The theory of expected Utility by J. Von Neumann and O. Morgenstern

1. Von Neumann–Morgenstern utility function, an extension of the theory of consumer preferences that incorporates a theory of behaviour toward risk variance. It was put forth by John von Neumann and Oskar Morgenstern in Theory of Games and Economic Behavior (1944) and arises from the expected utility hypothesis. It shows that when a consumer is faced with a choice of items or outcomes subject to various levels of chance, the optimal decision will be the one that maximizes the expected value of the utility (i.e., satisfaction) derived from the choice made. Expected value is the sum of the products of the various utilities and their associated probabilities. The consumer is expected to be able to rank the items or outcomes in terms of preference, but the expected value will be conditioned by their probability of occurrence.
The von Neumann–Morgenstern utility function can be used to explain risk-averse, risk neutral, and risk-loving behaviour. For example, a firm might, in one year, undertake a project that has particular probabilities for three possible payoffs of $10, $20, or $30; those probabilities are 20 percent, 50 percent, and 30 percent, respectively. Thus, expected payoff from the project would be $10(0.2) + $20(0.5) + $30(0.3) = $21. The following year, the firm might again undertake the same project, but in this example, the respective probabilities for the payoffs change to 25, 40, and 35 percent. It is easy to verify that the expected payoff is still $21. In other words, mathematically speaking, nothing has changed. It is also true that the probabilities of the lowest and highest payoffs rose at the expense of the middle one, which means there is more variance (or risk) associated with the possible payoffs. The question to pose to the firm is whether it will adjust its utility derived from the project despite the project’s having the same expected value from one year to the next. If the firm values both iterations of the project equally, it is said to be risk neutral. The implication is that it equally values a guaranteed payoff of $21 with any set of probabilistic payoffs whose expected value is also $21.
If the firm prefers the first year’s project environment to the second, it places higher value on less variability in payoffs. In that regard, by preferring more certainty, the firm is said to be risk averse. Finally, if the firm actually prefers the increase in variability, it is said to be risk loving. In a gambling context, a risk averter puts higher utility on the expected value of the gamble than on taking the gamble itself. Conversely, a risk lover prefers to take the gamble rather than settle for a payoff equal to the expected value of that gamble. The implication of the expected utility hypothesis, therefore, is that consumers and firms seek to maximize the expectation of utility rather than monetary values alone. Since utility functions are subjective, different firms and people can approach any given risky event with quite different valuations. For example, a corporation’s board of directors might be more risk loving than its shareholders and, therefore, would evaluate the choice of corporate transactions and investments quite differently even when all monetary values are known by all parties.
Preferences may also be affected by the status of an item. There is, for example, a difference between something possessed (i.e., with certainty) and something sought out (i.e., subject to uncertainty); thus, a seller may overvalue the item being sold relative to the item’s potential buyer. This endowment effect, first noted by Richard Thaler, is also predicted by the prospect theory of Daniel Kahneman and Amos Tversky. It helps explain risk aversion in the sense that the disutility of risking the loss of $1 is higher than the utility of winning $1. A classic example of this risk aversion comes from the famous St. Petersburg Paradox, in which a bet has an exponentially increasing payoff—for example, with a 50 percent chance to win $1, a 25 percent chance to win $2, a 12.5 percent chance to win $4, and so on. The expected value of this gamble is infinitely large. It could be expected, however, that no sensible person would pay a very large sum for the privilege of taking the gamble. The fact that the amount (if any) that a person would pay would obviously be very small relative to the expected payoff shows that individuals do account for risk and evaluate the utility derived from accepting or rejecting it. Risk loving may also be explained in terms of status. Individuals may be more apt to take a risk if they see no other way to improve a given situation. For example, patients risking their lives with experimental drugs demonstrate a choice in which the risk is perceived as commensurate with the gravity of their illnesses.
The von Neumann–Morgenstern utility function adds the dimension of risk assessment to the valuation of goods, services, and outcomes. As such, utility maximization is necessarily more subjective than when choices are subject to certainty.
Expected utility theory emerged as a by-product, an addition to game theory. In the second publication of his book (1947) as an introductory chapter preceding the description of game theory and its applications to economics, von Neumann and Morgenstern give a brief description of the main points economic theory, in which they propose to give an adequate mathematical toolkit based on game theory. It is here, in this chapter, supplementary to the general concept of the book, added only in second edition, the authors presented the main theses of their theory of expected utility. Von Neumann and Morgenstern note that the concept of rational behavior (maximizing utility or profit) underlying economic theory is insufficiently quantified. From Robinson, the common hero of the original marginalist models, “a participant in the economics of public exchange differs in that the result of his actions depends not only on them, but also on the actions of others. Each participant tries to maximize some function ... not all of whose elements are under his control. " In a situation of such uncertainty or risk, it is difficult to formulate a criterion
rational behavior. Von Neumann and Morgenstern moved from choosing between certain outcomes to choosing between lotteries, including several uncertain outcomes, and proved that the criterion rationality can be maximized here by maximizing the expected utility: rational an economic entity must choose a behavior option (lottery) that has the maximum value. Under some of the simplest axioms regarding the ordering of preferences, one can prove that the option chosen by the individual should have the highest expected utility value. The most important of the axioms is that preferences must be transitive: if A> B, and B> C, then A> C; any complex, multi-stage lottery should be decomposed into simple lotteries in in accordance with the rules of calculating probabilities; if A> B and B> C, then there must be a lottery with outcomes A and C, equal to guaranteed receipt of B. Thus, by arranging the options in accordance with the diminishing expected utility, we get
for a given individual (it is impossible to compare the expected utility of different individuals) the function the Neumann – Morgenstern utility. The concept and quantitative indicator of expected utility includes two main components: probability and utility. These components in different versions of expected utility theory different meanings were given. Let's consider them separately. As for the utility, first of all, it should be noted that the Neumann – Morgenstern theory breathed new life into the concept of cardinal utility. Approach from the standpoint of the theory of the expected utility allows us to make the concept of utility "operational" and to quantify it. Let the individual prefer good A to good B, and good B to good C (A> B> C). Let him be offered a choice between the lottery, in which there is an opportunity to choose the good A or the good C, and the reliable receipt of B. It is clear that if the probability of winning A is close to 1, our hero will choose the lottery. If the mentioned the probability is close to 0, a reliable receipt of B will be chosen. There is (in accordance with one of the Neumann-Morgenstern axioms) one probability of falling out A, in which the player is indifferent to the choice between a lottery or a guaranteed prize. However, it should be remembered that our decision only works in a risk situation. We do not have, for example, the possibility of asserting that in a situation of certainty, the difference between utility B and C will also be 2 times the difference between utility A and B. The fact is that the individual's attitude to reliable outcomes A, B, and C are inextricably intertwined with his attitude to risk. For example, if the individual is not very love risk, he can pay to avoid the lottery (insurance case). Besides, the value of utility follows from real choice, and not vice versa. This distinguishes the usefulness according to Neumann -Morgenstern from the neoclassical cardinalist concept of utility.