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.

翻译：势函数是理论计算机科学中的关键工具，其应用涵盖从算法与数据结构的运行时间分析、随机过程和搜索启发式算法预期行为分析，到证明策略博弈中均衡态存在性。通常，使用势函数的证明简短优雅且易于验证，却极具效力。更重要的是，势函数是构造性证明的核心要素，尤其在算法博弈论中。在该领域，均衡态存在性是一个关键问题，但最强大的定理——纳什定理——却非构造性的。对于许多策略博弈，势函数通过提供构造性证明来挽救局面，有时甚至能给出寻找均衡的有效算法。我们为此做出贡献，提出一类新颖的基于熵的对数多项式势函数，适用于理性代理类型各异、通过选择行动以最大化效用的自然博弈场景。特别地，我们考虑基于同类型与不同类型代理采取相同行动比例的效用函数。通过为两个近期博弈论模型提供简单的均衡存在性证明，我们展示了新势函数类的多功能性，而此前这些模型仅有复杂的技术性证明。更重要的是，新势函数类能为更广泛的模型提供构造均衡的有效算法，从而正面解决了多个开放问题。