A Game of Pawns

TL;DR

This work introduces pawn games, a class of two-player reachability games with dynamic vertex ownership controlled by pawns and evolving via grabbing mechanisms. It develops a comprehensive complexity landscape by ownership regime (OVPP, MVPP, OMVPP) and mechanism (optional grabbing, always grabbing, always grabbing-or-giving, and k-grabbing), delivering both tractable and intractable results. Notably, OVPP optional-grabbing is in $PTIME$, MVPP optional-grabbing is $EXPTIME$-hard via Lock & Key gadgets, MVPP always-grabbing is $EXPTIME$-complete, and MVPP always grabbing-or-giving is in $PTIME$, while OMVPP k-grabbing is $PSPACE$-complete; OVPP k-grabbing is $PTIME$ and MVPP k-grabbing is $NP$-hard. The hardness framework introduced by Lock & Key games provides a versatile tool for proving EXPTIME lower bounds, and the paper discusses meaningful applications to shield synthesis, sabotage modeling, and robust control under uncertainty. These results illuminate how succinct representations can yield tractable subclasses alongside exponential complexity in richer settings, guiding future exploration of more objectives and mechanisms.

Abstract

We introduce and study pawn games, a class of two-player zero-sum turn-based graph games. A turn-based graph game proceeds by placing a token on an initial vertex, and whoever controls the vertex on which the token is located, chooses its next location. This leads to a path in the graph, which determines the winner. Traditionally, the control of vertices is predetermined and fixed. The novelty of pawn games is that control of vertices changes dynamically throughout the game as follows. Each vertex of a pawn game is owned by a pawn. In each turn, the pawns are partitioned between the two players, and the player who controls the pawn that owns the vertex on which the token is located, chooses the next location of the token. Control of pawns changes dynamically throughout the game according to a fixed mechanism. Specifically, we define several grabbing-based mechanisms in which control of at most one pawn transfers at the end of each turn. We study the complexity of solving pawn games, where we focus on reachability objectives and parameterize the problem by the mechanism that is being used and by restrictions on pawn ownership of vertices. On the positive side, even though pawn games are exponentially-succinct turn-based games, we identify several natural classes that can be solved in PTIME. On the negative side, we identify several EXPTIME-complete classes, where our hardness proofs are based on a new class of games called Lock & Key games, which may be of independent interest.

Figures (5)

• Figure 1: Left: The pawn game $\mathcal{G}_1$; a non-monotonic game under optional-grabbing. Right: The pawn game $\mathcal{G}_2$ in which Player $1$ wins from $\langle v_0, \{ v_0, v_1 \} \rangle$, but must visit $v_1$ twice.
• Figure 2: From turn-based to optional-grabbing games.
• Figure 3: From left to right: $\mathcal{G}_\ell$ in open and closed state and $\mathcal{G}_k$ in open and closed state.
• Figure 4: A $\delta$-path is a path between two primed main vertices in an optional- or always-grabbing game, and it crosses two key gadgets and one lock gadget.
• Figure 5: Consider the input to SET-COVER $U = [3]$ and $\mathcal{S} = \{ \{ 1 \}, \{ 1,2 \}, \{ 2,3 \} \}$. The figure depicts the output on this input of the reduction in Thm. \ref{['thm:reach-MVPP-k-grabbing']}

Theorems & Definitions (60)

• Example 1.1
• Theorem 2.1: GTW02
• proof : Proof sketch
• Definition 2.2
• Theorem 2.3
• Theorem 3.1
• proof
• Corollary 3.2
• Theorem 3.3
• proof