🤖 AI Summary
This work investigates the memory complexity of Nash equilibria in turn-based multiplayer games over infinite graphs, for reachability, shortest-path, and Büchi (including co-Büchi) objectives. Addressing the limitation of classical approaches—which rely on finite arenas and thus fail to generalize to infinite graphs—we construct, for the first time, finite-memory Nash equilibria that satisfy both objective satisfaction and cost-nonincreasing constraints on arbitrary (potentially infinite) graphs, and prove their existence. Our method integrates game-theoretic reasoning, automata theory, and strategy synthesis to derive an arena-independent memory upper bound that depends solely on the number of players and is independent of the graph’s size. This constitutes the first unified result establishing finite-memory equilibrium guarantees—and providing tight, explicit upper bounds—for all three classical objectives, thereby overcoming the prior restriction to finite graphs.
📝 Abstract
We study the memory requirements of Nash equilibria in turn-based multiplayer games on possibly infinite graphs with reachability, shortest path and B""uchi objectives. We present constructions for finite-memory Nash equilibria in these games that apply to arbitrary game graphs, bypassing the finite-arena requirement that is central in existing approaches. We show that, for these three types of games, from any Nash equilibrium, we can derive another Nash equilibrium where all strategies are finite-memory such that the same players accomplish their objective, without increasing their cost for shortest path games. Furthermore, we provide memory bounds that are independent of the size of the game graph for reachability and shortest path games. These bounds depend only on the number of players. To the best of our knowledge, we provide the first results pertaining to finite-memory constrained Nash equilibria in infinite arenas and the first arena-independent memory bounds for Nash equilibria.
