Optimal coupling in probability theory and optimal cooperation in game theory

Abstract

In this paper, we extend the concept of binary optimal coupling. We obtain the result I: Suppose $X$ and $Y$ are Polish spaces, $\varphi:~X\times~Y\to~\mathbb{R}$is measurable, $\mu\in\mathscr~P(X),$ $~\nu\in\mathscr~P(Y)$. (i) If $\varphi$ is a lower semi-continuous function with a lower bound, then $\varphi$ optimal coupling$\gamma_\varphi$ exists; (ii) if $\varphi$ is an upper semi-continuous function with an upper bound, then $\varphi$ upper optimal coupling $\gamma^\varphi$ exists. In addition, we obtain the result II:Suppose $G_i$ $(i=1,~2)$ is a transition probability measure sequence. (i) If $\varphi:~X_1\times~X_2\to~\mathbb{R}$ is a lower semi-continuous function with a lower bound,then $\varphi$ optimal coupling of $G_1$ and $G_2$ exist; (ii) if $\varphi:~X_1\times~X_2\to~\mathbb{R}$ is an upper semi-continuous function with an upper bound,then $\varphi$ upper optimal coupling $G_1$ and $G_2$ exist. A concept of $n$-fold optimal coupling with constraints is presented and the existence of this optimal coupling isproved. Furthermore, an optimal cooperation equilibrium of Nash equilibrium in game theory is defined. This equilibrium is superior to Nash equilibrium as illustrated.