Scoring play games were first studied by Fraser Stewart for his PhD thesis. He showed that under the disjunctive sum, scoring play games are partially ordered, but do not have the same "nice" structure of normal play games. In this paper I will be considering scoring play games under three different operators given by John Conway and William Stromquist and David Ullman, namely the conjunctive sum, selective sum and sequential join.
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…
In Combinatorial Game Theory, the fundamental relation of game equivalence, denoted by $=$, is introduced early on and overrides the notion of set equality. We explore what happens if set equality is given its due before game equivalence is introduced.
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…
Genetic programming is the practice of evolving formulas using crossover and mutation of genes representing functional operations. Motivated by genetic evolution we develop and solve two combinatorial games, and we demonstrate some advantages and pitfalls of using genetic programming to investigate Grundy values. We conclude by investigating a combinatorial game whose ruleset and starting positions are inspired by genetic structures.
我们研究零和（组合）游戏，在所谓的Richman拍卖（Lazarus et al. 1996）的框架内，即我们将交替游戏评分规则累积减法（CS）（Cohensius et al. 2019）修改为离散投标方案（类似于Reynin和Payne 2010）。玩家出价移动，出价最高的玩家赢得移动，并将获胜的出价金额交给另一名玩家。这款游戏被称为Bidding Cumulative Subtraction（BCS）。在所谓的单一游戏中，玩家从一堆相同的物品中取出一件物品，直到堆为空，他们的行动产生一个共同的分数，该分数增加或减少一个单位，取决于最大化玩家是否赢得了回合。我们发现，有一个独特的投标均衡的一个更大的类的游戏，推广标准的评分发挥在文献中。我们证明，对于所有足够大的堆大小，酉BCS的均衡结果最终是周期性的，周期为2，我们表明，周期性出现在最新的堆的大小平方的总预算。
We define a natural equivalence relation on collections of cards from the card game SET, and enumerate some of the equivalence classes, vastly generalizing the standard game. On this basis, we describe several alternative games for the SET deck, including the game of STUN, in which the goal is to find three cards with exactly two values in each attribute.
This paper studies a single-suit version of the card game War on a finite deck of cards. There are varying methods of how players put the cards that they win back into their hands, but we primarily consider randomly putting the cards back and deterministically always putting the winning card before the losing card. The concept of a \emph{passthrough} is defined, which refers to a player playing through all cards in their hand from a particular point in the game. We consider games in which the second player wins during their first passthrough. We introduce several combinatorial objects related …
The $\mathscr{P}$-position sets of some combinatorial games have special combinatorial structures. For example, the $\mathscr{P}$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, $\mathcal{D}_{\text{sh}}$. There were, however, few known games related to Steiner systems like the hexad game. For a given Steiner system, we construct a game whose $\mathscr{P}$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $\mathcal{D}_{\text{s…
In a Take-Away Game on hypergraphs, two players take turns to remove the vertices and the hyperedges of the hypergraphs. In each turn, a player must remove either a single vertex or a hyperedge. When a player chooses to remove one vertex, all of the hyperedges that contain the chosen vertex are also removed. When a player chooses to remove one hyperedge, only that chosen hyperedge is removed. Whoever removes the last vertex wins the game. Following from the winning strategy for the Take-Away Impartial Combinatorial Games on only Oddly Uniform or only Evenly Uniform Hypergraphs, this paper is a…
We develop a theory of combinatorial games that is appropriate for describing positions in Hex and other monotone set coloring games. We consider two natural conditions on such games: a game is monotone if all moves available to both players are good, and passable if in each position, at least one player has at least one good move available. The latter condition is equivalent to saying that if passing were permitted, no player would benefit from passing. Clearly every monotone game is passable, and we prove that the converse holds up to equivalence of games. We give some examples of how this t…
We propose a unifying additive theory for standard conventions in Combinatorial Game Theory, including normal-, misère- and scoring-play, studied by Berlekamp, Conway, Dorbec, Ettinger, Guy, Larsson, Milley, Neto, Nowakowski, Renault, Santos, Siegel, Sopena, Stewart (1976-2019), and others. A game {\em universe} is a set of games that satisfies some standard closure properties. Here, we reveal when the fundamental game comparison problem, ``Is $G\su H$?'', simplifies to a constructive `local' solution, which generalizes Conway's foundational result in ONAG (1976) for normal-play games. This ha…
We consider three variants of a partisan combinatorial game between two players, Left and Right, played on an undirected simple graph. Left is able to delete vertices (and incident edges) while Right is able to delete edges. This natural extension of a similar impartial game gives a clear advantage to one player by allowing them the ability to play on a small subgraph which the other player can not. Our last variant removes this advantage by assuming a move is valid for one player if and only if the other player has a valid move on the same graph. In this case, we show that the ability to remo…
We introduce a one-person game that we call Padlock Solitaire which resembles the well-known clock solitaire card game. Analyzing variants of this game we obtain simple proofs of some classical results of combinatorics including ballot theorems and the enumeration of spanning trees in various graphs and hypergraphs.
We define an all-small ruleset, Bipass, within the framework of normal-play combinatorial games. A game is played on finite strips of black and white stones. Stones of different colors are swapped provided they do not bypass one of their own kind. We find a surjective function from the strips to integer atomic weights (Berlekamp, Conway and Guy 1982) that measures the number of units in all-small games. This result provides explicit winning strategies for many games, and in cases where it does not, it gives narrow bounds for the canonical form game values. We prove that the game value *2 does …
