Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER VI. THE MATHEMATICS OF HUMAN BEHAVIOR

48. EXPECTED UTILITY

This section is somewhat technical. Its purpose is to describe rational considerations that people accept in regard to lotteries, yet, as we shall see in the next section, they do not follow them when acting intuitively.

Game theory allows the decision maker to act according to his own subjective preferences. In the case of mixed strategies, that is, using lotteries as a means of decision making, the subjectivity is likely to reflect also the attitude of the decision maker to the lottery. Nevertheless, in the previous section we assumed that the expected payoffs determine the value of the lottery for each of the players. This assumption does not reflect reality. Some people are sure that luck always lets them down, and therefore they will not agree that the expectation of payoff should determine the value. Others love risk, and for them the lottery is worth more than the expected payoff.

John von Neumann and Oskar Morgenstern studied this question in the aforementioned book. They proposed the following solution. Try to replace the payoffs listed in the table of the game with other numerical values without altering the ranking of the payoffs such that the new values fulfill the expectation condition set previously. In other words, the value of the lottery, which von Neumann and Morgenstern called the utility of the lottery, will be the expectation of the new payoffs. There is no a priori reason that it will always be possible to find numbers whose expectation will reflect the preferences of the players with regard to the lotteries. Von Neumann and Morgenstern proved, however, that provided the players’ actions are in accordance with some simple characteristics that every sensible person would accept as reasonable, it is possible to find such a utility. We would note at the outset that people do not act in accordance with the characteristics identified by von Neumann and Morgenstern, and we will discuss that in the next section. Yet if we examine those traits in an abstract rational manner, it is apparent that they describe how we should conduct ourselves. In the context of our discussion, the characteristics, which in the spirit of Greek mathematics can be called axioms, are as follows.

1. The player knows which of any two possible payoffs, including the ones identified by lottery, is preferable to him, or he can decide that they are equal. This relation is transitive, that is, if option A is preferable to B and B is preferable to C, then A is preferable to C.

2. If in a certain lottery a player is offered the possibility of changing a payoff for another that is preferable to him, including a payoff that is a lottery, he will accept the offer.

3. The way the lottery is carried out, that is, the way in which the probabilities are formed, does not affect the value of the lottery, as long as the probabilities do not change.

4. For all three payoffs in which A is preferable to B that is preferable to C, there is a positive probability, which we denote by p and which may be very small, that getting C with a probability of p and getting A with a probability of (1 – p) is preferable to getting B.

The axioms are indeed convincing. For anyone not motivated by superstition there is no reason not to accept the second and third axioms. The first axiom is correct theoretically but perhaps not practical. Von Neumann and Morgenstern, however, suggested ways of calculating the new utility, if it exists. The fourth axiom is also reasonable. To anyone claiming that his reservation about getting payoff C is so strong that he is not prepared to take even the small risk p that he will receive C, we would point out that he does leave his house, travel by car or train, and even fly occasionally, despite the fact that those activities bear a risk, which may be small but is certainly not zero, that he will suffer severe or even fatal injury.

A utility that has the property we mentioned, that is, that the utility of the lottery is the expected utility, is named after those who developed the concept and is known as von Neumann–Morgenstern utility. As stated, if the axioms are fulfilled, the von Neumann–Morgenstern utility exists. The possibility of changing the actual payoffs for others such that the expectation of the new values reflects the value of the lottery to the participant was, in fact, proposed by Daniel Bernoulli in relation to the St. Petersburg paradox mentioned in section 39. Bernoulli's explanation of the paradox was that the value of very large monetary payoffs are not reflected in their face value but in another value, which Bernoulli already called utility. That utility is a function that increases very slowly when the amounts of money keep rising. The value of the lottery in the St. Petersburg paradox should be measured, according to Bernoulli, by the expected value of the function, which explains why people are not prepared to pay large sums to participate in that lottery.