群体博弈Nash平衡存在性定理与Brouwer不动点定理

Nash Equilibrium Existence Theorem of Population Games and Brouwer’s Fixed Point Theorem

首先,运用Brouwer不动点定理证明了群体博弈的Nash平衡存在性定理;结合群体博弈的Nash平衡存在性定理与变分不等式解的存在性定理的等价性,反过来,运用群体博弈的Nash平衡存在性定理又证明了Brouwer不动点定理。所得结果表明:群体博弈的Nash平衡存在性定理与著名的Brouwer不动点定理等价,这是一个深刻的结果。

First of all,Nash equilibrium existence theorem of population games is shown by Brouwer’s fixed point theorem.Furthermore,Brouwer’s fixed point theorem is conversely proven using Nash equilibria existence theorem of population games based on the equivalence between Nash equilibria existence theorem and solution existence of variational inequality.This result is significant that Nash equilibria existence theorem of population games is equivalent to Brouwer’s fixed point theorem.