepieces based on the results of 10,000 AlphaZero games,to provide insight into favourable exchange sequences andmake it easier for human players to understand the game.No-castling chess, being the first variant that we analysed(chronologically), has already been tried in an experimentalblitz grandmaster tournament in Chennai, as well as a coupleof longer grandmaster games. Our assessment suggeststhat several of the assessed chess variants might be quiteappealing to interested players, and we hope that this studywill prove to be a valuable resource for the wider chesscommunity.
Acknowledgements
We would like to thank chess grandmasters Peter HeineNielsen, and Matthew Sadler for their valuable feedbackon our preliminary findings and the early version of themanuscript. Oliver Smith and Kareem Ayoub have beenof great help in managing the project. We would also liketo thank the team of Chess.com for providing us with aplatform to announce and discuss No-castling chess andpresent annotated games.
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A.1. Proof of equation (13)Let p and q be two vectors with non-negative entries thatsum to one. Define r as a vector with elements r i = max( p i , q i ) (cid:80) i (cid:48) max( p i (cid:48) , q i (cid:48) ) . (18)We show below that e − (cid:80) i r i log r i ≤ e − (cid:80) i p i log p i + e − (cid:80) i q i log q i . (19)Let R = (cid:80) i max( p i , q i ) be the normalizing constant in(18). It is bounded by ≤ R ≤ . We write the entropy as − (cid:88) i r i log r i = − R (cid:88) i max( p i , q i ) log max( p i , q i ) + log R = − R (cid:88) i max( p i log p i , q i log q i ) + log R ≤ − (cid:88) i max( p i log p i , q i log q i ) + log R ≤ − (cid:88) i p i log p i − (cid:88) i q i log q i + log R (20)where the last inequality in (20) follows from max( a, b ) ≥ a + b . Exponentiating (20) and applying Jensen’s inequalityyields e − (cid:80) i r i log r i ≤ R e ( − (cid:80) i − p i log p i )+ ( − (cid:80) i q i log q i ) ≤ R (cid:18)
12 e − (cid:80) i p i log p i + 12 e − (cid:80) i q i log q i (cid:19) ≤ e − (cid:80) i p i log p i + e − (cid:80) i q i log q i . (21)The final line follows from R/ ≤ as ≤ R ≤ . Thebound is tight at R = 1 when p and q both put probabilitymass uniformly on two non-intersecting same-sized subsetsof elements. A.2. Additional figures An example of two vectors giving a tight bound in (19) is p = [ , , , , and q = [0 , , , , . Game length . . . . . . . D e n s i t y ClassicalNo-castlingNo-castling (10)Pawn one squareStalemate=win TorpedoSemi-torpedoPawn-backPawn-sidewaysSelf-capture (a) The game length distributions of the total number of pliesfor all self-play games for each variant.
Game length . . . . . . D e n s i t y ClassicalNo-castlingNo-castling (10)Pawn one squareStalemate=win TorpedoSemi-torpedoPawn-backPawn-sidewaysSelf-capture (b) The game length distributions of the total number of plies forthe subset of decisive (not drawn) self-games for each variant.
Figure 8.
The game length distributions of the total number of pliesof AlphaZero games in each chess variant, based on a sample of10,000 games played at 1 second per move. The experimentalsetup is described in Section 3.1.20 ssessing Game Balance with AlphaZero − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . D e n s i t y ( h i s t og r a m ) ClassicalNo-castling (a) No-castling and Classical chess − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . D e n s i t y ( h i s t og r a m ) ClassicalTorpedo (b) Torpedo and Classical chess − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . D e n s i t y ( h i s t og r a m ) ClassicalSelf-capture (c) Self-capture and Classical chess − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . D e n s i t y ( h i s t og r a m ) ClassicalNo-castling (10) (d) No-castling (10) and Classical chess − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . D e n s i t y ( h i s t og r a m ) ClassicalStalemate=win (e) Stalemate=win and Classical chess − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . D e n s i t y ( h i s t og r a m ) ClassicalSemi-torpedo (f) Semi-torpedo and Classical chess
Figure 9.
The density of (negative) log likelihoods for the prior opening lines for Classical chess and each of the variants. The mean ofeach histogram gives the entropy or average information content for each variant’s prior p ( s ) , as given in (8). The subfigures are orderedby entropy, following Table 3. Figure 9g continues on the next page.21 ssessing Game Balance with AlphaZero − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . D e n s i t y ( h i s t og r a m ) ClassicalPawn-back (g) Pawn-back and Classical chess − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . D e n s i t y ( h i s t og r a m ) ClassicalPawn-sideways (h) Pawn-sideways and Classical chess − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . . . D e n s i t y ( h i s t og r a m ) ClassicalPawn one square (i) Pawn one square and Classical chess − log p ( s ) when s ∼ p ( s ) at ply depth 20 . . . . . D e n s i t y ( h i s t og r a m ) ClassicalUniform random (j) Uniform random classical moves and Classical chess
Figure 9. (Continued from previous page.)
The density of (negative) log likelihoods for the prior opening lines for Classical chess andeach of the variants. The mean of each histogram gives the entropy or average information content for each variant’s prior p ( s ) , as givenin (8). The subfigures are ordered by entropy, following Table 3. −
10 0 10 20 30 40 log p cl ( s ) − log q var ( s ) when s ∼ p cl ( s ) at ply depth 20 . . . . . . . . D e n s i t y ( h i s t og r a m ) Stalemate=winNo-castling (10)No-castlingPawn one square (a) A decomposition of the entropy of subset variants of Classi-cal chess relative to Classical chess. log p var ( s ) − log q cl ( s ) when s ∼ p var ( s ) at ply depth 20 . . . . . . . . . D e n s i t y ( h i s t og r a m ) Stalemate=winSelf-captureSemi-torpedoPawn-backTorpedoPawn-sideways (b) A decomposition of the entropy of Classical chess relativeto its superset variants.
Figure 10.
Histograms of the density of terms log p ( s ) − log q ( s ) whose mean under p ( s ) is the Kullback-Leibler divergence in (10).22 ssessing Game Balance with AlphaZero Ply t . . . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Classical (w)Classical (b) (a) Classical chess
Ply t . . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s No-castling (w)No-castling (b) (b) No-castling chess
Ply t A v e r a g e nu m b e r o f c a nd i d a t e m o v e s No-castling (10) (w)No-castling (10) (b) (c) No-castling (10) chess
Ply t . . . . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Pawn one square (w)Pawn one square (b) (d) Pawn one square chess
Ply t A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Stalemate=win (w)Stalemate=win (b) (e) Stalemate=win chess
Ply t . . . . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Torpedo (w)Torpedo (b) (f) Torpedo chess
Figure 11.
The average number of candidate moves M ( t ) from (9) for each of the variants, as computed from their prior distributions p ( s ) . Figure 11g continues on the next page. 23 ssessing Game Balance with AlphaZero Ply t . . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Semi-torpedo (w)Semi-torpedo (b) (g) Semi-torpedo chess
Ply t A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Pawn-back (w)Pawn-back (b) (h) Pawn-back chess
Ply t . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Pawn-sideways (w)Pawn-sideways (b) (i) Pawn-sideways chess
Ply t . . . . . . . . A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Self-capture (w)Self-capture (b) (j) Self-capture chess
Ply t A v e r a g e nu m b e r o f c a nd i d a t e m o v e s Uniform random (w)Uniform random (b) (k) Uniform random moves under classical chess rules
Figure 11. (Continued from previous page.)
The average number of candidate moves M ( t ) from (9) for each of the variants, as computedfrom their prior distributions p ( s ) . 24 ssessing Game Balance with AlphaZero B. Appendix
Here we present a selection of instructive games for eachof the chess variations considered in the study, along with adetailed assessment of the variations by Vladimir Kramnik.Given that different rule changes that we examined hadled to a different degree of departure from existing chesstheory and patterns, we do not present an equal amount ofinstructive positions and games for each chess variation, andrather focus on those that have either been assessed to be ofgreater immediate interest or simply employ patterns thatare unfamiliar and novel and require more time to introduceand understand.The Appendix is organised into sections corresponding toeach of the chess variations and rule alterations examined inthis study, in the following order: No-castling chess (Page25), No-castling (10) chess (Page 31), Pawn one squarechess (Page 34), Stalemate=win chess (Page 37), Torpedo(Section 40), Semi-torpedo (Page 54), Pawn-back chess(Page 61), Pawn-sideways chess (Page 70) and Self-capturechess (Page 85).Each of the variants-specific sections first introduces therule change, sets out the motivation for why it seemed ofinterest to be tried out, gives a qualitative assessment and ahigh-level conceptual overview of the dynamics of arisingplay by Vladimir Kramnik and then concludes with severalinstructive games and positions, selected to illustrate thetypical motifs that arise in AlphaZero play in these varia-tions.
B.1. No-castling
In No-castling chess, the adjustment to the original rulesinvolved a full removal of castling as an option.B.1.1. M
OTIVATION
The motivation for the No-castling chess variant, as providedby Vladimir Kramnik: “ Adjustments to castling rules were chronologi-cally the first type of changes implemented andassessed in this study. Firstly, excluding a singleexisting rule makes it comparatively easy for hu-man players to adjust, as there is no need to learnan additional rule. Secondly, the right to castle isrelatively new in the long history of the game ofchess. Arguably, it stands out amongst the rulesof chess, by providing the only legal opportunityfor a player to move two of their own pieces atthe same time. ” B.1.2. A
SSESSMENT
The assessment of the no-castling chess variant, as providedby Vladimir Kramnik: “ I was expecting that abandoning the castling rulewould make the game somewhat more favorablefor White, increasing the existing opening advan-tage. Statistics of AlphaZero games confirmedthis intuition, though the observed difference wasnot substantial to the point of unbalancing thegame. Nevertheless, when considering humanpractice, and considering that players would findthemselves in unknown territory at the very earlystage of the game, I would expect White to havea higher expected score in practice than underregular circumstances.One of the main advantages of no-castling chessis that it eliminates the nowadays overwhelmingimportance of the opening preparation in profes-sional chess, for years to come, and makes playersthink creatively from the very beginning of eachgame. This would inevitably lead to a consider-ably higher amount of decisive games in chesstournaments until the new theory develops, andmore creativity would be required in order to win.These factors could also increase the followingof professional chess tournaments among chessenthusiasts.With late middlegame and endgame patterns stay-ing the same as in regular chess, there is a majordifference in the opening phase of a no-castlingchess game. The main conceptual rules of piecedevelopment and king safety are still valid, butmost concrete opening variations of regular chessno longer apply, as castling is usually an essentialpart of existing chess opening variations.For example, possibly opening a game with 1. f4,which is not a great idea in classical chess, mightbe one of the better options already, since it mightmake it easier to evacuate the king after Nf3, g3,Bg2, Kf2, Rf1, Kg1. Some completely new pat-terns of playing the openings start to make sense,like pushing the side pawns in order to developthe rooks via the “h” file or “a” file, as well as“artificial castling” by means of Ke2, Re1, Kf1 andothers. Many new conceptual questions arise inthis chess variation.For instance, one has to think about what oughtto be preferable: evacuating the king out of thecenter of the board as soon as possible or aim-ing to first develop all the pieces and claim spaceand central squares. Years of practice are likely ssessing Game Balance with AlphaZero required to give a clear answer on the guidingprinciples of early play and best opening strate-gies. Even with the help of chess engines, it wouldlikely take decades to develop the opening theoryto the same level and to the same depth as wehave in regular chess today. The engines can behelpful with providing initial recommendationsof plausible opening lines of play, but the rightunderstanding and timing of the implementationof new patterns is crucial in practical play.Studying the numerous no-castling games playedby AlphaZero, I have noticed one major concep-tual change. Since both kings have a harder timefinding a safe place, the dynamic positional fac-tors (e.g. initiative, piece activity, attack), seem tohave more importance than in regular chess. Inother words, a game becomes sharper, with bothsides attacking the opponent king at the sametime.I am convinced that because of the aforemen-tioned reasons we would see many interestinggames, and many more decisive games at the toplevel chess tournaments in case the organisersdecide to give it a try. Due to the simplicity of theadjustment compared to regular chess, it is alsoeasy to implement this variation at any other level,including the online chess playing platforms, asit merely requires an agreement between the twoplayers not to play castling in their game. ” B.1.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under No-castling chess, when playing with roughly one minute permove from a particular fixed first move. Note that theseare not purely deterministic, and each of the given lines ismerely one of several highly promising and likely options.Here we give the first 20 moves in each of the main lines,regardless of the position.
Main line after e4
The main line of AlphaZero after e4in No-castling chess is: e4 (book) c5 Nf3 Nc6 Nc3 Nf6 d4 cxd4 Nxd4e6 Ndb5 d6 Bf4 e5 Bg5 a6 Na3 b5
Nd5Be7
Bxf6 Bxf6 c4 Ne7
Nxf6+ gxf6 cxb5 h5
Qd2 Kf8
Bc4 Kg7
Rd1 d5 exd5 Qb6 bxa6Rd8
Nc2 Bxa6 rZ0s0Z0Z Z0Z0mpj0 bl0Z0o0Z Z0ZPo0Zp Z0Z0Z0Z0 PONL0OPO Z0ZRJ0ZR a b c d e f g h
Main line after d4
The main line of AlphaZero after d4in No-castling chess is: d4 (book) d5 c4 e6 Nc3 c5 cxd5 exd5 Nf3Nf6 g3 Nc6 Bg2 h6 Kf1 Be6 Bf4 Rc8 h4 a6
Rc1 Rg8 a3 c4
Ne5 Bd6 e4 dxe4
Nxe4Bxe5 dxe5 Nxe4
Bxe4 Qa5
Kg2 Rd8
Qe2Nd4
Qe3 Nf5 ZpZ0Zpo0 pZ0ZbZ0o l0Z0OnZ0 O0Z0L0O0 Z0S0Z0ZR a b c d e f g h
Main line after c4
The main line of AlphaZero after c4in No-castling chess is: c4 (book) e5 Nc3 Nf6 Nf3 Nc6 d4 exd4 Nxd4Bb4 Bf4 Bxc3 bxc3 d6 g3 Ne5 Bg2 Kf8 c5Ng6
Be3 dxc5
Nb3 Qe8 h4 h5
Bxc5+ Kg8
Qc2 a5
Bd4 Ne7
Bxf6 gxf6 a4 Kg7
Nd4Rb8
Kf1 Bd7 ssessing Game Balance with AlphaZero Zpobmpj0 o0Z0Z0Zp PZ0M0Z0O Z0O0Z0O0 S0Z0ZKZR a b c d e f g h
B.1.4. I
NSTRUCTIVE GAMES
Game AZ-1: AlphaZero No-castling vs AlphaZero No-castling
The first ten moves for White and Black havebeen sampled randomly from AlphaZero’s opening “book”,with the probability proportional to the time spent calculat-ing each move. The remaining moves follow best play, atroughly one minute per move. d4 d5 c4 e6 Nc3 c5 cxd5 exd5 Nf3 Nf6 g3Nc6 Bg2 h6 h4 Be6 Kf1 Rc8
Be3 Ng4
Qd2b5 o0Z0Zpo0 ZpopZ0Z0 Z0M0ANO0 PO0LPOBZ S0Z0ZKZR a b c d e f g h
Nxb5 Qb6 a4 a6 dxc5 Nxe3+
Qxe3 Bxc5
Nd6+ Z0Z0Zpo0 plnMbZ0o Z0apZ0Z0 PZ0Z0Z0O Z0Z0LNO0 S0Z0ZKZR a b c d e f g h
16. . .
Ke7
Nxc8+ Rxc8 a5 Qa7
Qb3 Bxf2
Bh3Rb8 l0Z0jpo0 pZnZbZ0o O0ZpZ0Z0 ZQZ0ZNOB S0Z0ZKZR a b c d e f g h
Qa3+ Bc5
Qd3 Nb4
Qh7 Qd7
Bxe6 Qxe6
Rc1 Be3
Rc3 d4
Rc5 Kd6
Re5 Qg4
Qf5Qxg3
Rh2 Z0Z0Zpo0 pZ0j0Z0o O0Z0SQZ0 Z0Z0aNl0 Z0Z0ZKZ0 a b c d e f g h
30. . .
Qg6
Rg2 Qxf5
Rxf5 Ke6
Rc5 Kd6
Rf5Ke6
Re5+ Kf6 h5 Rc8
Rg4 Rc1+
Kg2 Nc6 ssessing Game Balance with AlphaZero Re8 Rc2
Kh3 Rc5
Kh4 Bf2+
Kh3 Be3
Rh4 Rxa5
Kg3 Ra1
Rhe4 Kf5
Nh4+ Kf6
Rc8 Ne7
Re8 Nc6
Nf3 Kf5 b3 Rb1
Nh4+Kf6
Ra8 Ra1
Kh3 Ne5
Re8 Rh1+
Kg3Nc6
Ra8 Ra1 b4 Nxb4
Rd8 Rg1+
Kh3Rh1+
Kg2 Rg1+
Kh3 Rh1+
Kg3 Rg1+
Ng2Nc2
Kh2 Rf1
Rc8 Kf5
Nxe3+ Nxe3
Rxd4Kg5
Rc5+ f5
Kg3 Kxh5
Re4 Ng4
Kg2Rf2+
Kg1 Nf6
Re7 Rf4
Rxg7 Ng4
Rc3 Kh4
Re7 Kg5
Ra3 h5
Rxa6 Rb4
Ra5 h4
Ra3Nf6
Rg7+ Kh5
Rf7 Kg5
Rg7+ Kh5
Kh1Rb2
Ra5 Kh6
Rg2 Rb1+
Rg1 Rxg1+
Kxg1Kg5
Ra8 Ne4
Kg2 Kg4
Ra4 Kg5
Rb4 Kg4
Rd4 Kh5
Kh3 Ng5+
Kh2 Ne4
Kg2 Kg4
Rb4 Kg5
Kf3 Nd2+
Ke3 Ne4
Rb7 Kg4
Rg7+ Ng5
Rg8 h3
Kf2 f4
The first ten moves for White and Black havebeen sampled randomly from AlphaZero’s opening “book”,with the probability proportional to the time spent calculat-ing each move. The remaining moves follow best play, atroughly one minute per move. Nf3 e6 d4 d5 e3 c5 b3 h5 dxc5 Bxc5 Bb2Kf8 c4 Nf6 h4 Bd7 Nc3 Nc6
Be2 Rc8
Rc1Qa5 cxd5 Nxd5
Kf1 Bxe3 opZbZpo0 l0ZnZ0Zp ZPM0aNZ0 PA0ZBOPZ Z0SQZKZR a b c d e f g h
Rc2 Bh6
Ng5 Ncb4
Rc1 Ke7
Rh3 Rhd8 a3 Nxc3
Bxc3 Rxc3 opZbjpo0 l0Z0Z0Mp OPs0Z0ZR Z0SQZKZ0 a b c d e f g h
Rhxc3 Bc6
Qe1 Qxa3
Kg1 g6
Bf1 Bg7
Re3 Rd6
Rc4 Nd5
Rf3 Nf6 opZ0jpa0 Z0Z0Z0Mp lPZ0ZRZ0 Z0Z0LBJ0 a b c d e f g h
Rff4 Qxb3
Be2 Rd7
Rc1 Qb2
Bf3 Bxf3
Rxf3 Qd2
Qf1 Qd5
Qe1 Qd2
Qf1 Qd5
Qe2 Bh6
Qb2 Bxg5 hxg5 Ng4
Re1 Qd2
Qa3+ Rd6
Rb1 Kf8 g3 Ne5
Rf4 Qd3
Qxd3Nxd3
Ra4 Rd5
Rxb7 Rxg5
Ra3 Rd5
Rbxa7Ne5
R7a5 Rd1+
Kg2 Ng4
Ra1 Rd4
R5a4Rd3
R4a3 Rd4
Ra4 Rd3
R4a3 Rd2
R3a2Rd7
Ra7 Rd6
R7a6 Rd7
Ra7 Rd6
R7a6 Rd5
R6a5 Rd2
R5a2 Rd5
Ra5 Rd2
R5a2
The first ten moves for White and Black havebeen sampled randomly from AlphaZero’s opening “book”,with the probability proportional to the time spent calculat-ing each move. The remaining moves follow best play, atroughly one minute per move. c4 e5 Nc3 Nf6 Nf3 Nc6 d4 exd4 Nxd4 Bb4 g3 Ne4 Qd3 Nc5 Qe3+ Kf8 Bg2 Qf6
Ndb5Ne6
Kd1 Bc5
Qe4 d6
Nd5 ssessing Game Balance with AlphaZero rZbZ0j0s opo0Zpop ZNaNZ0Z0 Z0Z0Z0O0 PO0ZPOBO S0AKZ0ZR a b c d e f g h
13. . .
Qd8 f4 Ned4
Qd3 Bf5 e4 Bg4+
Ke1Be2 rZ0l0j0s opo0Zpop ZNaNZ0Z0 Z0ZQZ0O0 PO0ZbZBO S0A0J0ZR a b c d e f g h
Qc3 Nxb5 cxb5 Bxb5
Be3 Bxe3
Nxe3 Ne7
Kf2 h5 h4 Rh6
Rac1 c6
Rhd1 Qb6
Rd4a5
Rcd1 d5 exd5 cxd5
Qa3 rZ0Z0j0Z ZpZ0mpo0 obZpZ0Zp L0Z0M0O0 PO0Z0JBZ Z0ZRZ0Z0 a b c d e f g h
The game soon ended in a draw.
B.1.5. H
UMAN GAMES
Here we take a brief look at a couple of recently played blitzgames between professional chess players from the tour-nament that took place in Chennai in January 2020 (Shah,2020). We focus on new motifs in the opening stage ofthe game, and show how these might be counter-intuitivecompared to similar patterns in classical chess.
