Single Company Insuring Multiple Objects

In this section the insurance model of multiple objects $i=1,...,m$ is reduced to that of a single object. Some companies apply "zero-one" insurance policies: $x_i=0$ or $x_i=z_i$. In this case no equilibrium exist because only two discrete strategies can be applied². If a number of customers is large, for example a set of car owners, one approximates discrete set by the continuous one assuming that $x$ is a sum of insurance policies of $n$ customers that decided to insure their property $x=\sum_{i=1}^n x_i$. Here $n \le m$ is a number of customers that decided to insure their objects including $s$ insured survivors. Then $k=m-n$ is a number of not insured objects including $l$ of not insured survivors.

Assume as a first approximation that

$$x_i=z_i=1.$$ (26)

Denote by $u(y)$ some cumulative utility function for a set of customers where variable $y$ means their total wealth. In theory the cumulative utility $u(y)$ is determined by individual utilities $u_i(y_i)$. However, that is not a trivial computational problem. Statistical analysis of corresponding data is needed, too.

Under these assumptions the expected cumulative utility

$$U(x,a)=\sum_y u(y) p(y),$$ (27)

where $p(y)$ is a probability of the wealth $y$.

Suppose that total wealth of all customers
$y= c (x)$.
From (26)

$$c(x)= (1-a)n+l,$$ (28)

where $x=n$ is a number customers that decided to insure their property and $l$ is a number of not insured survivors.

From the assumption that survival probabilities $p$ of all the objects³ are equal and independent follows the binomial distribution. If $\lambda =qk$, where $q=1-p$, is not very large and not very small then one approximates the binomial distribution by the Poisson distribution . The probability that $l$ of $k$ objects survive and the rest $k-l$ do not

$$p(k,l,q)=e^{-\lambda} \frac {\lambda^l}{l!}$$ (29)

From here and assumption (26) the probability $p(y)$ of returned wealth $y=(1-a)n+l$ is

$$p(y)=p(k,l,q)=P\{y=(1-a)n+l\}.$$ (30)

This completes the definition of the expected cumulative utility $U(x,a)$.

The expected utility $V(x,a)$ of insurance company is defined in a similar way.

$$V(x,a)=\sum_y v(y) p(y),$$ (31)

where $p(y)$ is a probability of profit $y$,
$v(y)$ is an utility function of the profit $y$.

Suppose that
$y= d (x)$.
From (26)

$$d(x)= a n-(1-a)(n-s),$$ (32)

where $x=n$ is a number customers that decided to insure their property and $s$ is a number of insured survives.

From here and assumption (26) the probability $p(y)$ of profit $y=an-(1-a)(n-s)$

$$p(y)=p(n,s,q)=P\{y=sa-(1-a)(n-s)\}.$$ (33)

This defines the expected utility function $V(x,a)$ for an insurance company.

Here insurance policy $x$ is defined indirectly by the number of customers $n$ that insure their of objects maximizing their expected cumulative utility (15) that depends on the rate of insurance charge $a$. This is a correct assumption if the cumulative utility function $u(y)$ represents individual utilities $u_i(y_i)$ well enough. Therefore definition of $u(y)$ is important part of model that represents multiple customers as a single one.

The equilibrium between interests of the company and the customer is achieved when both insurance policy $x$ and insurance charge $a$ satisfies Nash conditions. One obtains the Nash equilibrium using the same expressions as in the previous section.