Keywords: sharesl

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## Single Company Insuring Multiple Objects

In this section the insurance model of multiple objects is reduced to that of a single object. Some companies apply "zero-one" insurance policies: or . 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 is a sum of insurance policies of customers that decided to insure their property . Here is a number of customers that decided to insure their objects including insured survivors. Then is a number of not insured objects including of not insured survivors.

Assume as a first approximation that

|  |  | (26)

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

Under these assumptions the expected cumulative utility

|  |  | (27)

where is a probability of the wealth .

Suppose that total wealth of all customers
.
From (26)

|  |  | (28)

where is a number customers that decided to insure their property and is a number of not insured survivors.

From the assumption that survival probabilities of all the objects3 are equal and independent follows the binomial distribution. If , where , is not very large and not very small then one approximates the binomial distribution by the Poisson distribution . The probability that of objects survive and the rest do not

|  |  | (29)

From here and assumption (26) the probability of returned wealth is

|  |  | (30)

This completes the definition of the expected cumulative utility .

The expected utility of insurance company is defined in a similar way.

|  |  | (31)

where is a probability of profit ,
is an utility function of the profit .

Suppose that
.
From (26)

|  |  | (32)

where is a number customers that decided to insure their property and is a number of insured survives.

From here and assumption (26) the probability of profit

|  |  | (33)

This defines the expected utility function for an insurance company.

Here insurance policy is defined indirectly by the number of customers that insure their of objects maximizing their expected cumulative utility (15) that depends on the rate of insurance charge . This is a correct assumption if the cumulative utility function represents individual utilities well enough. Therefore definition of 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 and insurance charge satisfies Nash conditions. One obtains the Nash equilibrium using the same expressions as in the previous section.

Next: Two Competing Companies Insuring Up: Nash Equilibrium Previous: Search for Equilibrium