Game H-1: Arjun, Kalyan (2477) vs D. Gukesh (2522)(blitz) d4 d5 c4 c6 Nc3 Nf6 Nf3 rmblka0s opZ0opop Z0ZpZ0Z0 Z0M0ZNZ0 PO0ZPOPO S0AQJBZR a b c d e f g h
Interestingly, even at an early stage we can see an exampleof a difference in patterns that originate in Classical chessand those that arise in No-castling chess. The positioning ofthe knight on f3 is very natural, but is in fact an imprecision.AlphaZero prefers keeping the option open of playing thepawn to f3 instead, in order to tuck the king away to safety.It gives the following line as its favored continuation: e3Bf5 Bd3 g6 h3 e6 Nge2 Be7 f3 Bxd3 Qxd3 Kf8
Kf2 Bg7
Rd1. rm0l0Z0s opZ0apjp Z0ZpZ0Z0 Z0MQOPZP PO0ZNJPZ S0ARZ0Z0 a b c d e f g hanalysis diagram
Yet, Nf3 was played in the game, which continued: ssessing Game Balance with AlphaZero
4. . . e6 e3 Nbd7 Qc2 Bd6 b3 b6 rZblkZ0s o0ZnZpop Z0ZpZ0Z0 ZPM0ONZ0 PZQZ0OPO S0A0JBZR a b c d e f g h
Here AlphaZero suggests that it was instead time to movethe king to safety. Deciding on when exactly to initiate theevacuation of the king from the centre and choosing the bestway of achieving it is one of the key motifs of No-castlingchess. This decision is less clear than the decision to castlein Classical chess, due to a larger number of options andthe fact that the sequence takes more moves that all need tobe staged accordingly. Instead of moving the pawn to b6,AlphaZero suggests the following instead:
7. . . h5 Bb2Kf8 Rd1 Kg8. rZbl0Zks opZnZpo0 Z0ZpZ0Zp ZPM0ONZ0 PAQZ0OPO Z0ZRJBZR a b c d e f g hanalysis diagram
Going back to the game continuation, after
7. . . b6 Whitehas the upper hand. The game continued: Bb2 Bb7 Bd3Qe7 e4 rZ0ZkZ0s obZnlpop Z0ZpZ0Z0 ZPMBZNZ0 PAQZ0OPO S0Z0J0ZR a b c d e f g h
This is another example of mistiming the evacuation of theking. Instead of playing e4, it was the right time to movethe king to safety instead, retaining a large plus for Whiteafter:
Kf1 Kf8 h4 h5 a4 Ng4
Rh3 Rh6 rZ0Z0j0Z obZnlpo0 Z0ZpZ0Zp PZPO0ZnO ZPMBONZR S0Z0ZKZ0 a b c d e f g hanalysis diagram
Going back to the position after e4, the game continua-tion goes as follows:
10. . . dxe4
Nxe4 (Giving away the advantage. Recap-turing with the bishop was correct, even though it mightseem as otherwise counter-intuitive.)
11. . .
Nxe4
Bxe4f5. (This is looking bad for Black;
12. . .
Nf6 is the pre-ferred move.)
Bd3 c5 (At this point, AlphaZero assessesthe position as winning for White.)
Kf1 (The advantagecould have been kept with d5.)
14. . .
Bxf3 gxf3 cxd4(
15. . .
Rf8 may have been equalizing)
Bxd4 (Gives theadvantage to Black. White ought to have captured on f5instead. The right way to respond to the game move wouldhave been
16. . .
Qh4.)
16. . .
Be5
Bxe5 Nxe5
Bxf5 ssessing Game Balance with AlphaZero rZ0ZkZ0s o0Z0l0op Z0Z0mBZ0 ZPZ0ZPZ0 PZQZ0O0O S0Z0ZKZR a b c d e f g h
A brilliant piece sacrifice.
18. . . exf5
Re1 Kd8
Qxf5 (
Qd2+ may have beenstronger)
20. . .
Re8 f4 Qb7
Rg1 Ng6 (The finalmistake, it appears that
Nf7 might hold)
Rd1+ Ke7
Rg3 Qh1+
Ke2 Qe4+
Re3 Qxe3+ fxe3 Rad8
Rxd8 Rxd8
Qe4+ Kf8
Qb7 f4 h5 Already Kramnik demonstrates a motif that is quitestrong in no-castling chess, pushing one of the side pawnsearly. rmblkans opopopo0 Z0Z0Z0Zp Z0Z0Z0Z0 POPOPZPO SNAQJBMR a b c d e f g h Nf3 e6 e3 Nf6 b3 (Interestingly, AlphaZero doesn’tlike this very normal-looking move, giving Black a slightplus after
4. . . c5 Bb2 Be7 Be2 d5 Rf1 Kf8 Kf2Nc6 Kg1 Kg8 a4 Bd7.)
4. . . b6 Bb2 Bb7 Bd3( Be2 might have been better.)
6. . . h4 (Not the mostprecise, according to AlphaZero, suggesting that
6. . . c5 Rf1 Be7 Kf2 h4 Ng5 Kf8
Kg1 Rh6
Be2 Nc6was still slightly better for Black.) h3 (This turns out tobe the wrong reaction, giving the advantage back to Blackagain.)
7. . .
Nh5 Kf2 Be7 (Here, there was an opportunityto play
8. . .
Bc5 instead: rm0lkZ0s obopZpo0 Z0a0Z0Zn ZPZBONZP PAPO0JPZ SNZQZ0ZR a b c d e f g hanalysis diagram which would have kept a big plus for Black.) Re1 Bf6
Bxf6 (
Nc3)
10. . .
Qxf6 (
10. . . gxf6 wasthe better recapture)
Nc3 Ng3
Kg1 d6 (
12. . .
Ke7was the correct plan)
Ng5 Nd7
Nce4 Nxe4
Bxe4Bxe4
Nxe4 Qg6
Ng5 Ke7 e4 (
Qe2)
18. . . e5 d4 exf4 Nf3 Kf8
Qd2 Qg3
Re2 Rh6 rZ0Z0j0Z o0onZpo0 Z0Z0Z0Z0 ZPZ0ZNlP PZPLRZPZ S0Z0Z0J0 a b c d e f g h
Rf1 (Black gains the upper hand.)
23. . .
Re6
Nh2(A mistake, e5 was required.)
24. . .
Rae8
Rxf4Nf6 e5 dxe5
Rf3 (Another mistake,
Rxe5 wascorrect.)
27. . .
Qg6 d5 (Taking on e5 was still a bettercontinuation.)
28. . .
R6e7 c4 e4
Rc3 Nh5
Nf1Kg8
Qe1 Nf4
Rd2 e3
Rxe3 Rxe3
Nxe3 Qe4
In the No-castling (10) variant of chess, castling is onlyallowed from move 11 onwards, both for the first and thesecond player. ssessing Game Balance with AlphaZero B.2.1. M
OTIVATION
When it comes to limit the impact of castling on the game, itis possible to consider different types of partial limitations,the easiest of which is disallowing it for a fixed numberof opening moves. In this variation, we have explored theimpact of disallowing castling for the first 10 moves, but anyother number could have been used instead. Each choiceleads to a slightly different body of opening theory, as par-ticular lines either become viable or stop being viable underdifferent circumstances.B.2.2. A
SSESSMENT
The assessment of the No-castling (10) chess variant, asprovided by Vladimir Kramnik: “ The main purpose of the partial restriction tocastling, as a hypothetical adjustment to the rulesof chess, would be to sidestep opening theory. Assuch, it is aimed at professional chess as an op-tion to potentially consider. The game itself doesnot change in other meaningful ways, and Alp-haZero usually aims at playing slower lines wherecastling does indeed take place after the first 10moves. This makes sense, given that castling is afast an powerful move, so aiming to take advan-tage of it if available makes for a good approach.Yet, the slowing down of the game could as aside-effect lead to an increased number of draws.Another disadvantage is the need to count andkeep track of the move number when consideringvariations. ” B.2.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under No-castling (10) chess, when playing with roughly one minuteper move from a particular fixed first move. Note that theseare not purely deterministic, and each of the given lines ismerely one of several highly promising and likely options.Here we give the first 20 moves in each of the main lines,regardless of the position.
Main line after e4
The main line of AlphaZero after e4in No-castling (10) chess is: e4 (book) e5 Bc4 Nc6 Nf3 Nf6 Qe2 Bc5 c3 Qe7 b4 Bb6 a4 a6 a5 Ba7 d3 d6 Na3 Be6
Nc2O-O
O-O h6
Be3 Qd7
Bxa7 Nxa7
Rfe1Nc6 h3 Rfe8
Bxe6 Qxe6
Ne3 d5
Qc2 Rad8
Rab1 Qd7 ZpoqZpo0 pZnZ0m0o O0Zpo0Z0 Z0OPMNZP ZRZ0S0J0 a b c d e f g h
Main line after d4
The main line of AlphaZero after d4in No-castling (10) chess is: d4 (book) d5 Nf3 Nf6 c4 dxc4 Nc3 e6 Qa4+c6 Qxc4 b5 Qd3 Bb7 e4 b4 Na4 Qa5 b3 c5
Ne5 cxd4
Qb5+ Qxb5
Bxb5+ Nfd7
Bb2 f6
Nxd7 Nxd7
Bxd4 Bxe4
O-O Bd6
Rfe1 Bd5
Nc5 Bxc5
Bxc5 Rb8 o0ZnZ0op ZBAbZ0Z0 ZPZ0Z0Z0 PZ0Z0OPO S0Z0S0J0 a b c d e f g h
Main line after c4
The main line of AlphaZero after c4in No-castling (10) chess is: c4 (book) e5 Nc3 Nf6 Nf3 Nc6 e4 Bb4 d3 d6 a3 Bc5 b4 Bb6 Be3 Bg4 Be2 Bxf3
Bxf3 Nd4
Na4 Nxf3+
Qxf3 Bxe3 fxe3 Nd7
O-O O-O
Nc3 c6 h3 Qb6
Rab1 Rae8 a4 Re6 a5Qd8
Qg3 Rf6 ssessing Game Balance with AlphaZero opZnZpop O0Z0o0Z0 Z0MPO0LP ZRZ0ZRJ0 a b c d e f g h
B.2.4. I
NSTRUCTIVE GAMES
Game AZ-4: AlphaZero No-castling (10) vs AlphaZeroNo-castling (10)
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. c4 e5 d4 exd4 Qxd4 Nc6 Qe3+ Nge7 Nf3 d5 cxd5 Qxd5 Nc3 Qa5 Qg5 Bf5 Bd2 f6
Qh5+g6
Qh4 Nb4
Rc1 O-O-O
Qxf6 Bh6 opo0m0Zp l0Z0ZbZ0 Z0M0ZNZ0 PO0APOPO Z0S0JBZR a b c d e f g h
A stunning move, offering up a piece on h6. Acceptingwould be disastrous for White, as Black pieces mobilisequickly via Ned5. The h8 rook can also potentially come toe8, and this justifies the material investment. e3 Rhe8
Qh4 Bg7
Nb5 Rxd2 opo0m0ap lNZ0ZbZ0 Z0Z0ONZ0 PO0s0OPO Z0S0JBZR a b c d e f g h
The fireworks continue. . .
Rxc7+ Qxc7
Nxc7 Rxb2
Nxe8 Rb1+ opZ0m0ap Z0Z0ZbZ0 Z0Z0ONZ0 PZ0Z0OPO ZrZ0JBZR a b c d e f g h
Leading to a draw by perpetual check.
The next game is less tactically rich, but rather interestingfrom the perspective of showcasing differences in openingplay and the overall approach, when castling is not possiblein the first ten moves.
Game AZ-5: AlphaZero No-castling (10) vs AlphaZeroNo-castling (10)
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. c4 e5 Nc3 Nf6 Nf3 Nc6 Qa4 ssessing Game Balance with AlphaZero rZblka0s opopZpop Z0Z0o0Z0 QZPZ0Z0Z Z0M0ZNZ0 PO0OPOPO S0A0JBZR a b c d e f g h
This is a slightly unusual move, showcasing that the styleof play in this variation of chess involves opting for movesthat do not necessarily achieve as much immediately andare somewhat less direct, potentially trying to wait for theright time to castle, when possible. In this game, however,castling does not end up being critical.
4. . . e4 Ng5 Qe7 c5 e3 rZbZka0s opoplpop Z0O0Z0M0 QZ0Z0Z0Z Z0M0o0Z0 PO0OPOPO S0A0JBZR a b c d e f g h dxe3 Qxc5 Nge4 Nxe4 Qxe4+ Qe5
Qxe5 Nxe5 e4 Bb4 f4 Nc4 e3 Nd6
Bd3 Bxc3 bxc3 f6
Ba3 Nf7
Bc4 b6 rZbZkZ0s o0opZnop Z0Z0Z0Z0 A0O0O0Z0 PZ0Z0ZPO S0Z0J0ZR a b c d e f g h
Bd5 c6
Bb3 Rb8
Bxf7+ Kxf7
Bd6 Ra8 e5c5
O-O-O Ba6 e4 h5 rZ0Z0Z0s o0ZpZko0 bo0A0o0Z Z0o0O0Zp Z0O0Z0Z0 PZ0Z0ZPO Z0JRZ0ZR a b c d e f g h
And the game eventually ended in a draw.
B.3.1. M
OTIVATION
Restricting the pawn movement to one square only is in-teresting to consider, as the double-move from the second(or seventh rank) seems like a “special case” and an excep-tion from the rule that pawns otherwise only move by onesquare. In addition, slowing down the game could make itmore strategic and less forcing.B.3.2. A
SSESSMENT
The assessment of the Pawn one square chess variant, asprovided by Vladimir Kramnik: “ The basic rules and patterns are still mostly thesame as in classical chess, but the opening theorychanges and becomes completely different. Intu-itively it feels that it ought to be more difficult ssessing Game Balance with AlphaZero for White to gain a lasting opening advantageand convert it into a win, but since new open-ing theory would first need to be developed, thiswould not pertain to human play at first. In mostAlphaZero games one can notice the rather typi-cal middlegame positions arise after the openingphase.This variation of chess can be a good pedagogicaltool when teaching and practicing slow, strategicplay and learning about how to set up and committo pawn structures. Since the pawns are unableto advance very fast, many attacking ideas thatinvolve rapid pawn advances are no longer rel-evant, and the play is instead much slower andultimately more positional. Additionally, this vari-ation of chess could simply be of interest for thosewishing for an easy way of side-stepping openingtheory. ” B.3.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under Pawnone square chess, when playing with roughly one minuteper move from a particular fixed first move. Note that theseare not purely deterministic, and each of the given lines ismerely one of several highly promising and likely options.Here we give the first 20 moves in each of the main lines,regardless of the position.
Main line after e3
The main line of AlphaZero after e3in Pawn one square chess is: e3 (book) Nf6 d3 d6 Nf3 h6 e4 b6 c3 Bb7 Qc2e6 c4 e5 g3 g6 Nc3 Bg7
Bg2 Nc6
Be3 Nd7
Ne2 Nc5 a3 Na5 rZ0lkZ0s obo0Zpa0 m0m0o0Z0 O0ZPANO0 S0Z0J0ZR a b c d e f g h
An instructive position, as it looks optically like Black isblundering material. In this variation of chess, however,b2-b4 is not a legal move, because pawns can only move one square. This justifies the move sequence.
Nd2 Nc6 b3 a6
Nf3 Ne6 h3 O-O
O-ONcd4
Nfxd4 exd4
Bd2 c6 rZ0l0skZ ZbZ0Zpa0 poponZpo Z0Z0Z0Z0 OPZPZ0OP S0Z0ZRJ0 a b c d e f g h
Main line after d3
The main line of AlphaZero after d3in Pawn one square chess is: d3 (book) d6 e3 Nf6 Nd2 e6 Ngf3 Nbd7 d4g6 Bd3 Bg7 O-O O-O h3 e5 c3 Re8 e4 b6 Re1 Bb7 a3 a6
Qc2 h6
Nb3 a5 dxe5 Nxe5
Nxe5 dxe5
Be3 Nh5
Bb5 Rf8
Nd2 Qf6 g3Rfd8 rZ0s0ZkZ Zbo0Zpa0 oBZ0o0Zn O0O0A0OP S0Z0S0J0 a b c d e f g h
Main line after c3
The main line of AlphaZero after c3in Pawn one square chess is: c3 (book) d6 d3 Nf6 Nf3 h6 d4 Bf5 c4 e6 Nc3c6 e3 d5 Bd3 dxc4 Bxc4 Bd6
O-O Nbd7
Re1Ne4
Bd3 Nxc3 bxc3 Bxd3
Qxd3 Qe7 c4 e5
Qf5 O-O
Rb1 b6 c5 Bc7
Ba3 b5 d5 cxd5 ssessing Game Balance with AlphaZero rZ0Z0skZ o0anlpo0 ZpOpoQZ0 A0Z0ONZ0 PZ0Z0OPO ZRZ0S0J0 a b c d e f g h
B.3.4. I
NSTRUCTIVE GAMES
Here we present some examples of AlphaZero play in Pawnone square chess.
Game AZ-6: AlphaZero Pawn One Square vs Alp-haZero Pawn One Square
The first ten moves for Whiteand Black have been sampled randomly from AlphaZero’sopening “book”, with the probability proportional to thetime spent calculating each move. The remaining movesfollow best play, at roughly one minute per move. d3 Nf6 Nd2 d6 e3 e6 Ngf3 g6 h3 Bg7 c3O-O c4 Nbd7 Rb1 e5 b3 c6 Bb2 Qe7
Be2 b6 b4 Bb7 a3 h6
O-O h5
Qc2 Rfd8
Rfd1 c5 rZ0s0ZkZ obZnlpa0 Z0o0o0Zp O0ZPONZP ZRZRZ0J0 a b c d e f g h
Here we have a rather normal middlegame position. Thegame continued:
Ne4 Rac8 b5 d5 cxd5 Nxd5
Nfd2 f6
Nc4Nf8 a4 Kh7 a5 Ne6
Ra1 Nb4
Qb1 Rb8 axb6 axb6
Nc3 Qe8
Ra7 Nc7
Na3 Rd7
Ba1 Nbd5
Na4 Ne6 e4 Ndf4
Bf1 Bc8
Rxd7 Qxd7
Nc4 Qa7
Naxb6 Rxb6
Nxb6Qxb6 g3 Z0Z0Z0ak ZPo0o0Zp Z0ZPZ0OP AQZRZBJ0 a b c d e f g h
38. . . c4 gxf4 Nxf4 Bc3 Bxh3
Bd2 Qe6
Bxf4exf4 f3 Z0Z0Z0ak ZPZ0Z0Zp Z0ZPZPZb ZQZRZBJ0 a b c d e f g h
43. . .
Bg4
Bg2 Bxf3
Bxf3 Qh3 dxc4 Qxf3
Qd3 Qg4+
Kf2 Qh4+
Ke2 Qh2+
Kf1 Qh1+
Kf2 Qh4+
Ke2 Qh2+
Ke1 Bf8
Qf3 Bc5
Kf1 Qg1+
Ke2 Qh2+
Kf1 Qg1+
Ke2 Qh2+
Kf1
The first ten moves for Whiteand Black have been sampled randomly from AlphaZero’sopening “book”, with the probability proportional to thetime spent calculating each move. The remaining movesfollow best play, at roughly one minute per move. d3 c6 e3 d6 c3 g6 d4 Nf6 Nf3 Bf5 Be2 e6 O-O Nbd7 c4 Bg7 b3 O-O Ba3 Ne4
Nfd2c5
Nxe4 Bxe4
Nd2 Bc6
Rc1 Qa5
Bb2 cxd4 exd4 d5 ssessing Game Balance with AlphaZero rZ0Z0skZ opZnZpap l0ZpZ0Z0 ZPZ0Z0Z0 PA0MBOPO Z0SQZRJ0 a b c d e f g h
This is a very normal-looking position, and one would behard-pressed to guess that it originated from a differentvariation of chess, as it looks pretty “classical”.
Re1 Rfe8 h3 Bh6
Bc3 Qc7
Bf1 b6
Bb2Qb7 a3 Rac8
Rc2 Bg7
Qc1 Bh6
Qd1 Bg7
Qc1 h6 c5 bxc5 dxc5 e5
Qb1 h5
Qa2 a6 b4 Ba4
Rcc1 Bh6
Ba1 e4
Rb1 Ne5
Nb3Kh7
Nd4 Nd3
Re3 ZqZ0ZpZk pZ0Z0Zpa Z0OpZ0Zp bO0MpZ0Z O0ZnS0ZP QZ0Z0OPZ ARZ0ZBJ0 a b c d e f g h
A very instructive position, reminiscent of a famous clas-sical game between Petrosian and Reshevsky from Zurichin 1953, where Petrosian was playing Black. The posi-tional exchange sacrifice allows White easy play on the darksquares.
37. . .
Bxe3 fxe3 f6
Be2 Rc7
Rf1 Rf7
Qd2Ne5
Qe1 Bb5
Nxb5 axb5 a4 Nd3
Qh4 bxa4
Bxh5 Re5
Be2+ Kg7
Qg3 Qc7
Bxe5 Qxe5
Qxe5 fxe5 c6 Rxf1+
Bxf1 a3 c7 a2 c8=Qa1=Q
Qb7+ Kh6
Qxd5 Qe1
Qf7 Qxb4
Qa2Qc5
Qd2 Nb4
Kf2 Nd5 g3 Qf8+
Kg1 Qc5
Kf2 Qf8+
Ke1 Nb4
Bc4 Kh7
Qd7+ Kh6
Qd2 Kg7
Qf2 Qe7
Kf1 Nd3
Qe2 Qf6+
Kg2 Qc6
Bb3 Qc5 h4 Qc1
Kh2 Ne1
Bd1 Nf3+
Kg2 Kh6
Kf1 g5 hxg5 Kxg5
Kf2 Qd2
Bc2 Qxe2+
Kxe2 Kf5
Kf2 Ng5
Kg2 Nh7
Kh3 Nf6
Bb3 Kg5
Be6 Kh5
Bb3 Kg5
Be6Kh5 g4+ Kg5
Kg3 Nh7
Kh3 Nf6
Kg3 Nh7
Kh3 Nf6
In this variation of chess, achieving a stalemate position isconsidered a win for the attacking side, rather than a draw.B.4.1. M
OTIVATION
The stalemate rule in classical chess allows for additionaldrawing resources for the defending side, and has beena subject of debate, especially when considering ways ofmaking the game potentially more decisive. Yet, due to itspotential effect on endgames, it was unclear whether such arule would also discourage some attacking ideas that involvematerial sacrifices, if being down material in endgames endsup being more dangerous and less likely to lead to a drawthan in classical chess.B.4.2. A
SSESSMENT
The assessment of the Stalemate=win chess variant, as pro-vided by Vladimir Kramnik: “ I was at first somewhat surprised that the decisivegame percentage in this variation was roughlyequal to that of classical chess, with similar lev-els of performance for White and Black. I waspersonally expecting the change to lead to moredecisive games and a higher winning percentagefor White.It seems that the openings and the middlegame re-main very similar to regular chess, with very fewexceptions, but that there is a significant differ-ence in endgame play since some basic endgamelike K+P vs K are already winning instead ofbeing drawn depending on the position. ssessing Game Balance with AlphaZero Z0Z0O0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0 a b c d e f g h
In the position above, with White to move, in clas-sical chess the position would be a draw due tostalemate after Ke6. Yet, the same move wins inthis variation of chess, so the defending side needsto steer away from these types of endgames.Similarly, the stalemates that arise in K+N+N vsK are now wins rather than draws, for example: Z0Z0Z0Z0 Z0Z0ZKZk Z0Z0ZNZ0 Z0Z0Z0Z0 a b c d e f g h
Looking at the games of AlphaZero, it seems thatthere are enough defensive resources in most mid-dlegame positions that certain types of inferiorendgame positions, now possible under this rulechance, could be avoided and defended. A strongplayer can in principle learn to navigate to thesepositions to take advantage of them, or find waysto escape them.In terms of the anticipated effect on human play,I would still expect this rule change to lead to ahigher percentage of wins in endgames where oneside has a clear advantage, but probably not asmuch as one would otherwise have been expecting.This may be a nice variation of chess for chessenthusiasts with an interest in endgame patterns. ” B.4.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under Stale-mate=win chess, when playing with roughly one minute permove from a particular fixed first move. Note that theseare not purely deterministic, and each of the given lines ismerely one of several highly promising and likely options.Here we give the first 20 moves in each of the main lines,regardless of the position.
