Strategically Robust Game Theory via Optimal Transport

TL;DR

This paper introduces strategically robust equilibria in game theory, leveraging optimal transport to protect against uncertainty in opponents' behavior, ensuring existence, computational efficiency, and improved payoffs in various strategic settings.

Contribution

It proposes a novel equilibrium concept, strategically robust equilibrium, operationalized via optimal transport, which guarantees existence, interpolates between Nash and security strategies, and maintains computational efficiency.

Findings

Strategically robust equilibria exist under standard assumptions.

These equilibria often yield higher payoffs than Nash equilibria.

The approach effectively guards against opponent behavior uncertainty.

Abstract

In many game-theoretic settings, agents are challenged with taking decisions against the uncertain behavior exhibited by others. Often, this uncertainty arises from multiple sources, e.g., incomplete information, limited computation, bounded rationality. While it may be possible to guide the agents' decisions by modeling each source, their joint presence makes this task particularly daunting. Toward this goal, it is natural for agents to seek protection against deviations around the emergent behavior itself, which is ultimately impacted by all the above sources of uncertainty. To do so, we propose that each agent takes decisions in face of the worst-case behavior contained in an ambiguity set of tunable size, centered at the emergent behavior so implicitly defined. This gives rise to a novel equilibrium notion, which we call strategically robust equilibrium. Building on its definition, we show that, when judiciously operationalized via optimal transport, strategically robust equilibria (i) are guaranteed to exist under the same assumptions required for Nash equilibria; (ii) interpolate between Nash and security strategies; (iii) come at no additional computational cost compared to Nash equilibria. Through a variety of experiments, including bi-matrix games, congestion games, and Cournot competition, we show that strategic robustness protects against uncertainty in the opponents' behavior and, surprisingly, often results in higher equilibrium payoffs - an effect we refer to as coordination via robustification.