Return-based mean

We are given a lottery introduced by Bernoulli – throw a coin repeatedly and you get if the first tail occurs on the n-th trial; this lottery can be presented as:

By accepting this gamble repeatedly, you can expect to get on average:

Yet people are reluctant to pay for it more than $10.

Let’s consider an amount of money , which you invest in such a lottery. Then we can express the lottery in terms of returns it brings:

By investing an amount , you can expect to get the average return of:

You want this average return to be at least equal to 1:

We know that if , then the following is true:

If we plug , then we can go back to our inequality:

We define as the cutoff investment amount for which the average return of the lottery is equal to 1.

Hence the investment in a Bernoulli lottery should not exceed 4, so that on average we get positive return (=1).

More generally we can ask for a cutoff investment value of a whole wealth position with some risk. Suppose the initial wealth is (it could be negative) and you accept a Bernoulli lottery . Then your wealth position is equal to:

And the cutoff investment value for the whole position has to satisfy the following condition:

1

I can take logarithms on both sides to get:

The situation above focused on a lottery in which you can only gain and you can never loose. Suppose now we analyzed a lottery in which you can lose. Then you need some initial wealth (or reserve) in order to cover losses in case you are unlucky. We will henceforth assume that borrowing is not allowed and hence to play lottery in which the minimal value (for which the probability is nonzero) is , you need a reserve (or initial wealth) not smaller than this maximal loss:

So we will always be talking about returns in terms of wealth positions:

Consider now a lottery , such that:

  • (arithmetic mean is positive)
  • (losses are possible)

We want to find a value of wealth/reserve such that lottery is neither desirable nor undesirable (the maximal investment we are willing to make for it is equal to )

If we take the logarithm on both sides we get:

Since function is continuous, there must be a value for which it is equal to 0:

Let’s take an example where . It has positive mean and negative values with positive probability. Hence there exists a cutoff value :

Solving gives .