Main line after e4
The main line of AlphaZero after e4in Stalemate=win chess is: e4 (book) e5 Nf3 Nc6 Bb5 Nf6 O-O Nxe4 Re1Nd6 Nxe5 Be7 Bf1 Nxe5 Rxe5 O-O d4 Bf6 Re1 Re8 c3 Rxe1
Qxe1 Ne8
Bf4 d5
Nd2Bf5
Qe2 Nd6
Re1 Qd7
Qd1 c6
Nb3 b6
Nd2 Ne4
Nf3 Bg4 rZ0Z0ZkZ o0ZqZpop Z0ZpZ0Z0 Z0O0ZNZ0 PO0Z0OPO Z0ZQSBJ0 a b c d e f g h
Main line after d4
The main line of AlphaZero after d4in Stalemate=win chess is: d4 (book) Nf6 c4 e6 g3 Bb4+ Bd2 Be7 Qc2c6 Bg2 d5 Nf3 b6 O-O O-O Bf4 Bb7
Rd1Nbd7
Ne5 Nh5
Bd2 Nhf6 cxd5 cxd5
Nc6Qe8
Nxe7+ Qxe7
Qc7 Ba6
Nc3 Rfc8
Qf4Nf8
Be1 h6
Qd2 Ng6 ssessing Game Balance with AlphaZero rZrZ0ZkZ o0Z0lpo0 bo0Zpmno Z0ZpZ0Z0 Z0M0Z0O0 PO0LPOBO S0ZRA0J0 a b c d e f g h
Main line after c4
The main line of AlphaZero after c4in Stalemate=win chess is: c4 (book) e5 g3 Nf6 Bg2 Bc5 d3 d5 cxd5 Nxd5 Nf3 Nc6 O-O O-O Nc3 Nxc3 bxc3 Rb8 Bb2Re8 d4 Bd6 e4 Bg4 h3 Bh5
Qc2 f6 d5Na5 c4 b6
Nh4 Nb7
Rae1 Rf8 f4 Nc5
Re3Qd7 o0oqZ0op Z0mPo0Zb Z0Z0S0OP PAQZ0ZBZ Z0Z0ZRJ0 a b c d e f g h
B.4.4. I
NSTRUCTIVE GAMES
The games in Stalemate=win chess are at the first glancealmost indistinguishable from those of classical chess, asthe lines are merely a subset of the lines otherwise playableand plausible under classical rules.
Game AZ-8: AlphaZero Stalemate=win vs AlphaZeroStalemate=win The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. Game AZ-8 was labelled wrongly in arXiv:2009.04374v1,and this game replaces it. e4 c6 d4 d5 e5 Bf5 Be2 h6 Nf3 e6 O-O a6 c4 dxc4 Bxc4 Nd7 Bd3 Ne7
Nc3 Nb6
Re1Nbd5
Ne4 Ng6
Bd2 Qb6 rZ0Zka0s ZpZ0Zpo0 plpZpZno Z0ZnObZ0 Z0ZBZNZ0 PO0A0OPO S0ZQS0J0 a b c d e f g h h3 Be7
Bc4 Ngf4
Bf1 g5
Ng3 Rg8
Bc3Qc7
Nh2 Bg6
Ne4 O-O-O Zpl0apZ0 pZpZpZbo Z0ZnO0o0 Z0A0Z0ZP PO0Z0OPM S0ZQSBJ0 a b c d e f g h g3 Nxc3 bxc3 Nd5
Rc1 Qa5
Qb3 Qa3
Qc2Kb8
Nf3 Bb4 ZpZ0ZpZ0 pZpZpZbo Z0ZnO0o0 l0O0ZNOP PZQZ0O0Z Z0S0SBJ0 a b c d e f g h ssessing Game Balance with AlphaZero cxb4 Qxf3 Rb1 Bxe4
Rxe4 Qf5
Bg2 Nc7
Rb2 Rd7 a4 Rgd8
Qc3 h5
Kh2 Qh7
Qd2Qg8 ZpmrZpZ0 pZpZpZ0Z Z0Z0O0op PO0ORZ0Z Z0Z0Z0OP Z0Z0Z0Z0 a b c d e f g h h4 gxh4
Rxh4 f5 exf6 e5 b5 cxb5
Qe3exd4
Qf4 Ka7
Qf3 Rb8
Rb4 Qd8
Bh3 Rd5 f7 bxa4
Rxa4 d3 jpm0ZPZ0 pZ0Z0Z0Z Z0ZrZ0Zp RZ0Z0Z0S Z0ZpZQOB Z0Z0Z0Z0 a b c d e f g h
Qe3+ Ka8
Rad4 Qf8
Be6 Rxd4
Rxd4 h4
Bc4 hxg3+ fxg3 Nb5
Bxb5 axb5
Rxd3 b4
Rd5 Rc8 kZrZ0l0Z ZpZ0ZPZ0 Z0ZRZ0Z0 Z0Z0L0O0 Z0Z0Z0Z0 a b c d e f g h
White is clearly winning here, and Ra5+ is good and tempt-ing. AlphaZero is only optimised for achieving an endresult. Even though a slower approach achieves the sameoutcome, a win is a win! This game ultimately finishes withcheckmate.
Rf5 Kb8 g4 Ka8
Rf2 b3
Qxb3 Qe7
Ra2+Kb8
Qg3+ Rc7
Rf2 Ka7 f8=Q Qh7+
Qh3Qb1
Qf5 Qa1
Qf1 Qh8+
Qh5 Qg7
Qh4Rc5
Kh1 Ra5
Qg3 Ra4
Rf3 Ra2
Qff2+Rxf2
Qxf2+ Ka6
Qg3 b5
Qf4 Qg8
Qf6+Ka5
Qf5 Ka4
Qf8 Qh7+
Kg2 b4
Qe8+ Ka5
Qh5+ Qxh5 gxh5 Ka4
Rf1 Ka3
Kf2 Kb2
Ke1 b3
Kd1 Kc3
Kc1 Kd4 h6 b2+
Kxb2Ke5
Re1+ Kf6
Rd1 Kg6
Rc1 Kxh6
Rc3Kg5
Rc5+ Kh4
Rc7 Kg3
Rb7 Kh4
Ra7 Kh3
Kc3 Kg3
Ra1 Kh4
Ra3 Kh3
Kd4+ Kg4
Ra5 Kf4
Ra7 Kg5
Ra1 Kf5
Ra3 Kg4
Ke5 Kh5
Ra5 Kg5
Rd5 Kh6
Rd7 Kg5
Rc7 Kh6
Kf5 Kh5
Rh7
In the variation of chess that we’ve named Torpedo chess,the pawns can move by either one or two squares forwardfrom anywhere on the board rather than just from the initialsquares, which is the case in Classical chess. We will referto the pawn moves that involve advancing them by twosquares as “torpedo” moves.We have also looked at a Semi-torpedo variant in our experi-ments, where we only add a partial extension to the originalrule and have the pawns be able to move by two squaresfrom the 2nd/3rd and 6th/7th rank for White and Black re-spectively. In this section we will focus on the universalmotifs of full Torpedo chess, and cover the sub-motifs andsub-patterns that correspond to Semi-torpedo chess in itsown dedicated section in Appendix B.6. ssessing Game Balance with AlphaZero B.5.1. M
OTIVATION
In a sense, having the pawns always be able to move by oneor two squares makes the pawn movement more consistent,as it removes a “special case” of them only being able todo the “double move” from their initial position. Increasingpawn mobility has the potential of speeding up all stages ofthe game. It adds additional attacking motifs to the openingsand changes opening theory, it makes middlegames morecomplicated, and changes endgame theory in cases wherepawns are involved.B.5.2. A
SSESSMENT
The assessment of the Torpedo chess variant, as providedby Vladimir Kramnik: “ The pawns become quite powerful in Torpedochess. Passed pawns are in particular a verystrong asset and the value of pawns changes basedon the circumstances and closer to the endgame.All of the attacking opportunities increase andthis strongly favours the side with the initiative,which makes taking initiative a crucial part of thegame. Pawns are very fast, so less of a strategicalasset and much more tactical instead. The gamebecomes more tactical and calculative comparedto standard chess.There is a lot of prophylactic play, which is whysome games don’t feature many “torpedo” moves– “torpedo” moves are simply quite powerful andthe play often proceeds in a way where eachplayer positions their pawn structure so as to dis-incentivise “torpedo” moves, either by the virtueof directly blocking their advance, or by placingtheir own pawns on squares that would be able tocapture “en passant” if “torpedo” moves were tooccur.This seems to favour the “classical” style of playin classical chess, which advocates for strongcentral control rather than conceding space tolater attack the center once established. It seemslike it is more difficult to play openings like theGrunfeld or the King’s Indian defence.In summary, this is an interesting chess variant,leading to lots of decisive games and a potentiallyhigh entertainment value, involving lots of tacticalplay. ” B.5.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under Torpedochess, when playing with roughly one minute per movefrom a particular fixed first move. Note that these are not purely deterministic, and each of the given lines is merelyone of several highly promising and likely options. Here wegive the first 20 moves in each of the main lines, regardlessof the position.
Main line after e4
The main line of AlphaZero after e4in Torpedo chess is: e4 (book) c5 Nf3 d6 d4 Nf6 Nc3 cxd4 Nxd4 a6 g3 h6 Bg2 e5 Nde2 Be7 Be3 Be6
Nd5 Nbd7 c4 Rc8 b3 Ng4
O-O Nxe3
Nxe3 h4
Nf5Kf8
Qd2 Nf6
Nc3 g6
Nxe7 Qxe7
Rad1 Rc6
Rc1 Kg7 ZpZ0lpj0 pZrobmpZ Z0Z0o0Z0 ZPM0Z0O0 PZ0L0OBO Z0S0ZRJ0 a b c d e f g h
Main line after d4
The main line of AlphaZero after d4in Torpedo chess is: d4 (book) d5 c4 e6 Nf3 Nf6 Nc3 a6 e3 b6 Bd3 Bb7 O-O Bd6 cxd5 exd5 Ne5 O-O a3Nbd7 f4 Ne4
Bd2 c5
Be1 cxd4 exd4 b5 h3 Rc8
Qe2 Ndf6 a4 b4
Nxe4 dxe4
Bxa6Bxa6
Qxa6 Bxe5 Z0Z0Zpop QZ0Z0m0Z Z0Z0a0Z0 Po0OpO0Z Z0Z0Z0ZP S0Z0ARJ0 a b c d e f g h ssessing Game Balance with AlphaZero Main line after c4
The main line of AlphaZero after c4in Torpedo chess is: c4 (book) c5 e3 e6 d4 d5 Nc3 Nc6 Nf3 Nf6 a3 h6 dxc5 Bxc5 cxd5 exd5 b4 Bd6 Bb2 O-O
Be2 a5 b5 Ne7
O-O Re8
Rc1 Be6
Bd3Ng6
Ne2 a4
Rc2 Qe7
Qa1 Nf8
Nfd4 N8d7
Ng3 Ng4 rZ0ZrZkZ ZpZnlpo0 ZPZpZ0Z0 pZ0M0ZnZ O0ZBO0M0 L0Z0ZRJ0 a b c d e f g h
B.5.4. I
NSTRUCTIVE GAMES
Here we showcase several instructive games that illustratethe type of play that frequently arises in Torpedo chess,along with some selected extracted game positions in caseswhere particular (endgame) move sequences are of interest.
Game AZ-9: AlphaZero Torpedo vs AlphaZero Tor-pedo
The first ten moves for White and Black have beensampled randomly from AlphaZero’s opening “book”, withthe probability proportional to the time spent calculatingeach move. The remaining moves follow best play, atroughly one minute per move. d4 d5 Nf3 Nf6 c4 e6 Nc3 c6 e3 Nbd7 g3 Ne4 Nxe4 dxe4 Nd2 f5 c5 Be7 h4 O-O rZbl0skZ opZna0op Z0O0ZpZ0 Z0Z0O0O0 PO0M0O0Z S0AQJBZR a b c d e f g h g5 b6 b4 a5
Bc4 axb4
Bxe6+ Kh8
Bb2Ne5 rZbl0s0j Z0Z0a0op Z0O0mpO0 Z0Z0O0Z0 PA0M0O0Z S0ZQJ0ZR a b c d e f g h
Bc4 Ng4 d6 cxd5 h6 Rg8 rZbl0Zrj Z0Z0a0op Z0OpZpO0 Z0Z0O0Z0 PA0M0O0Z S0ZQJ0ZR a b c d e f g h hxg7+ Rxg7 c7 Qd7
Bxd5 Qxd5
Nc4 rZbZ0Z0j Z0O0a0sp Z0ZqZpO0 Z0Z0O0Z0 PA0Z0O0Z S0ZQJ0ZR a b c d e f g h
22. . .
Qg8
Ne5 Nxe5
Bxe5 Bxg5
Qh5 ssessing Game Balance with AlphaZero rZbZ0Zqj Z0O0Z0sp Z0Z0ApaQ Z0Z0O0Z0 PZ0Z0O0Z S0Z0J0ZR a b c d e f g h
25. . . b2 axb3 Rxa1+ Bxa1 Be7 f4 exf3
Rg1Bf8
Qg5 Z0O0Z0sp Z0Z0ZpL0 ZPZ0OpZ0 A0Z0J0S0 a b c d e f g h
30. . . h6 Qxh6+ Qh7
Bxg7+ Bxg7
Qxh7+ Kxh7
Kf2 Be5
Rd1 Bb7
Rc1 Bc8
Kxf3 Kg6 e4b4
Rc4 Kf6
Rc6+ Kg5
Ke3 f4+
Kf2 Bd4+
Kg2 Be5
Rc5 Kf6
Kf3 Ke6
Rb5 Bd7
Rxb4Bxc7
Rd4 Ke7
Rd2 Be8
Rh2 Bd6
Rh7+Ke6
Rh6+ Ke7
Rh2 Kf6
Rh8 Ke7
Rh2 Kf6
Rh6+ Ke7
Rh1 Kf6
Rh8 Ke7
Rh6 Be5 b4Kd7
Kf2 Bc3 b6 Kc8
Rd6 Kb7
Kf3 Be5
Rd5 Bb8
Rd8 Bc6
Rf8 Ba4
Ke2 Be5
Rf5Bb8
Rf6 Be5
Rf5 Bb8 e5 Kxb6
Rxf4 Bb5+
Kd1 Bxe5
Rf5 Bh2
Rxb5+ Kxb5
The first ten moves for White and Black have beensampled randomly from AlphaZero’s opening “book”, withthe probability proportional to the time spent calculatingeach move. The remaining moves follow best play, atroughly one minute per move. d4 d5 c4 e6 Nf3 Nf6 Nc3 a6 e3 b6 Bd3 Bb7 O-O Bd6 cxd5 exd5 a3 O-O Ne5 c5 f4 Nbd7
Bd2 cxd4 exd4 Ne4
Be1 b5 h3 Rc8
Qe2Ndf6 ZbZ0Zpop pZ0a0m0Z ZpZpM0Z0 O0MBZ0ZP S0Z0ARJ0 a b c d e f g h
A normal-looking position arises in the middlegame (thisis one of AlphaZero’s main lines in this variation of chess),but the board soon explodes in tactics. a4 b4
Nxe4 dxe4
Bxa6 Bxa6
Qxa6 Bxe5 dxe5 Qd4+
Kh1 Qxb2
Bh4 e2 Z0Z0Zpop QZ0Z0m0Z Z0Z0O0Z0 Po0Z0O0A Z0Z0Z0ZP S0Z0ZRZK a b c d e f g h
Rfe1 Nd5 e7 Rfe8
Qd6 Qd2 a6 Ra8
Qc6b2 ssessing Game Balance with AlphaZero rZ0ZrZkZ Z0Z0Opop PZQZ0Z0Z Z0ZnZ0Z0 Z0Z0Z0ZP S0Z0S0ZK a b c d e f g h
A series of consecutive torpedo moves had given rise tothis incredibly sharp position, with multiple passed pawnsfor White and Black, and the threats are culminating, asdemonstrated by the following tactical sequence.
Qxe8+ Rxe8 a8=Q Nc7 QZ0ZrZkZ Z0m0Opop Z0Z0Z0Z0 Z0Z0Z0ZP S0Z0S0ZK a b c d e f g h
Qa5 bxa1=Q
Qxd2 Qa4
Rxe2 f6
Re3 Kf7
Bf2 Qb5
Qd6 Qf1+
Bg1 Qc4 f5 g6
Rg3gxf5
Qd1 Z0m0OkZp Z0Z0ZpZ0 Z0Z0Z0SP Z0ZQZ0AK a b c d e f g h
Here Black utilizes a torpedo move to give back the pawnand protect h5 via d5.
40. . . f3 Qxf3 Qd5
Qg4 Ne6
Be3 Rb8
Qa4Kxe7
Bc1 Kf7
Qc2 f5
Kh2 Rb5
Qa4 h5
Qa7+ Qb7
Qa4 Qd5
Ba3 f4
Rf3 Ra5
Qb4Rc5 Z0Z0ZkZ0 Z0sqZ0Zp A0Z0ZRZP Z0Z0Z0Z0 a b c d e f g h
Rf2 Qf5
Qb2 Rd5
Qb7+ Kg6
Qc6 Kh7
Bb2 Rd8
Qb7+ Kg8
Rf3 Qg6
Be5 Qf5
Ba1 Rd3
Qb1 Rd5
Qb8+ Rd8
Qb2 Nd4
Rf2 Ne6
Qb3 Kh7
Qb7+ Kg8
Qa6 Kh7
Qa7+ Kg6
Qb7 Rd1
Qa8 Rd8
Qc6 Kh7
Qb7+ Kg6
Bb2 Rd1
Qb8 Re1
Bc3 Re3
Qg8+ Kh6
Qh8+ Kg6
Qe8+ Kh7
Qd7+ Kg6
Qc6 Kh7
Qb7+ Kg8
Qa8+ Kh7
Qb7+ Kg8
Qc8+ Kh7
Qh8+ Kg6
Qg8+ Kh6
Qc8 Kh7
Qd7+ Kg6
Qe8+ Kh7
Bd2 Rd3
Qb8 Rd8
Qb7+ Kg8
Qc6 Qg6
Qc3 Rd4
Qa5 Kh7
Kg1 Re4 h4 Nd4
Qc7+ Qg7
Qa5 Qf7
Qa6 f3
Qh6+ Kg8
Qg5+ Kh7
Be3 Ne2+ ssessing Game Balance with AlphaZero Z0Z0ZqZk Z0Z0Z0Lp Z0Z0ApZ0 Z0Z0Z0J0 a b c d e f g h
And the game soon ends in a draw.
Kh2 Qc7+
Kh1 Ng3+
Kg1 Ne2+
Kf1Ng3+
Kg1 Ne2+
Rxe2 fxe2
Qxh5+ Kg8
Qxe2 Rxh4
Qa6 Qe7
Qc8+ Kh7
Qf5+Kg7
Qg5+ Qxg5
Bxg5 Re4
Kf2 Kg6
Kf3 Re8
Bf4 Re7 g4 Rf7
Kg3 Rg7
Bb8 Kg5
Bd6 Rb7
Bf4+ Kg6
Kf3 Ra7
Bb8 Rb7
Bf4 Ra7
Bb8 Rb7
Bf4
The first ten moves for White and Black have beensampled randomly from AlphaZero’s opening “book”, withthe probability proportional to the time spent calculatingeach move. The remaining moves follow best play, atroughly one minute per move. d4 d5 Nf3 Nf6 c4 e6 Nc3 a6 e3 b6 Bd3 Bb7 a3 g6 cxd5 exd5 h4 Nbd7 O-O Bd6
Nxd5 rZ0lkZ0s ZbonZpZp po0a0mpZ Z0ZNZ0Z0 O0ZBONZ0 S0AQZRJ0 a b c d e f g h
An interesting tactical motif, made possible by torpedomoves. One has to wonder, after
11. . .
Nxd5 e4, whathappens on
12. . .
Nf4? The game would have followed e5 Nxd3 exd6 Nxc1 dxc7 Qxc7 rZ0ZkZ0s ZblnZpZp po0Z0ZpZ Z0Z0Z0Z0 O0Z0ZNZ0 S0mQZRJ0 a b c d e f g hanalysis diagram and here, White would have played d6, a torpedo move –gaining an important tempo while weakening the Black king.
16. . .
Qc4
Rxc1, followed by Re1+ once the queen hasmoved. AlphaZero evaluates this position as being stronglyin White’s favour, despite the material deficit.Going back to the game continuation,
11. . .
Nxd5 e4 O-O exd5 Bxd5
Bg5 Qb8
Re1Re8
Nd2 Rxe1+
Qxe1 Bf8
Qe3 c6 rl0Z0akZ Z0ZnZpZp popZ0ZpZ Z0ZbZ0A0 O0ZBL0Z0 S0Z0Z0J0 a b c d e f g h
Now we see several torpedo moves taking place. First Whitetakes the opportunity to plant a pawn on h6, weakening theBlack king, then Black responds by a4 and b4, getting thequeenside pawns in motion and creating counterplay on theother side of the board. h6 a4
Re1 Qa7
Bf5 b4 ssessing Game Balance with AlphaZero rZ0Z0akZ l0ZnZpZp Z0ZbZBA0 po0O0Z0Z O0Z0L0Z0 Z0Z0S0J0 a b c d e f g h
Bg4 Qb7
Bh3 Rc8
Qd3 Ra8
Qe3 Nb6
Rc1Nd7 g3 b3
Bg4 Qc7
Re1 Nb6
Ne4 Bxe4
Qxe4 Qd6
Qd3 Qd5
Re5 Qc4
Qe4 c5
Re8Rxe8
Qxe8 cxd4 Z0Z0ZpZp Z0Z0Z0A0 pZqo0ZBZ OpZ0Z0O0 Z0Z0Z0J0 a b c d e f g h
The position is getting sharp again, with Black havinggained a passed pawn, and White making threats around theBlack king.
