Transverse Wave: an impartial color-propagation game inspired by Social Influence and Quantum Nim
aa r X i v : . [ c s . CC ] J a n TRANSVERSE WAVE: AN IMPARTIAL COLOR-PROPAGATIONGAME INSPRIED BY SOCIAL INFLUENCE AND QUANTUM NIMKyle Burke
Department of Computer Science, Plymouth State University, Plymouth, NewHampshire, https://turing.plymouth.edu/~kgb1013/
Matthew Ferland
Department of Computer Science, University of Southern California, Los Angeles,California, United States
Shang-Hua Teng Department of Computer Science, University of Southern California, Los Angeles,California, United States https://viterbi-web.usc.edu/~shanghua/
Abstract
In this paper, we study a colorful, impartial combinatorial game played on a two-dimensional grid,
Transverse Wave . We are drawn to this game because of itsapparent simplicity, contrasting intractability, and intrinsic connection to two othercombinatorial games, one inspired by social influence and another inspired by quan-tum superpositions. More precisely, we show that
Transverse Wave is at theintersection of social-influence-inspired
Friend Circle and superposition-based
Demi-Quantum Nim . Transverse Wave is also connected with Schaefer’s logicgame
Avoid True . In addition to analyzing the mathematical structures and com-putational complexity of
Transverse Wave , we provide a web-based version of thegame, playable at https://turing.plymouth.edu/~kgb1013/DB/combGames/transverseWave.html.
Furthermore, we formulate a basic network-influence inspired game, called
Demo-graphic Influence , which simultaneously generalizes
Node-Kyles and
Demi-Quantum Nim (which in turn contains as special cases
Nim , Avoid True , and
Transverse Wave .). These connections illuminate the lattice order , induced byspecial-case/generalization relationships over mathematical games, fundamental toboth the design and comparative analyses of combinatorial games. Supported by the Simons Investigator Award for fundamental & curiosity-driven research andNSF grant CCF1815254.
1. The Game Dynamics of
Transverse Wave
Elwyn Berlekamp, John Conway, and Richard Guy were known not only for theirdeep mathematical discoveries but also for their elegant minds. Their love of math-ematics and their life-long efforts of making mathematics fun and approachablehad led to their master piece, “Winning Ways for your Mathematical Plays,” [2], abook that has inspired many. The field that they pioneered— combinatorial gametheory —reflects their personalities. Illustrious combinatorial games are usually: • Approachable : having a simple, easy to remember and easy to understandruleset, and • Elegant : having attractive game boards, yet • Intriguing & Challenging : having rich strategy structures and requiringnontrivial efforts to play well (optimally).
The last property has a characterization using computational complexity. As thedimensions of games grow—illustrated by popular games from Go [17] to Hex [21,19]—the underlying computational problem for determining the outcome of theirgame positions and for selecting winner moves becomes computational intractable.Thus, elegant combinatorial games with simple, easy to understand & rememberrulesets yet intractable complexity are the gold standard for combinatorial gamedesign [8, 4].The computational complexity of determining the winnability of a ruleset is acommon problem. For games where the winner can be determined algorithmicallyin polynomial-time, optimal players can be programmed that will run efficiently.In a match with one of these players, there is no need to play the game out todetermine whether you can win; just run the algorithm and see what it tells you.In order to make the competition interesting , we want the winnability to be computationally intractable , meaning there’s no known efficient algorithm to alwayscalculate a position’s outcome class. One way to argue that this is the case is toshow that the problem of finding the outcome class is hard for a common complexityclass. Many such combinatorial games are found to be PSPACE-hard, meaningthat finding a polynomial-time algorithm automatically leads to a polynomial-timesolution to all problems in PSPACE.This argument does not entirely settle the debate about whether a ruleset is”interesting”. Indeed, it could be the case that from common starting positions,there is a strategy for the winning player to avoid the computationally-hard posi-tions. Finding computational-hardness for general positions in a ruleset is only amininum-requirement. Improvements can be made by finding hard positions morelikely to result after game play from the start. The best proofs of hardness yieldpositions that are starting positions themselves . In this paper, we consider a simple, colorful, impartial combinatorial game over two-dimensional grids. We call this game
Transverse Wave . We become interestedin this game during our study of quantum combinatorial games [4], particularly, inour complexity-theoretical analysis of a family of games formulated by superpositionof the classical
Nim . In addition to Schaefer’s logic game,
Avoid True , Trana-verse Wave is also fundamentally connected with several social-influence-inspiredcombinatorial games. As we shall show,
Transverse Wave , is PSPACE-hard onsome possible starting positions.
Ruleset 1 ( Transverse Wave) . For a pair of integer parameters m, n > , agame position of Transverse Wave is an m by n grid G , in which the cells arecolored either green or purple . For the game instance starting at this position, two players take turns selectinga column of this colorful grid. A column j ∈ [ n ] is feasible for G if it contains atleast one green cell. The selection of j transforms G into another colorful m by n grid G ⊗ [ j ] by recoloring column j and every row with a purple cell at column j with purple . In the normal-play convention, the player without a feasible moveloses the game.
Note that purple cells cannot change to green, and each move increases thenumber of purple cells in the grid by at least one. In fact, each move turns at leastone column into one with all purple cells. Thus, any position with dimension m by n must end in at most n turns, and height of Transverse Wave ’s game tree is atmost n . Consequently, Transverse Wave is solvable in polynomial space.Note also that due to the transverse wave , the selection of column j could makesome other feasible columns infeasible. In Section 4, we will show that the in-teraction among columns introduce sufficiently rich mathematical structures for Transverse Wave to efficiently encode any PSPACE-complete games such as
Hex [13, 18, 9, 19],
Avoid True [21],
Node Kalyes [21], Go [17], Geography [17]. In other words,
Transverse Wave is a PSPACE-complete impartial game. This requires some variance on these starting positions. ”Empty” or well-structured initialboards do not have a large enough descriptive size to be computationally hard in the expectedmeasures. We tested the approachability of this game by explaining its ruleset to a bilingual eight-year-old second-grade student—in Chinese—and she turned to her historian mother and flawlesslyexplained the ruleset in English. or any pair of easily-distinguishable colors. Think of purple paint cascading down column j and inducing a purple ”transverse wave”whenever the propagation goes through an already-purple cell.
