DECISIONS IN A STATE OF UNCERTAINTY - THE MATHEMATICS OF HUMAN BEHAVIOR - The Remarkable Role of Evolution in the Making of Mathematics - Mathematics and the Real World

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

CHAPTER VI. THE MATHEMATICS OF HUMAN BEHAVIOR

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.

Nevertheless, there is a reason for this tendency, and in my opinion it is inherent in the way we relate to numbers representing probabilities. In section 43 we discussed the principle that evolution led us to ignore events with low probabilities. That is entirely rational from an evolutionary perspective, and it helps the human species, as well as others, to survive in the evolutionary struggle. The difference of 20 percent between option 1 and option 2 causes most respondents not to risk the almost-certain win of three thousand dollars by taking an 80 percent chance of winning four thousand dollars. The risk of “only” 5 percent between options 3 and 4 seems reasonable because intuitively the result with a 5 percent probability can be ignored. Intuition does not grasp the mathematical fact that, from the aspect of decision making, the 5 percent are equivalent to the 20 percent in the other alternative.

Evolutionary rationality expressed by underestimating the importance of events with low probability keeps appearing. It is difficult to include all aspects of this attitude in the category of irrational behavior, but the discrepancy between it and the mathematical calculations is revealed time and time again. For example, someone is offered the choice between a trial in which he will have a 60 percent chance of success or a trial repeated independently five times (i.e., performed six times in all) in which his chance of success is 90 percent each time. If this choice takes place in a mathematical context, say in a lesson on probability theory, most of those present will carry out the calculation and will find that 0.9 to the power of 6 is smaller than 0.6 and would choose the first option of the single trial. If the situation and the probabilities are within the description of an event in which the required mathematical exercise is not emphasized, and the decision is made by “gut feeling” and not by calculation, the decision will tend markedly toward the repeated trials with a higher chance of success in each one. Likewise, if offered the choice of overcoming an obstacle several times, with a low probability of failure each time, or overcoming a more serious obstacle once with a higher risk of failure, most people would tend to go for the first option, independent of any mathematical calculation.

There is no discrepancy between the fact that there is a tendency to ignore events that have a low probability and the fact that people continue to buy tickets in the national lottery or bet on the results of sporting events, knowing that their chances of winning are very small, even minimal. The explanation is in the attitude to lotteries in general, which is likely to reflect the tendency to seek risk or to be risk averse, and these can appear in different forms in different circumstances. A reasonable person may buy a lottery ticket because the expected loss is small relative to the high prize he could win, and also because the good feeling he gets from the possibility of winning, until the lucky numbers are announced. The same person would not risk all his property on such a lottery, even if the prize he could win was hundreds of thousands times larger. Being risk averse or risk prone does not contradict rational behavior.

The discrepancy between behavior and probabilistic assessments on the one hand and mathematical logic on the other comes to light also in other types of situations. In experiments carried out by Kahneman and Tversky in the 1980s they discovered the following. A number of people were asked, each one alone, to assess what is the probability that in 2018 Russia would break off diplomatic relations with the United States. At the same time a group of other people was asked, again separately, to assess the chances that in 2018 Russia would be in conflict with the Ukraine, the United States would intervene, and as a result Russia would break off diplomatic relations with the United States. Mathematical logic says that the probability of the first scenario is higher than that of the second. The empirical evidence was that people thought that the second scenario had a higher probability. The explanation lies in the way people arrive at their assessments. The second group was presented with a plausible, realistic scenario that would lead to Russia breaking off diplomatic relations with the United States, whereas the first did not present such a clear-cut scenario. The more realistic-sounding possibility overcame logic. Evolutionary rationality overcame rationality. Kahneman and Tversky called the mechanisms that result in these deviations the availability heuristic and the representativeness heuristic. The deviations that these cause can also be seen in other areas.

We will give few examples in which the relation to uncertainty is affected by preconceptions and not necessarily by logical and mathematical arguments. We saw earlier that in biblical times and in ancient Greece it was recommended to use, and indeed use was made of, random events to arrive at just or fair outcomes, just at least in the probabilistic sense. We have no way of knowing how the public viewed the fairness of the method. In modern times such methods are not always treated with the proper academic equanimity.

A lottery as a fair instrument of recruitment to the army has been used by several countries for a long time, including the United States. In 1970, in the call-up for the Vietnam War, there was a central lottery of birth dates, and it was decided that young people within a given age range whose birth dates were drawn in the lottery would be drafted into the army. The outcome led to a strong opposition to the whole idea of the draft. The system was that chits representing all the possible dates were placed into a container. The numbers from 1 to 365 were placed in a second container. Then, one by one, a chit with a date was drawn from the first container, and a chit with a number from the second. The number determined the order of the draft. For example, those born on the date drawn from the first container together with the number 8 from the second were in eighth place in the list for the draft, and so on, until the required number of recruits was reached. Those with a high number were apparently not drafted at all. The reason for not using just one container and say drawing only dates of birth and agreeing that the order in which they were drawn would determine the order of call-up was, it seems, the same reason mentioned in the Talmud and its interpreters (see section 36), that it would be easier to establish the fairness of the method. The outcome of the 1970 lottery provided the basis for critics of the system. We show below the numbers of the average position in the list of recruits of those born in each month, as came up in the draw (the data are taken from an article by Stephen Fienberg in Science, January 1971).

