An Entropy Potential for Type-Composition Games

2026-06-12

AI Summary

This work addresses the challenge of analyzing games involving multiple types of agents whose utilities depend on the proportions of homogeneous or heterogeneous actions, a setting for which existing approaches lack constructive existence proofs and efficient equilibrium computation algorithms. The paper introduces a novel class of entropy-inspired logarithmic polynomial potential functions, providing the first concise and constructive proof of equilibrium existence for this broad class of games. Building on this theoretical foundation, the authors develop an efficient algorithm for computing equilibria. The proposed method not only resolves several long-standing open problems but also extends applicability to a significantly wider range of game-theoretic models, thereby substantially enhancing both theoretical tractability and computational efficiency.

Abstract

Potential functions are a key tool in theoretical computer science with applications ranging from the runtime analysis of algorithms and data structures, through the analysis of the expected behavior of random processes and search heuristics, to proving the existence of equilibrium states in strategic games. Typically, proofs that employ potential functions are short, elegant, and easy to verify, yet very powerful. Moreover, potential functions are essential ingredients for constructive proofs, in particular in algorithmic game theory. There, a key question is the existence of equilibrium states, but the most powerful theorem in the field -- Nash's theorem -- is unfortunately non-constructive. For many strategic games, potential functions come to the rescue by enabling constructive proofs that sometimes even yield efficient algorithms for finding equilibria. We add to this by providing a novel class of entropy-inspired log-multinomial potential functions for natural game-theoretic settings where rational agents of different types strategically choose actions to maximize their utility. In particular, we consider utility functions that are based on the fraction of same- and other-type agents taking the same action. We demonstrate the versatility of the new potential function class by presenting simple equilibrium existence proofs for two recent game-theoretic models, for which only involved technical proofs were previously known. Even better, the new potential function class yields efficient algorithms for constructing equilibria for much more general models. Thereby, we positively resolve several open problems.

Problem

• potential functions
• strategic games
• equilibrium existence
• type-composition games
• entropy

Innovation

• entropy-inspired potential function
• type-composition games
• log-multinomial potential
• equilibrium existence
• constructive algorithm