Be7 Qc1+
Kg2 Qxh6
Bc5 Z0Z0ZpZp Z0A0Z0Z0 pZ0o0ZBZ OpZ0Z0O0 Z0Z0Z0Z0 a b c d e f g h
A critical moment, and a decision which shows just howvaluable the advanced pawns are in this chess variation. Nor-mally it would make sense to save the knight, but AlphaZerodecides to keep the pawn instead, and rely on promotionthreats coupled with checks on d5.
39. . . d3 Bxb6 Qg5
Bd1 Qd5+
Kh2 Qe6 Z0Z0ZpZp Z0Z0Z0Z0 pZ0Z0Z0Z OpZpZ0O0 Z0ZBZ0Z0 a b c d e f g h
Being a piece down, Black offers an exchange of queens,an unusual sight, but tactically justified – Black is alsothreatening to capture on a3, and that threat is hard to meet.White can’t passively ignore the capture and defend the b2pawn with the bishop, because Black could capture on b2,offering the piece for the second time – and then follow upby an immediate a3, knowing that bxa3 would allow forb1=Q. In addition, Black could retreat the bishop instead ofcapturing on b2, to make room for a2 bxa3 and again b1=Q.So, it’s again a torpedo move that makes a difference andjustifies the tactical sequence.
Qb5 Qf6
Kg2 h5
Be3 Qxb2
Qxa4 Qxa3
Qxb3 Qxb3
Bxb3 ssessing Game Balance with AlphaZero Z0Z0ZpZ0 Z0Z0Z0Zp ZBZpA0O0 Z0Z0Z0Z0 a b c d e f g h
White is a piece up for two pawns, and has the bishop pair.Yet, Black is just in time to use a torpedo move to shut theWhite king out and exchange a pair of pawns on the h-file(by another torpedo move).
48. . . g4 Bd2 Kg7
Kf1 f5
Ke1 Be7
Bc4 h3 Z0Z0a0j0 Z0Z0ZpZ0 Z0ZpZ0Op Z0Z0J0Z0 a b c d e f g h gxh4 Bxh4
Kf1 Bg5
Bc3+ Bf6
Bd2 Bg5
Bc3+ Bf6
Bxf6 Kxf6
Bxd3 f4
Be4 g3
Bg2gxf2
Ba8 Kf5
Kxf2 Kg4
Bb7 Kh5
Ba6 f3
Kxf3
Playing from a predefined Nimzo-Indian openingposition (the first 3 moves for each side). The remainingmoves follow best play, at roughly one minute per move. d4 (book) Nf6 (book) 2. c4 (book) e6 (book) 3. Nc3 (book)
Bb4 (book) 4. e3 Bxc3 bxc3 d6 Nf3 O-O Ba3 Re8 e5 rmblrZkZ opo0Zpop Z0Z0O0Z0 A0O0ZNZ0 PZ0Z0OPO S0ZQJBZR a b c d e f g h
Already we see the first torpedo move, keeping the initiative.
8. . . dxe5 Nxe5 Nbd7
Bd3 c5
O-O Qa5
Bb2Nxe5 dxe5 Nd7
Re1 f5
Qh5 Re7 rZbZ0ZkZ opZns0op l0o0OpZQ Z0OBZ0Z0 PA0Z0OPO S0Z0S0J0 a b c d e f g h
Qh4 Re8
Qg3 Nf8
Bc1 Kh8
Be2 Bd7
Rd1Bc6
Bh5 h6
Bxe8 Rxe8
Rd6 f3 opZ0Z0o0 l0o0O0Z0 Z0O0ZpL0 PZ0Z0OPO S0A0Z0J0 a b c d e f g h
Here we see an effect of another torpedo move, after the ssessing Game Balance with AlphaZero exchange sacrifice earlier, taking over the initiative andcreating a dangerous pawn. Bd2 fxg2 f4 Qc7
Be3 Qf7
Bxc5 Qf5
Rxc6bxc6
Bxa7 Ng6
Be3 Ra8 a4 Qd3 rZ0Z0Z0j Z0Z0Z0o0 Z0Z0O0Z0 PZPZ0O0Z Z0OqA0L0 S0Z0Z0J0 a b c d e f g h
Re1 Kh7 a5 Rxa5
Bb6 Qxg3 hxg3 Ra2 f5exf5 Z0Z0Z0ok Z0Z0OpZ0 Z0O0Z0O0 rZ0Z0ZpZ Z0Z0S0J0 a b c d e f g h
The following move shows the power of advanced pawns – e6!, in order to create a threat of e8=Q, so Black hasto block with the knight. If instead e7, Black respondsby first giving the knight for the pawn –
37. . .
Nxe7, andthen after
Rxe7 follows it up with
38. . . h4!, similar tothe game continuation. e6 Ne7
Bc5 h4
Bxe7 hxg3
Re3 f4 Z0Z0A0ok Z0Z0Z0Z0 Z0O0S0o0 rZ0Z0ZpZ Z0Z0Z0J0 a b c d e f g h and Black manages to force a draw, as the pawns are justtoo threatening.
Rf3 Ra1+
Kxg2 Ra2+
Kg1 Ra1+
Kg2 Ra2+
Kh1 Ra1+
Kg2
The game starts from a predefined Ruy Lopez open-ing position (the first 5 plies). The remaining moves followbest play, at roughly one minute per move. e4 (book) e5 (book) 2. Nf3 (book)
Nc6 (book) 3.
Bb5 (book) a6 Bxc6 dxc6 O-O f6 d4 exd4 Nxd4 Bd6 Be3 c5 Ne2 Ne7
Nbc3 b6
Qd3 Be6
Rad1Be5
Nd5 O-O
Bf4 c6
Nxe7+ Qxe7
Bxe5 fxe5 a3 rZ0Z0skZ Z0Z0l0op popZbZ0Z Z0o0o0Z0 O0ZQZ0Z0 Z0ZRZRJ0 a b c d e f g h
Here comes the first torpedo move (b6-b4), gaining spaceon the queenside.
17. . . b4 a4 h6 Qe3 Rad8 f3 a5 f5 ssessing Game Balance with AlphaZero Z0Z0l0o0 o0o0oPZ0 Po0ZPZ0Z Z0Z0L0Z0 Z0ZRZRJ0 a b c d e f g h
Here we see an effect of another torpedo move, f3-f5, ad-vancing towards the Black king.
21. . .
Bc4
Rde1 Qd6
Rd1 Qxd1
Rxd1 Rxd1+
Kf2 Rd4 g3 Rfd8 Z0Z0Z0o0 o0o0oPZ0 PobsPZ0Z Z0Z0L0O0 Z0Z0Z0Z0 a b c d e f g h
Nxd4 cxd4
Qd2 Rd6 g5 Z0Z0Z0o0 o0Z0oPO0 PoboPZ0Z Z0Z0Z0Z0 Z0Z0Z0Z0 a b c d e f g h
White uses a torpedo move to generate play on the kingside.
29. . . hxg5 b3 Bd5 Z0Z0Z0o0 o0ZboPo0 Po0oPZ0Z ZPZ0Z0Z0 Z0Z0Z0Z0 a b c d e f g h
The Black bishop can’t be taken, due to a torpedo threate3+!
Qxg5 Bxe4 f7+ Z0Z0ZPo0 o0Z0o0L0 Po0obZ0Z ZPZ0Z0Z0 Z0Z0Z0Z0 a b c d e f g h
And yet another torpedo strike, in order to capture on e5.
32. . .
Kxf7
Qxe5 Rf6+
Ke1 Bxc2
Qxd4 Bxb3
Qd7+ Kg8
Qd8+ Kh7
Qd3+ g6
Qxb3 c5 ssessing Game Balance with AlphaZero Z0Z0Z0Zk o0o0Z0Z0 Po0Z0Z0Z ZQZ0Z0Z0 Z0Z0J0Z0 a b c d e f g h
White ends up with the queen against the rook and twopawns, but this ends up being a draw, as the pawns aresimply too fast and need to remain blocked. Normally thequeen on b3 would prevent the c5 pawn from moving, but ac5-c3 torpedo move shows that this is no longer the case!
Kd1 c3
Qc4 Rf5
Kc2 Rf2+
Kb1 Rb2+
Kc1Rd2
Kb1 Kh6 h3 Rd1+
Kc2 Rd2+
Kb1 Rd1+
Kc2 Rh1
Qf4+ Kh7
Qc7+ Kh6
Qb8 Rf1
Qh8+ Kg5
Qd8+ Kh6 h4 Rf2+
Kb1 Rf1+
Kc2 Rf2+
Kb1 Rf1+
Kc2
The position below, with Black to move, is takenfrom a game that was played with roughly one minute permove: Z0Z0Z0ok QZ0Z0Z0o Z0Z0OpZ0 pZ0ZpZqZ Z0Z0O0O0 PZ0Z0O0O A0a0Z0J0 a b c d e f g h
A dynamic position from an endgame reached in one of theAlphaZero games. White has an advanced passed pawn,which is quite threatening – and Black tries to respond bycreating threats around the White king. To achieve that,Black starts with a torpedo move:
31. . . h4 e6 hxg3 hxg3 Bxe3 Z0Z0Z0ok QZ0ZPZ0Z Z0Z0ZpZ0 pZ0ZpZqZ Z0Z0a0O0 PZ0Z0O0Z A0Z0Z0J0 a b c d e f g h
White is one torpedo move away from queening, but has tofirst try to safeguard the king.
Be5 Qd1+
Qf1 Bxf2+ Z0Z0Z0ok Z0Z0ApZ0 pZ0ZpZ0Z Z0Z0Z0O0 PZ0Z0a0Z Z0ZqZQJ0 a b c d e f g h
Black is in time, due to the torpedo threats involving thee-pawn.
Kxf2 e3+
Kxe3 Qxf1 Z0Z0Z0ok Z0Z0ApZ0 pZ0Z0Z0Z Z0Z0J0O0 PZ0Z0Z0Z Z0Z0ZqZ0 a b c d e f g h ssessing Game Balance with AlphaZero Black captures White’s queen, but White creates a new one,with a torpedo move. e8=Q Qe1+
Kd3 Qb1+
Kc3 Qa1+
Kb4 Qxa2 Z0Z0Z0ok Z0Z0ApZ0 pJ0Z0Z0Z Z0Z0Z0O0 qZ0Z0Z0Z Z0Z0Z0Z0 a b c d e f g h
An interesting endgame arises, where White is up a piece,given that Black had to give away its bishop in the tacticsearlier, and Black will soon only have a single pawn inreturn. Yet, after a long struggle, AlphaZero manages todefend as Black and achieve a draw.
Qe7 Qb3+
Ka5 Qg8
Kxa4 Kg6
Bf4 Qc4+
Ka5 Qd5+
Kb4 Qd4+
Kb3 Qd3+
Kb2 Qd4+
Kc2 Qd5
Qe3 Qc4+
Kd2 Qb4+
Kd3 Qb3+
Kd4 Qb4+
Kd5 Qb5+
Kd6 Qa6+
Kd7 Qb5+
Ke7 Qb7+
Kd8 Qd5+
Kc7 Qc4+
Kb6 Qb4+
Kc6 Qa4+
Kb7 Qd7+
Kb6 Kh5
Qf3+ Kg6
Kc5 Qa7+
Kb4 Qb6+
Kc3 Qa5+
Kc2 Qa4+
Kd2 Qa5+
Ke2 Qb5+
Kf2 Qb2+
Kg1 Qb1+
Kg2 Qb2+
Kh3 Qc2
Bd6 Qc1
Bf4 Qc2
Qe3 Qc6
Qe1 Kf6
Kh4 Kg6
Kh3 Kf6
Kh2Qc2+
Bd2 Qd3
Bf4 Qd5
Qe2 Qd4
Bd2 Kf7
Bg5 Qd5
Bc1 Qc6
Bf4 Kf6
Qd3 Qe6
Bd2Kf7
Kh3 Qf6
Qd5+ Ke7 g5 fxg4+
Kxg4Qe6+
Qxe6 Kxe6
The position below, with Black to move, is takenfrom a game that was played with roughly one minute permove: Z0oqZ0op ZpZPZpZ0 Z0ZQO0O0 Z0SRZ0J0 a b c d e f g h
A position from one of the AlphaZero games, illustratingthe utilization of pawns in a heavy piece endgame. Theb-pawn is fast, and it gets pushed down the board via atorpedo move.
26. . . h5 h4 b3 Qc4 Rb7
Rb1 Qb5
Qd4 c6 dxc6 ZrZ0Z0o0 ZqZ0ZpZp ZpZ0O0O0 ZRZRZ0J0 a b c d e f g h
Unlike in Classical chess, this capture is possible, eventhough it seemingly hangs the queen. If Black were to cap-ture it with the rook, the c-pawn would queen with check ina single move! The threat of c8=Q forces Black to recapturethe pawn instead.
31. . .
Rxc6 e5 fxe4
Qxe4 Rb8
Rb2 Qc4
Qe5and the game soon ended in a draw.
The first ten moves for White and Black were sam-pled randomly from AlphaZero’s opening “book”, with theprobability proportional to the time spent calculating eachmove. The remaining moves follow best play, at roughlyone minute per move. d4 Nf6 c4 e6 Nc3 d5 Nf3 a6 e3 b6 g3 dxc4 ssessing Game Balance with AlphaZero e5 Nd5 Bxc4 Be7 O-O Bb7
Re1 h6 a3 b5
Bb3 Nxc3 bxc3 a4 rm0lkZ0s Zbo0apo0 ZpZ0O0Z0 pZ0O0Z0Z OBO0ZNO0 S0AQS0J0 a b c d e f g h
In the early stage of the game, we see White using a torpedoe3-e5 move to expand in the center and Black respondingby an a6-a4 torpedo move to gain space on the queenside.
Bc2 Bd5
Qe2 c6
Nd2 Qa5
Ne4 Nd7
Bd2Qc7
Qg4 g6 h3 O-O-O
Qf4 Nb6 c5 Z0l0apZ0 ZpObO0Z0 pZ0ONL0Z O0Z0Z0OP S0Z0S0J0 a b c d e f g h
White moves forward with a c3-c5 torpedo move.
22. . .
Nc4
Bc3 Rdf8
Nd6+ Bxd6 exd6 g5 Z0l0ZpZ0 ZpObZ0o0 pZnO0L0Z O0A0Z0OP S0Z0S0J0 a b c d e f g h
Qf6 Qd8
Qxd8+ Rxd8
Bd3 Rdg8 f3 h5
Be4 Re8
Bd3 Rh6
Rf1 f5
Bxc4 Bxc4
Rf2Bd5
Kh2 Rg6
Rg1 Reg8
Bd2 R6g7
Bb4 Kd7 f4 gxf4
Rxf4 Kc8
Be1 b3 Z0Z0Z0s0 Z0ObZpZp pZ0O0S0Z OpZ0Z0OP Z0Z0A0S0 a b c d e f g h
Black uses two consecutive torpedo moves (b5-b3, a4-a2)on the queenside to create a dangerous passed pawn on a2. axb4 a2
Bc3 Ra7
R4f1 Kd8
Ra1 Kd7
Bb2Ra4
Bc3 Ra3
Rac1 Be4 h4 Rg4
Bd2 f3 ssessing Game Balance with AlphaZero Z0ZkZ0Z0 Z0O0Z0Zp s0Z0ZpO0 pZ0A0Z0J Z0S0Z0S0 a b c d e f g h
Black uses another torpedo move (f5-f3) to advance furtheron the kingside and create another passed pawn.
Rf1 Rg8
Ra1 Rga8
Rf2 Rb3
Kh3 Rb1
Bc3Bd5 g4 Rb3
Be1 hxg4+
Kg3 Rb1
Bc3 Rb3
Bd2 Rb1
Bc3 Rb3
Bd2 Rb2 h6 rZ0Z0Z0Z Z0ZkZ0Z0 Z0ObZ0Z0 Z0Z0ZpJ0 ps0A0S0Z S0Z0Z0Z0 a b c d e f g h
White advances the h-pawn with an h4-h6 torpedo move,seeking counterplay.
63. . .
Rg8
Raf1 Bc4 h7 Rf8
Rh1 Rh8
Bc3Rxf2
Kxf2 g2 Z0ZkZ0ZP Z0O0Z0Z0 Z0A0ZpZ0 pZ0Z0JpZ Z0Z0Z0ZR a b c d e f g h
The torpedo move g4-g2 forces the White rook away fromthe h-file.
Re1 Rxh7 b6 Z0ZkZ0Zr Z0O0Z0Z0 Z0A0ZpZ0 pZ0Z0JpZ Z0Z0S0Z0 a b c d e f g h
White needs to generate immediate counterplay, and doesso via b4-b6, another torpedo move. White then uses a b6-b8=Q torpedo move to promote to a queen in the next move,demonstrating how fast the pawns are in this variation ofchess.
70. . .
Rh1 b8=Q Rf1+ ssessing Game Balance with AlphaZero Z0ZkZ0Z0 Z0O0Z0Z0 Z0A0ZpZ0 pZ0Z0JpZ Z0Z0SrZ0 a b c d e f g h
Rxf1 gxf1=Q+
Kg3 and the game eventually endedin a draw due to mutual threats and ensuing checks.
This game was an experi-ment combining the No-castling chess with Torpedo chess,resulting in a highly tactical position. The first ten moves forWhite and Black were sampled randomly from AlphaZero’sopening “book”, with the probability proportional to thetime spent calculating each move. The remaining movesfollow best play, at roughly one minute per move. rZ0Z0j0Z o0Z0m0Zb Z0ZpZ0o0 rOpO0A0Z Z0Z0ZPZ0 Z0Z0Z0SR a b c d e f g h
Here White executes a stunning ’double attack’:27. Qc2!! Kg8Black can’t afford to capture the Queen, due to the powerfulattack following 27... Bxc2 28. h8=Q+. White also had toassess the consequences of 27... gxf428. Qxa4 Qxd4+ 29. Kxg3 gxf4+ 30. Kh2 Qf2 31. Rf1Qg3+ 32. Kg1 rZ0Z0ZkZ o0Z0m0Zb Z0ZpZ0Z0 QOpZ0o0Z Z0Z0ZPl0 Z0Z0ZRJR a b c d e f g h rZ0Z0Z0j Z0L0Z0Z0 pZ0Z0onl Z0Z0ZbZ0 Z0ZpZPZ0 Z0Z0S0Z0 a b c d e f g h
43. Qc1 Qxc1 44. Rxc1 Ne7 45. Ne3 Bg6 46. Ra1 Nc6 47.Rh4+ Kg7 48. b5 Nb8 49. Rc4 Bf7 50. Rc7 f4 51. Nd1 a452. Nc2 a2 53. Nxd3 Kf6 54. Rc8 Ra3 55. Nxf4 Nd7 56.Ne2 Ne5 57. b7 Rxf3+ 58. Kg2 Rb3 59. b8=Q Rxb8 60.Rxb8 and White went on to win the game easily. 1-0
B.6. Semi-torpedo
In Semi-torpedo chess, we consider a partial extension to therules of pawn movement, where the pawns are allowed tomove by two squares from the 2nd/3rd and 6th/7th rank forWhite and Black respectively. This is a restricted version ofanother variant we have considered (Torpedo chess) wherethe option is extended to cover the entire board. Yet, eventhis partial extension adds lots of dynamic options and herewe independently evaluate its impact on the arising play.B.6.1. M
OTIVATION
As with Torpedo chess, the motivation in extending the pos-sibilities for rapid pawn movement lies in adding dynamic, ssessing Game Balance with AlphaZero attacking options to the middlegame. Yet, given that it isonly a partial extension, adding an extra rank for each sidefrom which the pawns can move by two squares, its impacton endgame patterns is much more limited.B.6.2. A SSESSMENT
The assessment of the Semi-torpedo chess variant, as pro-vided by Vladimir Kramnik: “ Compared to Classical chess, the pawns thathave been played to the 3rd/6th rank becomemuch more useful, which manifests in severalways. First, prophylactic pawn moves to h3/h6and a3/a6 now allow for a subsequent torpedopush. Having played h3 for example, it is now pos-sible to play the pawn to h5 in a single move. Thisalso means, if the goal was to push the pawn to h5in two moves, that there are two ways of achiev-ing it – either via h4 and h5 or via h3 and h5 –and doing the latter does not expose a weaknesson the g4 square and can thus be advantageous.Secondly, fianchetto setups now allow for addi-tional dynamic options. The g3 pawn can now bepushed to g5 in a single move, to attack a knighton f6 – and vice versa. Thirdly, openings whereone of the central pawns is on the 3rd/6th rankchange – consider the Meran for example – thee3 pawn can now go to e5 in a single move.Theory might change in other openings as well,like for instance the Ruy Lopez with a7-a6, giventhat there would be some lines where the tor-pedo option of playing a6-a4 might force Whiteto adopt a slightly different setup. AlphaZero alsolikes playing g6 early for Black, with a threat ofg4 in some lines, aimed against a knight on f3 ifWhite starts expanding in the center. As anotherexample, consider a pretty standard opening se-quence in the Sicilian defence: 1. e4 c5 2. Nf3Nc6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 e5 6. Ndb5 d6– it turns out that here 7. Bg5 no longer keeps theadvantage, because of 7. . . a6 8. Na3 followed upby a torpedo move 8. . . d4: rZblka0s ZpZ0Zpop pZnZ0m0Z Z0Z0o0A0 M0M0Z0Z0 POPZ0OPO S0ZQJBZR a b c d e f g h
Here, the game could continue 9. exd5 Bxa310. bxa3 Nd4 11. Bd3 Qa5, and the position isassessed as equal by AlphaZero. This variationillustrates nicely how the torpedo moves providenot only an additional attacking option for White,but also additional equalizing options for Black,depending on the position.Semi-torpedo chess seems to be more decisivethan Classical chess, and less decisive than Tor-pedo chess. It is an interesting variation, to bepotentially considered by those who like the gen-eral middlegame flavor of Torpedo chess, but areunwilling to abandon existing endgame theory. ” B.6.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under Semi-torpedo chess, when playing with roughly one minute permove from a particular fixed first move. Note that theseare not purely deterministic, and each of the given lines ismerely one of several highly promising and likely options.Here we give the first 20 moves in each of the main lines.
