When game comparison becomes play: Absolutely Categorical Game Theory
aa r X i v : . [ m a t h . C O ] S e p WHEN GAME COMPARISON BECOMES PLAY:ABSOLUTELY CATEGORICAL GAME THEORY
URBAN LARSSON, RICHARD J. NOWAKOWSKI, AND CARLOS P. SANTOS
Abstract.
Absolute Universes of combinatorial games, as defined in a recentpaper by the same authors, include many standard short normal- mis`ere- andscoring-play monoids. In this note we show that the class is categorical, byextending Joyal’s construction of arrows in normal-play games. Given G and H in an Absolute Universe U , we study instead the Left Provisonal Game[ G, H ], which is a normal-play game, independently of the particular AbsoluteUniverse, and find that G −→ H (implying G < H ) corresponds to the setof winning strategies for Left playing second in [ G, H ]. By this we define thecategory
LNP ( U ). Introduction
This is the study of game comparison in Combinatorial Game Theory (CGT),specifically, Combinatorial Game Spaces, and their sub spaces (universes of games).The concept of a Combinatorial Game Space allows for a general frame work, whichincludes many standard classes of terminating games. One of the most elegantdiscoveries of normal-play CGT, [1], is that Left wins playing second in the game G if and only if G ≥
0. Since normal-play games constitute a group structure, thisleads to a constructive (subordinate) general game comparison, G ≥ H if and onlyif Left wins the game G − H playing second. Joyal [3] proved that games, underthe normal-play convention, form a category where H −→ G if Left wins playingsecond in G − H . That is, Left has good replies against any Right moves G R − H and G − H L and so forth.More generally, for any winning convention in CGT, game comparison is axiom-atized by: Left prefers G to H if, for all games X , Left does at least as well in G + X as in H + X . Each different winning convention, possibly coupled with otherconstraints gives a different partial order.The authors recently demonstrated [4] that there is a set of properties thatdefine Absolute Universes and together these properties reduce game comparisonsto considering only a certain
Proviso , and a
Common Normal Part (correspondingto Theorem 2.4 in this paper). Except for normal-play, typically Absolute Universesonly have a monoid structure (group structure is not common in scoring-play andnon-existent in mis`ere-play), so we cannot use the ‘inverse’ of any game freely. Itis generally believed that game comparison in normal-play is a special case, whichdoes not apply to other monoids of combinatorial games.Here, we construct a normal-play game, called the
Left Provisonal Game , [
G, H ]which is essentially playing G − H (as if H were invertible) but where Left’s options Mathematics Subject Classification.
Primary 91A46, Secondary 18B99.
Key words and phrases.
Keywords: Absolute Universe, Category theory, Combinatorial gamespace, Dicot mis`ere, Game comparison. are restricted by the Proviso, and where the games G and H belong to any AbsoluteUniverse. The previous work [4] implies that in any Absolute Universe, the games G and H satisfy G < H if and only if Left wins the normal-play game [ G, H ]whenever Right starts (Theorem 2.5). This allows for a construction of arrows,similar to Joyal’s, which shows that Absolute Universes are categorical.We give the relevant background on Absolute Combinatorial Game Theory [4]in Appendix A at the end of this paper. Appendix B contains code for CGsuite0.7, which ‘compares’ mis`ere dicot games by, instead, analyzing the Left ProvisonalGame.2.
Absolute game comparison and the Left Provisional Game
First we recall the Proviso for a pair of games in a given Absolute Universe [4],and we remind the reader that relevant background on outcomes, left-atomic gamesand so on, is also given in Appendix A.
