Kakutani's Fixed Point theorem is a fundamental theorem in topology with numerous applications in game theory and economics. Computational formulations of Kakutani exist only in special cases and are too restrictive to be useful in reductions. In this paper, we provide a general computational formulation of Kakutani's Fixed Point Theorem and we prove that it is PPAD-complete. As an application of our theorem we are able to characterize the computational complexity of the following fundamental problems: (1) Concave Games. Introduced by the celebrated works of Debreu and Rosen in the 1950s and 60s, concave $n$-person games have found many important applications in Economics and Game Theory. We characterize the computational complexity of finding an equilibrium in such games. We show that a general formulation of this problem belongs to PPAD, and that finding an equilibrium is PPAD-hard even for a rather restricted games of this kind: strongly-concave utilities that can be expressed as multivariate polynomials of a constant degree with axis aligned box constraints. (2) Walrasian Equilibrium. Using Kakutani's fixed point Arrow and Debreu we resolve an open problem related to Walras's theorem on the existence of price equilibria in general economies. There are many results about the PPAD-hardness of Walrasian equilibria, but the inclusion in PPAD is only known for piecewise linear utilities. We show that the problem with general convex utilities is in PPAD. Along the way we provide a Lipschitz continuous version of Berge's maximum theorem that may be of independent interest.

翻译：角谷不动点定理是拓扑学中的基本定理，在博弈论与经济学中应用广泛。角谷定理的计算公式仅在特殊情形下存在，且因过于受限而难以用于归约。本文提出角谷不动点定理的通用计算公式，并证明其为PPAD完备问题。作为该定理的应用，我们刻画了以下基本问题的计算复杂性：（1）凹博弈。由Debreu与Rosen在20世纪50至60年代开创性工作提出的凹n人博弈，已在经济学与博弈论领域获得重要应用。我们刻画了此类博弈中均衡求解的计算复杂性，证明该问题的通用形式属于PPAD类，且即使对相当受限的博弈类型（可用常度数多元多项式表示、带轴对齐盒约束的强凹效用函数），求解均衡也是PPAD困难的。（2）瓦尔拉斯均衡。利用角谷不动点与Arrow-Debreu定理，我们解决了与瓦尔拉斯定理（关于一般经济中价格均衡存在性）相关的开放问题。现有诸多结果显示瓦尔拉斯均衡为PPAD困难问题，但其属于PPAD类的结果仅对分段线性效用函数成立。我们证明具有一般凸效用函数的问题属于PPAD类。在此过程中，我们给出了Berge最大值定理的Lipschitz连续版本，该结论可能具有独立研究价值。