Poset Positional Games

TL;DR

This work extends positional games by introducing poset positional games, where legal moves must respect a given poset on the board, and analyzes the Maker-Breaker outcome across structural parameters such as height, width, and the winning-set family. The authors provide a comprehensive complexity landscape: MB poset games are PSPACE-hard at width $2$ with WS size up to $3$, and NP-hard at height $3 even with a single WS of size $1$, while they identify polynomial-time solvable cases including height $2$ with a unique WS of size $1$ and specific disjoint-chain configurations. They also develop a width-parameterized dynamic-programming algorithm with time $O(|X|^w 2^m w^4)$ and an $O(|X|^4)$ algorithm for height-2 chain unions with WS size at most $2$, together with a complete disjoint-union outcome characterization. The results illuminate how zugzwang phenomena shape strategy and computation in restricted-order games, offering both theoretical insight and practical algorithmic techniques for restricted-move variants of classical positional games.

Abstract

We propose a generalization of positional games, supplementing them with a restriction on the order in which the elements of the board are allowed to be claimed. We introduce poset positional games, which are positional games with an additional structure -- a poset on the elements of the board. Throughout the game play, based on this poset and the set of the board elements that are claimed up to that point, we reduce the set of available moves for the player whose turn it is -- an element of the board can only be claimed if all the smaller elements in the poset are already claimed. We proceed to analyse these games in more detail, with a prime focus on the most studied convention, the Maker-Breaker games. First we build a general framework around poset positional games. Then, we perform a comprehensive study of the complexity of determining the game outcome, conditioned on the structure of the family of winning sets on the one side and the structure of the poset on the other.

Figures (7)

• Figure 1: A poset on nine vertices.
• Figure 2: Possible outcomes for a disjoint union of two poset positional games depending on the parity of the number of vertices in both games. We write "${all }$" when all four outcomes are possible.
• Figure 3: Reduction from Set Cover to MB Poset Positional Game. A solid arc means that all the possible relations are in the poset. The winning set is represented by a red circle.
• Figure 4: The poset positional game obtained by reduction of $\forall x_1 \exists x_2 \forall x_3 \exists x_4 (x_1 \vee x_2 \vee x_3) \wedge (\neg x_2 \vee x_3 \vee \neg x_4)$
• Figure 5: Reduction of a game ${\mathcal{G} }$ containing a black winning set of size 1 to a game ${\mathcal{G} }'$. The winning set $\{x_{1,4},x_{2,2}\}$ disappears since it contains a black vertex in $Y$. The winning set $\{x_{1,3},x_{2,3}\}$ becomes the winning set $\{x_{2,3}\}$ in ${\mathcal{G} }'$ since $x_{1,3}$ is white. By Lemma \ref{['lem:reducWS1']}, the two games have the same outcome. In this case, since ${\mathcal{G} }'$ has a white winning set of size 1, both games are winning for Maker.

Theorems & Definitions (48)

• Remark 1
• Lemma 2
• proof
• Remark 3
• Lemma 4
• proof
• Theorem 5
• proof
• Theorem 7
• proof