We introduce the game INFLUENCE, a scoring combinatorial game, played on a directed graph where each vertex is either colored black or white. The two players, Black and White play alternately by taking a vertex of their color and all its successors (for Black) or all its predecessors (for White). The score of each player is the number of vertices he has taken.
We prove that INFLUENCE is a nonzugzwang game, meaning that no player has interest to pass at any step of the game, and thus belongs to Milnor's universe. We study this game in the particular class of paths where black and white are alternated. We give an almost tight strategy for both players when there is one path. More precisely, we prove that the first player always gets a strictly better score than the second one, but that the difference between the score is bounded by 5. Finally, we exhibit some graphs for which the initial proportion of vertices of the color of a player is as small as possible but where this player can get almost all the vertices.
Partizan subtraction games are combinatorial games where two players, say Left and Right, alternately remove a number n of tokens from a heap of tokens, with $n \in S_L$ (resp. $n \in S_R$) when it is Left's (resp. Right's) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i.e. an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current paper, we investigate the other kinds of behaviors for the outcome sequence. In addition to dominan…
Much progress has been made in misere game theory using the technique of restricted misere play, where games can be considered equivalent inside a restricted set of games without being equal in general. This paper provides a survey of recent results in this area, including two particularly interesting properties of restricted misere games: (1) a game can have an additive inverse that is not its negative, and (2) a position can be reversible through an end (a game with Left but not Right options, or vice versa). These properties are not possible in normal play and general misere play, respectiv…
In Combinatorial Game Theory, we study the set of games G, whose elements are mapped from positions of rulesets. In many case, given a ruleset, not all elements of G can be given as a position in the ruleset. It is an intriguing question what kind of ruleset would allow all of them to appear. In this paper, we introduce a ruleset named turning tiles and prove the ruleset is a universal partizan ruleset, that is, every element in G can occur as a position in the ruleset. This is the second universal partizan ruleset after generalized konane.
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.
We consider Flipping Coins, a partizan version of the impartial game Turning Turtles, played on lines of coins. We show the values of this game are numbers, and these are found by first applying a reduction, then decomposing the position into an iterated ordinal sum. This is unusual since moves in the middle of the line do not eliminate the rest of the line. Moreover, when $G$ is decomposed into lines $H$ and $K$, then $G=(H:K^R)$. This is in contrast to Hackenbush Strings where $G= (H:K)$.
Generalized Geography is a combinatorial game played on a directed graph. Players take turns moving a token from vertex to vertex, deleting a vertex after moving the token away from it. A player unable to move loses. It is well known that the computational complexity of determining which player should win from a given position of Generalized Geography is PSPACE-complete. We introduce several rule variants to Generalized Geography, and we explore the computational complexity of determining the winner of positions of many resulting games. Among our results is a proof that determining the winner …
Univ Lyon, CNRS, INSA Lyon, UCBL, Centrale Lyon, Univ Lyon 2, LIRIS, UMR5205（里昂大学）；Université Savoie Mont Blanc, CNRS UMR5127, LAMA（萨瓦-蒙布朗大学）；Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院数学与信息学系）
Positional games have been introduced by Hales and Jewett in 1963 and have been extensively investigated in the literature since then. These games are played on a hypergraph where two players alternately select an unclaimed vertex of it. In the Maker-Breaker convention, if Maker manages to fully take a hyperedge, she wins, otherwise, Breaker is the winner. In the Maker-Maker convention, the first player to take a hyperedge wins. In both cases, the game stops as soon as Maker has taken a hyperedge. By definition, this family of games does not handle scores and cannot represent games in which pl…
We study zero-sum (combinatorial) games, within the framework of so-called Richman auctions (Lazarus et al. 1996) namely, we modify the alternating play scoring ruleset Cumulative Subtraction (CS) (Cohensius et al. 2019), to a discrete bidding scheme (similar to Develin and Payne 2010). Players bid to move and the player with the highest bid wins the move, and hands over the winning bidding amount to the other player. The new game is dubbed Bidding Cumulative Subtraction (BCS). In so-called unitary games, players remove exactly one item out of a single heap of identical items, until the heap i…
The game INFLUENCE is a scoring combinatorial game that has been introduced in 2020 by Duchene et al. It is a good representative of Milnor's universe of scoring games, i.e. games where it is never interesting for a player to miss his turn. New general results are first given for this universe, by transposing the notions of mean and temperature derived from non-scoring combinatorial games. Such results are then applied to INFLUENCE to refine the case of unions of segments. The computational complexity of the score of the game is also solved and proved to be PSPACE-complete. We finally focus on…
