Mathematics and the Real World (42 page)

The model we have analyzed is only one of a multitude of possibilities suggested by game theory to reflect and analyze circumstances in which decisions must be made either in confrontation with or in cooperation with other players whose interests might be opposed to ours. One fundamental
model is of
games in cooperative form
. We shall not elaborate on this approach but will just say that here games are not determined by strategies that the players must take, but by the payoffs to coalitions that the players may form. Different formations of coalitions give rise to different allocations of, say, wealth. The theory studies, for instance, what would be stable outcomes, namely, outcomes that would not result in the disintegration of a given coalition; or how to measure the strength of a player based on the various coalitions he can join. The theory has been promoted to a great extent by John von Neumann and his colleague Oskar Morgenstern (1902–1977). They displayed much of the theory in the book
Theory of Games and Economic Behavior
, published in 1944. Many contributions to the theory of games in cooperative form have been added since then. It may be mentioned that von Neumann himself saw in this theory the right building block for a mathematical analysis of human behavior, parallel to infinitesimal calculus as the building block for the analysis of the physical world (in particular, he was not too fond of games in strategic form). It seems that in recent years games in strategic form were found to be more attractive; yet the cooperative game models, including market design relying on what cooperative game theory tells us (recall the aforementioned Nobel Prize to Alvin Roth) has not been abandoned and continues to be a fruitful field of study.

Mathematics plays a crucial role in analyzing the situations we addressed. Yet unlike the mathematics of the natural sciences, which can predict what is likely to happen, the mathematics of decision makers in a situation of conflict does not indicate what may occur, nor does it advise how to act. The current models are far from accurate depictions of the complexity of confrontations in real life. The product of this mathematics consists of proposed concepts or methods that will give the decision maker a better understanding of what confronts him. Sometimes mathematics may indicate a way to arrive at a criterion that expresses what is meant by “best for all.” In that case, mathematics may lead to the discovery of the best action. Mathematics may sometimes indicate what properties in the model limit the decision maker and, if possible, may lead to a change in the rules of the
game. The mathematical analysis shown above is based on the assumption that the participants are rational and that they are acting to achieve the best, the best according to their subjective preferences, of course. Whether decision makers, either individuals or those making cardinal decisions at the national level, do in fact act in such a rational manner is a serious question that we will address in the next sections.

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.

49. DECISIONS IN A STATE OF UNCERTAINTY

In this section we focus on the question of how people make decisions in states of uncertainty and on the fallacies we discussed in sections 40, 42, and 43. Also in this section we discuss the general question of how decisions are made intuitively as opposed to using mathematical analyses. It is perhaps not surprising to discover that the way decisions are made is not always consistent with what mathematics and orderly logical analysis would recommend. We will try to understand some of the reasons for that. Some questions will still remain unanswered, such as: Is it possible to develop mathematics that will describe human conduct that is not always rational? Can people be taught to behave rationally? Is it worthwhile trying to do so? Will decision makers behave rationally when faced with really important questions?

It is worth clarifying what we mean when we declare that certain conduct is irrational. Different people have different objectives. It would be wrong to state that someone who decides to inflict pain on himself or to lose money is
irrational. The desire to own assets is a subjective characteristic, and throwing money away is clearly rational for someone who detests money. Likewise, choosing to hurt himself is rational behavior if the person likes doing so. A common expression used to describe actual subjective preferences is
revealed preferences
, that is, preferences revealed by your actions. According to this approach, everything you do is rational from your point of view.

The irrationality we are trying to identify here is different. We find that sometimes someone's behavior deviates from basic assumptions or axioms that are not subjective and that the decision maker himself agrees are the guidelines for conduct. Nevertheless, he sometimes acts in a manner opposed to those axioms. Why? We claim that to a great extent the reason for this type of irrationality is evolutionary. The way we think and respond is molded by millions of years of evolution that brought us to ways of making decisions that in the terms we have just described are irrational. Yet underlying this behavior there is a logic, and I therefore suggest describing such conduct as reflecting
evolutionary rationality
. In many cases of irrational behavior, it is possible to discover the underlying evolutionary rationality.

