Nash equilibrium in a stochastic model of two competing athletes

摘要: We propose a toy model for stochastic description of the competition between two athletes unequal strength, whose average strength difference is represented by parameter $d$. The interact through choice their strategies $x,y\in[0,1]$. These variables denote amount energy each invests in competition, and determine performance athlete. Each athlete picks his strategy based on knowledge own competitor's distribution, evaluation danger exhaustion, which increases with invested energy. formulate this problem as zero-sum game. Mathematically it class "discontinuous games" Nash equilibrium not guaranteed advance. demonstrate explicit construction that has mixed $\big( f(x),g(y) \big)$ arbitrary $0