Definition 2.1 (Proviso) . Consider an Absolute Universe U , and let G, H ∈ U .The ordered pair of games [ G, H ] ∈ Proviso( U ) if o L ( G + X ) > o L ( H + X ) for all left-atomic games X ∈ U ; o R ( G + X ) > o R ( H + X ) for all right-atomic games X ∈ U .From now onwards, pairs of games in an Absolute Universe will combine toanother (normal-play) game. Definition 2.2 (Left Provisonal Game) . Let U be an Absolute Universe. The LeftProvisonal Game (LPG) is defined on U × U as follows.(1) The positions are ordered pairs of games [ G, H ] ∈ U × U ;(2) The Left options of [ G, H ] are of the form:(a) [ G L , H ] ∈ Proviso( U );(b) [ G, H R ] ∈ Proviso( U ).(3) The Right options are games of the form [ G R , H ] or [ G, H L ];(4) A player who cannot move loses.That is Right cannot move and loses playing first if both G R and H L are empty.For Left the situation is more intricate. If for all G L , [ G L , H ] Proviso( U ) andfor all H R , [ G, H R ] Proviso( U ), then Left cannot move and loses. Thus, LeftProvisonal Game [ G, H ] is in fact a normal-play game, regardless of U . Using thestandard notation of normal-play, thus [ G, H ] = { [ G, H ] L | [ G, H ] R } = { [ G L , H ] ∈ Proviso( U ) , [ G, H R ] ∈ Proviso( U ) | [ G R , H ] , [ G, H L ] } , where G L ranges over all G ’sLeft options, H R ranges over H ’s all Right options, etc. Definition 2.3 (Left’s Maintenance) . Consider an Absolute Universe U , and let G, H ∈ U . The Left Provisional Game [ G, H ] ∈ Maintain( U ) ⊂ U × U if, forall Right options [ G, H ] R ∈ [ G, H ] R , there is a Left option [ G, H ] RL , such that[ G, H ] RL ∈ Maintain( U ).Let us recall the main theorem for comparing games in an Absolute Universe, nowstated as an equivalence involving Left Provisonal Games (see also Appendix A). Theorem 2.4 (Basic order of CGT, [4]) . Consider an Absolute Universe U and let G, H ∈ U . Then G < H if and only if Left Provisonal Game [ G, H ] ∈ Proviso ( U ) ∩ Maintain ( U ) . HEN GAME COMPARISON BECOMES PLAY 3 [ & , ,
0] [ ∗ , ,
0] [0 , Figure 1.
The game tree of [ & , Theorem 2.5.
Let
G, H be games in an Absolute Universe U . Then G < H if andonly if [ G, H ] ∈ Proviso ( U ) and the Left Provisonal Game [ G, H ] ≥ .Proof. By Theorem 2.4, it suffices to prove that [
G, H ] ≥ G, H ] ∈ Maintain( U ). This follows precisely because the inequality means Leftwins playing second in normal-play, which is Defininition 2.2 (3) combined with thedefinition of Maintain( U ). (cid:3) To the authors’ knowledge, in each studied Absolute Universe, Proviso( U ) isconstructive, in the sense that the condition in Definition 2.1 can be simplified tocompare only (variations of) the outcome of the actual games G and H , omittingthe potentially infinite class of atomic distinguishing games X . For example, in theuniverse of dicot mis`ere-play games, Proviso( U ) = { [ G, H ] : o ( G ) > o ( H ) } . Example 2.6.
