Technology Category
Game Theory and Economic Paradigms: Mechanism DesignMultiagent Systems: Mechanism DesignKnowledge Representation and Reasoning: Computational Complexity of Reasoning
Token positional games
🤖 AI Summary
This study investigates a novel variant of Maker-Breaker positional games under token constraints, introducing the "token sliding positional game" model. In this setting, players possess a limited number of tokens and compete over a hypergraph by sliding tokens rather than merely placing them. Combining techniques from combinatorial game theory and computational complexity, the work establishes a sharp threshold phenomenon for the minimum number of tokens required for Maker to win on k-uniform hypergraphs: exactly k tokens suffice when k = 2 or 3, whereas Ω(n) tokens are necessary for k ≥ 4. Furthermore, the paper presents a polynomial-time algorithm for Breaker when equipped with only a single token and proves that the token-sliding variant of the game is EXPTIME-complete.
Technology Category
Application Category
📝 Abstract
The classical Maker-Breaker positional game is played on a board which is a hypergraph $\mathcal{H}$, with two players, Maker and Breaker, alternately claiming vertices of $\mathcal{H}$ until all the vertices are claimed. When the game ends, Maker wins if she has claimed all the vertices of some edge of $\mathcal{H}$; otherwise, Breaker wins. Playing this game in real life can be done by placing tokens on the vertices of the board. In this paper, we study the unfortunate case in which one or both players do not have enough tokens to cover all the vertices and, as such, will have to move their tokens around at some point instead of placing new ones. There may be a bias, in that Maker and Breaker do not necessarily have the same amount of tokens. The present paper initiates the study of this generalization of positional games, called token positional games. A particularly interesting case is when Maker has a winning strategy in the classical game: what is the lowest number of tokens with which she still wins against Breaker's unlimited stock? We notably show that, for $k$-uniform hypergraphs on an arbitrarily large number $n$ of vertices, this number equals $k$ if $k \in\{2,3\}$ but can vary from $k$ to $\Omega(n)$ if $k \geq 4$. From an algorithmic point of view, PSPACE-hardness in general is inherited from classical positional games, but we get a polynomial-time algorithm to solve the case where Breaker only has one token. We also establish EXPTIME-completeness for a"token sliding"variation of the game.
Problem
Research questions and friction points this paper is trying to address.
positional games
token placement
Maker-Breaker games
hypergraphs
combinatorial game theory
Innovation
Methods, ideas, or system contributions that make the work stand out.
token positional games
Maker-Breaker games
resource-limited strategy
computational complexity
hypergraph
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