Infinite cyclic impartial games

We define the family of {\it locally path-bounded} digraphs, which is a class of infinite digraphs, and show that on this class it is relatively easy to compute an optimal strategy (winning or nonlosing); and realize a win, when possible, in a finite number of moves. This is done by proving that the Generalized Sprague-Grundy function exists uniquely and has finite values on this class.Comment: To appear in Proc. Computer Games 199

Cumulative Games: Who is the current player?

Combinatorial Game Theory (CGT) is a branch of game theory that has developed almost independently from Economic Game Theory (EGT), and is concerned with deep mathematical properties of 2-player 0-sum games that are defined over various combinatorial structures. The aim of this work is to lay foundations to bridging the conceptual and technical gaps between CGT and EGT, here interpreted as so-called Extensive Form Games, so they can be treated within a unified framework. More specifically, we introduce a class of $n$-player, general-sum games, called Cumulative Games, that can be analyzed by both CGT and EGT tools. We show how two of the most fundamental definitions of CGT---the outcome function, and the disjunctive sum operator---naturally extend to the class of Cumulative Games. The outcome function allows for an efficient equilibrium computation under certain restrictions, and the disjunctive sum operator lets us define a partial order over games, according to the advantage that a certain player has. Finally, we show that any Extensive Form Game can be written as a Cumulative Game.Comment: 54 pages, 4 figure

Two-Player Tower of Hanoi

The Tower of Hanoi game is a classical puzzle in recreational mathematics (Lucas 1883) which also has a strong record in pure mathematics. In a borderland between these two areas we find the characterization of the minimal number of moves, which is $2^n-1$, to transfer a tower of $n$ disks. But there are also other variations to the game, involving for example real number weights on the moves of the disks. This gives rise to a similar type of problem, but where the final score seeks to be optimized. We study extensions of the one-player setting to two players, invoking classical winning conditions in combinatorial game theory such as the player who moves last wins, or the highest score wins. Here we solve both these winning conditions on three heaps.Comment: 16 pages, 6 figures, 1 tabl

Three-player impartial games

Past efforts to classify impartial three-player combinatorial games (the theories of Li and Straffin) have made various restrictive assumptions about the rationality of one's opponents and the formation and behavior of coalitions. One may instead adopt an agnostic attitude towards such issues, and seek only to understand in what circumstances one player has a winning strategy against the combined forces of the other two. By limiting ourselves to this more modest theoretical objective, and by regarding two games as being equivalent if they are interchangeable in all disjunctive sums,as far as single-player winnability is concerned, we can obtain an interesting analogue of Grundy values for three-player impartial games.Comment: 8 pages, 10 tables; to appear in Theoretical Computer Scienc

Playing Games with Algorithms: Algorithmic Combinatorial Game Theory

Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.Comment: 42 pages, 18 figures. Major revision to survey, and new author. To appear in Games of No Chance II

Conway games, algebraically and coalgebraically

Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on non-losing strategies. Hypergames are a fruitful metaphor for non-terminating processes, Conway's sum being similar to shuffling. We develop a theory of hypergames, which extends in a non-trivial way Conway's theory; in particular, we generalize Conway's results on game determinacy and characterization of strategies. Hypergames have a rather interesting theory, already in the case of impartial hypergames, for which we give a compositional semantics, in terms of a generalized Grundy-Sprague function and a system of generalized Nim games. Equivalences and congruences on games and hypergames are discussed. We indicate a number of intriguing directions for future work. We briefly compare hypergames with other notions of games used in computer science.Comment: 30 page

An Extension of the Normal Play Convention to NN-player Combinatorial Games

We examine short combinatorial games for three or more players under a new play convention in which a player who cannot move on their turn is the unique loser. We show that many theorems of impartial and partizan two-player games under normal play have natural analogues in this setting. For impartial games with three players, we investigate the possible outcomes of a sum in detail, and determine the outcomes and structure of three-player Nim.Comment: 45 pages; Presented at Integers Conference 2018, submitted to Proceeding

The ⋆\star-operator and Invariant Subtraction Games

We study 2-player impartial games, so called \emph{invariant subtraction games}, of the type, given a set of allowed moves the players take turn in moving one single piece on a large Chess board towards the position $\boldsymbol 0$. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, $\star$ of the old game, by taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One such game is \W^\star= (Wythoff Nim)$^\star$, where the moves are defined by complementary Beatty sequences with irrational moduli. Here we give a polynomial time algorithm for infinitely many $P$-positions of \W^\star. A repeated application of $\star$ turns out to give especially nice properties for a certain subfamily of the invariant subtraction games, the \emph{permutation games}, which we introduce here. We also introduce the family of \emph{ornament games}, whose $P$-positions define complementary Beatty sequences with rational moduli---hence related to A. S. Fraenkel's `variant' Rat- and Mouse games---and give closed forms for the moves of such games. We also prove that ($k$-pile Nim)$^{\star\star}$ = $k$-pile Nim.Comment: 30 pages, 5 figure

Algebraic games - Playing with groups and rings

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group $A$, a move consists of picking some nonzero element $a \in A$. The game then continues with the quotient group $A/ \langle a \rangle$. We prove that under the normal play rule, the second player has a winning strategy if and only if $A$ is a square, i.e. $A$ is isomorphic to $B \times B$ for some abelian group $B$. Under the mis\`ere play rule, only minor modifications concerning elementary abelian groups are necessary to describe the winning situations. We also compute the nimbers, i.e. Sprague-Grundy values, of $2$-generated abelian groups. An analogous game can be played with arbitrary algebraic structures. We study some examples of non-abelian groups and commutative rings such as $R[X]$, where $R$ is a principal ideal domain.Comment: 31 pages; complete revision; added computations of nimbers and a section about polynomial ring

The spectrum of nim-values for achievement games for generating finite groups

We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is $\{0,1,2,3,4\}$. This positively answers two conjectures from a previous paper by the last two authors.Comment: 11 pages, 5 figures, 2 table