Is there an algorithm that takes a game in normal form as input, and outputs a Nash equilibrium? If the payoffs are integers, the answer is yes, and lot of work has been done in its computational complexity. If the payoffs are permitted to be real numbers, the answer is no, for continuity reasons. It is worthwhile to investigate the precise degree of non-computability (the Weihrauch degree), since knowing the degree entails what other approaches are available (eg, is there a randomized algorithm with positive success change?). The two player case has already been fully classified, but the multiplayer case remains open and is addressed here. Our approach involves classifying the degree of finding roots of polynomials, and lifting this to systems of polynomial inequalities via cylindrical algebraic decomposition.

翻译：是否存在一个算法，输入标准形式的博弈，输出纳什均衡？若收益为整数，答案是肯定的，且已有大量工作研究其计算复杂性。若收益可为实数，则出于连续性原因，答案是否定的。研究其精确的不可计算程度（Weihrauch度）具有价值，因为了解该程度意味着可获知其他可行方法（例如，是否存在具有正成功概率的随机化算法？）。两人情形已被完整分类，但多人情形仍悬而未决，本文即对此展开研究。我们的方法涉及对多项式求根的程度进行分类，并通过柱形代数分解将其推广至多项式不等式系统。