Berge equilibria – an algebraic approach
用多项式方程组刻画有限正规型博弈中的伯格均衡，借助格罗布纳基算法判断其存在性并计算，同时给出完全混合伯格均衡存在时博弈集合的维数上界。
Abstract 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.