Main line after e4
The main line of AlphaZero after e4in Semi-torpedo chess is: e4 (book) c5 c3 Nf6 e5 Nd5 Bc4 e6 Nf3 Be7 d4 d6 O-O O-O Re1 Nc6 exd6 Qxd6 dxc5Qxc5 Nbd2 b6 b4 Qd6
Qc2 Bb7 a3 Nf6
Ne4 Qc7
Bd3 h6 c5 bxc5
Nxc5 Bxc5 bxc5Na5
Ne5 Rac8 ssessing Game Balance with AlphaZero obl0Zpo0 m0O0M0Z0 O0ZBZ0Z0 S0A0S0J0 a b c d e f g h and after
Bb2 White would have compensation for thepawn. There are also tactical resources in this position, forinstance White could consider a more forcing line of play –
Bxh6!? gxh6
Qd2 Kg7
Re3 Rh8
Rg3+ Kf8
Rae1 h4
Rg7! Kxg7
Qg5+ Kf8
Qxf6 Rg8
Ng6+ Rxg6
Bxg6 – potentially leading to a draw byperpetual check.
Main line after d4
The main line of AlphaZero after d4in Semi-torpedo chess is: d4 (book) Nf6 c4 e6 e3 d5 cxd5 exd5 Nc3 Bd6 Bd3 O-O Nge2 a6 O-O Re8 b3 Nc6 Ng3 Bg4 f3 Bc8 a3 Ne7
Bb2 h6
Qd2 c6 e5 dxe4
Ncxe4 Ned5
Nxd6 Qxd6
Rae1 Qd8
Rxe8+Nxe8
Re1 Bd7 rZ0lnZkZ ZpZbZpo0 pZpZ0Z0o Z0ZnZ0Z0 OPZBZPM0 Z0Z0S0J0 a b c d e f g h
Main line after c4
The main line of AlphaZero after c4in Semi-torpedo chess is: c4 (book) c5 g3 g6 Bg2 Bg7 e3 e6 d4 cxd4 exd4 Ne7 Nc3 O-O Nge2 d5 cxd5 Nxd5 h4Bd7 Nxd5 exd5
Be3 Re8
Nc3 Nc6
O-O Be6 h5 h6 hxg6 fxg6
Qd2 Kh7
Ne2 Qf6
Nf4Bf7
Nxd5 Bxd5 rZ0ZrZ0Z opZ0Z0ak Z0ZbZ0Z0 Z0Z0A0O0 PO0L0OBZ S0Z0ZRJ0 a b c d e f g h
B.6.4. I
NSTRUCTIVE GAMES
Game AZ-18: AlphaZero Semi-torpedo vs AlphaZeroSemi-torpedo
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. d4 Nf6 c4 e6 e3 d5 cxd5 exd5 Nc3 Bd6 Nb5c6 Nxd6+ Qxd6 Bd3 Ne4 f3 Qb4+ Bd2 Nxd2
Qxd2 Qd6
Ne2 O-O
O-O Nd7 g4 Nf6
Kg2Bd7
Ng3 Kh8
Rae1 Rae8
Bb1 Ng8 h3 Ne7 f5 opZbmpop Z0ZpZPZ0 Z0Z0O0MP PO0L0ZKZ ZBZ0SRZ0 a b c d e f g h
Here we see the first torpedo move of the game, f3-f5, claim-ing space before Black has the chance to play f5.
20. . . f6 a3 b6 Nh5 Rb8
Qf2 b4 ssessing Game Balance with AlphaZero o0Zbm0op Z0ZpZPZN O0Z0O0ZP ZBZ0SRZ0 a b c d e f g h
Black utilizes a torpedo move of its own, b6-b4, to initiatecounterplay on the queenside.
Qf4 Nc8 a4 b3
Qf3 Nb6
Nf4 Rbe8
Re2c4And c6-c4 comes as another torpedo move, speeding up thequeenside expansion. White chooses not to take en passant ,but to play a5 instead in reply. a5 Nc8 o0ZbZ0op O0ZpZPZ0 ZpZ0OQZP ZBZ0ZRZ0 a b c d e f g h
Rfe1 Ne7
Rd1 Rc8 e5 o0Zbm0op O0ZpOPZ0 ZpZ0ZQZP ZBZRZ0Z0 a b c d e f g h
White expands in the center with another torpedo move,e3-e5.
32. . . dxe4
Bxe4 Rfe8
Ne6 Nc6
Bxc6 Bxc6 d5 Ba8
Re3 Bb7 h5 obZ0Z0op O0ZPZPZP ZpZ0SQZ0 Z0ZRZ0Z0 a b c d e f g h
Here comes another torpedo advance, h3-h5, creating threatson the kingside.
38. . . h6 Kh3 Qd7
Rc3 Re7
Qf1 Qb5 d6 Rd7
Qf4 Ba6 ssessing Game Balance with AlphaZero o0ZrZ0o0 bZ0ONo0o OqZ0ZPZP ZpS0Z0ZK Z0ZRZ0Z0 a b c d e f g h
Qd2 Qe5
Rg3 Rc6
Nf8 o0ZrZ0o0 bZrO0o0o O0Z0lPZP ZpZ0Z0SK Z0ZRZ0Z0 a b c d e f g h
46. . .
Rcxd6
Qb4 Qb5
Qxd6 Rxd6
Rxd6 Qb8
Ng6+ Kh7
Rxa6 Qb7 oqZ0Z0ok RZ0Z0oNo O0Z0ZPZP ZpZ0Z0SK Z0Z0Z0Z0 a b c d e f g h
Re3 Qh1+
Kg3 Qg1+
Kf3 Qf1+
Kg3 Qg1+
Kf3 Qf1+
Ke4 Qg2+
Kf4 Qf2+
Ke4 Qg2+
Kf4 Qf2+
Ke4 Qg2+
The position below, with Black to move, istaken from a game that was played with roughly one minuteper move: ZbZnZ0a0 o0ZPoPZn PZ0Z0Z0Z Z0MBA0OP Z0ZRJ0ZR a b c d e f g h
18. . .
Bxd5
Nde4 Bxe4
Qb3+ Kh8
Nxe4 d4 Z0ZnZ0a0 o0Z0oPZn PZ0oNZ0Z ZQZBA0OP Z0ZRJ0ZR a b c d e f g h
Here, a torpedo move (d6-d4) unleashes a tactical sequence.
Nd6 Rf8
Be2 Rc7 fxg6 Nc5 ssessing Game Balance with AlphaZero Z0s0Z0a0 o0m0o0Zn PZ0o0Z0Z ZQZ0A0OP Z0ZRJ0ZR a b c d e f g h
Qxb6 Nxa4
Nf7+ Z0s0ZNa0 o0Z0o0Zn nZ0o0Z0Z Z0Z0A0OP Z0ZRJ0ZR a b c d e f g h
26. . .
R8xf7
Qxa5 Rfd7
Qxa4 dxe3 Z0srZ0a0 Z0Z0o0Zn QZ0Z0Z0Z Z0Z0o0OP Z0ZRJ0ZR a b c d e f g h
Bxh5 Rxd1+
Qxd1 exf2+
Kxf2 Rd7
Qc1Rd2+
Ke1 Rd3
Kf2 Rd2+
Ke1 Rd3
Rg1 e4 Z0Z0Z0a0 Z0Z0Z0ZB Z0ZrZ0OP Z0L0J0S0 a b c d e f g h
Rg2 Qa5+
Kf1 Qf5+
Qf4 Qxf4+ gxf4 Rxh3
Bd1 Rh4
Kf2 Rxf4+
Ke3 Rf1
Bg4 Rf6 witha draw soon to follow.
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. d4 Nf6 Nf3 d5 c4 e6 a3 dxc4 e3 c6 Bxc4 b5 Bd3 Bb7 Nc3 a6 e5 rm0lka0s ZbZ0Zpop pZpZpm0Z ZpZ0O0Z0 O0MBZNZ0 S0AQJ0ZR a b c d e f g h
Here we see another typical central torpedo move (e3-e5),claiming space.
9. . .
Nd5
Be4 Be7 h3 Nxc3 bxc3 Nd7
O-ORb8
Qe2 c4 ssessing Game Balance with AlphaZero ZbZnapop pZ0ZpZ0Z ZpZ0O0Z0 O0O0ZNZP S0A0ZRJ0 a b c d e f g h
Black uses a torpedo move as a counter (c6-c4), expandingon the queenside.
Bxb7 Rxb7
Qe4 Rc7
Qg4 g6 a5 Z0snapZp pZ0ZpZpZ OpZ0O0Z0 Z0O0ZNZP S0A0ZRJ0 a b c d e f g h
Another torpedo move follows (a3-a5), giving rise to a the-matic pawn structure.
18. . . h5 Qg3 Nb8 d5 Qxd5
Bg5 Qd8
Rad1Rd7
Bxe7 Qxe7
Ng5 O-O
Ne4 Rxd1
Rxd1Rd8
Rd6 Rxd6 exd6 Qd8
Qe5 Nd7
Qd4 Qh4and the game eventually ended in a draw.
The position below, with White to move,is taken from a game that was played with roughly oneminute per move: rZ0l0skZ ZpZbZpZ0 pZ0apZ0o Z0ZpMno0 Z0OBA0ZN PO0Z0OPO S0ZQS0J0 a b c d e f g h f3 Bxe5 dxe5 Nxe3
Rxe3 f5 rZ0l0skZ ZpZbZ0Z0 pZ0ZpZ0o Z0ZpOpo0 Z0OBSPZN PO0Z0ZPO S0ZQZ0J0 a b c d e f g h exf6 Qb6
Qc1 Nxf6
Nf2 e4 rZ0Z0skZ ZpZbZ0Z0 pl0Z0m0o Z0ZpZ0o0 Z0OBSPZ0 PO0Z0MPO S0L0Z0J0 a b c d e f g h
Here we see a torpedo move e6-e4 being used in a tacticalsequence in center of the board. fxe4 Rae8 e5 Ng4
Nxg4 Bxg4
Kh1 Rf2 b4 Ref8 ssessing Game Balance with AlphaZero ZpZ0Z0Z0 pl0Z0Z0o Z0ZpO0o0 Z0OBS0Z0 PZ0Z0sPO S0L0Z0ZK a b c d e f g h
Qe1 Be6
Re2 R2f4 a3 Kg7 h3 Qd8
Re3h5
Rd1 g4
Rd2 h4 hxg4 Qg5 ZpZ0Z0j0 pZ0ZbZ0Z Z0ZpO0l0 O0OBS0Z0 Z0Z0L0ZK a b c d e f g h
Rh3 Rxg4
Qe3 d4 cxd4 R8f4
Rf3 Bd5 ZpZ0Z0j0 pZ0Z0Z0Z Z0ZbO0l0 O0ZBLRZ0 Z0Z0Z0ZK a b c d e f g h
Rxf4 Qxf4
Qxf4 Rxf4
Kg1 Rxd4 and the gamesoon ended in a draw.
In the Pawn-back variation of chess, the pawns are allowedto move one square backwards, up to the 2nd/7th rank forWhite and Black respectively. In addition, if the pawn movesback to its starting rank, it is allowed to move by two squaresagain on its next move. In this particular implementation,the two-square pawn move is always allowed from the 2ndor the 7th rank, regardless of whether the pawn has movedbefore. A different implementation of this variation of chessmight consider disallowing this, though it is unlikely tomake a big difference. Because the pawns are allowed tomove backwards and pawn moves are now reversible in thisimplementation of chess, the 50 move rule is modified sothat 50 moves without captures lead to a draw, regardless ofwhether any pawn moves were made in the meantime.B.7.1. M
OTIVATION
In Classical chess, pawns that move forwards leave weak-nesses behind. Some of these remain long-term weaknesses,resulting in squares that can be easily occupied by the op-ponent’s pieces. If the pawns could move backwards, theycould come back to help fight for those squares and thereforereduce the number of weaknesses in a position. Allowingthe pawns to move backwards would therefore make it easierto push them forward, as the effect would not be irreversible.This might make advancing in a position easier, but equally,it could provide defensive options for the weaker side, suchas retreating from a less favourable situation and covering aweaknesses in front of the king.B.7.2. A
SSESSMENT
The assessment of the Pawn-back chess variant, as providedby Vladimir Kramnik: “ There are quite a few educational motifs in thisvariation of chess. The backward pawn movescan be used to open the diagonals for the bishops,or make squares available for the knights. Thebishops can therefore become more powerful, asthey are easier to activate. The pawns can bepushed in the center more aggressively than inclassical chess, as they can always be pulled back.Exposing the king is not as big of an issue, as thepawns can always move back to protect. Weaksquares are much less important for positionalassessment in this variation, given that they canalmost always be protected via moving the pawnsback.It was interesting to see AlphaZero’s strong pref-erence for playing the French defence underthese rules, the point being that the light-squaredbishop is no longer bad, as it can be developed ssessing Game Balance with AlphaZero via c8-b7 followed by a timely d5-d6 back-move.Other openings change as well. After the standard1. e4 e5 2. Nf3 Nc6, there comes a surprise: 3. c4! rZblkans opopZpop Z0Z0o0Z0 Z0Z0ZNZ0 PO0O0OPO SNAQJBZR a b c d e f g h
It is followed by 3. . . Bc5 4. e3 (a back-move!)Bb6 5. d4 d6 rZblkZns opo0Zpop Z0Z0o0Z0 Z0Z0ONZ0 PO0Z0OPO SNAQJBZR a b c d e f g h
Who would have guessed that we are on move 5,after the game having started with e4 e5?The Pawn-back version of chess allows for morefluid and flexible pawn structures and could po-tentially be interesting for players who like suchstrategic manoeuvring. Given that Pawn-backchess offers additional defensive resources, win-ning with White seems to be slightly harder, so thevariant might also appeal to players who enjoydefending and attackers looking for a challenge. ” B.7.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under Pawn-back chess, when playing with roughly one minute per move from a particular fixed first move. Note that these are notpurely deterministic, and each of the given lines is merelyone of several highly promising and likely options. Here wegive the first 20 moves in each of the main lines, regardlessof the position.
Main line after e4
The main line of AlphaZero after e4in Pawn-back chess is: e4 (book) e6 Nc3 d5 d4 Nf6 e5 Nfd7 f4 c5 Nf3a6 Be3 b5 f5 Nc6 fxe6 fxe6 e4 cxd4 Nxd4Nxd4
Qxd4 b4
Ne2 Nf6 exd5 Qxd5
Nf4Qxd4
Bxd4 Bd6
Nd3 a5
Be5 Ke7
Bxd6+Kxd6
O-O-O Ke7 rZbZ0Z0s Z0Z0j0op o0Z0Z0Z0 Z0ZNZ0Z0 POPZ0ZPO Z0JRZBZR a b c d e f g h
Main line after d4
The main line of AlphaZero after d4in Pawn-back chess is: d4 (book) d5 e3 Nf6 Nf3 e6 c4 Be7 b3 O-O cxd5 exd5 Bd3 Re8 Bb2 a5 O-O Bf8
Nc3c6
Qc2 b6
Ne2 Ra7
Rac1 Rc7
Rfe1 Bb4
Nc3 Ba6
Bxa6 Nxa6 h4 b7 a3 Bf8
Ne2Rc8
Nf4 Nc7 Zpm0Zpop o0ZpZ0Z0 OPZ0ONZ0 Z0S0S0J0 a b c d e f g h ssessing Game Balance with AlphaZero Main line after c4
The main line of AlphaZero after c4in Pawn-back chess is: c4 (book) e5 e3 c5 Nc3 Nc6 Nf3 f5 d4 e4 Nd2Nf6 d5 Ne5 Be2 g6 d4 Nf7 dxc5 Bxc5 a3 Bf8 b4 Bg7 Bb2 O-O
O-O d6 a4 Be6
Qb3 a5
Rfd1 b6 bxa5 bxa5
Qa3 e5 c5 Qb8 rl0Z0skZ Z0Z0Znap o0O0opZ0 PZ0Z0Z0Z L0M0O0Z0 S0ZRZ0J0 a b c d e f g h
B.7.4. I
NSTRUCTIVE GAMES
Game AZ-22: AlphaZero Pawn-back vs AlphaZeroPawn-back
The first ten moves for White and Black havebeen sampled randomly from AlphaZero’s opening “book”,with the probability proportional to the time spent calculat-ing each move. The remaining moves follow best play, atroughly one minute per move. e4 c6 d4 d5 e5 Bf5 h4 h5 c4 d6 rm0lkans opZ0opo0 Z0Z0ObZp Z0Z0Z0Z0 PO0Z0OPZ SNAQJBMR a b c d e f g h
Here we see d5-d6 as the first back-move of the game,challenging White’s (over)extended center – an option thatwould not have been available in classical chess. exd6 exd6 d5 Be7 Nc3 Bxh4 Be3 Qe7 g3 Bf6
Rxh5 Rxh5
Qxh5 Bg6
Qe2 Bxc3+ bxc3 Nd7 f3 O-O-O
Kf2 Ngf6 opZnlpo0 Z0ZPZ0Z0 Z0O0APO0 PZ0ZQJ0Z S0Z0ZBM0 a b c d e f g h
Black is putting pressure on d5, so White uses the back-move d5-d4 option to reconfigure the central pawn structure,rather than release the tension. d4 d5 c5 opZnlpo0 Z0OpZ0Z0 Z0O0APO0 PZ0ZQJ0Z S0Z0ZBM0 a b c d e f g h
Black and White repeat back-moves a couple of times. Eachtime that Black challenges the c5 pawn via a d5-d6 back-move, White responds by c5-c4, refusing to exchange onthat square.
18. . . d6 c4 d5 Rc1 Rh8 c5 d6 c4 d5
Bf4Qxe2+
Bxe2 dxc4
Bxc4 b5
Bf1 Nd5
Bd2N7b6 ssessing Game Balance with AlphaZero o0Z0Zpo0 ZpZnZ0Z0 Z0O0ZPO0 PZ0A0J0Z Z0S0ZBM0 a b c d e f g h
Here we see an example of how back-moves can help coverweak squares. Black is threatening to invade on the lightsquares on the queenside at an opportune moment, but Whiteutilizes a back-move d4-d3 and protects c4. This, however,enables Black to go forward and Black takes the opportunityto play c6-c5. d3 c5 f4 c4 o0Z0Zpo0 ZpZnZ0Z0 Z0OPZ0O0 PZ0A0J0Z Z0S0ZBM0 a b c d e f g h
White decides to keep retreating here and not give up thelight squares with a back-move c3-c2. c2 Rh2+ o0Z0Zpo0 ZpZnZ0Z0 Z0ZPZ0O0 PZPA0J0s Z0S0ZBM0 a b c d e f g h
At this point it should come as no surprise how White shouldrespond to the rook invasion using a back-move g3-g2! g2 c3
Be1 Bf5
Nf3 Rh6 g3 Rd6
Bg2 a6 a3 Na4
Ng5 f6
Ne4 Rd7
Kf3 Kc7
Bf2 Z0jrZ0o0 pZ0Z0o0Z ZpZnZbZ0 nZ0ZNO0Z O0oPZKO0 Z0S0Z0Z0 a b c d e f g h
40. . .
Bxe4
Kxe4 Kd6
Re1 Rc7
Kf5 Ne7+
Kg4 c4 d2 Z0s0m0o0 pZ0j0o0Z ZpZ0Z0Z0 nZpZ0OKZ O0Z0Z0O0 Z0Z0S0Z0 a b c d e f g h ssessing Game Balance with AlphaZero Here we see both Black and White having retreated from theinteraction on the queenside, Black via a back-move c3-c4and White by playing the d-pawn back to d2. The gamesoon ended in a draw.
45. . . f5+
Kf3 c3 d3 c4 d2 c3 d3 c4 g4cxd3 cxd3 Rc3 gxf5 Rxd3+
Kg4 Rd2
Re6+Kd7
Kf3 Rd3+
Kg4 Rd2
Kf3 Rd3+
Kg4 Rd2
The position below, with Black to move, istaken from a game that was played with roughly one minuteper move: rZblka0s ZpZ0Zpo0 oNZpZPZ0 PZ0ZpA0Z ZNZ0Z0O0 S0ZQZRJ0 a b c d e f g h
White is targeting c7 with the bishop and the knight, buthere Black plays a back-move, e4-e5. It initiates a longforced tactical sequence, showcasing that things can indeedget quite tactical in this variation of chess, depending on theline of play.
13. . . e5 e4 rZblka0s ZpZ0Zpo0 oNZpoPZ0 PZ0ZPA0Z ZNZ0Z0O0 S0ZQZRJ0 a b c d e f g h
AlphaZero decides to sacrifice a piece for the initiative!
14. . . exf4 exd5 Qb6+
Kh1 Na7
Qe1+ rZbZka0s mpZ0Zpo0 oNZPZPZ0 PZ0Z0o0Z ZNZ0Z0O0 S0Z0LRZK a b c d e f g h
17. . .
Kd8
N5d4 Bd7
Nxa5 fxg3
Rd1 Bb4
Nxb7+ rZ0j0Z0s mNZbZpo0 Z0ZPZPZ0 Pa0M0Z0Z Z0Z0Z0o0 Z0ZRLRZK a b c d e f g h
Sacrificing another piece!
21. . .
Qxb7
Qxg3 Rg8
Ne6+ rZ0j0ZrZ mqZbZpo0 Z0ZPZPZ0 Pa0Z0Z0Z Z0Z0Z0L0 Z0ZRZRZK a b c d e f g h
Third consecutive piece sacrifice by White! ssessing Game Balance with AlphaZero
23. . . fxe6 dxe6 Qc7
Bxa8 Qxg3 hxg3 Kc7
Bg2 Bc6 m0j0Z0o0 Z0Z0ZPZ0 Pa0Z0Z0Z Z0Z0Z0O0 Z0ZRZRZK a b c d e f g h
It’s time to take stock – White has a rook and 4 pawns for 3pieces, a very unusual material imbalance.
Rf4 Rb8
Rc4 Bd6
Rb1 Kb6
Re1 Nh5 g4Nf6
Re3 Rc8
Rec3 Be5 a5+ Ka6
Rxc6+ Rxc6
Rxc6+ Nxc6
Bxc6 Bxb2 e7 Kxa5 Z0Z0O0o0 j0Z0ZPZ0 Z0Z0Z0Z0 Z0Z0Z0ZK a b c d e f g h
And the game soon ended in a draw.
The first ten moves for White and Black havebeen sampled randomly from AlphaZero’s opening “book”,with the probability proportional to the time spent calculat-ing each move. The remaining moves follow best play, atroughly one minute per move. e4 e6 Nc3 d5 d4 Nf6 e5 Nfd7 f4 c5 Nf3 a6 a3 Nc6 Be3 b5 Ne2 Bb7 c3 rZ0lka0s ZbZnZpop pZnZpZ0Z ZpopO0Z0 O0O0ANZ0 S0ZQJBZR a b c d e f g h
This looks like a pretty normal French position, but herecomes Black’s main equalizing resource, a back move d5-d6! Maybe that’s all that was needed to make the French anundeniably good opening for Black?