The Proviso simplifies to o ( G ) > o ( H ) in dicot mis`ere play, sincethe only atomic games are the purely atomic ones. Take U as the dicot mis`ereuniverse and let G = & = h , ∗ | ∗i (“mup”, that means “mis`ere up”, the simplestdicotic game strictly larger than zero) and H = 0. In the Left Provisional Game[ & , ∗ , P = o ( ∗ ) > U o (0) = N gives that theProviso is not satisfied. The game tree of the Left Provisonal Game position [ & , G, H ] = ↑ > G < H . Still in dicotmis`ere, the Left Provisional Game [ ↑ ,
0, because R = o ( ↑ ) > o (0) = N , by theProviso. 3. Categories
Joyal’s construction for a category of normal-play games G and H uses that G ≥ H if and only if G − H ≥ G − H (Left’s set of winning strategies is “the arrow”). Inour terminology, this corresponds to the Left Maintenance for the free space ofnormal-play games. This follows since, for normal-play, the Proviso is implied by Where o ( X ) = ( o L ( X ) , o R ( X )) ∈ { ( − , −
1) = R, ( − , +1) = P, (+1 , −
1) = N, (+1 , +1) = L } , X ∈ U , inducing a partial order of outcomes. URBAN LARSSON, RICHARD J. NOWAKOWSKI, AND CARLOS P. SANTOS the Maintenance part, which is the condition G ≥ H in normal-play. We showthat each Absolute Universe is categorical by extending Joyal’s construction to theLeft Provisonal Game.In a category, the Hom(
H, G ) is a collection of morphisms that link the object H to the object G in a, for the given structure, specific and meaningful way. Themorphisms can be functions but it is not a requirement, as we saw with for exampleJoyal’s winning strategies. The arrows preserve some important property of thegiven structure, such as “winning” in Joyal’s example. We write H −→ G ifHom( H, G ) is not empty (and H f / / G if we want to particularize an element f ∈ Hom(
H, G )). To have a categorical structure, three properties must hold:(1) Identity: G −→ G for every object G ;(2) Composition: given f ∈ Hom(
H, J ) and g ∈ Hom(
J, G ) there is a naturalcomposition g ◦ f ∈ Hom(
H, G );(3) Associativity: the defined composition is associative.We will give a categorical construction based on a calculus of defined
Left Main-tenance Strategies of the LPG. Joyal’s and Conway’s “winning” is merely a conse-quence of being able to maintain an advantage, specificly, being able to move whenit is your turn. By using the LPG rather than the actual games, our “arrows”will contain all information of how Left maintains the ability to move, in particularwhile facing the additional burden of the Proviso part.
Definition 3.1. A play in a Left Provisonal Game X = [ G, H ] is a chain ofpositions X ❀ X ❀ · · · ❀ X n where the ‘moves ❀ ’ correspond to alternatingLeft and Right (or Right and Left) moves, and where n > Definition 3.2. A Left Maintenance Strategy in a given LPG is a play with thefollowing property: consider any stage of the play, where Right is to move; if Righthas a move, then Left has a response to this move. We write L R ( H, G ) for the setof all Left Maintenance Strategies in the game [
G, H ], assuming that Right starts,and L L ( H, G ) for all Left Maintenance Strategies, assuming that Left starts.Note 1: The reason that we reverse the order of the games in the sets of mainte-nance strategies is that these will correspond to the homorphisms of the categories,and the order of categorical objects related to “arrows” is reversed as comparedwith the conventions in game theory.Note 2: Since the LPG is a normal-play game, if you have a maintenance strategyyou will eventually win. The particular winning convention of the component gamesinside the LPG is irrelevant as long as the universe is absolute.Choosing a strategy f ∈ L R ( H, G ) is equivalent to choosing a strategy f G R ∈L L ( H, G R ) for each position G R ∈ G R and a strategy f H L ∈ L L ( H L , G ) for eachposition H L ∈ H L . Therefore L R ( H, G ) ∼ = [ G R L L ( H, G R ) ∪ [ H L L L ( H L , G ) . For normal-play games we use the standard notation for inequality ≥ , whereas in any other(general) universe we write < . HEN GAME COMPARISON BECOMES PLAY 5
The concept of a residual strategy is crucial in obtaining the composition of mor-phisms. The fundamental idea is the swivel-chair strategy, using the terminologyof [1] (or strategy stealing), see also [3, 2].
