The Verification Problem for the Subgame Perfect Equilibrium and the Nash Equilibrium in Finite-Horizon Probabilistic Concurrent Game Systems

Abstract

Finite-horizon probabilistic multiagent concurrent game systems, also known as finite multiplayer stochastic games, are a well-studied model in artificial intelligence due to their ability to represent a wide range of real-world scenarios involving strategic interactions among agents over a finite amount of iterations (given by the finite-horizon). The analysis of these games typically focuses on evaluating and computing which strategy profiles (functions that represent the behavior of each agent) qualify as equilibria. The two most prominent equilibrium concepts are the \emph{Nash equilibrium} and the \emph{subgame perfect equilibrium}, with the latter considered a conceptual refinement of the former. However, computing these equilibria from scratch is often computationally infeasible. Therefore, recent attention has shifted to the verification problem, where a given strategy profile must be evaluated to determine whether it satisfies equilibrium conditions. In this paper, we demonstrate that the verification problem for subgame perfect equilibria lies in PSPACE, while for Nash equilibria, it is EXPTIME-complete. This is a highly counterintuitive result since the subgame equilibria are often seen as a strict strengthening of the Nash equilibrium and are intuitively seen as more complicated.