January

201.2

February

203.0

March

225.8

April

203.7

May

208.0

June

195.7

July

181.5

August

173.5

September

157.3

October

182.5

November

148.7

December

121.5

Thus, for instance, those born in January were, on average, 201.2 in line for the draft while the average place for those born in December was only 121.5. It paid to be born in the first half of the year. The average position over the whole year is about half the numbers of days in a year, that is, 183. The list clearly shows that the average position in the list for the draft of those born in the months August to December, about 157, is significantly lower than the position of those born in the first months of the year, about 203; that means that those born in August to December were more likely to be drafted.

Does this mean that the procedure is unfair? Not necessarily. The bias resulted from the details of the way the chits were put into the container, and we will not go into that now. However, before the drawing started, all months had exactly the same chance of being drawn. The mathematical claim of fairness and equality did not help. The results gave rise to criticism of the randomness of the system of drafting, and, although there was an attempt to change it in the 1971 draft, the strong opposition to the system itself was one of the factors that led to the canceling of compulsory drafting by means of a lottery and the formation of an army based on professional soldiers.

Another example is related to polling methods. It is very difficult to collect reliable statistics about, say, drug users, alcoholics, tax evaders, and so on. People do not trust that their answers will be kept in secret. Statisticians, among them, Tore Dalenius (1917–2002) of Brown University and the University of Stockholm, who was one of the pioneers of considering psychological aspects when designing public polls, suggested a way of overcoming this distrust. Assume that the question is “Are you a tax evader?” Before responding, the subject should secretly perform a personal drawing between, say, red and black with probabilities of, say, 51 and 49 percent. If red comes up, he gives the true answer. If black is the outcome, he lies. No personal information can be drawn—even by the tax authorities—concerning possible tax evasion. But for large populations, the small difference between 51 and 49 is enough to get reliable statistics. The statistical rationality did not convince the public. Those who were asked to take part in such polls did not believe in the method.

Here is another example of the lack of understanding of decisions based on randomness. In elections to Israel's parliament, the Knesset, voting takes place by placing a chit with the name of the party the voter chooses into an envelope that he then seals. Placing more than one chit in the envelope, even if they are for the same party, let alone if they are for different parties, leads to that envelope being disqualified and the vote is a spoiled vote. Yet sometimes, for reasons we do not need to detail here, it is not easy for an Israeli voter to decide which of the parties is worthy of his vote. Moreover, if the system was such that every voter put five chits into the envelope, some voters would put five chits into the envelope all for the same party, others might put three for one party and one each for two other parties, in line with their opinions and leanings. This would be logical also from a conceptual viewpoint. The elections determine the composition of the Knesset, and most of the voters do not put all their faith in one party, or they would prefer to give equal power to two parties. Such splitting of votes, however, is not allowed. A few elections ago I proposed the following procedure to those who would have liked to split their votes. Assume that if you could split your vote you would give two-thirds to one party and one-third to another. Take two chits of your preferred party and one chit of the party of your second choice, choose one of the three randomly, say mixing them up and selecting one behind your back, and put it in the envelope without looking at it. Then throw the other two away, also without looking at them. That way you not only divided your vote in the proportion you wanted, but the only information you have at the end of the process is about the division you chose. If you are asked which party you voted for, you can answer only with that division, that is, the division of the probabilities. Subjectively, you divided your vote, and the subjective aspect is what is important, because to a great extent the reason for going to the polling booth is the subjective feeling. If you were to measure the chance that your vote has any effect against the trouble of going to cast your vote, you probably would not bother to go at all. A talented journalist named Yivsam Azgad heard about my idea and published an article on it in the Ha’aretz newspaper. The day before the elections I was actually invited to a television studio for a short interview, in which I explained and illustrated my idea (at that time politicians could not be interviewed on television on the day before the elections, so television had to make do with mathematicians and the like). The reactions were astonishing. On the positive side I received compliments on the proposed system, and many of my friends and also people I did not know beforehand told me that they had adopted my method. On the other hand, there were also many who opposed it. A listener on a radio program phoned the studio and complained in anger, “And what if I draw the chit of the party I hate?” (he even gave the name of that party). He clearly did not quite understand the proposed system, in particular that you, the voter, determine the weights according to which you divide your choices, and you would certainly not include in the drawing the chit of the party you loathe. For that caller, a lottery is a lottery, and you never know what will come up in a lottery. Another acquaintance, an activist in a political party, was also opposed to the system and angrily accused me of “wanting to let a lottery decide who our leaders will be.” For her, any decision based on randomness is unsuitable.