Definition 3.3 (Left’s Residual Strategy) . Given two maintenance strategies g ∈L R ( J, G ) and f ∈ L R ( H, J ), we construct
Left’s residual strategy g ⊛ f as follows.Consider a Right move from [ G, H ] to [ G R , H ]. We will find a Left’s maintenanceresponse, given the candidate morphisms f and g .Set up the two games [ G, J ] and [
J, H ], corresponding to the maintenance strate-gies f and g respectively; see the columns of Figure 2.If Left’s maintenance response in [ G R , J ] is to [ G RL , J ], then adapt this mainte-nance strategy for the game [ G R , H ], as [ G RL , H ].If Left’s maintenance response in [ G R , J ] is to [ G R , J R ] then Left considers in-stead her maintenance response to the Right move in the game [ J R , H ]. If this is[ J R , H R ], then her response in [ G R , H ] is to [ G R , H R ]. If, instead, the response isto some [ J RL , H ], she swivels back to the game [ G R , J RL ], and finds a response tothis Right move, and so on.In case [ G R , J R ] is a terminal position, then, because [ J R , H ] is a Right’s movein a Left Maintenance Strategy, there must exist a Right move in H R , and so theplay will terminate in [ G R , H R ], with a Left win (recall the Left Maintenance Gameis normal-play).In either case, by continuing this idea, because J is finite and because f and g are maintenance strategies, eventually Left’s response shifts to either of the forms[ G RL , J α ], with α = RL . . . L or [ J α , H R ] with α = RL . . . R (i.e. α is a finitesequence of alternating moves). In the first case, the response in [ G R , H ] will be to[ G RL , H ] and in the second case it will be to [ G R , H R ]. Unless this is a terminalposition, we may iterate the argument.The construction of g ⊛ f in the case of the Right move [ G, H L ] is analogous.By the definition of f and g it is clear that the residual strategy g ⊛ f is welldefined. As an immediate consequence we get Lemma 3.4.
Consider
G, J, H ∈ U and suppose that f ∈ L R ( H, J ) and g ∈L R ( J, G ) are Left Maintenance Strategies in the games [ J, H ] and [ G, J ] respectively.Then the residual strategy f ⊛ g ∈ L R ( H, G ) is a Left Maintenance Strategy in thegame [ G, H ] .Proof. By the swivel-chair construction in the definition of a residual strategy forthe game [
G, H ], Left has a response to any Right move at each stage of play. Thus f ⊛ g ∈ L R ( H, G ). (cid:3) Lemma 3.5.
The operator ⊛ is associative.Proof. Given f ∈ L R ( H, J ), h ∈ L R ( J, W ), and g ∈ L R ( W, G ), we construct thecomposite residual strategy g ⊛ h ⊛ f ∈ L R ( G, H ) in analogy with the swivel chairconstruction in Definition 3.3. Against, say, a Right move from (
G, H ) to ( G R , H ),Left executes the stealing procedure over the strategies f , h and g , getting, after afinite number of steps, an option G RL or H R . That g ⊛ h ⊛ f = g ⊛ ( h ⊛ f ) = ( g ⊛ h ) ⊛ f is then trivial. (cid:3) Definition 3.6 (Mimic strategy) . Consider the Left Provisonal Game position[
G, G ]. We define the mimic strategy m ∈ L R ( G, G ) (or copy-cat) as the strategy
URBAN LARSSON, RICHARD J. NOWAKOWSKI, AND CARLOS P. SANTOS [ G, J ] [ G R , J ][ G R , J R ] g [ J, H ] [ J R , H ] f [ J RL , H ][ G R , J RL ] g [ G R , J RLR ] [ J RLR , H ]( . . . ) Choice of G RL or Choice of H R f Figure 2.
Strategy stealingwhere Left replies to [ G R , G ] and [ G, G L ] with [ G R , G R ] and [ G L , G L ] respectively,and repeats this mimic process during the play. Lemma 3.7.
The mimic strategy is a Left Maintenance Strategy.Proof.
In any game of the form [
X, X ], the proviso is trivially satisfied, so Left hasthe same options as Right, and, as a required response, can thus imitate each Rightmove. (cid:3)
By using maintenance strategies in the Left Provisonal Game as the morphisms,we generalize Joyal’s results on categories for normal-play, to any Absolute Universeof combinatorial games.
Theorem 3.8.
Consider an Absolute Universe U and G, H ∈ U . If G < H , then,the structure ( U , f, ◦ ) , where f ∈ L R ( H, G ) = Hom(
H, G ) , and g ◦ f = g ⊛ f , iscategorical.Proof. By Theorem 2.4, G < H implies [ G, H ] ∈ Maintain( U ) ∩ Proviso( U ), whichin particular implies that, in the LPG, the set of Left’s maintenance strategies L R ( H, G ) is nonempty. Moreover, we have seen that the operator is consistent withthe residual strategy as composition and the mimic strategy as identity. Indeed,that the following diagram commutes was explained in Lemma 3.4.