3b 0 1 2 3cFigure 1: An example move for
Transverse Wave . a : The player chooses column2, which has a green cell. b : Indigo cells denote those that will become purple.These include the previously-green cells in column 2 as well as the green cells inrows where column 2 had purple cells. c : The new position after all cells are changedto be purple.We have implemented Transverse Wave in HTML/Javascript. Transverse WaveTransverse Wave uses only two colors; a position can be expressed naturally witha Boolean matrix. Furthermore, making a move can be neatly captured by basicBoolean functions. Let consider the following two combinatorial games over Booleanmatrices that are isomorphic to
Transverse Wave . While these logic associationsare straightforward, they set up stimulating connections to combinatorial gamesinspired by social influence and quantum superposition.We use the following standard notation for matrices: For an m × n matrix A , i ∈ [ m ] and j ∈ [ n ], let A [ i, :], A [: , j ], A [ i, j ] denote, respectively, the i th row, j th column, and the ( i, j ) th entry in A . Ruleset 2 ( Crosswise AND) . For integers m, n > , Crosswise AND plays onan m × n Boolean matrix B .During the game, two players alternatively select j ∈ [ n ] , where j is feasible for B if B [: , j ] = ~ . The move with selection j then changes the Boolean matrix to oneas the following: ∀ i = j ∈ [ m ] , the i th row takes a component-wise AND with its j th Web version: https://turing.plymouth.edu/~kgb1013/DB/combGames/transverseWave.html bit, B [ i, j ] , then the ( i, j ) th entry is set to 0. Under normal play, the player withno feasible column to choose loses the game. By mapping purple cell to Boolean 0 (i.e., false ) and green cell to Boolean 1(i.e., true ), and purple transverse wave to crosswise-AND-with-zero, we have:
Proposition 1.
Transverse Wave and
Crosswise AND are isomorphic games.
Ruleset 3 ( Crosswise OR) . For integer parameters m, n > , Crosswise OR plays on an m × n Boolean matrix B .During the game, two players alternatively select j ∈ [ n ] , where j is feasible for B if B [: , j ] = ~ . The move with selection j then changes the Boolean matrix to oneas the following: ∀ i = j ∈ [ m ] , the i th row takes a component-wise OR with its j th bit, B [ i, j ] , then the ( i, j ) th entry is set to 1. Under normal play, the player withno feasible column to choose loses the game. By mapping purple cell to Boolean 1, and green cell to Boolean 0, and purpletransverse wave to crosswise-OR-with-one, we have:
Proposition 2.
Transverse Wave and
Crosswise OR are isomorphic games.
Transverse Wave
Game Values
As we will discuss later, in general,
Transverse Wave is PSPACE-complete. So,we have no hope for an efficient complete characterization for the game values. Inthis subsection, we show that the game values for a specific case of
TransverseWave , where each column with has no more than 1 purple tile, can be fully char-acterized. Because we are able to classify the moves into two types of options, andby extension can define the game by two parameters, a fun and interesting nimberversion of Pascal-Like triangle arises.
Theorem 1 (Pascal-Like Nimber Triangle) . Let p be the number of rows with atleast 1 purple tile and k be the number of rows with an odd number of purple tiles,and q = 0 if there are an even number of columns with only green tiles, and 1otherwise. We define G ′ to be G ′ = , ( k is even and p > k ) or ( k is odd and p < k ) ∗ , ( k is even and p < k ) or ( k is odd and p > k ) ∗ , p = 2 k If G is a Transverse Wave position where for every column we can select,there is no more than one purple tile (discounting rows with only purple tiles), then, G = G ′ + ∗ q Proof.
We will call the rows with an odd number of purple tiles the odd parity rowsand the ones with an even number of purple tiles the even parity rows. We claimthat G ′ is the game value without including the columns with only green tiles asoptions, and G to be the game value with them.Assuming G ′ is as we claim, it isn’t difficult to see that G is correct. For eachall-green column, they can be selected at any time, and don’t effect the game atall (since they can always be selected). Thus, each all-green column is just a ∗ , so G = G ′ if there are an even number of all-green columns and G = G ′ + ∗ if thereare an odd number of all-green columns.We have an illustrative triangle of cases of G ′ of up to 8 rows in Figure 2.87654321 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Figure 2: A Pascal-Like Nimber Triangle of the values of
Transverse Wave positions with no overlapping literals.If the player chooses a column where the purple tile is in an odd parity row, thenan even number of other rows share that single purple cell. Later selecting any ofthose rows makes no other change to the game state, so they can each be consideredto contribute (additively) a ∗ , for a total of 2 k ∗ = 0. Then the resulting option’svalue is just the same as one with p − k − ∗ is added (2 k + 1) ∗ = ∗ added to the option. Thus, theresulting game is the value above and right in the table (the same number of oddrows and one less row overall) plus ∗ .By inspection, note that Table 1 represents the correct value for each possibleparents in the game tree.Now, we have 5 cases which invoke those table cases.Case Above left Above right Valuea 0b 0 0c 0 ∗ d 0 0 ∗
2e 0 ∗ ∗ f 0 ∗ ∗ g ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Table 1: Values for a game with a given position based on what is above and leftin the triangle and what is above and right1. k even and p > k : case a, b, h, or j (thus value 0)2. k odd and p < k : case a, g, h, or k (thus value 0)3. k even and p < k : case c, e, or l (thus value ∗ )4. k odd and p > k : case e or f (thus value ∗ )5. p = 2 k : case d or i (thus in ∗ p − k even and p > k . This is either it is thebase case (case a ), because there are 0 odd rows (case b ), or k some other evennumber less than p . It’s left parent will have k − p − > k − ∗ . It’s right parent has p − ≥ k , and thus is either 0 or ∗
2. Thus, it must be either case h or j .Now, let’s look at k odd and p < k . Then, either it is the base case (case a ), k = p (then it must be case g , since it only has a single left parent with even k ),or it is some other odd k such that p < k . Then, it has a right parent with odd k and p < k , which is 0. And the left parent has even k and p − ≤ k − ∗ or ∗
2, which is case h and k , respectively.If k is even and p < k , then either k = p , in which case it has a single left parentwhich is in case 2, which is case c , or it is some other even k with p < k . In thatcase, the top right parent will have the same k and thus be case 3 and thus ∗ . Thetop left parent will have k − n − ≤ k − ∗
2. This is e or l .If k is odd and p − > k − p > k − p − ≥ k , and is thus 0 or ∗
2, and is thus e or f .Finally, if p = 2 k , then k can either be odd or even. If it is odd, then the leftparent is has even k and p − > k −
1) and is thus 0, and the right parent haseven k and p < k and is thus 0, putting this in case d . If k is even, then the leftparent is odd and thus ∗ , and the right parent is even and thus ∗ , putting us in case i . We also have some game miscellaneous game values, and have examples of up to ∗
7, as shown in in Table 2.Nimber Rows (shorthand) Other Columns0 (0) ∗ (0) (01) ∗ ∗ ∗ ∗ ∗ ∗ ∗
7. The shorthand uses parenthesis to indicatedrows and numbers to indicate purple columns So, for example, the ∗ Avoid True and
Demi-Quantum Boolean Nim .Although we can only characterize
Transverse Wave in very special cases,the Pascal-like formation of these game values provides us a glimpse of a potentialelegant structure. Something interesting to explore in the future is whether one cancleanly characterize the game values when the game is restricted to have only twopurple tiles in each column. And then if so, one would like to see how large of theparameter one can characterize and when we encounter intractability. Answeringthese questions can also tell us about the values for the other related games as well.
