Token positional games

TL;DR

This paper extends Maker-Breaker positional games by introducing token budgets, defining the $\,(a,b)\,$-game on a hypergraph $\mathcal{H}$ (with moves involving placement and movement of tokens) and exploring both unlimited-token and single-token-Breaker regimes, plus a token-sliding variant. It establishes sharp structural results for $k$-uniform hypergraphs: for $k\in\{2,3\}$ Maker can win with exactly $k$ tokens, whereas for $k\ge4$ the minimal token budget can grow up to $\Omega(n)$, revealing a drastic shift in complexity and strategy. The paper proves PSPACE-hardness in general, provides a polynomial-time algorithm when Breaker has one token, and shows EXPTIME-completeness for token sliding, along with an XP algorithm parameterized by token counts. It also develops deep structural results for unlimited tokens, including tight thresholds $\theta(\mathcal{H})$ and $\tau(\mathcal{H})$ in 3-uniform and higher-uniform regimes, and constructs hypergraphs that exhibit maximal gaps between these parameters. Overall, the work elucidates how token budgets influence both the combinatorial structure of winning strategies and the computational complexity of determining outcomes, offering a framework and several benchmarks for future token-based game investigations.

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 $Ω(n)$ 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.

Figures (10)

• Figure 1: The construction from Proposition \ref{['prop:k-unif-kvs1']} for $k = 7$.
• Figure 2: Left: an edge of size 3. Middle: an edge of size 3 on which Maker and Breaker each have a token. Right: an edge of size 4.
• Figure 3: Left: a nunchaku of length 5. Right: a necklace of length 6.
• Figure 4: A token-free "equivalent" of the nunchaku.
• Figure 5: The construction for Proposition \ref{['prop:gap']}, with $L=11$. The two dotted lines at the bottom symbolize the edges in $E_1$ and $E_2$ respectively.

Theorems & Definitions (54)

• Proposition 1
• proof
• Proposition 2: Subhypergraph monotonicity
• proof
• Proposition 3: Pairing strategy
• proof
• Proposition 4
• proof
• Proposition 5
• proof