Two Competing Companies Insuring Multiple Objects

Following the model of previous section we will approximate discrete set of insurance policies $x_i$ by the continuous one and shall use similar expressions of the expected cumulative utility function $U(x,a)$ of multiple customers.

The expected utilities $V_j(x,a_j)$ of two competing insurance companies $j=1,2$ are defined as in the previous section.

$$V_j(x,a)=\sum_{y_j} v_j(y_j) p(y_j),$$ (34)

where $p(y_j)$ is a probability of profit $y_j$,
$v_j(y_j)$ is a utility of the profit $y_j$.

Suppose that
$y_j= d (x_j)$.
From (26)

$$d(x_j)= a_j s_j-(1-a_j)(n_j-s_j),$$ (35)

where $x_j=n_j$is a number customers that decided to insure their property at the company $j$and $s_j$ shows how many of them survive.

From here and assumption (26) the probability $p(y_j)$ of profit $y_j=s_j a_j-(1-a_j)(n_j-s_j)$

$$p(y_j)=p(n_j,s_j,q)=P\{y=s_j a_j-(1-a_j)(n_j-s_j)\}.$$ (36)

This defines the expected utility function $V_j(x_j,a_j)$ for the insurance company $j$.

Note that insurance policies $x_j, \ j=1,2$ are defined indirectly by the number of customers $n_j$ that insure their objects at the company $j$. It is supposed that customers maximize expected cumulative utility that depends on the rates of insurance charges $a_j, \ j=1,2$.

The equilibrium between interests of companies and customers is achieved if insurance policies $x_j$ and insurance charges $a_j$ satisfy Nash conditions. Here search for the Nash equilibrium is performed minimizing differences between the fraud and contract vectors.