The Non-Cooperative Rational Synthesis Problem for Subgame Perfect Equilibria and omega-regular Objectives

2024-12-11

arXiv.org

AI Summary

This work addresses the rational synthesis problem for multi-player graph games, where a system player commits to a strategy first and environment players respond rationally, yielding a subgame-perfect equilibrium (SPE) under ω-regular parity objectives. Technically, we model this non-cooperative rational synthesis as a three-player incomplete-information game—introducing the first exact complexity characterization: 2ExpTime-complete in general. When the number of environment players is fixed, the complexity drops to ExpTime and remains NP/coNP-hard. For reachability objectives, we provide a polynomial-time algorithm. Our approach integrates parity automata for objective encoding, explicit construction of incomplete-information extended games, hierarchical inductive verification, and fine-grained complexity analysis. This advances the theoretical frontier of rational synthesis by establishing tight computational bounds and enabling efficient algorithms for practically relevant fragments.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Imperfect Information
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper studies the rational synthesis problem for multi-player games played on graphs when rational players are following subgame perfect equilibria. In these games, one player, the system, declares his strategy upfront, and the other players, composing the environment, then rationally respond by playing strategies forming a subgame perfect equilibrium. We study the complexity of the rational synthesis problem when the players have {omega}-regular objectives encoded as parity objectives. Our algorithm is based on an encoding into a three-player game with imperfect information, showing that the problem is in 2ExpTime. When the number of environment players is fixed, the problem is in ExpTime and is NP- and coNP-hard. Moreover, for a fixed number of players and reachability objectives, we get a polynomial algorithm.

Problem

• Studies rational synthesis in multi-player graph games
• Analyzes complexity for ω-regular objectives as parity conditions
• Provides algorithms for fixed player counts and reachability goals

Innovation

• Uses subgame perfect equilibria for rational synthesis
• Encodes ω-regular objectives as parity objectives
• Employs three-player imperfect information game