Search for Equilibrium

First we fix the initial values, the "Contract-Vector" $(x^0,a^0)$. The transformed values, the "Fraud-Vector" $(x^1,a^1)$, are obtained by maximizing the utilities $U(x,a)$ and $V(x,a)$ respectively. The maximization is performed under the assumption that a partner honors the contract $(x^0,a^0$

$$x^1 = arg \max_{x} U(x, a^0),$$ (21)

$$a^1 = arg \max_{a} V(x^0, a_),$$ (22)

Formally, condition (22) transforms the vector
$w^n=(x^n,a^n) \in B ,\ n=0,1,2,...$ into the vector $w^{n+1}$. To make expressions shorter denote this transformation by $T$

$$w^{n+1} = T(w^n),\ n=0,1,2,...$$ (23)

One may obtain the equilibrium at the fixed point $w^n$, where

$$w^{n} = T(w^n).$$ (24)

The fixed point $w^n$ exists, if the feasible set $B$ is convex and all the profit functions are convex [2]. We obtain the equilibrium directly by iterations (23), if the transformation $T$ is contracting [4]. If not, then we minimize the square deviation

$$\min_{w \in B} \parallel w - T(w) \parallel^2.$$ (25)

The equilibrium is achieved, if the minimum (25) is zero. If the minimum (25) is positive then the equilibrium does not exist. That is a theoretical conclusion. In statistical modeling, some deviations are inevitable. Therefore, we assume that the equilibrium exists, if the minimum is not greater then modeling errors.