Constrained Existence Problem for Weak Subgame Perfect Equilibria with $\omega$-Regular Boolean Objectives

TL;DR

This paper analyzes the computational complexity of the constrained existence problem for weak subgame perfect equilibria in multiplayer games with omega-regular objectives, providing a detailed complexity classification and algorithmic insights.

Contribution

It offers a complete complexity classification for the constrained existence problem of weak SPEs across various objectives and introduces a fixpoint algorithm for computing payoff profiles.

Findings

P-complete for Explicit Muller objectives

NP-complete for Co-B"uchi, Parity, Muller, Rabin, and Streett objectives

PSPACE-complete for Reachability and Safety objectives

Abstract

We study multiplayer turn-based games played on a finite directed graph such that each player aims at satisfying an omega-regular Boolean objective. Instead of the well-known notions of Nash equilibrium (NE) and subgame perfect equilibrium (SPE), we focus on the recent notion of weak subgame perfect equilibrium (weak SPE), a refinement of SPE. In this setting, players who deviate can only use the subclass of strategies that differ from the original one on a finite number of histories. We are interested in the constrained existence problem for weak SPEs. We provide a complete characterization of the computational complexity of this problem: it is P-complete for Explicit Muller objectives, NP-complete for Co-B\"uchi, Parity, Muller, Rabin, and Streett objectives, and PSPACE-complete for Reachability and Safety objectives (we only prove NP-membership for B\"uchi objectives). We also show that the constrained existence problem is fixed parameter tractable and is polynomial when the number of players is fixed. All these results are based on a fine analysis of a fixpoint algorithm that computes the set of possible payoff profiles underlying weak SPEs.