HEN GAME COMPARISON BECOMES PLAY 7 ∗ ↑ ∗ ↓ && ∗ .. ∗ ↑↑ ↑↑
00 0 0 {↑ | ↓ ∗}{↑ | ↓ ∗} {↑ || { , ↓ ∗ | , ↓ ∗}}{↑ || { , ↓ ∗ | , ↓ ∗}} b bbbb bb bb Figure 3.
Games of rank 2 in dicot mis`ere-play. H f / / g ⊛ h ❅❅❅ ❅❅❅ J g (cid:15) (cid:15) G m Z Z That the defined composition (residual strategy) is associative was explained inLemma 3.5. (cid:3)
For any Absolute Universe U , call this category LNP ( U ), Left Normal Play over U . We finish off by continuing Example 2.6, the dicot mis`ere-play application. Example 3.9.
We compare the games of rank 2 in the dicot mis`ere-play universe.The Proviso is o ( G ) > o ( H ). The order is given in Figure 3, the value of the LPG[ G, H ] where G covers H in the partial order is written by the appropriate edge.In the picture, the dicot mis`ere-play game values (literal forms) are ↑ = h | ∗i , ↓ = h∗ | i , & = h , ∗ | ∗i , . = h∗ | ∗ , i (“mown”), & ∗ = h , ∗ | i , and . = h | ∗ , i . Acknowledgement . We thank Darien DeWolf for suggesting our category’s name.
URBAN LARSSON, RICHARD J. NOWAKOWSKI, AND CARLOS P. SANTOS
Appendix A
The following is a shortened introduction to Absolute CGT [4]. Combinatorialgames have two players, usually called
Left (female) and
Right (male) who movealternately. Both players have perfect information, and there are no chance devices.Thus these are games of pure strategy with no randomness. Combinatorial gamesare commonly represented by a rooted tree called the game tree. The nodes arepositions that can be reached during the play of the game and the root is thepresent position. The children of a node are all the positions that can be reachedin one move and these are called options . We distinguish between the left-options,those positions that Left can reach in one move, and the right-options, denoted by G L and G R respectively. Any game G can be represented by two such lists and wewrite G = h G L | G R i . Thus, G can be expanded in terms of elements of its terminal positions (those positions with no options). The rank of a game is the depth ofthe game tree (see also Definition 3.10). This gives the common proof technique‘induction on the options’ since the depth of the game tree of an option is at leastone less than that of the original position (we study games without cycles).Let ( A , +) be a totally ordered, additive group. A terminal position will be ofthe form h∅ ℓ | ∅ r i where ℓ, r ∈ A . The intuition, adapted from scoring game theory isthat, if Left is to move, then the game is finished, and the ‘score’ is ℓ , and similarlyfor Right, where the ‘score’ would be r . In general, if G is a game with no Leftoptions then we write G L = ∅ ℓ for some ℓ ∈ A and if Right has no options then wewrite G R = ∅ r for some r ∈ A .We refer to ∅ a as an atom and a ∈ A as the adorn . Positions in which Left (Right)does not have a move are called left- (right-) atomic . A purely-atomic position isboth left- and right-atomic. It is useful to identify a = h∅ a | ∅ a i for any a ∈ A . Forexample, = h∅ | ∅ i where 0 is the identity of A . Definition 3.10.
Let A be a totally ordered group and let Ω = {h∅ ℓ | ∅ r i | ℓ, r ∈ A } .For n >
0, the set Ω n is the set of all games with finite sets of options in Ω n − ,including games which are left- and/or right-atomic, and the set of games of rank n is Ω n \ Ω n − . Let Ω = ∪ n ≥ Ω n . Then (Ω , A ) is a free space of games.Many combinatorial games decompose into independent sub-positions as playprogresses. A player must choose exactly one of these sub-positions and play in it.This is known as the disjunctive sum. Here, and elsewhere, an expression of thetype G L + H denotes the list of games of the form G L + H , G L ∈ G L . Definition 3.11.
