# PROBABILITY OF SURVIVAL FOR INCOME AS A RANDOM VARIABLE

If income is a random variable we have two probability distributions to deal with — the income probability distribution and the agent survival probability distribution. By maximizing the probability of survival we arrive at a criterion reflecting the agent's risk aversion.

From the survival probability perspective, income (a random variable *d* having mean ͞*d* and variance σ2) that has a higher mean and a higher variance can be less advantageous than income that has a lower mean and a lower variance.

Let us assume, for example, a subsistence level of *b =* 100 money units, and let us compare income *d* = 500 with a "lottery" in which there is a 0.5 probability of winning 2000 money units (and a 0.5 probability of winning nothing). Even though the expected value of the lottery E(*d*) is double, a survival-probability-maximizing agent will opt forcertainty, because if he exchanges his entire income to take part in the lottery the expected value of his probability of survival *p(d)* will decrease:

b) *d* Î {0; 2000} with 50% probability for both alternatives:

The survival-probability-maximizing agent is therefore risk-averse. We will explore how much he is willing to pay to enter the St. Petersburg casino. First, though, we will analyse his willingness to enter a less attractive casino which, however, also has an infinite expected payoff.

# FOR MULATION OF THE LENINGRAD CASINO PROBLEM

We start by modifying the rules slightly. In our "Leningrad casino" a player gets nothing if the coin does not land heads at least 31 times in a row. Otherwise the rules are the same as in the St. Petersburg casino. Even this disadvantageous game has an infinite expected payoff of 2-31 • (1 + 1 + ••• + 1 + •••), so for an expected-income-maximizing agent it should be attractive whatever the entry fee is.

Let us denote the payoff by *v* and the Leningrad casino entry fee by y. Let us first assume that y ≤ *d - b,* i.e. the player may only gamble the excess of his income over the subsistence level.

If the player loses, paying the Leningrad casino entry fee will have reduced his probability of survival by

A rational decision-taker will compare this fall in the agent's probability of survival in the no-win scenario with the probability of winning, which would mean guaranteed survival (as a consequence of the astronomically high payoff). We ask: What is the maximum casino entry fee *Y* which the agent is willing to pay? For a survival-probability-maximizing agent the following must hold:

Let us denote the ratio of income to the subsistence level by. Therefore, *k* is a measure of the agent's economic situation. After a simple rearrangement we obtain:

We will begin by estimating the fee that is acceptable to agents with income lying between the subsistence level and double that level, i.e. 1 < *k < 2 . For* these poor people [let's say that *k =* 2 is the poverty line) the maximum acceptable fee is less than 4*b* • *2*-31, i.e. a tiny fraction of the subsistence level ( for example, less than 1 cent given a subsistence level of $10,000). Normal poor people, therefore, will not be frequent the Leningrad casino.

What is the approximate fee for an extremely rich person (let's say someone with an income that is 100 times the subsistence level, i.e. *k =* 100)? Even for him the figure is not astronomically high — the maximum acceptable fee is less than one one-hundred-thousandth of the subsistence level (10 cents given a subsistence level of $10,000). Even the rich will not visit the Leningrad casino in this model.

Nevertheless, we can theoretically consider agents who will find the Leningrad casino attractive. If we drop the assumption of *y ≤ d-b,* we can take into account a desperate man whose only hope is to win, as his income is below the subsistence level: *d ≤ b,* i.e. *k <* 1. For this agent, risky action is his only chance of survival. Moreover, this is in fact a case of moral hazard, because he is gambling with someone else's money (for example, a governmental or municipal support fund for the down and out). Even a low probability of survival is better than a zero probability, so he is willing to pay his entire income *d* (if he doesn't play, he won't survive anyway). The relation between the maximum acceptable entry fee and the agent's income is shown in Figure 4.

**Figure 4: The maximum acceptable Leningrad casino entry fee ****¥**** versus the agent's income ****d (b**** is the subsistence level)**

So, only down-and-out individuals with income *d e (y, b)* will enter the model Leningrad casino. If the entry fee is increased above the subsistence level, i.e. if *y > b,* no one will enter at all.^{[1]}

So, if we disregard the possibility of entry by down-and-outs with a negligible probability of economic survival, we can say that the Leningrad casino is not attractive at any price.

- [1] We could also theoretically consider an agent with income d = b + δ, where d is so small that the survival probability δ/b is smaller than the payout probability 2–31. This case, however, can be ruled out because even for the smallest indivisible money unit (let's say one cent) it holds that 0.01/b >2–31.