10. . . d6 rZ0lka0s ZbZnZpop pZnopZ0Z Zpo0O0Z0 O0O0ANZ0 S0ZQJBZR a b c d e f g h
This completely changes the nature of the position, asthe center is suddenly not static and Black’s light-squaredbishop can find good use on the a8-h1 diagonal.
Ng3 dxe5 fxe5 Qb6
Bf2 Rd8
Qb1 cxd4 cxd4 b4
Be2 bxa3 bxa3 Qa5+
Kf1 Rb8 h4 Be7
Kg1 O-O
Qd3 Rfc8
Kh2 Qd8 ssessing Game Balance with AlphaZero ZbZnapop pZnZpZ0Z Z0Z0O0Z0 O0ZQZNM0 S0Z0Z0ZR a b c d e f g h
Here AlphaZero prefers a solid back-move h4-h3 to a furtherexpansion with h5. h3 Na5
Rhc1 Nf8
Qe3 ZbZ0apop pZ0ZpZ0Z m0Z0O0Z0 O0Z0LNMP S0S0Z0Z0 a b c d e f g h
The a6 pawn is under pressure from the e2 bishop, andsimply moves back to a7. The game soon fizzles out to adraw.
25. . . a7 a4 Ng6 Rab1 Rxc1
Qxc1 Rc8
Qd1Ba8
Ba6 Rb8
Bf1 Rxb1
Qxb1 Bc6
Bb5 Qb8
Qd3 Qb7
Ne2 h6
Bg3 Be4
Qe3 Bb4
Bf2Bd5
Qd3 Be4
Qe3 Bc6
Qd3 Be7
Bg3 Be4
Qe3 Bb4
Bf2
The first ten moves for White and Black havebeen sampled randomly from AlphaZero’s opening “book”,with the probability proportional to the time spent calculat-ing each move. The remaining moves follow best play, atroughly one minute per move. e4 e6 Nc3 d5 d4 Nf6 e5 Nfd7 Be3 c5 f4 a6 Nf3 b5 f5 Nc6 fxe6 fxe6 Bd3 g6
O-O cxd4
Nxd4 Ndxe5
Kh1 Ne7
Rf6 Bg7 rZblkZ0s Z0Z0m0ap pZ0ZpSpZ ZpZpm0Z0 Z0MBA0Z0 POPZ0ZPO S0ZQZ0ZK a b c d e f g h
Nxe6 Bxe6
Rxe6 O-O
Bg5 Ra7
Be2 Nf7
Bh4 g5 s0Z0mnap pZ0ZRZ0Z ZpZpZ0o0 Z0M0Z0Z0 POPZBZPO S0ZQZ0ZK a b c d e f g h
Here we see that moves like g5, that would potentially other-wise be quite weakening, are perfectly playable, given thatthe g-pawn can (and soon will) move back to g6, and in themeantime the threatening bishop is forced to move back andunpin the Black knight on e7.
Bf2 d4
Ne4 Nf5
Qd3 g6 ssessing Game Balance with AlphaZero s0Z0Znap pZ0ZRZpZ ZpZ0ZnZ0 Z0ZQZ0Z0 POPZBAPO S0Z0Z0ZK a b c d e f g h
After moving the pawn back to g6 with a back-move, Blacksafeguards the kingside, justifying the previous g5 pawnpush, which was helpful in achieving development. g4 Ne3
Bxe3 dxe3
Qxe3 Re7
Rxe7 Qxe7 a4 Nd6
Bd3 Bxb2
Rb1 Qe5 axb5 axb5
Qe2 Ba3 Z0Z0Z0Zp ZpZ0l0Z0 a0ZBZ0Z0 ZRZ0Z0ZK a b c d e f g h
As a mirror-motif to Black’s g5-g6, here White plays g4-g3to improve the safety of its king. g3 Nxe4
Qxe4 Qxe4
Bxe4 b4 and the game soonended in a draw.
The first ten moves for White and Black havebeen sampled randomly from AlphaZero’s opening “book”,with the probability proportional to the time spent calculat-ing each move. The remaining moves follow best play, atroughly one minute per move. d4 Nf6 c4 e6 Nf3 a6 Nc3 d5 cxd5 exd5 b3 Bb4 Bd2 Be7 e3 O-O Bc1 Bf5
Bd3 Bxd3
Qxd3 c6
Qc2 Re8
O-O a5 h4 Na6
Ne2 Nb4
Qd1Bd6
Bb2 h6 rZ0lrZkZ ZpZ0Zpo0 o0ZpZ0Z0 ZPZ0ONZ0 PA0ZNOPZ S0ZQZRJ0 a b c d e f g h
Here we see the first back-move of the game, opening thediagonal for the White bishop – d4-d3! d3 Nd7 a4 c5 rZ0lrZkZ ZpZnZpo0 o0opZ0Z0 Pm0Z0Z0O ZPZPONZ0 S0ZQZRJ0 a b c d e f g h
Just having played a4 on the previous move, White plays aback-move a4-a3 to challenge the b4 knight, given that thecircumstances have changed due to Black having played c5. a3 Na6 d4 rZ0lrZkZ ZpZnZpo0 nZ0a0Z0o o0opZ0Z0 OPZ0ONZ0 S0ZQZRJ0 a b c d e f g h ssessing Game Balance with AlphaZero White goes back to the previous plan and plays the pawn tod4 again, despite having moved it back before, showcasingthe fluidity of pawn structures Black responds by movingthe c-pawn back, to avoid having an isolated pawn.
21. . . c6 g3 Nc7 a4 Ne6 Kg2 Nf6
Rc1 Bf8 d3 rZ0lrakZ ZpZ0Zpo0 o0ZpZ0Z0 PZ0Z0Z0O ZPZPONO0 Z0SQZRZ0 a b c d e f g h
White opens the Bishop’s diagonal with a back-move, again.
26. . .
Rc8
Nfd4 Nxd4
Bxd4 c5
Ba1 Nh5
Ng1g6
Nf3 Ng7
Bxg7 Bxg7 d4 c6
Qd2 Bf8 h5 g5 ZpZ0ZpZ0 o0ZpZ0oP PZ0O0Z0Z ZPZ0ONO0 Z0S0ZRZ0 a b c d e f g h
Having just played h5, White plays h5-h4 now, to attackBlack’s g-pawn again. They repeat once before continuingwith other plans. h4 g6 h5 g5
Qd3 Rc7
Qf5 Qc8
Qxc8Rcxc8 h4 f6 g4 Re4 ZpZ0Z0Z0 o0ZpZ0o0 PZ0OrZPO ZPZ0ONZ0 Z0S0ZRZ0 a b c d e f g h
Black is attacking White’s pawn on g4, so it just moves backto g3. g3 Kf7
Rh1 g4
Ne1 Bd6 h3 g5 ZpZ0ZkZ0 o0ZpZ0o0 PZ0OrZ0Z ZPZ0O0OP Z0S0M0ZR a b c d e f g h
After having been challenged by a h4-h3 back-move, Blackretreats with g4-g5 as well.
Nd3 Ke8 h4 g4 h3 g5 h4 g6
Kf3 Kd7
Nf4 Rg8
Ne2 h5
Nf4 Bxf4 gxf4 b6 a3 f7
Rc2 Ra8
Rb1 Re6
Ke2 Rf6
Kf2d6 a4 Re8
Rbc1 b7 ssessing Game Balance with AlphaZero ZpZkZpZ0 o0Z0ZrZp PZ0O0O0O ZPZ0OPZ0 Z0S0Z0Z0 a b c d e f g h
White takes aim at the c6 pawn, but Black simply plays b6-b7, guarding it. With no clear way forward in this position,and after many more pawn structure reconfigurations, thegame unsurprisingly ended in a draw.
In the Pawn-sideways version of chess, pawns are allowedan additional option of moving sideways by one square,when available.B.8.1. M
OTIVATION
Allowing the pawns to move laterally introduces lots of newtactics into chess, while keeping the pawn structures veryflexible and fluid. It makes pawns much more powerful thanbefore and drastically increases the complexity of the game,as there are many more moves to consider at each juncture –and no static weaknesses to exploit.B.8.2. A
SSESSMENT
The assessment of the Pawn-sideways chess variant, as pro-vided by Vladimir Kramnik: “ This is the most perplexing and “alien” of allvariants of chess that we have considered. Evenafter having looked at how AlphaZero plays Pawn-side chess, the principles of play remain somewhatmysterious – it is not entirely clear what each sideshould aim for. The patterns are very differentand this makes many moves visually appear verystrange, as they would be mistakes in Classicalchess.Lateral pawn moves change all stages of the game.Endgame theory changes entirely, given that thepawns can now “run away” laterally to the edgeof the board, and it is hard to block them and pinthem down. Consider, for instance, the followingposition, with White to move: ZPZ0Z0Z0 Z0Z0Z0Z0 ZrZ0Z0Z0 Z0Z0Z0Z0 a b c d e f g h
In classical chess, White would be completely lost.Here, White can play b7-a7 or b7-c7, changingfiles. The rook can follow, but the pawn can al-ways step aside. In this particular position, afterb7-c7, Rc3, c7-d7 – Black has no way of stoppingthe pawn from queening, and instead of losing –White actually wins!It almost appears as if being a pawn up might givebetter chances of winning than being up a piecefor a pawn. In fact, AlphaZero often chooses toplay with two pawns against a piece, or a mi-nor piece and a pawn against a rook, suggestingthat pawns are indeed more valuable here than inclassical chess.This variant of chess is quite different and at timeshard to understand, but could be interesting forplayers who are open to experimenting with fewattachments to the original game! ” B.8.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under Pawn-sideways chess, when playing with roughly one minute permove from a particular fixed first move. Note that theseare not purely deterministic, and each of the given lines ismerely one of several highly promising and likely options.Here we give the first 20 moves in each of the main lines,regardless of the position.
Main line after e4
The main line of AlphaZero after e4in Pawn-sideways chess is: e4 (book) c5 c3 b6 dd4 Bb7 Nd2 g6 Bd3 Bg7 Ngf3 a5 ssessing Game Balance with AlphaZero rm0lkZns ZbZpopap o0o0Z0Z0 Z0OBZNZ0 PO0M0OPO S0AQJ0ZR a b c d e f g h
The previous move (a5) seems very unusual to a Classicalchess player’s eye. Black chooses to disregard the cen-tre, while creating a glaring weakness on b5. Yet, thereis method to this “madness”. It seems that rushing to grabspace early is not good in this setup, so White’s most promis-ing plan according to AlphaZero is to prepare b4. Apartfrom fighting against that advance, a5 prepares for playinga5-b5! later in this line, as we will see. Yet, this whole lineof play is hard to grasp as it violates the Classical chessprinciples. O-O d6 Rb1 Nf6 a3 O-O b4 rm0l0skZ ZbZ0opap o0o0Z0Z0 O0OBZNZ0 ZRAQZRJ0 a b c d e f g h
White has achieved the desired advance, to which Blackresponds with a lateral move – c5-d5!
10. . . cd5 rm0l0skZ ZbZ0opap o0ZpZ0Z0 O0OBZNZ0 ZRAQZRJ0 a b c d e f g h
Qc2 Nxe4
Nxe4 dxe4
Bxe4 Bxe4
Qxe4 Nd7
Be3 ab5 rZ0l0skZ Z0Znopap ZpZ0Z0Z0 O0O0ANZ0 ZRZ0ZRJ0 a b c d e f g h
As mentioned earlier, the a5 pawn finds a new purpose – onb5! The b6 pawn will soon move to c6, in the process ofreconfiguring the pawn structure. ab3 Nf6
Qd3 bc6 cc4 Qb8 a4 b4 c3 Rxa4 Z0Z0opap Z0Z0Z0Z0 roPO0Z0Z Z0OQANZ0 ZRZ0ZRJ0 a b c d e f g h ssessing Game Balance with AlphaZero Main line after d4
The main line of AlphaZero after d4in Pawn-sideways chess is: d4 (book) d5 e3 e6 cc4 dxc4 Bxc4 a6 a4 c5 Nf3 Nc6 Be2 cxd4 exd4 g6 b3 Nge7 Bb2 Bg7
Na3 rZblkZ0s ZpZ0mpap pZnZpZpZ Z0Z0Z0Z0 PZ0O0Z0Z MPZ0ZNZ0 S0ZQJ0ZR a b c d e f g h
Here Black has a way of opening the light-squared bishopwhile safeguarding the e5 square, by playing:
11. . . d6 O-O O-O c3 d5
Re1 Qc7
Bf1 Be6 h3 rZ0Z0skZ Zpl0mpap pZnZbZpZ Z0ZpZ0Z0 PZ0O0Z0Z M0O0ZNZP S0ZQSBJ0 a b c d e f g h
In this position, Black utilizes a rather unique defensiveresource:
16. . . gf6 rZ0Z0skZ Zpl0mpap pZnZbo0Z Z0ZpZ0Z0 PZ0O0Z0Z M0O0ZNZP S0ZQSBJ0 a b c d e f g h
Nc2 Rfd8
Qb1 Rab8 hg3 b5 b4 Ra8 rZ0s0ZkZ Z0l0mpap pZnZbo0Z ZpZpZ0Z0 Z0O0ZNO0 SQZ0SBJ0 a b c d e f g h
Main line after c4
The main line of AlphaZero after c4in Pawn-sideways chess is: c4 (book) c5 Nc3 Nc6 g3 g6 Bg2 e6 e4 a6 a3Rb8 Nge2 Bg7 Rb1 dd6 bb4 ZpZ0Zpap pZnopZpZ Z0o0Z0Z0 O0M0Z0O0 ZRAQJ0ZR a b c d e f g h
Here, Black responds with a typical lateral move. ssessing Game Balance with AlphaZero
9. . . c7 Z0o0Zpap pZnopZpZ Z0o0Z0Z0 O0M0Z0O0 ZRAQJ0ZR a b c d e f g h
O-O Nge7 bxc5 Rxb1 cxd6 Z0o0mpap pZnOpZpZ Z0Z0Z0Z0 O0M0Z0O0 ZrAQZRJ0 a b c d e f g h
White fights for the advantage by going for this kind of amaterial imbalance, an exchange down.
12. . .
Rb8 dxe7 Qxe7 dd4 O-O h4 Rd8 d5Qc5 Z0o0Zpap pZnZpZpZ Z0lPZ0Z0 O0M0Z0O0 Z0AQZRJ0 a b c d e f g h
Here another lateral move proves useful: b4 Qc4
Bf4 e5
Bg5 gf6
Be3 e6 Z0o0Zpap pZnZpZ0Z Z0ZPo0Z0 O0M0A0O0 Z0ZQZRJ0 a b c d e f g h
Black moves the g6 pawn first to f6 and then to e6, reachingthis position. The continuation shown here is not forced, andin some of its games, AlphaZero opts for slightly differentlines with Black, as this seems to be a very rich opening.B.8.4. I
NSTRUCTIVE GAMES
Game AZ-27: AlphaZero Pawn-sideways vs AlphaZeroPawn-sideways
The game is played from a fixed openingposition that arises after: e4 e5 Nf3 Nc6 Bc4. Theremaining moves follow best play, at roughly one minuteper move. e4 (book) e5 (book) 2. Nf3 (book)
Nc6 (book) 3.
Bc4 (book) d6 O-O Be6 Bb3 g5 dd4 rZ0lkans opo0ZpZp Z0Z0o0o0 ZBZ0ZNZ0 POPZ0OPO SNAQZRJ0 a b c d e f g h
6. . .
Bxb3 axb3 g4 Nxe5 ssessing Game Balance with AlphaZero rZ0lkans opo0ZpZp Z0Z0M0Z0 ZPZ0Z0Z0 SNAQZRJ0 a b c d e f g h
Already, things are getting very tactical and very unortho-dox.
8. . . dxe5 d5 rZ0lkans opo0ZpZp Z0ZPo0Z0 ZPZ0Z0Z0 SNAQZRJ0 a b c d e f g h
Black leaves the knight on c6 and goes on with creatingcounter-threats.
9. . . hg7
Qxg4 Nf6
Qf3 Ne7
Re1 Ng6 d4 rZ0lka0s opo0Zpo0 Z0ZPo0Z0 ZPZ0ZQZ0 SNA0S0J0 a b c d e f g h
White uses a lateral move (e4-d4) to create threats on thee-file.
13. . . e4 cc4 Bd6 g3 Kf8 Qg2 Ng4 rZ0l0j0s opo0Zpo0 Z0ZPZ0Z0 ZPZ0Z0O0 SNA0S0J0 a b c d e f g h
Black goes for the attack.
Rxe4 Nxh2
Nd2 f5
Re1 Bf4 rZ0l0j0s opo0Z0o0 Z0ZPZpZ0 ZPZ0Z0O0 S0A0S0J0 a b c d e f g h
Offering a piece on f4. gxf4 Nxf4
Qg3 gg5 ssessing Game Balance with AlphaZero rZ0l0j0s opo0Z0Z0 Z0ZPZpo0 ZPZ0Z0L0 S0A0S0J0 a b c d e f g h
White uses a lateral pawn move to safeguard the king. g2 Qd6
Nf1 Nxf1
Qxg5 Nh3+ rZ0Z0j0s opo0Z0Z0 Z0ZPZpL0 ZPZ0Z0Zn S0A0SnJ0 a b c d e f g h gxh3 Rg8
Re5 Rxg5+
Bxg5 Ng3
Be7+ Qxe7
Rxe7 Kxe7 rZ0Z0Z0Z opo0j0Z0 Z0ZPZpZ0 ZPZ0Z0mP S0Z0Z0J0 a b c d e f g h
Finally the dust has settled: White having two pawns for thepiece. c3 a6
Re1+ Kd7
Kg2 Ne4 e5 Ke6 d3Rg8+
Kf3 Ng5+
Kg3 c6
Re3 Rd8 ed5+ Kf6 h4 ZpZ0Z0Z0 pZpZ0j0Z Z0ZPZpm0 Z0ZPS0J0 Z0Z0Z0Z0 a b c d e f g h
Here Black decides to take on d5 rather than try to movethe knight, and White recaptures on d5 as well rather thantaking on g5!
39. . . cxd5 cxd5 Rxd5 e4 fxe4 dxe4 Nxe4
Rxe4 a5 ZpZ0Z0Z0 o0ZrZ0Z0 Z0Z0Z0J0 Z0Z0Z0Z0 a b c d e f g h
And now the game moves towards a draw.
Ra4 Rd2 a2 ab5
Rf4+ Ke5
Rb4 c5
Rxb7Rxa2 h5 Ra6
Rb5 d5
Kg4 Ke4
Kg5 d4with a draw to follow soon.
The game is played from a fixed openingposition that arises after c4 c5. The remaining movesfollow best play, at roughly one minute per move. c4 (book) c5 (book) 2. Nc3 Nc6 g3 g6 Bg2 e6 e4a6 a3 dd6 Nge2 Bg7 Rb1 Nge7 O-O Rb8 bb4c7 bxc5 Rxb1 cxd6 Rb8 dxe7 Qd7 c5 b6 ssessing Game Balance with AlphaZero Z0oqOpap Z0O0Z0Z0 O0M0Z0O0 Z0AQZRJ0 a b c d e f g h
To Black’s a6-b6, White responds with c5-b5, another lateralmove. b5 Nd4
Nxd4 Bxd4
Ne2 Bg7 a4 Qxe7 dd4O-O
Qb3 a6
Be3 axb5 axb5 Ba6 Z0o0lpap bZ0ZpZpZ ZPZ0Z0Z0 ZQZ0A0O0 Z0Z0ZRJ0 a b c d e f g h
White uses a lateral move to protect the pawn c4 c6 b6 c5 ed4 d5 cxd5 Z0Z0lpap bO0ZpZpZ Z0ZPZ0Z0 ZQZ0A0O0 Z0Z0ZRJ0 a b c d e f g h
Not minding to give up the piece, for getting strong passedpawns in return.
26. . .
Bxe2
Re1 Z0Z0lpap Z0ZPZ0Z0 ZQZ0A0O0 Z0Z0S0J0 a b c d e f g h
And yet, Black agrees and decides to return the piece in-stead.
27. . . exd5
Rxe2 Bxd4
Bxd4 Qxe2
Bxd5White opts to have the bishop pair and a pawn for twoexchanges, an unbalanced position.
30. . . gf6 h4 hg7 b7 Qa6
Bg2 Rfe8
Bc5 gg6
Qf3 Kg7 a7 Rbd8
Be3 Rh8 O0Z0Zpj0 qZ0Z0opZ Z0Z0Z0Z0 Z0Z0AQO0 Z0Z0Z0J0 a b c d e f g h
Qf4 Rd7
Qb8 Rdd8
Qc7 Qa1+
Kh2 g5
Qc4 Qe5
Kh3 Rc8
Qg4 Qe6
Bb7 Qxg4+
Kxg4 Rc4+
Kf3 gxh4 a8=R Rxa8
Bxa8 g4+ ssessing Game Balance with AlphaZero BZ0Z0Z0Z Z0Z0Zpj0 Z0Z0Z0Z0 Z0Z0AKO0 Z0Z0Z0Z0 a b c d e f g h
Ke2 e6 ff3 gxf3+
Bxf3 f5
Kd3 Ra4
Bd1Ra3+
Ke2 ee5 f3 Kf6
Bc1 Ra2+
Bd2 g5
Bb3 Ra3
Bd5 ef5
Be3 f4
Bd4+ Kf5
Be4+Ke6 e3 fg4
Kf2 f5
Bb7 e5
Bc8+ Kd5
Bxe5Kxe5
Bxg4and the game soon ended in a draw.
Position from an AlphaZero game playedat roughly one minute per move, from a predefined position. rZ0Zkans opo0lpZp Z0ZPo0Z0 M0Z0ZNZ0 POPZ0OPO S0AQS0J0 a b c d e f g h
8. . . gxf3 Qxf3 Bd7
Nb5Instead of capturing the knight, White has something elsein mind. . . rZ0Zkans opoblpZp ZNZPo0Z0 Z0Z0ZQZ0 POPZ0OPO S0A0S0J0 a b c d e f g h
10. . .
Nd4
Nxd4 exd4
Bg5 rZ0Zkans opoblpZp Z0ZPZ0A0 Z0Z0ZQZ0 POPZ0OPO S0Z0S0J0 a b c d e f g h with a motif of a lateral (e4-f4) discovery! In the game,Black didn’t take the bishop. So, how would have the gameproceeded if Black took the bishop? Here is one possiblecontinuation from AlphaZero:
12. . .
Qxg5 f4+ Qe7
Rxe7+ Nxe7 c5 dxc5
Qxb7 Rc8
Re1 Kd8
Qxa7 Nc6
Qa4 hg7 c3 Rh6
Bb5 Rb8 g3Rd6 d3 f6 h4 e6 h5 f7
Rb1 Rb6
Kg2. Thecontinuation is assessed as better for White.