Consider a totally ordered group A and G, H ∈ (Ω , A ). Thedisjunctive sum of G and H is given by: G + H = h ∅ ℓ + ℓ | ∅ r + r i , if G = h ∅ ℓ | ∅ r i and H = h ∅ ℓ | ∅ r i ;= h ∅ ℓ + ℓ | G R + H, G + H R i , if G = h ∅ ℓ | G R i , H = h ∅ ℓ | H R i , and at least one of G R and H R is not empty;= h G L + H, G + H L | ∅ r + r i , if G = h G L | ∅ r i , H = h G L | ∅ r i , and at least one of G L and H L is not empty;= h G L + H, G + H L | G R + H, G + H R i , otherwise. Definition 3.12.
A combinatorial game space is the structureΩ = ((Ω , A ) , S , ν L , ν R , +) , HEN GAME COMPARISON BECOMES PLAY 9 where ‘+’ is the disjunctive sum in the free space (Ω , A ), S is a totally ordered set ofgame results, and ν L : A → S and ν R : A → S are order preserving maps. Moreover,if | A | > ν ( a ) = ν L ( a ) = ν R ( a ), for all a ∈ A .Suppose a, b ∈ S with a > b , the standard convention is that Left prefers a andRight prefers b . The three winning conventions usually considered in the literatureare: • normal-play corresponds to: (i) the trivial group A = { } and the set S = {− , +1 } ; (ii) the maps ν L (0) = − ν R (0) = +1, • mis`ere-play corresponds to: (i) the trivial group A = { } and the set S = {− , +1 } ; (ii) the maps ν L (0) = +1, ν R (0) = − • scoring-play usually corresponds to the adorns being the group of real num-bers, with its natural order and addition, and moreover S = A = R , andwhere ν is the identity map.The conjugate denotes the position where Left and Right have ‘switched roles’. Definition 3.13.
The conjugate of G ∈ Ω is ↔ G = h∅ − b | ∅ − a i , if G = h∅ a | ∅ b i , a, b ∈ A h ↔ G R | ∅ − a i , if G = h∅ a | G R ih∅ − a | ↔ G L i , if G = h G L | ∅ a ih ↔ G R | ↔ G L i , otherwise , where ↔ G R denotes the list of games ↔ X , for X ∈ G R , and similarly for G L .By the recursive definition of the free space (Ω , A ), each combinatorial gamespace is closed under conjugation. In normal-play, the games form an ordered groupand each game G has an additive inverse, appropriately called − G and − G = ↔ G .However, there are other spaces of games, for example scoring and mis`ere games,where ↔ G is not necessarily − G (e.g. [5]). Definition 3.14. A universe of games, U ⊆ Ω, is a subspace of a given combina-torial game space Ω = ((Ω , A ) , S , ν L , ν R , +), with:(1) a = h∅ a | ∅ a i ∈ U for all a ∈ A ;(2) options closure: if A ∈ U and B is an option of A then B ∈ U ;(3) disjunctive sum closure: if A, B ∈ U then A + B ∈ U ;(4) conjugate closure : if A ∈ U then ↔ A ∈ U ;The mapping of adorns in A to elements of S is extended to positions in generalvia two recursively defined (optimal play) outcome functions . Definition 3.15.
Let G ∈ U ⊆ Ω and consider given maps ν L : A → S and ν R : A → S , where S is a totally ordered set. The left - and right-outcome functions are o L : Ω → S , o R : Ω → S , where o L ( G ) = ( ν L ( ℓ ) if G = h∅ ℓ | G R i ,max L { o R ( G L ) } otherwise o R ( G ) = ( ν R ( r ) if G = h G L | ∅ r i ,min R { o L ( G R ) } otherwise, where the max L (min R ) ranges over all Left (Right) options.From this we conclude that each universe is a partially ordered commutativemonoid with as the additive identity.Let G ∈ U . From Definition 3.15 we have that o L ( G ) = ν L ( ℓ ) and o R ( G ) = ν R ( r )for some ℓ, r ∈ A . Therefore we may always assume that the set of (left- and right-)outcomes is S = { ν L ( a ) : a ∈ A } ∪ { ν R ( a ) : a ∈ A } . Definition 3.16.
