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，在博弈论和经济学中有众多应用。目前仅在某些特殊情况下存在角谷定理的计算公式化表述，且这些表述限制过严，难以用于归约。本文给出了角谷不动点定理的一般计算公式化表述，并证明它是PPAD完全的。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和Rosen在20世纪50-60年代的著名工作引入的n人凹博弈，已在经济学和博弈论中取得重要应用。我们刻画了此类博弈中寻找均衡点的计算复杂性，证明该问题的一般形式属于PPAD类，且即使对于此类博弈中相当受限的情形——可表示为常数次多元多项式且带有轴对齐盒约束的强凹效用函数——寻找均衡点仍是PPAD难的。(2) 瓦尔拉斯均衡。利用角谷不动点定理，Arrow和Debreu解决了关于一般经济中价格均衡存在性的瓦尔拉斯定理相关开放问题。虽然已有众多关于瓦尔拉斯均衡PPAD难度的结果，但仅对分段线性效用函数已知其属于PPAD类。我们证明具有一般凸效用函数的问题也属于PPAD类。在此过程中，我们给出了Berge最大值定理的Lipschitz连续版本，这可能具有独立研究价值。