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？若收益为整数，则答案是肯定的，且其计算复杂性已有大量研究。若允许收益为实数，则由于连续性原因，答案是否定的。探究其不可计算性的精确度（魏赫劳赫度）是有价值的，因为了解该度意味着其他哪些方法是可行的（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未解决，本文即致力于此。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。