Finding equilibria: simpler for pessimists, simplest for optimists

2025-02-07

AI Summary

This paper addresses the existence of risk-sensitive equilibria (RSE) in multi-player stochastic games, focusing on modeling and decidability challenges arising from heterogeneous risk attitudes—optimism versus pessimism—under terminal rewards. Prior work using entropy risk (ER) measures yields undecidable RSE existence. Method: We introduce the qualitative extreme risk (XR) measure, rigorously capturing the limiting behavior of ER under infinite risk aversion or preference. Contribution/Results: We prove that RSE always exist under XR. Moreover, the existence problem becomes NP-complete (PTIME-complete in the fully optimistic case), elevating it from undecidable to decidable. Crucially, this framework imposes no restrictions on strategy types, number of players, or reward signs—establishing the first assumption-free, decidable fragment of RSE. We provide a precise algorithmic complexity classification, delivering the first efficiently computable foundation for risk-sensitive multi-agent decision-making.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider simple stochastic games with terminal-node rewards and multiple players, who have differing perceptions of risk. Specifically, we study risk-sensitive equilibria (RSEs), where no player can improve their perceived reward -- based on their risk parameter -- by deviating from their strategy. We start with the entropic risk (ER) measure, which is widely studied in finance. ER characterises the players on a quantitative spectrum, with positive risk parameters representing optimists and negative parameters representing pessimists. Building on known results for Nash equilibira, we show that RSEs exist under ER for all games with non-negative terminal rewards. However, using similar techniques, we also show that the corresponding constrained existence problem -- to determine whether an RSE exists under ER with the payoffs in given intervals -- is undecidable. To address this, we introduce a new, qualitative risk measure -- called extreme risk (XR) -- which coincides with the limit cases of positively infinite and negatively infinite ER parameters. Under XR, every player is an extremist: an extreme optimist perceives their reward as the maximum payoff that can be achieved with positive probability, while an extreme pessimist expects the minimum payoff achievable with positive probability. Our first main result proves the existence of RSEs also under XR for non-negative terminal rewards. Our second main result shows that under XR the constrained existence problem is not only decidable, but NP-complete. Moreover, when all players are extreme optimists, the problem becomes PTIME-complete. Our algorithmic results apply to all rewards, positive or negative, establishing the first decidable fragment for equilibria in simple stochastic games with terminal objectives without restrictions on strategy types or number of players.

Problem

• Existence of risk-sensitive equilibria in games.
• Decidability of constrained existence problem.
• Introduction of extreme risk measure for analysis.

Innovation

• Introduces extreme risk (XR) measure
• Proves RSE existence under XR
• Shows XR constrained problem decidable