2. Connection to a Social-Influence-Inspired Combinatorial Game
Because mathematical principles are ubiquitous, combinatorial game theory is afield intersecting many disciplines. Combinatorial games have drawn inspirationwidely from logic to topology, from military combat to social sciences, from graphtheory to game theory. Because the field cherishes challenging games with simplerulesets and elegant game boards, combinatorial game design is also a distillationprocess, aiming to derive elementary moves and transitions in order capture theessence of complex phenomena that inspire the game designers.In this and the next sections, we discuss two games whose intersection contains
Transverse Wave . While they have found a common ground, these games rootedfrom different research fields. The first game, called
Friend Circle is motivatedby viral marketing [20, 16, 6], while the second, called
Demi-Quantum Nim , wasinspired by quantum superpositions [14, 7, 4]. In this Section, we first focus on
Friend Circle . Friend Circle
In many ways, viral marketing itself is a game of financial optimization. It aims toconvert more people, through network-propagation-based social influence, by strate-gically investing in a seed group of people [20, 16]. In the following combinatorialgame inspired by viral marketing,
Friend Circle , we use a high-level perspec-tive of social influence and social networks. Consider a social-network universe likeFacebook, where people have their “circles” of friends. They can broadcast to allpeople in their friend circles (with) a single post), or they can individually interactwith some of their friends (via various personalized means). We will use individualinteraction to set up the game position. In game
Friend Circle , only broadcast-type of interaction is exploited. We will use the following traditional graph-theorynotation: In an undirected graph G = ( V, E ), for each v ∈ V , the neighborhood of v in G is N G ( v ) = { u | ( u, v ) ∈ E } . Ruleset 4 ( Friend Circle) . For a ground set V = [ n ] (of n people), a FriendCircle position is defined by a triple ( G, S, w ) , where • G = ( V, E ) is an undirected graph. An edge between two vertices represents afriendship between those people. • S ⊂ V denotes the seed set, and • w : E → { f , t } (false and true) represents whether those friends have alreadyspoken about the target product (with at least one recommending it to theother).To choose their move, a player–a viral marketing agent–picks a person from the seed set, v ∈ S , such that ∃ e = ( v, x ) ∈ E where w ( e ) = f . This represents choosingsomeone who hasn’t spoken about the product to at least one of their friends.The result of this move is a new position ( G, S, w ′ ) , where w ′ is the same as w except that ∀ x ∈ N G ( v ) : • w ′ (( v, x )) = t , and • if w (( v, x )) = t then ∀ y ∈ N G ( x ) : w ′ (( x, y )) = t . An example of a
Friend Circle move is shown in Figure 2.1. v v v v v t ftf ff v v v v v t t t t ff Figure 3: Example of a
Friend Circle move. In the position on the left, let theseed set S = { v , v , v , v } , all of which are acceptable to choose because they allhave an incident false edge. If a player chooses v , then the result is the right-handposition. In the second position, v has had all of it’s incident edges become true.In addition, since ( v , v ) was true, all of v ’s incident edges have also changed totrue. The altered edges in the figure are represented in bold. Note that in theresulting position, the next player can only choose to play at either v and v , as v and v have only true edges and v / ∈ S .By inducing people in the seed’s friend circle who had existing intersection withthe chosen seed to broadcast, Friend Circle emulates an elementary two-stepnetwork cascading in social influence.
Friend Circle
We first connect
Friend Circle to the classical graph-theory game
Node-Kayles . Ruleset 5 (( Node-Kayles)) . The starting position of
Node-Kayles is an undi-rected graph G = ( V, E ) .During the game, two players alternate turns selecting vertices, where a vertex v ∈ V is feasible if neither it has already been selected in the previous turns norany of its neighbors has already been selected. The player who has no more feasiblemove loses the game. maximal independent set of G , the next playercannot make a move, and hence loses the game. It is well-known that Node-Kayles is PSPACE complete [21].
Theorem 2 ( Friend Circle is PSPACE-complete) . The problem of determiningwhether a
Friend Circle position is winnable is
PSPACE -complete.Proof.
First, we show that
Friend Circle is PSPACE-solvable. During a game of
Friend Circle starting at (
G, S, w ), once a node s ∈ S is selected by one of theplayers, s all edges incident to s become t . Since true edges can never later become f , s can never again be chosen for a move and the height of the game tree is at most | S | . Then, by the standard depth-first-search (DFS) procedure for evaluating thegame tree for ( G, S, w ) in
Friend Circle , we can determine the outcome class ofin polynomial space.To establish that
Friend Circle is a PSPACE-hard game, we reduce
Node-Kayles to Friend Circle . vv and N G ( v ) v t v tttt t f Figure 4: Example of the reduction from
Node Kayles to Friend Circle . Onthe left is a
Node Kayles vertex and it’s neighborhood. On the right is thosesame vertices, along with t v with t weights on all the old edges and f on the newedge with ( v, t v ).Suppose we have a Node-Kayles instance at graph G = ( V , E ). For thereduced Friend Circle position, we create a new graph G = ( V, E ) as the follow-ing. First, for each v ∈ V , we introduce a new vertex t v . Let T = { t v | v ∈ V } , so V = V ∪ T . In addition, let E = { ( v, v t ) | v ∈ V } , so E = E ∪ E . Next, we setthe weights: • ∀ e ∈ E : w ( e ) = t , and • ∀ e ∈ E : w ( e ) = f as shown in Friend Circle . Last, we set S = V .2We now prove that Friend Circle is winnable at (
G, S, w ) if and only
NodeKayles is winnable at G . Note that because ∀ v ∈ V , w ( v, t v ) = f , all verticesin V are feasible choices for the current player in Friend Circle . As the gameprogresses, vertices in V are no longer able to be chosen when their edge to the t vertex becomes true. From here the argument is very simple: each play on vertex v ∈ V in Node Kayles corresponds exactly to the play on v in Friend Circle .In
Node Kayles , when v is chosen, itself and all its neighbors, N G ( v ) are removedfrom future consideration. In Friend Circle , v is also removed because the edge( v, t v ) becomes t . In addition, since all neighboring vertices x ∈ N G ( v ) share a t edge with v , their edge with t x will also become t , but no other vertices will beremoved from future choice. Transverse Wave in Friend Circle
We now show that
Friend Circle contains
Transverse Wave as special cases .In the proposition below and the rest of the paper, we say that two game instancesare isomorphic to each other if there exists a bijection between their moves suchthat their game trees are isomorphic under this bijection. Proposition 3 (Social-Influence Connection of
Transverse Wave ) . For anycomplete bipartite graph G = ( V , V , E ) over two disjoint ground sets V and V (i.e., with E = V × V ), any weighting w : E → { f , t } , and seeds S = V , FriendCircle position ( G, S, w ) is isomorphic to Crosswise OR over a pseudo-adjacencymatrix A G for G with V as columns and V as rows. In this matrix, we will havethe entry at column x ∈ V and row y ∈ V be 0 if w (( x, y )) = f and 1 if the weightis t . Note that by varying w : E → { f , t ] } , one can realize any | V | × | V | Booleanmatrix with A G . Thus, Friend Circle generalizes
Transverse Wave . Proof.
