🤖 AI Summary
This paper introduces “pawn games,” a novel class of two-player zero-sum graph games where vertex control dynamically shifts during play, governed by contestable “pawns” that determine move rights. The work formally defines this dynamic control model, specifying control-transfer rules based on a grabbing mechanism, and introduces the Lock&Key game—a new hardness tool with independent significance. Using formal game-theoretic modeling, complexity analysis (classifying variants within PTIME or EXPTIME), and carefully constructed reductions, the paper precisely characterizes the computational complexity of multiple pawn game variants: it identifies natural subclasses solvable in polynomial time and proves EXPTIME-completeness for several settings. These results demonstrate that dynamic ownership fundamentally affects solution complexity, revealing a previously unexplored dimension of difficulty in graph games and substantially extending the theoretical landscape of infinite-state and turn-based graph games.
📝 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.
