Econ Papers Daily / 期刊论文
伯格均衡——一种代数方法
Berge equilibria – an algebraic approach
摘要
Berge equilibrium offers an alternative to Nash equilibrium in game theory, emphasizing cooperative stability rather than individual optimization. Despite recent interest, a systematic study of Berge equilibria in finite normal form games is still lacking, with fundamental questions like existence remaining open. This paper characterizes Berge equilibria through a polynomial system of equations, enabling computational algebra and algebraic geometry methods to analyze them. Algorithms based on Gröbner bases determine the existence and computation of Berge equilibria. Furthermore, we show that the set of games admitting completely mixed Berge equilibria is contained within a determinantal variety, whose dimension we explicitly bound from above.
中文摘要
伯格均衡为博弈论中的纳什均衡提供了一种替代方案，强调合作稳定性而非个体最优化。尽管近期研究兴趣渐增，但有限正规型博弈中伯格均衡的系统性研究仍显不足，诸如存在性等基本问题尚未解决。本文通过多项式方程组刻画伯格均衡，从而能够运用计算代数和代数几何方法对其进行分析。基于格罗布纳基的算法可判定伯格均衡的存在性并实现其计算。此外，我们证明存在完全混合伯格均衡的博弈集合包含于某一行列式簇中，并给出了该簇维数的显式上界。