12. . . f6 f4 de6 ssessing Game Balance with AlphaZero rZ0Zkans opobl0Zp Z0ZPZ0A0 Z0Z0ZQZ0 POPZ0OPO S0Z0S0J0 a b c d e f g h
Black uses lateral moves to cover the file as well. dxe6 Bc6
Bd5 O-O-O
Bxc6 bxc6
Qxc6 o0o0l0Zp Z0Z0Z0A0 Z0Z0Z0Z0 POPZ0OPO S0Z0S0J0 a b c d e f g h
White has gained several pawns for the piece, has a danger-ous attack and a substantial advantage, according to Alp-haZero. Yet, Black uses a lateral pawn move here to preventimmediate disaster:
17. . . ab7
Qa4 Kb8
Bh4 Qb4
Qb3 g7
Bg3Bd6 c3 Qxb3 axb3 dxc3 bxc3 Nh6 h3 Nf5
Bh2 Rhe8
Re2 Ne7 gg3 g5 Zpo0m0Z0 Z0Z0Z0o0 ZPO0Z0OP S0Z0Z0J0 a b c d e f g h fxg5 fxg5 h4 gxh4 gxh4 Rh8
Bxd6 cxd6
Ra4 Rc8
Rg4 Nf5 e7 Kc7
Rf4 Nh6 g4Kd7 f3 Rhe8
Rh2 Rxc3
Rxh6 Rxe7
Kh2 d5 b4 e5
Rf8 Rb3 g5 e4 fxe4 Rxb4 f4 Rb3
Kg1 Re2
Rh7+ Kd6
Rd8+ Kc5
Rd1 Rg3+
Kh1 Re4
Rf1 Rg4
Rxb7 Rexf4
Rxf4 Rxf4
Kh2 Rg4
Rg7 Kd6
Kh3 Rg1
Kh4 Z0Z0Z0S0 Z0Z0Z0O0 Z0Z0Z0Z0 Z0Z0Z0s0 a b c d e f g h and White soon won the game.
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. c4 c5 Nc3 Nc6 g3 g6 Bg2 e6 e4 dd6 Rb1 a6 a3 Bg7 Nge2 Rb8 bb4 c7 O-O Nge7 bxc5Rxb1 cxd6 Rb8 dxe7 Qd7 ssessing Game Balance with AlphaZero Z0oqOpap pZnZpZpZ Z0Z0Z0Z0 O0M0Z0O0 Z0AQZRJ0 a b c d e f g h
In this game (unlike in the main lines section before), Blackdecides to recapture on e7 with the knight instead.
Re1 Bb7 b3 Nxe7 dd4 O-O
Be3 Zboqmpap pZ0ZpZpZ Z0Z0Z0Z0 ZPM0A0O0 Z0ZQS0J0 a b c d e f g h
Here Black plays a lateral move (a6-b6) to improve its pawnstructure:
17. . . b6 h4 Ba8 Qc2But the pawn marches on, although not forward, openingthe line for the rook with:
19. . . bc6 bs0Z0skZ Z0oqmpap Z0Z0Z0Z0 ZPM0A0O0 Z0Z0S0J0 a b c d e f g h
Bh3 Qc8
Rd1 Qa6
Bg2 Rfd8
Rb1 cd6 d5exd5
Nxd5 Nxd5 exd5 bs0s0ZkZ Z0o0Zpap qZ0o0ZpZ Z0ZPZ0Z0 ZPZ0A0O0 ZRZ0Z0J0 a b c d e f g h
The d5 pawn is locking out the a8 bishop, so Black chal-lenges the center with a lateral move, only to decide to pushforward on the next move. This perhaps reveals a fluidity ofplans as well as structures.
26. . . e6 Nf4 e5
Ne2 Bf8
Nc3 c6The center is challenged again, this time from the other side,but White has a lateral response to keep things locked: dc5 ssessing Game Balance with AlphaZero bs0s0akZ Z0Z0ZpZp qZpZ0ZpZ Z0O0o0Z0 ZPM0A0O0 ZRZ0Z0J0 a b c d e f g h
And Black responds with a lateral move as well, bringingthe h-pawn towards the center.
30. . . hg7
Na4 Qa5 hg4 gf6 g5 fg6 f5 gf6 bs0s0akZ Z0Z0Zpo0 l0O0oPZ0 NZPZ0Z0Z ZPZ0A0O0 ZRZ0Z0J0 a b c d e f g h
After a sequence of lateral moves, the situation has settledon the kingside.
Rb2 d5
Bf4 e5
Bd2 Qc7
Be3 Qa5
Ra2Qb4
Rb2 Qa5
Kh2 d5
Bf4 e5
Bd2 Qc7
Be3 d6 d5 bs0s0akZ Z0l0Zpo0 Z0ZPoPZ0 NZPZ0Z0Z ZPZ0A0O0 Z0Z0Z0Z0 a b c d e f g h
Black and White keep reconfiguring the central pawns.
45. . . c6 dc5 d6 cxd6 Rxd6 c5 Rdd8 b4 Bxg2 Kxg2 Qc6+ f3 d5
Bd4 e5
Bf2 d5
Bd4e5
Bf2 d5
Qb3 Qb5
Nc3 Qc4 bb5 Rdc8
Nxd5 Qxb3
Rxb3 Bxc5 b6 Bd6 Z0Z0Zpo0 Z0ZNZPZ0 ZRZ0ZPO0 Z0Z0Z0Z0 a b c d e f g h
An interesting endgame arises.
Nc3 Bc5
Nd5 Bd6
Rb2 e6 e5 Z0Z0Zpo0 Z0ZNO0Z0 Z0Z0ZPO0 Z0Z0Z0Z0 a b c d e f g h ssessing Game Balance with AlphaZero Both sides using lateral move to create threats.
65. . .
Bf8
Nf4 Bc5 b7 Rc7
Rc2 Bb6
Rxc7Bxc7 ZPa0Zpo0 Z0Z0O0Z0 Z0Z0ZPO0 Z0Z0Z0Z0 a b c d e f g h
But the pawn can switch files! a7 O0a0Zpo0 Z0Z0O0Z0 Z0Z0ZPO0 Z0Z0Z0Z0 a b c d e f g h
70. . .
Ra8 d5 g5 b7 rZ0Z0ZkZ ZPa0ZpZ0 Z0ZPZ0o0 Z0Z0ZPO0 Z0Z0Z0Z0 a b c d e f g h
72. . .
Rb8 a7 Ra8
Nd3 exd5
Nb4 e5 b7 Rb8 a7 Ra8
Na6 Bd6
Bc5 rZ0Z0ZkZ O0Z0ZpZ0 NZ0a0Z0Z Z0A0o0o0 Z0Z0ZPO0 Z0Z0Z0Z0 a b c d e f g h
79. . .
Bxc5 b7 rZ0Z0ZkZ ZPZ0ZpZ0 NZ0Z0Z0Z Z0a0o0o0 Z0Z0ZPO0 Z0Z0Z0Z0 a b c d e f g h
80. . .
Rd8
Nxc5 f6
Ne6 Rb8 c7 Ra8
Nd8Rc8
Ne6 Kf7 d7 Z0ZPZkZ0 Z0Z0o0o0 Z0Z0ZPO0 Z0Z0Z0Z0 a b c d e f g h
86. . .
Rb8 d8=Q Rxd8
Nxd8+ Ke7
Nc6+ Kd6 ssessing Game Balance with AlphaZero Nd8 Ke7
Nb7 ff5 e3 g4
Kf2 e4
Ke2 Ke6
Nd8+ Ke7
Nc6+ Kf6
Nd4 Z0Z0Z0Z0 Z0Z0ZpZ0 Z0Z0O0O0 Z0Z0Z0Z0 a b c d e f g h
However, this position is a draw!
97. . . g5 Kf1 Ke5
Ne2 f5
Kg1 Kd5
Kf2Ke5
Kf1 Kd5
Kf2 Ke5
Kf1 Kd5
Kg2Kc4
Nd4 e5
Nc6 Kd5
Ne7 Ke6
Nc8 f5
Na7 Ke5
Nc6+ Kd5
Nd4 e5
Nf5 Ke6
Ng7+ Kf7
Nf5 Ke6
Nh6 f5
Kf1 Kf6
Ke1 Kg6
Ng8 Kf7
Nh6+ Kg6
Nxg4fxg4
Kd2 Kf5
Kc3 ef4 exf4 h4 e4+Kxe4 gxh4 Kf4
Kc2 Kg4
Kc1 Kxh4
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. c4 c5 e3 e6 dd4 cxd4 exd4 g6 Nc3 Bg7 Nb5bc7 Bf4 Na6 Nf3 Nf6 h3 d5 Bd3 O-O
O-Oab7
Re1 c6
Nd6 cc5
Be5 cxd4
Nxd4 Nc5
Bf1 Nce4
N4b5 rZbl0skZ ZpZ0Zpap ZNZpA0Z0 Z0Z0Z0ZP PO0Z0OPZ S0ZQSBJ0 a b c d e f g h
Here we see a new kind of tactic, made possible by a lateralpawn move!
17. . .
Nxf2
Kxf2 e7 rZbl0skZ ZpZ0o0ap ZNZpA0Z0 Z0Z0Z0ZP PO0Z0JPZ S0ZQSBZ0 a b c d e f g h
Kg1 exd6
Nxd6 Nh5
Bxg7 Nxg7
Nxc8 Qxc8 cxd5 Qc5+
Kh1 exd5
Qb3 b6 rZ0Z0skZ Z0Z0Z0mp Z0lpZ0Z0 ZQZ0Z0ZP PO0Z0ZPZ S0Z0SBZK a b c d e f g h
The dust has settled, and the game soon ended in a draw. g4 Qd6
Bg2 Rad8
Rac1 Ne6
Qxd5 Qxd5
Bxd5 Rxd5
Rxe6 Rf2
Rxb6 Rdd2 g5 hg7 a4 Rh2+
Kg1 Rdg2+
Kf1 Rf2+
Kg1 Rfg2+
Kf1 Rf2+
Ke1 Rfg2
Rb8+ Kh7
Kf1 Rf2+
Kg1 Rfg2+
Kf1 Rf2+
Kg1 Rfg2+
Kf1
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. c4 c5 Nc3 g6 e3 e6 dd4 bc7 dxc5 Bxc5 g4 ssessing Game Balance with AlphaZero rmblkZns o0opZpZp Z0a0Z0Z0 Z0M0O0Z0 PO0Z0O0O S0AQJBMR a b c d e f g h
Now that is an unusual sight, the early advance of the g-pawn.
6. . . hg7 Bg2 c6 Nf3 d5 O-O Qc7 d4 rmbZkZns o0l0Zpo0 Z0apZ0Z0 Z0M0ONZ0 PO0Z0OBO S0AQZRJ0 a b c d e f g h
White plays c4-d4, a lateral move, to reinforce the center.
10. . .
Bd6 h3 f5 f4 rmbZkZns o0l0Z0o0 Z0ZpZpZ0 Z0M0ONZP PO0Z0OBZ S0AQZRJ0 a b c d e f g h
The g-pawn, advanced earlier in what seemed to be weaken- ing, now finds its place on f4, where it shuts out the activityon the b8-h2 diagonal.
12. . .
Nf6 a3 b7
Rb1 f7 b4 O-O
Bb2 Rd8
Rc1 Bf8
Qb3 Bd7 dc4 dxc4
Qxc4 Be8 g3Bg7
Qb3 Qb6
Nd4 Nbd7 aa4 Bf8
Ba3 ee5 fxe5 Nxe5
Rfd1 Neg4 gf3 f4 rZ0sbakZ ZpZ0ZpZ0 Z0Z0Z0Z0 PO0M0onZ AQM0OPZ0 Z0SRZ0J0 a b c d e f g h
The game gets quite tactical here. a5 Qc7 exf4 Qxf4 fxg4 Rxd4
Rxd4 Qxd4 g5 Ng4
Ne4 Qe5
Bb2 Qh2+
Kf1 Bd7 f3Qf4
Re1 Re8
Qc4 Nh2+ ZpZbZpZ0 O0Z0Z0O0 Z0Z0ZPZ0 Z0Z0SKZ0 a b c d e f g h
Kg1 Nxf3+
Bxf3 Qxf3
Nf6+ ssessing Game Balance with AlphaZero ZpZbZpZ0 O0Z0Z0O0 Z0Z0ZqZ0 Z0Z0S0J0 a b c d e f g h
Black needs to give away its queen to stop the attack.
42. . .
Qxf6
Bxf6 Rxe1+
Kf2 Rd1
Qf4 Bf5
Qb8 Rd7 ZpZrZpZ0 O0Z0ZbO0 Z0Z0Z0Z0 Z0Z0Z0Z0 a b c d e f g h
Is this a fortress? As we will see, the question is slightlymore complicated by the fact that the pawn structure isn’tfixed, and things will eventually open up.
Be5 Rd2+
Ke1 Rd7
Bc3 Bd3 a4 Bc2
Qc8Re7
Kd2 Bf5
Qb8 Rd7+
Kc1 Rd3
Bf6 Rd7
Bb2 e7 ZpZro0Z0 O0Z0ZbO0 PZ0Z0Z0Z Z0Z0Z0Z0 Z0J0Z0Z0 a b c d e f g h
This resource is what Black was keeping in reserve, asa potential way of responding to the threats on the a3-f8diagonal while the f8 bishop was pinned.
Qh2 Bg7 b4 Bxb2+
Kxb2 f7 ZpZrZpZ0 O0Z0ZbO0 Z0Z0Z0Z0 Z0Z0Z0Z0 a b c d e f g h
The pawn has served its purpose on e7 and moves back. c4 Be6
Qb8+ Kh7
Kb3 Kg7
Kc3 f6 gxf6+Kxf6
Qf8+ Bf7
Kb4 g5
Qh6+ Bg6
Qh8+Kf7
Qh3 Re7
Kc5 f5
Kd6 Re6+
Kd7 Re7+
Kd8 Re8+
Kc7 Re7+
Kb6 e5 ssessing Game Balance with AlphaZero ZpZ0skZ0 O0Z0o0Z0 Z0Z0Z0ZQ Z0Z0Z0Z0 a b c d e f g h
Qf3+ Kg7 a6 bxa6
Kxc6 e4
Qe3 Rf7 c5f4
Qf3 Bf5
Kd6 Rf6+
Ke5 Bd7
Qg2+ Kf7
Qh1 Kg7
Qb7 Rf7
Qg2+ Kh7
Qf3 Kg7
Kd6 Kh6
Qb3 Kg7
Qc3+ Kh7
Qd3+ Kg7
Qd4+ Kh7
Qd5 Kg7
Qg2+ Kh7
Qh2+ Kg8
Qg1+ Kh7
Qb1+ Kg7
Qb2+ Kg8
Qb8+ Kg7
Qb2+ Kg8
Qg2+ Kh7
Qc2+ Kg8
Qg6+Kf8
Qh5 Kg7
Qe5+ Kg8
Qg5+ Kh7
Qd5Kg7
Qe5+ Kg8
Qg5+ Kh7
Qh5+ Kg7
Qf3 Kh6 c6 Z0ZbZrZ0 pZPJ0Z0j Z0Z0Z0Z0 Z0Z0ZQZ0 Z0Z0Z0Z0 a b c d e f g h
Bxc6
Kxc6 Kg5
Kd6 Rf5
Ke6 Rf6+
Ke5 Rf5+
Ke4 Rf7
Kd4 Rd7+
Kc4 b6
Qg2+ g4
Qf1 Rd6
Qc1+ f4
Qg1+ g4
Qe3+ Kf5
Qf2+ Kg5
Qe3+ Kf5
Qg3Rf6
Qh4 c6
Kd4 d6
Kd5 c6+
Kc5 Rg6
Qg3 Rf6
Qh4 Rg6
Qg3 Rf6
Kb6 Kg5
Kc7 Rf3
Qe5+ Rf5
Qe1 c5
Kd6 Z0Z0Z0Z0 Z0o0Zrj0 Z0Z0Z0Z0 Z0Z0L0Z0 a b c d e f g h c4 Qe7+ Kf4
Qe2 g3
Ke6 Kg5
Qxc4 Rf6+
Ke5 Rf5+
Kd6 Rf6+
Ke5Rf5+
Ke6 Rf6+
Kd7 Rf4
Qe2 Kh4
Kd6Kh3
Ke5 Rf2
Qh5+ Kg2
Ke4 Kg1
Ke3g2
Qh4 Rf8
Ke2 Rf1
Qg3 Kh1
Qh3+Kg1
Qh4 h2
Qd4+ Kh1
Qh4 Kg1
Qg5+g2
Qh6 Rf2+
Ke3 Rf1
Ke2 Rf2+
Ke3Rf1
Qh3 Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0J0ZQ Z0Z0Zrj0 a b c d e f g h
And the game ended in a draw in a couple of moves.
In Self-capture chess, we have considered extending therules of chess to allow players to capture their own pieces.B.9.1. M
OTIVATION
The ability to capture one’s own pieces could help break“deadlocks” and offer additional ways of infiltrating theopponent’s position, as well as quickly open files for theattack. Self-captures provide additional defensive resourcesas well, given that the King that is under attack can consider ssessing Game Balance with AlphaZero escaping by self-capturing its own adjacent pieces.B.9.2. A SSESSMENT
The assessment of the Self-capture chess variant, as pro-vided by Vladimir Kramnik: “ I like this variation a lot, I would even go as faras to say that to me this is simply an improvedversion of regular chess.Self-captures make a minor influence on the open-ing stage of a chess game, though we have seenexamples of lines that become possible under thisrule change that were not possible before. For ex-ample, consider the following line 1. e4 e5 2. Nf3Nc6 3. Bb5 a6 4. Ba4 Nf6 5. 0-0 Nxe4 6. d4 exd47. Re1 f5 8. Nxd4 Qh4 9. g3 in the Ruy Lopez. rZbZka0s ZpopZ0op pZnZ0Z0Z Z0Z0ZpZ0 BZ0MnZ0l Z0Z0Z0O0 POPZ0O0O SNAQS0J0 a b c d e f g h
While not the main line, it is possible to play inSelf-capture chess and AlphaZero assesses it asequal. In classical chess, however, this positionis much better for White. The key difference isthat in self-capture chess Black can respond tog3 by taking its own pawn on h7 with the queen,gaining a tempo on the open file. In fact, Whitecan gain the usual opening advantage earlier inthe variation, by playing 8. Ng5 d5 9. f3 Bd610. fxe4 dxe4, which AlphaZero assesses as givingthe 60% expected score for White after about aminute’s thought, which is usually possible to de-fend with precise play. In fact, there are multipleimprovements for both sides in the original line,but discussing these is beyond the scope of thisexample. It is worth noting that AlphaZero prefersto utilise the setup of the Berlin Defence, similarto its style of play in classical chess.Regardless of its relatively minor effect on theopenings, self-captures add aesthetically beau-tiful motifs in the middlegames and provide additional options and winning motifs in theendgames.Taking one’s own piece represents another way ofsacrificing in chess, and material sacrifices makechess games more spectacular and enjoyable bothfor public and for the players. Most of the timesthis is used as an attacking idea, to gain initiativeand compromise the opponent’s king.For example, consider the Dragon Sicilian, as anexample of a sharp opening. After 1. e4 c5 2. Nf3d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 g6 6. Be3 Bg77. f3 0-0 8. Qd2 Nc6 9. 0-0-0 d5 something like10. g4 e5 11. Nxc6 bxc6 is possible, at which pointthere is already Qxh2, a self-capture, opening thefile against the enemy king. Of course, Black can(and probably should) play differently. rZbl0skZ o0Z0Zpap Z0Zpo0Z0 Z0M0APZ0 POPZ0Z0L Z0JRZBZR a b c d e f g h
The possibilities for self-captures in this exampledon’t end, as after 12. . . d4, White could evenconsider a self-capture 13. Nxe4, sacrificing an-other pawn. This is not the best continuationthough, and AlphaZero evaluates that as beingequal. It is just an illustration of the ideas whichbecome available, and which need to be takeninto account in tactical calculations.In terms of endgames, self-captures affect a widespectrum of otherwise drawish endgame positionswinning for the stronger side. Consider the fol-lowing examples: ssessing Game Balance with AlphaZero ZPZ0Z0Z0 Z0Z0ZBZ0 Z0Z0Z0Z0 Z0Z0Z0Z0 a b c d e f g h
In this position, under Classical rules, the gamewould be an easy draw for Black. In Self-capturechess, however, this is a trivial win for White, whocan play Bc8 and then capture the bishop with theb7 pawn, promoting to a queen! Z0Z0akZ0 o0oPZPo0 PoPZ0ZPo ZPZ0ZKZP Z0Z0Z0Z0 a b c d e f g h
This endgame, which represents a fortress in clas-sical chess, becomes a trivial win in self-capturechess, due to the possibilities for the White kingto infiltrate the Black position either via e4 and aself-capture on d5 or via e2, d3 and a self-captureon c4.To conclude, I would highly recommend this vari-ation for chess lovers who value beauty in thegame on top of everything else. ” B.9.3. M
AIN L INES
Here we discuss “main lines” of AlphaZero under Self-capture chess, when playing with roughly one minute permove from a particular fixed first move. Note that theseare not purely deterministic, and each of the given lines is merely one of several highly promising and likely options.Here we give the first 20 moves in each of the main lines,regardless of the position.