Two of the major contributors to understanding human behavior in the context of uncertainty and decision making in general are Amos Tversky and his colleague Daniel Kahneman, who started their work at the Hebrew University of Jerusalem and continued at Stanford University and Princeton University. Kahneman was awarded the Nobel Prize in Economics in 2002, and his cooperation with Tversky was mentioned in the citation of the prize committee (Tversky died in 1996). We cannot summarize here the findings and explanations of what Tversky, Kahneman, Maya Bar-Hillel of the Hebrew University of Jerusalem, and others discovered, but we will cite some examples.

We will start with a result presented by the French economist Maurice Allais (1911–2010), Nobel Prize winner in Economics in 1988. Allais performed an experiment that can easily be reproduced, and I myself have used it in several of my lectures. The experiment reveals behavior that can readily be agreed deviates from basic principles of rationality. To
understand the deviation from rationality we will first discuss a somewhat-abstract example.

A person is asked to choose between two options:

(i)   To participate in a drawing for a gift A with a 75 percent probability or gift B with a 25 percent probability.

(ii)  To participate in a drawing for a gift A with a 75 percent probability or a gift C with a 25 percent probability.

In addition we know that the person prefers gift C to gift B. Which of the two options will he choose?

A rational person would opt for the second alternative, and most if not all people do that. Note that we did not say, for example, that C represents a more valuable asset or more money, as the decision maker's preference might be to lose money. We only said that in the order of priorities of the decision maker, C ranks higher than B; in other words, in a choice between B and C he would choose C. This was von Neumann and Morgenstern's second axiom in the previous section. If we improve the situation of the decision maker (improve in his eyes, that is) in one of the components of the lottery without worsening the payoff in any of the other components, a rational person will choose the improved payoff. Note that we do not discuss considerations about the expectation of the lottery. The gifts mentioned may have no numerical measurement.

The conduct that Allais found deviated from the rational choice we just described. His example was as follows.

A number of people were asked to choose between the following two options:

1. To participate in a drawing for three thousand dollars with a 100 percent probability of winning or for zero dollars with a 0 percent probability.
   
2. To participate in a drawing for four thousand dollars with an 80 percent probability of winning or for zero dollars with a 20 percent probability.
   

   The same people were then asked to choose between the following options:
   

3. To participate in a drawing for three thousand dollars with a 25 percent probability of winning or for zero dollars with a 75 percent probability.
   
4. To participate in a drawing for four thousand dollars with a 20 percent probability of winning or for zero dollars with an 80 percent probability.
   

(We put the probability of 100 percent in the first option to emphasize that the certainty of winning three thousand dollars plays no part in the example. Indeed, an event with zero probability can occur.) Most respondents chose the first option of the first two, and option 4 of the second two. In accordance with the revealed preferences principle, we do not determine which of the possibilities is better, and we certainly do not presume to state that anyone who chose differently than we would have chosen is not acting rationally.

This is where the surprise comes in. Those who chose options 1 and 4 were acting contrary to what we agreed above, which was that the better option, that is, option (ii) in the abstract example above prior to the presentation of this concrete example, is the rational choice. In the same way, those who chose options 2 and 3 in the above example are also deviating from the conclusion above, to which they agree, that option (ii) in the abstract example is the one to choose. We would emphasize that it is not irrational to choose either option 1 or option 2, but to choose option 1 and then option 4 reflects irrationality in the sense that we have indicated. The argument is that in a probabilistic sense option 3 consists of 25 percent of option 1 and another 75 percent chance of receiving zero, while option 4 consists of 25 percent of option 2 and 75 percent chance of receiving zero (it takes some calculation, which we leave out, to verify that). Thus, if you preferred option 4 to option 3 (and if in the abstract example you always opt for choice (ii)), you must prefer option 2 to option 1. One can try to excuse the deviation from rationality by claiming that the example is confusing, the calculations are complicated, and so it is difficult for the decision maker to explain his deviation from the principle to which he agrees
theoretically. This argument does not explain why participants consistently choose options 1 and 4.