Imagine these two games are played in tandem. We map the selection of avertex v ∈ S = V to the selection of the column associated with v in the matrix A G of G . Because G is a complete bipartite graph, v is feasible for Friend Circle if there exists u ∈ V such that w ( u, v ) = f . Thus, the column associated with v in A G is not all 1s. This is precisely the condition that v is feasible in CrosswiseOR over A G . The Direct Influence at v in Friend Circle over G changes all v ’s edges to t and the subsequent Cascading Influence on v ’s initially t neighbors in V is isomorphic to crosswise ORs. Thus, Friend Circle on G is isomorphic to Crosswise OR over A G .3 V V abcd abcd Figure 5: On the left is a
Friend Circle position on the complete bipartite graphbetween V and V , where the seed set S = V . Instead of labelling edges, wehave removed all false edges and are including only true edges. On the right is theequivalent Transverse Wave position. The purple cells correspond to the (true)edges in the bipartite graph.
3. Connection to Quantum Combinatorial Game Theory
In this section, we discuss the connection of
Transverse Wave to a basic quantum-inspired combinatorial game.Quantum computing is inspirational not only because the advances of quantumtechnologies have the potential to drastically change the landscape of computing anddigital security, but also because the quantum framework—powered by superposi-tions, entanglements, and collapses—has fascinating mathematical structures andproperties. Not surprisingly, quantumness has already found their way to enrichcombinatorial game theory. In early 2000s, Allan Goff introduced the basic quan-tum elements into
Tic-Tac-Toe , as conceptual illustration of quantum physics[14]. The quantum-generalization of
Tic-Tac-Toe expands players strategy spacewith superposition of classical moves, creating game boards with entangled compo-nents. Consistency-based conditions for collapsing can then reduce the degree ofpossible realizations in the potential parallel game scenarios. In 2017, Dorbec andMhalla [7] presented a general framework, motivated by Goff’s concrete adventures,for quantum-inspired extension of classical combinatoral games. Their framework4enabled our recent work [4] on the structures and complexity of quantum-inspiredgames, which also led us to make the connection with logic and social-influenceinspired
Transverse Wave and
Friend Circle in this paper.
In this subsection, we briefly discuss quantum combinatorial game theory (QCGT)to introduce needed concepts and notations for this paper. More detailed discussionsof QCGT can be found in [14, 7, 4]. • Quantum Moves : A quantum move is a superposition of two or more dis-tinct classical moves. The superposition of w classical moves σ , ..., σ w —calleda w -wide quantum move—is denoted by h σ | ... | σ w i . • Quantum Game Position : A quantum position is a superposition of twoor more distinct classical game positions. The superposition of s classicalpositions b , ..., b s —called an s -wide quantum superposition–is denoted by B = h b | ... | b s i . We call b , ..., b s realizations of B .We sometimes refer to classical moves and positions as 1-wide superpositions.Classical/quantum moves can be applied to classical/quantum positions. Vari-ants of Dorbec-Mhalla framework differ in the condition when classical moves areallowed to engage with quantum positions. In this paper, we will focus on theleast restrictive flavor—referred to by variant D in [7, 4]—in which moves, classi-cal or quantum, are allowed to interact with game positions, classical or quantum,provided that they are feasible. There are some subtle differences between thesevariants, and we direct interested readers to [7, 4]. In this least restrictive flavor, asuperposition of moves (including 1-wide superpositions) is feasible for a quantumposition (including 1-wide superpositions) if each move is feasible for some realiza-tion in the quantum position. A superposition of moves and a quantum position ofrealizations creates a ”tensor” of classical interactions in which infeasible classicalinteractions introduce collapses in realizations.Quantum moves can have impact on the outcome class of games, even on classicalpositions. We borrow an illustration (see Figure 6) from [4] showing that quantummoves can change the outcome class of a basic Nim position. (2 ,
2) becomes a fuzzy( N , a first-player win) position instead of a zero ( P , a second-player win) position.Quantumness matters for many combinatorial games, as investigated in [4]. Ruleset 6 (( Nim)) . A Nim position is described by a non-negative integer vector,e.g. (3 , ,
7) = G , representing heaps of objects (here pebbles). A turn consists ofremoving pebbles from exactly one of those heaps. We describe these moves as anon-positive vector with exactly one non-zero element, e.g. (0 , − , . Each movecannot remove more pebbles from a heap than already exist there. Thus, the move (2 , Nh (1 , | (2 , iPh (0 , | (1 , | (1 , iN h (0 , | (1 , iN (0 , N h (0 , | (1 , | (0 , iN h (0 , | (1 , | (2 , iN (0 , P h ( − , | (0 , − ih ( − , | (0 , − i ( − ,
0) ( − , h ( − , | ( − , i h ( − , | (0 , − i (0 , −
2) (0 , −
2) (0 , −
1) (0 , −
2) ( − , Figure 6: Illustration from [4]: Winning strategy for Next player in
Quantum Nim (2 , h (1 , | (2 , i that are not shown because they aresymmetric to moves given.)( − , , is not a legal move from G , above. When all heaps are zero, there are nolegal moves. To readers familiar with combinatorial game theory, it may seem odd that weexplicitly define the description of moves in the game. However, it is integral forplaying quantum combinatorial games, as move description effects quantum col-lapse. For more information, see [4].Quantum interactions between moves and positions, as demonstrated in [14, 7],can have highly non-trivial impact to Nimber Arithmetic. In additon, as shown in[4], quantum moves can also fundamentally impact the complexity of combinatorialgames.
