Mathematics and the Real World (43 page)

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).

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.

50. EVOLUTIONARY RATIONALITY

We will now expand the analysis of how people make decisions not necessarily in situations of randomness. Here too we see the effect of evolution on the way we decide and act and on how our brains think and analyze. The decisions are not always rational, but we can recognize evolutionary rationality. In this section we will again benefit from the contributions of
Amos Tversky and Daniel Kahneman and others, who identified the structures according to which the human mind works.

Here is a description of a tendency that Tversky and Kahneman called
anchoring
. A roulette wheel, with the numbers from 1 to 100, is spun in front of a group of people, and it is clear to everyone that the number the ball falls on is completely random. Say it falls on 80. The people are then asked to estimate whether the number of people, in millions, living in a certain geographic area, say West Java Province of Indonesia, is greater than or smaller than the number where the ball fell on the roulette wheel. In other words, is the population of West Java Province greater than or smaller than 80 million? We chose somewhere that the people may have heard of, but would certainly have no idea of the correct answer to the question. Their answers are then based on sensible estimates, or guesses, each with his or her own approach. At the second stage, the same people are asked to give a figure of what they thought the population of that province was. Here again the answer can be only an intelligent guesstimate. One surprising fact emerges. The numbers given in answer to the second question are affected by the random number that the people were exposed to in the spin of the roulette wheel. In other words, if the number that came up on the wheel was high, say 80 as in our example, the guesstimates of the population of West Java will be higher than if the number that came up was a low number, say 2. That is so despite the fact that all the participants saw and could confirm that the number that came up was completely random. The very fact that they were exposed to a particular number when first asked about the population resulted in that number affecting the figures they gave as their estimates of the population.