Keywords: sharesl

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### Search for Equilibrium

First we fix the initial values, the "Contract-Vector" . The transformed values, the "Fraud-Vector" , are obtained by maximizing the utilities and respectively. The maximization is performed under the assumption that a partner honors the contract

|  |  | (21)
|  |  | (22)

Formally, condition (22) transforms the vector
into the vector . To make expressions shorter denote this transformation by

|  |  | (23)

One may obtain the equilibrium at the fixed point , where

|  |  | (24)

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

|  |  | (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.

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