Demi-Quantum Nim : Superposition of
Nim
Positions
The combinatorial game that contains
Transverse Wave as a special case isderived from
Nim [3, 12], in a framework motivated by practical implementation of6quantum combinatorial games [4].For integer s >
1, then a s -wide quantum Nim position of n heaps can be specifiedby an s × n integer matrix, where each row defines a realization of Nim position.For example, the 4-wide quantum
Nim position with 6 piles, h (5 , , , , , | (1 , , , , , | (0 , , , , , | (4 , , , , , i , can be expressed in the matrix form as: (1)Like the quantum generalization of combinatorial games, this demi-quantum gen-eralization systematically extends any combinatorial game by expanding its gamepositions [4]. The intuitive difference here is that players may not introduce newquantum moves, they may only make classical moves, which apply to all (and maycollapse some) of the realizations in the current superposition. Definition 1 (Demi-Quantum Generalization of Combinatorial Games) . For anygame ruleset R , the demi-quantum generalization of R , denoted by, Demi-Quantum-R , is a combinatorial game defined by the interaction of classical moves of R withquantum positions in R .Central to the demi-quantum transition is the rule for collapses . Given a quantumsuperposition B and a classical move σ of R , σ is feasible if it is feasible for at leastone realization in B , and σ collapses all realizations in B for which σ is infeasible,meanwhile transforming each of the other realizations according to ruleset R .For example, the move (0 , , , , − ,
0) applied to the quantum
Nim position inEquation (1) collapses realizations 2 and 4, and transforms realizations 1 and 3,according to
Nim as: { − } ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ { − } ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ = (cid:18) (cid:19) Note that for any impartial ruleset R , Demi-Quantum-R remains impartial.We now show that
Demi-Quantum-Nim contains
Crosswise AND , and thus,
Transverse Wave , as a special case.
Proposition 4 (QCGT Connection of
Transverse Wave ) . Let
Boolean Nim denote
Nim in which each heap has either one or zero pebbles.
Demi-QuantumBoolean Nim is isomorphic to
Crosswise AND , and hence isomorphic to
Trans-verse Wave . Proof.
The proof uses the following equivalent “numerical interpretation” of col-lapses in (demi-)quantum generalization of
Nim . When a realization collapses, wecan either remove it from the superposition or replace it with a
Nim position inwhich all piles have zero pebbles. For example, the following two
Nim superposi-tions are equivalent for subsequent game dynamics. ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ≡ In Boolean Nim , each move can only remove one pebble from a pile. So, wecan simplify the specification of the move by the index i alone. Note also that eachquantum Boolean Nim position can be specified by a Boolean matrix. Let B de-note the Boolean matrix of Demi-Quantum-Boolean-Nim under consideration.With the above numerical interpretation of collapses in demi-quantum generaliza-tion, the collapse of a realization of B when applying a move i corresponding to thecase that the corresponding row in B has 0 at i th entry, and the row is replaced bythe crosswise AND with that column selection. Thus, Demi-Quantum BooleanNim with position B is isomorphic to Crosswise AND with position B .As an aside, notice that positions with all green tiles are trivial. Interesting gamesneed be primed with some arbitrary purpletiles. Thus the hard positions given bythe reduction could be natural starting positions. Thus the hardness statement isparticularly meaningful for Transverse Wave .
4. The Graph Structures Underlying
Demi-Quantum Nim
As the basis of Nimbers and Sprague-Grundy theory [2, 22, 15],
Nim holds a uniqueplace in combinatorial game theory. It is also among the few non-trivial-lookinggames with polynomial-time solution. Over the past decades, multiple efforts havebeen made to introduce graph-theoretical elements into the game of
Nim [11, 23, 5].In 2001, Fukuyama [11] introduced an edge-version of
Graph Nim , with
Nim pilesplaced on edges of undirected graphs. Stockman [23] analyzed several versions withpiles on the nodes. Both use the graph structure to capture the locality of thepiles players can nim from. Burke and George [5] then formulated a version called
Neighboring Nim , for which classical
Nim corresponds to
Neighboring Nim overthe complete graph, where each vertex hosts a pile.The graph structures have profound impact to the game of
Nim both mathemati-cally [11, 23] and computationally [5]. By a reduction from
Geography , Burke andGeorge proved that
Neighboring Nim on some graphs is PSPACE-hard while on8others (such as complete graph) is polynomial-time solvable [5]. However,
Neigh-boring Boolean Nim , the graph
Nim where each pile has at most one pebble,is equivalent to
Undirected Geography , and thus can be solved in polynomialtime [10].In contrast,
Demi-Quantum Boolean Nim is intractable.
Theorem 3 (Intractability of
Demi-Quantum Boolean Nim ) . Demi-QuantumBoolean Nim , and hence
Transverse Wave ( Crosswise AND ; CrosswiseOR ) , is a PSPACE-complete game. Demi-Quantum Boolean Nim
The intractability follows from the next theorem, which connects
Demi-QuantumBoolean Nim to Schaefer’s elegant PSPACE-complete game,
Avoid True [21].The reduction also reveals the bipartite and hyper-graph structures of
Demi-QuantumBoolean Nim . Ruleset 7 ( Avoid True) . A game position of
Avoid True is defined by a positiveCNF F ( and of a set of or -clauses of only positive variables) over a ground set V and a subset T ⊂ V , the ”true” variables, (which is usually the empty set at thebeginning of the game) .A turn consists of selecting one variable from V \ T , where a variable x ∈ V \ T is feasible for position ( F, V, T ) if assigning all variables in T ∪ { x } to true doesnot make F true . If x is feasible, then the position resulting from that move is ( F, V, T ∪ { x } ) . Under normal play, the next player loses if the position has nofeasible move. Theorem 4 ([4]) . Demi-Quantum Boolean Nim and
Avoid True are isomor-phic games.Proof.
