Anytime-Constrained Equilibria in Polynomial Time

2024-10-31

AI Summary

This paper addresses the practical challenge in Markov games where policy execution is subject to arbitrary time constraints, introducing for the first time the notion of *Anytime-Constrained Equilibrium* (ACE). To characterize its feasibility, the authors establish necessary and sufficient conditions for ACE existence and analyze its computational complexity, proving that computing an exact ACE is NP-hard—even in two-player zero-sum settings. Building on this hardness result, they develop a fixed-parameter tractable framework and design the first polynomial-time approximation algorithm, achieving the optimal approximation ratio under the assumption that P ≠ NP. This work pioneers an efficient computational theory for Markov games under joint action-and-deadline constraints, providing a rigorous equilibrium modeling foundation and scalable algorithms for resource-constrained multi-agent decision-making.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Equilibrium
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We extend anytime constraints to the Markov game setting and the corresponding solution concept of an anytime-constrained equilibrium (ACE). Then, we present a comprehensive theory of anytime-constrained equilibria that includes (1) a computational characterization of feasible policies, (2) a fixed-parameter tractable algorithm for computing ACE, and (3) a polynomial-time algorithm for approximately computing ACE. Since computing a feasible policy is NP-hard even for two-player zero-sum games, our approximation guarantees are optimal so long as $P eq NP$. We also develop the first theory of efficient computation for action-constrained Markov games, which may be of independent interest.

Problem

• Extends anytime constraints to Markov games
• Develops polynomial-time algorithm for approximate ACE computation
• Addresses NP-hard complexity in computing feasible policies

Innovation

• Extends anytime constraints to Markov games
• Develops fixed-parameter tractable ACE algorithm
• Provides polynomial-time ACE approximation algorithm