Search for Equilibrium

First we fix the initial values, the "Contract-Vector" $(x_j^0,a_j^0,\ j=1,2)$. The transformed values, the "Fraud-Vector" $(x_j^1,a_j^1,\ j=1,2)$, are obtained by maximizing the utilities $U(x_j,a_j)$ and $V_j(x_j,a_j)$ respectively. The maximization is performed under the assumption that partners honor the contract $(x_j^0,a_j^0,\ j=1.2)$

$$(x_1^1,x_2^1) = arg \max_{x_1,x_2} U(x_1,x_2, a_1^0,a_2^0),$$ (37)

$$a_j^1 = arg \max_{a_j} V_j(x_j^0, a_j,a_l^0,\ l \ne j),\ j=1,2$$ (38)

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

$$w^{n+1} = T(w^n),\ n=0,1,2,...$$ (39)

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

$$w^{n} = T(w^n).$$ (40)

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

$$\min_{w \in B} \parallel w - T(w) \parallel^2.$$ (41)

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.

The problem can be directly extended to $M$ competing companies and $N$ individually insured objects. The only obstacle is dimensionality of related optimization problem. For example, in this case one searches for equilibrium of four variables $(x_1,x_2,a_1,a_2)$ where variables $(x_1,x_2)$ are controlled by customers responding to insurance charges $(a_1,a_2)$ that are defined by coresponding insurance companies.