A universe U of combinatorial games is parental if, for each pairof finite non-empty lists, A , B ⊂ U , then hA | Bi ∈ U . Definition 3.17.
A universe U of combinatorial games is dense if, for all G ∈ U ,for any x, y ∈ S , there is a H ∈ U such that o L ( G + H ) = x and o R ( G + H ) = y . Definition 3.18.
A universe U of combinatorial games is an Absolute Universe ifit is both parental and dense.A partial order is defined on any universe of additive combinatorial games.
Definition 3.19.
Let U be any universe of combinatorial games. For G, H ∈ U , G < H modulo U if and only if o L ( G + X ) > o L ( H + X ) and o R ( G + X ) > o R ( H + X ),for all games X ∈ U .The main results for Absolute Combinatorial Game Theory [4] are the followingimprovements of general game comparison. (The “Common Normal Part” corre-sponds to the Maintenance part in this paper.) Theorem 3.20 (Basic order of games [4]) . Consider games
G, H ∈ U , an AbsoluteUniverse. Then G < H if and only if the following two conditions hold.Proviso: o L ( G + X ) > o L ( H + X ) for all left-atomic X ∈ U ; o R ( G + X ) > o R ( H + X ) for all right-atomic X ∈ U ;Common Normal Part:For all G R , there is H R such that G R < H R or there is G RL such that G RL < H ;For all H L , there is G L such that G L < H L or there is H LR such that G < H LR . Corollary 3.21 (Subordinate game comparison [4]) . Let
G, H ∈ U , an AbsoluteUniverse. Then G < U H if the Common Normal Part holds and if U is the • normal-play universe; • dicot mis`ere-play universe, and o ( G ) > o ( H ) ; • free mis`ere-play space, and H L = ∅ ⇒ G L = ∅ and G R = ∅ ⇒ H R = ∅ ; • dicot scoring-play universe, and o ( G ) > o ( H ) ; • guaranteed scoring-play universe, and o L ( G ) > o L ( H ) and o R ( G ) > o R ( H ) ,where o and o denotes Right’s and Left’s pass allowed left- and right-outcomes respectively [4] . Appendix B
One of the benefits of the Left Provisonal Game is that it allows for game com-parison in any
Absolute Universe in CG-suit. We attach code for version CG-suit0.7 (coded by C. Santos). The procedure CompareDM requires input Left Pro-visonal Game as a pair of literal form (dicot mis`ere-play) games. We begin byillustrating how to run the below code.
HEN GAME COMPARISON BECOMES PLAY 11
EXAMPLE:G=literally({0,*|*})H=literally({0,*|{0|0,*}})CompareM([G,H])Moutcome:=proc (G)local a,b,c,j,w,k,l,r,i;option remember;l:=LeftOptions(G);r:=RightOptions(G);b:=Length(l);c:=Length(r);if (G==0) thenk:=11;fi;if (G!