Part of the proof in [4] showing that
Quantum Nim is Σ -hard also estab-lishes the above theorem. Because establishing this theorem is not the main focusof [4], we reformulate the proof here to make this theorem more explicit as well asto provide a complete background of our discussion in this section.We first establish the direction from Demi-Quantum Boolean Nim to AvoidTrue . Given a position B in Demi-Quantum Boolean Nim , we can create a o r-clause from each realization in B . Suppose B has m realizations and n piles.We introduce n Boolean variables, V = { x , ..., x n } . For each realization in B , the or -clause consists of all variables corresponding to piles with zero pebbles. Thereduced CNF F B is the and of all these or -clauses. Because taking a pebble from apile collapses a realization for which the pile has no pebble is mapped to selecting thecorresponding Boolean variable making the or -clause associated with the realization true , playing Demi-Quantum Boolean Nim starting position B is isomorphic to9playing Avoid True starting at position ( F B , V, ∅ ). Note that the reduction canbe set up in polynomial time.For example, consider this Demi-Quantum Boolean Nim position (with heaps(columns) labelled by their indices and realizations (rows) labelled A , B , and C ): A B C This reduces to the
Avoid True position with formula:( x ∨ x ∨ x ∨ x ) | {z } A ∧ ( x ∨ x ∨ x ) | {z } B ∧ ( x ∨ x ∨ x ) | {z } C and T = ∅ . The three clauses are labelled by their respective realization. Thosevariables that appear in each clause are those with a zero in the matrix. Noticethat: • The third heap is empty in all
Nim realizations, so no player can legally playthere. That is the same in the resulting
Avoid True position; no player canpick x as it is in all clauses and would make the formula true. • Heaps 4 and 5 have a pebble in all three realizations in
Nim , so a player canplay in either of them without any collapses. Because of this, those Booleanvariables don’t occur in any of the
Avoid True clauses.For the reverse direction, consider an
Avoid True position (
F, V, T ). Assume V = { x , ..., x n } , and F has m clauses, C , ..., C m . We reduce it to a BooleanNim superposition B ( F,V,T ) with m realizations and n piles. In the realizationfor C i , we set piles corresponding to variables in C i zero to set up the mappingbetween collapsing the realization with making the clause true . We also set all pilesassociated with variables in T to zero, to set up the mapping between collapsing therealization with selecting a selected variable. Again, we can use these two mappingsto inductively establish that the game tree for Demi-Quantum Boolean Nim at B ( F,V,T ) is isomorphic to the game tree for Avoid True at (
F, V, T ). Note that thereduction also runs in polynomial time.We demonstrate this reduction on the following
Avoid True position, withformula:( x ∨ x ∨ x ∨ x ) | {z } A ∧ ( x ∨ x ∨ x ∨ x ) | {z } B ∧ ( x ∨ x ∨ x ) | {z } C ∧ ( x ∨ x ∨ x ) | {z } D T = { x } . Following the reduction, we producethe following Demi-Quantum Boolean Nim position: A B C Note, since x has already been made-true ( x ∈ T ): • The 8th column is all zeroes, and • Since x appears in clause D , that clause does not have a correspondingrealization in the quantum superposition (i.e. row in the matrix).Theorem 4 presents the following “rechargeable” bipartite-graph interpretationof Demi-Quantum Boolean Nim . Ruleset 8 ( Rechargeable Bipartite Boolean Nim) . The game is defined bya bipartite graph G = ( U, V, E ) (with edges E between vertex sets U and V ) and avertex s ∈ U , where there is a Nim pile of one pebble at each vertex in U \ { s } ; pilesat s have pebbles. The game start at s . During the game, the players takes turnsto search a pebble reachable from the current vertex (initially s ) via a node in V (wecall it a “station”): They first cross to a node in V in order to reach to a vertex in s ′ ∈ U whose Nim pile contains a pebble. However, the selection of a pebble only“recharge” the stations connected to s ′ . All other stations of V got powered-off andpermanently lost their connections to piles in U . The player who cannot find thenext “power pebble”/“energy pebble” loses the game. Theorem 4 also gives the following simple hyper-graph interpretation of
Demi-Quantum Boolean Nim . Recall a hyper-graph H over a groundset V = [ n ] is acollection of subsets in V . We write H = ( V, E ), where for each hyper-edge e ∈ E , e ⊆ V . Ruleset 9 ( Rechargeable Hypergraph Boolean Nim) . The game position isdefined by hyper-graph H = ( V, E ) , in which each vertex v ∈ V has a Nim pile of pebble, and a vertex c ∈ V . The next player needs to move to a non-empty vertex v connected to c by an hyper-edge, taking its pebble which also removes all hyper-edges without v from the hyper-graph (the “energy pebble” only recharge hyper-edgesincident to v ). The player who cannot find the next ‘pebble loses the game. Because both
Rechargeable Bipartite Boolean Nim and
RechargeableHypergraph Boolean Nim of simply the graph-theoretical interpretation of theproof for Theorem 4, the proof also establish the following:1
Corollary 1.
Rechargeable Bipartite Boolean Nim and
RechargeableHypergraph Boolean Nim are isomorphic to
Transverse Wave and are there-fore PSPACE-complete combinatorial games.
Demi-Quantum Nim
The connection between
Demi-Quantum Boolean Nim and
Friend Circle mo-tivates the following social-influence-inspired game,
Demographic Influence ,which mathematically generalizes
Demi-Quantum Nim . In a nutshell, the settinghas a demographic structure over a population, in which individuals have their ownfriend circles. Members in the population are initially un-influenced but receptive to ads, and (viral marketing) influencers try to target their ads at demographicgroups to influence the population. People can be influenced either by influencers’ads or by “enthusiastic endorsement” cascading through friend circles.The following combinatorial game is distilled from the above scenario.
Ruleset 10 ( Demographic Influence) . , and D = { D , ..., D m } is a collection of subsets of V , representing the demographicgroups, which can overlap. if Θ( v ) < , then v is strongly influenced . A Demo-graphic Influence position is defined by a tuple Z = ( G, Θ , D ) , where • G = ( V, E ) is an undirected graph representing a symmetric social network ona population. • Θ : V → Z represents how resistant each individual is to the product. (I.e.,their threshold to being influenced.) – If Θ( v ) > , then v is uninfluenced , – if Θ( v ) = 0 , then v is weakly influenced , and – if Θ( v ) < , then v is strongly influenced . • D = { D , . . . , D m } is the set of demographics, each a subset of V .A player’s turn consists of choosing a demographic, D k and the amount theywant to influence, c > where ∃ v ∈ D k where Θ( v ) ≥ c . (Since c > , there mustbe an uninfluenced member of D k .) • Θ( v ) decreases by c (all individuals are influenced by c ) and • If Θ( v ) became negative by this subtraction (if it went from Z + ∪ { } to Z − ),then ∀ x ∈ N G ( v ) : Θ( x ) = − . (Thematically, if one individual becomesstrongly influenced directly by the marketing campaign, then they enthusiasti-cally recommend it to their friends and strongly influence them as well). Importantly, we perform all the subtractions and determine which individuals arenewly-strongly influenced, before they go and strongly influence their friends.Note that when influencing a demographic, D k , since c cannot be greater thanthe highest threshold, that highest-threshold individual will not be strongly influencedby the subtraction step. (If one of their neighbors does get strongly influenced, thenthey will be strongly influenced in that manner.)Since a player needs to make a move on a demographic group with at least oneuninfluenced individual, the game ends when there are no remaining groups to in-fluence. v − v v v − v v v − v -1-2 v -1 cFigure 7: A Demographic Influence move, influencing D = { v , v , v } by 4.(a) shows G , Θ, and D prior to making the move. (b) shows the first part of themove: subtracting from the thresholds of v , v , and v . (c) since v went negative,it’s neighbors are set to − Demographic Influence generalizes
Demi-Quantum Nim . Theorem 5 ( Demi-Quantum Nim
Generalization: Social-Influence Connection) . Demographic Influence contains
Demi-Quantum Nim as a Special Case. There-fore,
Demographic Influence is a
PSPACE -complete game.Proof.
