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 $Ω(n)$ 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.

翻译：经典的Maker-Breaker位置博弈在一个作为超图$\mathcal{H}$的棋盘上进行，由两名玩家Maker和Breaker轮流占据$\mathcal{H}$的顶点，直至所有顶点被占据。游戏结束时，若Maker占据了$\mathcal{H}$某条边的所有顶点则获胜；否则Breaker获胜。在实际游戏中，玩家可通过在棋盘顶点放置令牌来实现。本文研究一种特殊情况：一方或双方玩家没有足够令牌覆盖所有顶点，因此必须在某些时刻移动已有令牌而非放置新令牌。博弈可能存在偏置，即Maker与Breaker持有的令牌数量未必相同。本文首次系统研究这类位置博弈的推广形式，称为令牌位置博弈。一个特别有趣的场景是：当Maker在经典博弈中拥有必胜策略时，面对持有无限令牌的Breaker，她仍能获胜所需的最低令牌数量是多少？我们特别证明：对于具有任意多$n$个顶点的$k$-均匀超图，该数量在$k \in\{2,3\}$时等于$k$，但当$k \geq 4$时可在$k$到$Ω(n)$之间变化。从算法角度，一般情况的PSPACE困难性继承自经典位置博弈，但我们给出了Breaker仅持有一枚令牌时的多项式时间算法。此外，针对游戏的"令牌滑动"变体，我们建立了EXPTIME完全性。