=0) thenj:=0;for i from 1 to b doif (Moutcome(l[i])==0 or Moutcome(l[i])==1) then j:=1; fi;od;w:=0;for i from 1 to c doif (Moutcome(r[i])==0 or Moutcome(r[i])==-1) then w:=1; fi;od;if (j==0 and w==0) then k:=0; fi;if (j==0 and w==1) then k:=-1; fi;if (j==1 and w==0) then k:=1; fi;if (j==1 and w==1) then k:=11; fi;fi;return k;end;Dual := proc (pos)local l,r,l1,r1,l2,r2,ll1,ll2,rr1,rr2,i,aux;option remember;l := [];r := [];l1 := LeftOptions(pos[1]);r1 := RightOptions(pos[1]);l2 := LeftOptions(pos[2]); r2 := RightOptions(pos[2]);ll1:=Length(l1);rr1:=Length(r1);ll2:=Length(l2);rr2:=Length(r2);for i from 1 to rr1 doaux:=[r1[i],pos[2]];Add(r,Dual(aux));od;for i from 1 to ll2 doaux:=[pos[1],l2[i]];Add(r,Dual(aux));od;for i from 1 to ll1 doif (Moutcome(l1[i])==1) thenaux:=[l1[i],pos[2]];Add(l,Dual(aux));fi;if (Moutcome(l1[i])==11 and (Moutcome(pos[2])==11 or Moutcome(pos[2])==-1)) thenaux:=[l1[i],pos[2]];Add(l,Dual(aux));fi;if (Moutcome(l1[i])==0 and (Moutcome(pos[2])==0 or Moutcome(pos[2])==-1)) thenaux:=[l1[i],pos[2]];Add(l,Dual(aux));fi;if (Moutcome(l1[i])==-1 and Moutcome(pos[2])==-1)thenaux:=[l1[i],pos[2]];Add(l,Dual(aux));fi;od;for i from 1 to rr2 doif (Moutcome(pos[1])==1)thenaux:=[pos[1],r2[i]];Add(l,Dual(aux));fi;if (Moutcome(pos[1])==11 and (Moutcome(r2[i])==11 or Moutcome(r2[i])==-1))thenaux:=[pos[1],r2[i]];Add(l,Dual(aux));
HEN GAME COMPARISON BECOMES PLAY 13 fi;if (Moutcome(pos[1])==0 and (Moutcome(r2[i])==0 or Moutcome(r2[i])==-1))thenaux:=[pos[1],r2[i]];Add(l,Dual(aux));fi;if (Moutcome(pos[1])==-1 and Moutcome(r2[i])==-1)thenaux:=[pos[1],r2[i]];Add(l,Dual(aux));fi;od;return {l | r};end;CompareDM := proc (pos)local l,r,l1,r1,l2,r2,ll1,ll2,rr1,rr2,i,a,b,s;option remember;l := [];r := [];l1 := LeftOptions(pos[1]);r1 := RightOptions(pos[1]);l2 := LeftOptions(pos[2]);r2 := RightOptions(pos[2]);ll1:=Length(l1);rr1:=Length(r1);ll2:=Length(l2);rr2:=Length(r2);a:=0; b:=0;if ((Moutcome(pos[1])==1) or (Moutcome(pos[1])==11 and(Moutcome(pos[2])==11 or Moutcome(pos[2])==-1)) or(Moutcome(pos[1])==0 and (Moutcome(pos[2])==0 orMoutcome(pos[2])==-1)) or (Moutcome(pos[1])==-1 andMoutcome(pos[2])==-1)) and (Dual(pos)>=0) thena:=1;fi;if ((Moutcome(pos[2])==1) or (Moutcome(pos[2])==11 and(Moutcome(pos[1])==11 or Moutcome(pos[1])==-1)) or(Moutcome(pos[2])==0 and (Moutcome(pos[1])==0 or
Moutcome(pos[1])==-1)) or (Moutcome(pos[2])==-1 andMoutcome(pos[1])==-1)) and (Dual([pos[2],pos[1]])>=0) thenb:=1;fi;if (a==1) and (b==1) then s:="G=H"; fi;if (a==1) and (b==0) then s:="G>H"; fi;if (a==0) and (b==1) then s:="G References [1] E. R. Berlekamp, J. H. Conway, and R. K. Guy. Winning Ways for your Mathematical Plays ,volume 1–4. A K Peters, Ltd., 2001–2004. 2nd edition: vol. 1 (2001), vols. 2, 3 (2003), vol. 4(2004).[2] J. R. B. Cockett, G. S. H. Cruttwell, and K. Saff. Combinatorial game categories. Preprint.[3] A. Joyal. Remarques sur la th´eorie des jeux ´a deux personnes. Gazette des SciencesMath´ematiques du Qu´ebec , 4:46–52, 1977.[4] U. Larsson, R. J. Nowakowski, C. P. Santos. Absolute Combinatorial Game Theory.http://arxiv.org/abs/1606.01975[5] R. Milley. Partizan kayles and mis`ere invertibility. Integers , (1309.1631), 2015. Dalhousie University, Canada E-mail address : [email protected] Dalhousie University, Canada E-mail address : [email protected] Universidade de Lisboa, Campo Grande, Portugal E-mail address ::