For every
Demi-Quantum Nim instance Z with m realizations of n piles,we construct the following Demographic Influence instance Z ′ , in which, (1)3 V = { ( r, c ) | r ∈ [ m ] , c ∈ [ n ] } . (2) E = { (( r , c ) , ( r , c )) | r = r } (i.e. verticesfrom all piles from the same realization are a clique), (3) ∀ ( r, c ) ∈ V , Θ(( r, c )) isset to be the number of pebbles that c th pile has in r th realization of Nim . (4) D = ( D , ..., D n ), where D c = { ( r, c ) | r ∈ [ m ] } , i.e., nodes associated with the c th Nim pile.We claim that Z and Z ′ are isomorphic games. Imagine the two games areplayed in tandem. Suppose the player in Demi-Quantum Nim Z makes a move( k, q ), that is, removing q pebbles from pile k . In its Demographic Influence counterpart, Z ′ , the corresponding player also plays ( k, q ), meaning to invest q units in demographic group k . Note that in Z , ( k, q ) is feasible iff in at leastone of the realizations, the k th Nim pile has at least q pebbles. This is same as q ≤ max i ∈ [ m ] Θ(( i, q )). Therefore, ( k, q ) is feasible in Z iff ( k, q ) is feasible in Z ′ .When ( k, q ) is feasible, then for any realization i ∈ [ m ], there are three cases:(1) if the k th Nim pile has more pebbles than q , then in that realization, a classicaltransition is made, i.e., the q pebbles are removed from the pile. This corresponds tothe reduction of the threshold at node ( i, k ) by q . (2) if the k th Nim pile has exactly q pebbles, then all pebbles are removed from the pile. This corresponds to the casethat node ( i, k ) becomes weakly influenced. (3) if q is more than the number ofpebbles in the k th Nim pile, then the move collapses realization i . This correspondsto the case in Demographic Influence where ( i, k ) become strongly influenced,and then strongly influences all other vertices in the same row ( i ). Therefore, Z and Z ′ are isomorphic games, with connection between the collapse of a realizationin the quantum version and the cascading of influence by endorsement in friendcircle.The proof of Theorem 5 illustrates that Demographic Influence can beviewed as a graph-theoretical extension of
Nim . Recall Burke-George’s
Neigh-boring Nim , which extends both the classical
Nim (when the underlying graph isa clique) and
Undirected Geography (when all
Nim heaps have at most oneitem in them, i.e.
Boolean Nim ). The next theorem complements Theorem 5 byshowing that
Demographic Influence also generalizes
Node-Kayles . Theorem 6 (Social-Influence Connection with
Node-Kayles ) . DemographicInfluence contains
Node-Kayles as a special case.Proof.
Consider a
Node-Kayles instance defined by an n -node undirected graph G = ( V , E ) with V = [ n ]. We define a Demographic Influence instance Z = ( G, Θ , D ) as the following. (1) For each v ∈ V , we introduce a new vertex t v . Let V = V ∪ T , where T := { t v | v ∈ V } . (2) For all v ∈ V , Θ( v ) = 0and Θ( t v ) = 1. (3) For all v ∈ V , C ( v ) = N G ( v ) ∪ { t w | w ∈ N G ( v ) } ∪ { t v } and C ( t v ) = { v } . (4) D = { D , ..., D n } , where D v = { v, t v } , ∀ v ∈ V .Note that because Θ( v ) ∈ { , } , ∀ v ∈ V , the space of moves in this Demo-graphic Influence is { ( v, | v ∈ [ n ] } , whereas in Node-Kayles , a move consists4of selecting one of the vertices from [ n ].We now show that Demographic Influence over Z is isomorphic to Node-Kayles over G under the mapping of moves ( v, ⇔ v .For a Node Kayles move at v , it removes v and all x ∈ N G ( v ) from futuremove choices. In Demographic Influence , choosing the ( v,
1) move means thatΘ( t v ) becomes 0 and Θ( v ) becomes -1. v is now strongly influenced and theystrongly influence their neighbors at N G ( v ) = N G ( v ) ∪ { t x | x ∈ N G ( v ) } , so allthose vertices also gain a threshold of -1. Those include both the x and t x verticesfor each x ∈ N G ( v ), so it removes all those neighboring demographics from futuremoves, ( x, Node Kayles move.Therefore, with induction, we establish that
Demographic Influence over Z is isomorphic to Node-Kayles over G under the mapping of moves from v ⇔ ( v, vx x v t v D v x t x D x x t x D x Figure 8: Example of reduction from
Node Kayles to Demographic Influence .Therefore,
Demographic Influence simultaneously generalizes
Node-Kayles and
Demi-Quantum Nim (which in turn generalizes the classical
Nim , AvoidTrue , and
Transverse Wave ).We can also establish that
Demographic Influence
Generalizes
Friend Cir-cle defined in the earlier section.
Theorem 7 ( Demographic Influence
Generalizes
Friend Circle ) . Demo-graphic Influence contains
Friend Circle as a special case.Proof.
For a
Friend Circle position, Z = ( G, S, w ), where G = ( V, E ), we con-struct the following
Demographic Influence instance Z ′ = ( G ′ , Θ , D ) as thefollowing:5 • For each edge e ∈ E , we create a new vertex v e . Then V ′ = { v e | e ∈ E } • E ′ = { ( v e , v e ) | ∃ v ∈ V : e , e both incident to v } • G ′ = ( V ′ , E ′ ) • Θ : V ′ → { , } , where if w ( e ) = f , then Θ( e ) = 1; otherwise, if w ( e ) = t ,then Θ( e ) = 0. • D = { D s } s ∈ S , where D s = { v e | e is incident to s } In other words, the connections are built on the line graph of the underlying graphin
Friend Circle . Each seed vertex defines the demographic group and associateswith all edges incident to it. Targeting this demographic group influences all theseedges and edges adjacent to t -edges in this set.We complete the proof by showing that a play on Friend Circle position s is isomorphic to a play on Demographic Influence ( D s , D s and investing c = 1. In Friend Circle , playing at s means that ∀ e incident to s : (1) w ( e ) becomes t , and (2) if w ( e ) was already t , then ∀ f adjacentto e : w ( f ) becomes t .In Demographic Influence , the corresponding play, ( D s ,
1) means that ∀ v e ∈ D s : • Θ( v e ) is reduced by 1. This corresponds to setting w ( e ) to t . • If Θ( v e ) becomes -1, then ∀ v f ∈ N G ′ ( v e ) : Θ( v f ) also becomes −
1. By ourdefinition of E ′ , these v f are exactly those where both w ( e ) was previously t and e and f are adjacent in G .Thus, following analagous moves, w ( e ) = t iff Θ( v e ) ≤
0. A seed vertex, s ′ ∈ S issurrounded by t edges (and inelligible as a move) exactly when all vertices v e ∈ D s are influenced, also making D s inelligible as a move. Thus, this mapping of movesshows that the games are isomorphic.See Figure 9 for an example of the reduction.
