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：是否存在一种算法，以标准形式的博弈作为输入，并输出一个纳什均衡？如果收益是整数，答案是肯定的，并且其计算复杂性已有大量研究。如果允许收益为实数，则由于连续性原因，答案是否定的。有必要研究其不可计算性的精确度（Weihrauch度），因为了解该度意味着可以知道有哪些其他方法可用（例如，是否存在具有正成功概率的随机算法？）。两人博弈的情形已得到完全分类，但多人博弈的情形仍然悬而未决，本文正致力于此。我们的方法涉及对多项式求根问题的度进行分类，并通过柱形代数分解将其提升至多项式不等式系统。