Main line after e4
The main line of AlphaZero after e4in Self-capture chess is: e4 (book) e5 Nf3 Nc6 Bb5 Nf6 O-O Nxe4 Re1Nd6 Nxe5 Be7 Bf1 Nxe5 Rxe5 O-O Nc3 Ne8
Nd5 Bd6
Re1 c6
Ne3 Be7 c4 Nc7 d4 d5 cxd5 Bb4
Bd2 Bxd2
Qxd2 Nxd5
Nxd5 Qxd5
Re5 Qd6
Bc4 Bd7 rZ0Z0skZ opZbZpop Z0Z0S0Z0 Z0Z0Z0Z0 PO0L0OPO S0Z0Z0J0 a b c d e f g h
Main line after d4
The main line of AlphaZero after d4in Self-capture chess is: d4 (book) d5 c4 e6 Nc3 Nf6 cxd5 exd5 Bg5 c6 Qc2 Nbd7 e3 Be7 Nf3 Nh5 Bxe7 Qxe7
Be2O-O
O-O Ndf6
Ne5 g6
Qa4 Be6 b4 a6
Qb3 Ng7
Na4 Ne4
Qb2 Qg5
Nf3 Qe7
Ne5Qg5
Nf3 Qe7 rZ0Z0skZ ZpZ0lpmp pZpZbZpZ Z0ZpZ0Z0 NO0OnZ0Z Z0Z0ONZ0 PL0ZBOPO S0Z0ZRJ0 a b c d e f g h
Main line after c4
The main line of AlphaZero after c4in Self-capture chess is: ssessing Game Balance with AlphaZero c4 (book) e5 g3 d5 cxd5 Nf6 Bg2 Nxd5 Nc3 Nb6 b3 Nc6 Bb2 f6 Rc1 Bf5 Bxc6+ bxc6
Nf3 Qd7
O-O Be7 d3 a5
Ne4 O-O
Qc2 a4
Qxc6Qxc6
Rxc6 Nd5
Nc3 Nxc3
Rxc3 axb3 axb3Rfb8
Rxc7 Bd8 rs0a0ZkZ Z0S0Z0op Z0Z0obZ0 ZPZPZNO0 Z0Z0ZRJ0 a b c d e f g h
B.9.4. I
NSTRUCTIVE GAMES
Game AZ-33: AlphaZero Self-capture vs AlphaZeroSelf-capture
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. d4 Nf6 c4 e6 Nc3 d5 cxd5 exd5 Bg5 c6 Qc2Nbd7 Nf3 h6 Bh4 Be7 e3 O-O Bd3 Re8
O-ONe4
Bxe4 Bxh4
Bh7+ Kh8
Nxh4 Qxh4
Bd3Qe7 a3 Nf6 b4 Bd7 h3 Kg8
Rfb1 Rec8
Qd1 Be6
Ne2 a6
Nf4 Ne8 a4 Nd6
Qb3Qd7
Be2 Bf5
Rc1 Qd8
Qb2 Be4
Rc5 Bf5
Rc3 Ra7
Rcc1 Raa8
Rc5 Qh4
Bf1 Re8
Rcc1 g5
Nd3 Bxd3
Bxd3 g4 hxg4 Re6
Qe2 rZ0Z0ZkZ ZpZ0ZpZ0 pZpmrZ0o Z0ZpZ0Z0 PO0O0ZPl Z0ZBO0Z0 S0S0Z0J0 a b c d e f g h
And here we see the first self-capture of the game, creatingthreats down the h-file:
37. . .
Rxh6
Qf3 Qh1+ rZ0Z0ZkZ ZpZ0ZpZ0 pZpm0Z0s Z0ZpZ0Z0 PO0O0ZPZ Z0ZBOQZ0 S0S0Z0Jq a b c d e f g h
The end? Not really. In self-capture chess the king canescape by capturing its way through its own army, andhence here it just takes on f2 and gets out of check.
Kxf2 Qh4+
Ke2 Re8
Rh1 Qxh1
Rxh1 Rxh1
Qf4 Ne4
Bxe4 Rxe4
Qb8+ Kg7
Qxb7 Rh6
Qxa6 Rxg4
Kf1 Rh1+
Kf2 Rg6 b5 Rh2 b6Rgxg2+
Kf3 Rf2+ Z0Z0Zpj0 QOpZ0Z0Z Z0ZpZ0Z0 PZ0O0Z0Z Z0Z0OKZ0 Z0Z0Z0Z0 a b c d e f g h
Unlike in classical chess, White can still play on here, andAlphaZero does, by advancing the king forward with a self-capture!
Kxe3 Rb2 a5 Rb3+ ssessing Game Balance with AlphaZero Z0Z0Zpj0 QOpZ0Z0Z O0ZpZ0Z0 ZrZ0J0Z0 Z0Z0Z0Z0 a b c d e f g h
And, as if one pawn was not enough, White self-capturesanother one by taking on d4.
Kxd4 Ra2
Ke5 Rb5 b7 Raxa5
Qxa5 Rxa5 b8=Q Ra2 Z0Z0Zpj0 Z0ZpJ0Z0 Z0Z0Z0Z0 rZ0Z0Z0Z Z0Z0Z0Z0 a b c d e f g h
White manages to get a queen, but in the end, Black’s de-fensive resources prove sufficient and the game eventuallyends in a draw.
Kd6 Re2
Kxc6 Re6+
Kd7 Rg6
Qa8 Re6
Qxd5 Kg8
Qa8+ Kg7With draw soon to follow.
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. d4 d5 c4 e6 Nc3 Nf6 Nf3 c6 Bg5 h6 Bh4 dxc4 e4 g5 Bg3 b5 Be2 Bb7
Ne5 Nbd7
Qc2 Bg7
Rd1 Qe7 h4 Nxe5
Bxe5 a6 a4 Rg8 hxg5hxg5 rZ0ZkZrZ ZbZ0lpa0 pZpZpm0Z ZpZ0A0o0 PZpOPZ0Z Z0M0Z0Z0 Z0ZRJ0ZR a b c d e f g h
Qc1 O-O-O
Qxg5 Nd5
Qxe7 Nxe7 g3 Bxe5 dxe5 Rxd1+
Kxd1 Rd8+
Kc1 b4 ZbZ0mpZ0 pZpZpZ0Z Z0Z0O0Z0 PopZPZ0Z Z0M0Z0O0 Z0J0Z0ZR a b c d e f g h
Here we come to the first self-capture of the game, Whitedecides to give up the a4 pawn in order to get the knight toan active square.
Nxa4 ssessing Game Balance with AlphaZero ZbZ0mpZ0 pZpZpZ0Z Z0Z0O0Z0 NopZPZ0Z Z0Z0Z0O0 Z0J0Z0ZR a b c d e f g h
And Black responds in turn with a self-capture of its own,on c6!
24. . .
Nxc6 ZbZ0ZpZ0 pZnZpZ0Z Z0Z0O0Z0 NopZPZ0Z Z0Z0Z0O0 Z0J0Z0ZR a b c d e f g h
Nb6+ Kc7
Nxc4 Nd4
Bd3 Nf3
Bc2 Rd4
Nd6 Nxe5
Nxb7 Kxb7 f4 Nd3+
Bxd3 Rxd3
Rh7 Rxg3
Rxf7+ Kc6
Rf6 Kd7
Rf7+ Kc6
Rf6 Kd7 f5 exf5 exf5 Rf3 Z0ZkZ0Z0 pZ0Z0S0Z Z0Z0ZPZ0 Z0Z0ZrZ0 Z0J0Z0Z0 a b c d e f g h
And the game eventually ended in a draw.
The first ten moves for White and Black havebeen sampled randomly from AlphaZero’s opening “book”,with the probability proportional to the time spent calculat-ing each move. The remaining moves follow best play, atroughly one minute per move. d4 e6 Nf3 Nf6 c4 d5 Bg5 dxc4 Nc3 a6 e4 b5 e5 h6 Bh4 g5 Nxg5 hxg5
Bxg5 Nbd7 rZblka0s Z0onZpZ0 pZ0Zpm0Z ZpZ0O0A0 Z0M0Z0Z0 PO0Z0OPO S0ZQJBZR a b c d e f g h
In this highly tactical position, self-captures provide addi-tional resources, as AlphaZero quickly demonstrates, bya self-capture on g2, developing the bishop on the longdiagonal at the price of a pawn.
Bxg2 rZblka0s Z0onZpZ0 pZ0Zpm0Z ZpZ0O0A0 Z0M0Z0Z0 PO0Z0OBO S0ZQJ0ZR a b c d e f g h
Yet, Black responds in turn by a self-capture on a6:
11. . .
Rxa6 ssessing Game Balance with AlphaZero Z0onZpZ0 rZ0Zpm0Z ZpZ0O0A0 Z0M0Z0Z0 PO0Z0OBO S0ZQJ0ZR a b c d e f g h exf6 Rg8 h4 Nxf6
Nxb5 Be7
Qc2 Nd5
Qh7 Z0o0apZQ rZ0ZpZ0Z ZNZnZ0A0 Z0Z0Z0Z0 PO0Z0OBZ S0Z0J0ZR a b c d e f g h
16. . .
Rf8
Bh6 Nf6
Qc2 Rg8
Bf3 c6
Nc3Qxd4
Be3 Qe5
O-O-O Nd5 Z0Z0apZ0 rZpZpZ0Z Z0Znl0Z0 Z0M0ABZ0 POQZ0O0Z Z0JRZ0ZR a b c d e f g h
Kb1 Nxc3+ bxc3 c5
Rhg1 Rh8
Rg4 Qf5
Qxf5 exf5
Rxc4 Be6
Bd5 Rd6
Rxc5 Rb6+
Kc2 Bxc5
Bxc5 Ra6 a3 Bxd5
Rxd5 Rxh4
Rxf5 Z0Z0ZpZ0 rZ0Z0Z0Z Z0A0ZRZ0 O0O0Z0Z0 Z0Z0Z0Z0 a b c d e f g h and the game eventually ended in a draw.
The first ten moves for White and Blackhave been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move.In this game, self-captures happen towards the end, but thegame itself is pretty tactical and entertaining. We thereforeincluded the full game. Nf3 d5 d4 Nf6 c4 e6 Nc3 c6 Bg5 h6 Bh4dxc4 e4 g5 Bg3 b5 Be2 Bb7
O-O Nbd7
Ne5h5
Nxd7 Qxd7 rZ0Zka0s obZqZpZ0 ZpZ0Z0op Z0M0Z0A0 PO0ZBOPO S0ZQZRJ0 a b c d e f g h
In the game, White played the pawn to a3, but it’s interestingto note that potential self-captures factor in the lines thatAlphaZero is calculating at this point. AlphaZero is initiallyconsidering the following line:
Qd2 Be7
Qxg5 b4
Na4 Qxc6 ssessing Game Balance with AlphaZero rZ0ZkZ0s obZ0apZ0 Z0Z0Z0Lp NopOPZ0Z Z0Z0Z0A0 PO0ZBOPO S0Z0ZRJ0 a b c d e f g hanalysis diagram where Black has just self-captured its c6 pawn!
Nc5Nxe4
Qe5 with exchanges to follow. Going back to thegame: a3 Rh6
Qc1 h4 rZ0Zka0Z obZqZpZ0 ZpZ0Z0o0 O0M0Z0A0 S0L0ZRJ0 a b c d e f g h
Be5 h3
Qxg5 hxg2
Rd1 Rg6
Qf4 Qe7
Qf3Bg7 h4 O-O-O obZ0lpa0 ZpZ0A0Z0 O0M0ZQZ0 S0ZRZ0J0 a b c d e f g h
Bg3 Bh6 h5 Rgg8 b3 Rxg3 obZ0lpZ0 ZpZ0Z0ZP OPM0ZQs0 S0ZRZ0J0 a b c d e f g h
Qxg3 Rg8
Qh3 Bf4
Qh4 Bg5
Qh2 cxb3
Rd3 a6
Rb1 c5 dxc5 Nd7 ZbZnlpZ0 pZ0ZpZ0Z ZpO0Z0aP OpMRZ0Z0 ZRZ0Z0J0 a b c d e f g h
Rg3 Rg7
Qxg2 f5
Rxb3 Nxc5
Rb4 Qf6
Bf1 fxe4 ZbZ0Z0s0 pZ0Zpl0Z Zpm0Z0aP O0M0Z0S0 Z0Z0ZBJ0 a b c d e f g h
Nxe4 Nxe4
Rxe4 Bxe4
Qxe4 Bf4
Rg6 Rxg6+ ssessing Game Balance with AlphaZero hxg6 Qg5+ Bg2 Be5 f4 Bxf4
Qb7+ Kd8 g7 Qc5+ ZQZ0Z0O0 pZ0ZpZ0Z Zpl0Z0Z0 O0Z0Z0Z0 Z0Z0Z0J0 a b c d e f g h
What happens next is a rather remarkable self-capture,demonstrating that it’s not only the pawns that can justi-fiably be self-captured, as the least valuable pieces. Indeed,White self-captures the bishop on g2, in its attempt at avoid-ing perpetuals!
Kxg2 Qg5+
Kf1 ZQZ0Z0O0 pZ0ZpZ0Z ZpZ0Z0l0 O0Z0Z0Z0 Z0Z0ZKZ0 a b c d e f g h
Yet, Black responds in turn, by capturing its own bishop!The game ultimately ends in a draw.
46. . .
Qxf4+
Kg2 Qg4+
Kf2 Qf4+
Ke2 Qe5+
Kf3 Qf5+
Ke3 Qe5+
Kd3 Qd6+
Ke4 Qxe6+
Kf4 Qf6+
Ke3 Qe5+
Kd3 Qd6+
Ke4 Qe6+
Kd4 Qf6+
Kd5 Qf3+
Kd6 Qf6+
Kc5 Qf2+
Kb4 Qd2+
Kb3 Qd3+
Kb2 Qd2+
Kb1 Qd3+
Kb2 Qd2+
Kb3 Qd3+
Kb4 Qd2+
Kxa3 Qc3+
Ka2 Qc2+
Ka1 Qc1+
Ka2 Qc2+
Ka1 Qc1+
Ka2 Qc2+
The first ten moves for White and Black have been sampled randomly from AlphaZero’s opening“book”, with the probability proportional to the time spentcalculating each move. The remaining moves follow bestplay, at roughly one minute per move. d4 Nf6 c4 e6 Qc2 c5 dxc5 h6 Nf3 Bxc5 a3O-O Bf4 Qa5+ Nbd2 Nc6 e3 Re8 Bg3 e5
Bh4g5 rZbZrZkZ opZpZpZ0 l0a0o0o0 O0Z0ONZ0 S0Z0JBZR a b c d e f g h
Nxg5 hxg5
Bxg5 Re6
O-O-O Bf8 h4 d5 rZbZ0akZ opZ0ZpZ0 l0Zpo0A0 O0Z0O0Z0 Z0JRZBZR a b c d e f g h
Here we see the first self-capture move of the game, creatingthreats along the h-file:
Rxh4 ssessing Game Balance with AlphaZero rZbZ0akZ opZ0ZpZ0 l0Zpo0A0 O0Z0O0Z0 Z0JRZBZ0 a b c d e f g h
It’s interesting to note that White could have also tried open-ing the h-file a move earlier, by playing
Rxh2 instead of h4, but AlphaZero prefers provoking
15. . . d5 first andhaving its rook on the 4th rank, where it stands more activeand controls additional squares.
16. . .
Bg7
Nb3 Qb6 cxd5 Nxd5
Rxd5 Rg6 rZbZ0ZkZ opZ0Zpa0 Z0ZRo0A0 ONZ0O0Z0 Z0J0ZBZ0 a b c d e f g h
Here comes another self-capture:
Bxe3 rZbZ0ZkZ opZ0Zpa0 Z0ZRo0Z0 ONZ0A0Z0 Z0J0ZBZ0 a b c d e f g h
20. . .
Qc7
Rc5 b6
Rc3 Bb7
Bd3 Qd8 g3Rd6
Bc4 Kf8
Qh7 Qf6
Nd2 Ne7
Rg4 Rad8
Bg5 Qxf2 obZ0mpaQ Z0Z0o0A0 O0S0Z0O0 Z0J0Z0Z0 a b c d e f g h
Bxe7+ Kxe7
Rxg7 Qe1+
Kc2 Be4+ o0Z0jpSQ Z0Z0o0Z0 O0S0Z0O0 Z0Z0l0Z0 a b c d e f g h
Nxe4 Qd1+ is what is played and made possible by aself-capture, avoiding mate: ssessing Game Balance with AlphaZero Kxb2 o0Z0jpSQ Z0Z0o0Z0 O0S0Z0O0 Z0ZqZ0Z0 a b c d e f g h
Here Black responds by a self-capture on b6:
34. . .
Rxb6+ o0Z0jpSQ Z0Z0o0Z0 O0S0Z0O0 Z0ZqZ0Z0 a b c d e f g h
The game soon ends in a draw.
Rb3 Rxb3+
Bxb3 Qe2+
Kb1 Qf1+
Ka2 Qe2+
Ka1 Qe1+
Ka2 Qe2+
Kb1 Qe1+
Kc2 Qe2+
Kc3 Qe3+
Kb2 Qe2+
Kxa3 Qa6+
Ba4 Qd3+
Ka2 Qe2+
Ka1 Qe1+
Kb2 Qe2+
Ka1 Qe1+
Ka2 Qe2+
Ka3 Qd3+
Bb3 Qa6+
Kb2 Qe2+
Kb1 Qf1+
Ka2 Qa6+
Kb2 Qe2+
Ka1 Qf1+
Ka2 Qa6+
Kb1 Qf1+
Kb2 Qe2+
The following position, with Black to play,arose in an AlphaZero game, played at roughly one minuteper move. rZ0l0skZ o0Znapo0 Z0o0Z0Zp Z0Z0ZQZ0 PZ0ZNOPZ Z0ZRZRJ0 a b c d e f g h
In this position, with Black to play, in classical chess Blackwould struggle to find a good plan and activity. Yet, here inself-capture chess, Black plays the obvious idea – sacrificingthe a7 pawn to open the a-file for its rook and initiate activeplay!
19. . .
Rxa7
Nc3 Qa8
Qg3 Rfd8 qZ0s0ZkZ s0Znapo0 Z0o0Z0Zp Z0M0Z0L0 PZ0Z0OPZ Z0ZRZRJ0 a b c d e f g h
Black soon managed to equalize and eventually draw thegame.
The following position, with White to play,arose in an AlphaZero game, played at roughly one minuteper move. ssessing Game Balance with AlphaZero oBo0Z0j0 Z0Z0Z0Z0 Z0Z0Z0Z0 PZ0Z0ZPZ S0Z0Z0ZK a b c d e f g h
In the previous moves, AlphaZero had manoeuvred its light-squared bishop to b7 via a6, with a clear intention of settingup threats to self-capture on b7 and promote the pawn on b8.Yet, if attempted immediately, Black can respond in turn byplaying c6, c5, or even self-capturing on c7 with the bishop.If the bishop moves away from the b8-h2 diagonal, Whitecan proceed with the plan. This explains why White playsthe following next:
Rc1 oBo0Z0j0 Z0Z0Z0Z0 Z0Z0Z0Z0 PZ0Z0ZPZ Z0S0Z0ZK a b c d e f g h
The rook can now be taken on c1, but this would allow thepromotion of the c-pawn via a self-capture.
34. . .
Be6
Rf1 Bd6
Rd1 Bf4
Rd4 Bg3
Rxh4 oBo0Z0j0 Z0Z0Z0Z0 Z0Z0Z0a0 PZ0Z0ZPZ Z0Z0Z0ZK a b c d e f g h
38. . .
Bxh4 cxb7 oPo0Z0j0 Z0Z0Z0Z0 Z0Z0Z0Z0 PZ0Z0ZPZ Z0Z0Z0ZK a b c d e f g h
And White went on to eventually win the game.
The following position, with White to play,arose in an AlphaZero game, played at roughly one minuteper move. RZ0Z0Z0Z ZNZ0Z0Z0 PZ0Z0a0Z Z0Z0Z0j0 rZ0Z0Z0o Z0Z0Z0Z0 Z0Z0Z0Z0 a b c d e f g h ssessing Game Balance with AlphaZero In this position, White plays a self-capture, axb7, givingaway the knight, for an immediate threat of promoting onb8. This is a common pattern in endgames in this variation,where pieces can be used to help promote the passed pawns.
Game AZ-41: AlphaZero Self-capture vs AlphaZeroSelf-capture
The following position, with Black to play,arose in an AlphaZero game, played at roughly one minuteper move. ZpZ0ZpZ0 s0ZpZpZP PZ0OnJ0Z Z0ZBO0O0 ZRZ0Z0Z0 a b c d e f g h
In this position, AlphaZero as Black plays another self-capture motif:
75. . . fxe4+, self-capturing its own knightwith check, while attacking White’s bishop on d3. Thishighlights novel tactical opportunities where self-capturescan be utilised not only as dynamic material sacrifices forthe initiative, but rather a key part of tactical sequenceswhere material gets immediately recovered.
Game AZ-42: AlphaZero Self-capture vs AlphaZeroSelf-capture
The following position, with White to play,arose in a fast-play AlphaZero game, played at roughly onesecond per move. o0Z0Zqop Z0m0m0Z0 Z0Z0LPZ0 PA0ZBZ0S Z0Z0Z0ZK a b c d e f g h
At the moment, White is two pawns down for the attack and has very strong threats against the Black king. In Clas-sical chess, those might prove fatal, but here Black uses aself-capture as a defensive resource, as can be seen in thefollowing forcing sequence:
Rxh7+ Kxh7
Rh4+ Kxg8 – Black is forced to captureits own rook to avoid checkmate – f4 Ng6
Rh2 Qxa2
Qc1 Qa4
Qc4+ Qxc4
Bxc4+ o0Z0Z0o0 Z0m0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0ZK a b c d e f g h
And here Black uses the second self-capture in this sequence,
40. . .
Kxg7, to secure the king.
Game AZ-43: AlphaZero Self-capture vs AlphaZeroSelf-capture
The following position, with White to play,arose in a fast-play AlphaZero game, played at roughly onesecond per move. ZQZ0Z0Z0 Z0Z0ZpZ0 Z0Z0Z0Z0 Z0Z0ZqAK a b c d e f g h
With White to play, in Classical chess this would result in amate in one move, on h7. Yet, in Self-capture chess Blackcan escape by self-capturing its rook on f8, at Which pointWhite has to attend to its own king’s safety.
Qh7+ Kxf8
Rxf6+ Nxf6
Qg6 Qf3+
Qg2Qxf4, leading to a simplified position. ssessing Game Balance with AlphaZero Game AZ-44: AlphaZero Self-capture vs AlphaZeroSelf-capture
The following position, with White to play,arose in a fast-play AlphaZero game, played at roughly onesecond per move. opZbZpop Z0Z0O0Z0 O0Z0Z0L0 Z0Z0ZRZK a b c d e f g h
In this position, with White to move, White self-capturesa pawn to open up dynamic possibilities against the Blackking on the f-file.
Qxf2 d3
Qxf7+ Kd8
Bxd3 Qxe5
Rd1 Be7
Bc4 opZbaQop Z0Z0l0Z0 O0Z0Z0Z0 Z0ZRZ0ZK a b c d e f g h
24. . .
Rf8
Ng5 opZbaQop Z0Z0l0M0 O0Z0Z0Z0 Z0ZRZ0ZK a b c d e f g h
25. . .
Qxg5
Qxe6 opZba0op Z0Z0Z0l0 O0Z0Z0Z0 Z0ZRZ0ZK a b c d e f g h
Here, Black utilizes a self-capture for defensive purposes,giving up the e7 bishop
26. . .
Qxe7
Qh3 Rf6
Qg3 Kc8
Re1 Qd6
Qxg7Bc6 opZ0Z0Lp Z0Z0Z0Z0 O0Z0Z0Z0 Z0Z0S0ZK a b c d e f g h with a roughly equal position.with a roughly equal position.