5. Conclusions and Future Work
One of the beautiful aspects of Winning Ways [2] is the relationships between games,especially when positions in one ruleset can be transformed into equivalent instancesof another ruleset. As examples,
Dawson’s Chess positions are equivalent to
Node Kayles positions on paths,
Wythoff’s Nim is the one-queen case of
WytQueens , and
Subtraction - { , , , } positions exist as instances of many rulesets,6 s s s x f e t e f e f e e e e e D s D s D x D s Figure 9: Example of the reduction. On the left is a
Friend Circle instance. Onthe right is the resulting
Demographic Influence position.including
Adders and Ladders with one token after the top of the last ladderand last snake.Transforming instances of one ruleset to another (reductions) is a basic partof Combinatorial Game Theory , just as it is vital to computational complexity. Transverse Wave arose not only by reducing to other things, but more concretelyas a special case of other games we explored.”Ruleset A is a special case of ruleset B ” (i.e. ” B is a generalization of A ”) notonly proves that computational hardness of A results in the computational hardnessof B , but also: • If A is interesting, then it is a fundamental part of B ’s strategies, and • If B ’s rules are straightforward, then A could be a block to create other funrulesets.Relevant special-case/generalization relationships between games presented hereinclude: • Demi-Quantum Nim is a generalization of
Nim . (See Section 3.) (This istrue of any ruleset R and Demi-Quantum R .) • Demi-Quantum Nim is a generalization of
Transverse Wave . (Via
Demi-Quantum Boolean Nim , see section 3.2.) • Friend Circle is a generalization of both
Transverse Wave (Section 2.3)and
Node Kayles (Section 2.2). ”Change the Game!” is the title of a section in Lessons in Play, Chapter 1, ”Basic Techniques”[1]. • Demographic Influence is a generalization of both
Demi-Quantum-Nim and
Friend Circle (Section 4.2).We show these relationships in a lattice-manner in Figure 10. Understanding
Transverse Wave is a key piece of the two other rulesets, which also include theimpartial classics
Nim and
Node Kayles . Nim TransverseWave NodeKaylesDemi-Quantum Nim FriendCircleDemographicInfluence
Figure 10: Generalization relationships of the rulesets in this paper. A → B meansthat A is a special case of B and B is a generalization of A .Furthermore, several of the relationships outside Figure 10 that were discussedin this paper were completely isomorphic, preserving the the game values not justwinnability (as in Section 1.4). More explicitly, any new findings on the Grundyvalues for Transverse Wave also give those exact same results for
CrosswiseOR , Crosswise AND , Demi-Quantum Boolean Nim , Avoid True , Rechar-gable Bipartite Boolean Nim , and
Rechargeable Hypergraph BooleanNim .We have been drawn to
Transverse Wave not only because it is colorful,approachable, and intriguing, but also because its relationships with other gameshave inspired us to discover more connections among games. Our work offers usa glimpse of the lattice order induced by special-case/generalization relationships over mathematical games, which we believe is an instrumental framework for boththe design and comparative analysis of combinatorial games. In one direction ofthis lattice, when given two combinatorial games A and B , it is a stimulating andcreative process to design a game with the simplest ruleset that generalizes both A and B . For example, in generalizing both
Nim and
Undirected Geography , Neighboring Nim highlights the role of “self-loop” in
Graph-Nim . In our work, It is also a relevant pedagogical question to ask when introducing students to combinatorialgame theory.
Node Kalyes and
Demi-Quantum Nim has contributedto our design of
Demographic Influence . In the other direction, identifying awell-formulated basic game at the intersection of two seemingly unconnected gamesmay greatly expand our understanding of game structures. It is also a refinementprocess for identifying intrinsic building blocks and fundamental games. By ex-ploring the lattice order of game relationships, we will continue to improve ourunderstanding of combinatorial game theory and identify new fundamental gamesinspired by the rapidly evolving world of data, network, and computing.
References [1] M. H. Albert, R. J. Nowakowski, and D. Wolfe.
Lessons in Play: An Introduc-tion to Combinatorial Game Theory . A. K. Peters, Wellesley, Massachusetts,2007.[2] Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy.
Winning Waysfor your Mathematical Plays , volume 1. A K Peters, Wellesley, Massachsetts,2001.[3] Charles L. Bouton. Nim, a game with a complete mathematical theory.
Annalsof Mathematics , 3(1/4):pp. 35–39, 1901.[4] Kyle Burke, Matthew Ferland, and Shang-Hua Teng. Quantum combinato-rial games: Structures and computational complexity.
CoRR , abs/2011.03704,2020.[5] Kyle W. Burke and Olivia George. A pspace-complete graph nim.
CoRR ,abs/1101.1507, 2011.[6] Wei Chen, Shang-Hua Teng, and Hanrui Zhang. A graph-theoretical basis ofstochastic-cascading network influence: Characterizations of influence-basedcentrality.
Theor. Comput. Sci. , 824-825:92–111, 2020.[7] Paul Dorbec and Mehdi Mhalla. Toward quantum combinatorial games. arXivpreprint arXiv:1701.02193 ∼ eppstein/cgt/hard.html.[9] S. Even and R. E. Tarjan. A combinatorial problem which is complete inpolynomial space. J. ACM , 23(4):710–719, October 1976.[10] Aviezri S. Fraenkel, Edward R. Scheinerman, and Daniel Ullman. Undirectededge geography.
Theor. Comput. Sci. , 112(2):371–381, 1993.9[11] Masahiko Fukuyama. A nim game played on graphs.
Theor. Comput. Sci. ,1-3(304):387–399, 2003.[12] David Gale. A curious nim-type game.
American Mathematical Monthly ,81:876–879, 1974.[13] David Gale. The game of Hex and the Brouwer fixed-point theorem.
AmericanMathematical Monthly , 10:818–827, 1979.[14] Allan Goff. Quantum tic-tac-toe: A teaching metaphor for superposition inquantum mechanics.
American Journal of Physics , 74(11):962–973, 2006.[15] P. M. Grundy. Mathematics and games.
Eureka , 2:198—211, 1939.[16] David Kempe, Jon Kleinberg, and Eva Tardos. Maximizing the spread ofinfluence through a social network. In
KDD ’03 , pages 137–146. ACM, 2003.[17] David Lichtenstein and Michael Sipser. Go is polynomial-space hard.
J. ACM ,27(2):393–401, 1980.[18] John F. Nash.
Some Games and Machines for Playing Them . RAND Corpo-ration, Santa Monica, CA, 1952.[19] S. Reisch. Hex ist PSPACE-vollst¨andig.
Acta Inf. , 15:167–191, 1981.[20] Matthew Richardson and Pedro Domingos. Mining knowledge-sharing sitesfor viral marketing. In
Proceedings of the 8th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining , KDD ’02, pages 61–70,2002.[21] Thomas J. Schaefer. On the complexity of some two-person perfect-informationgames.
Journal of Computer and System Sciences , 16(2):185–225, 1978.[22] R. P. Sprague. ¨Uber mathematische Kampfspiele.
Tˆohoku Mathematical Jour-nal , 41:438—444, 1935-36.[23] Gwendolyn Stockman. Presentation: The gameof nim on graphs: NimG, 2004. Available at