Decision Making In Prisoner’s Dilemma

4. Prospect theory

Prospect theory emerged as a critique of classical expected-utility theory, experimentally disproving some basic tenets the classical theory held. Here are some examples:

(a) Expected-utility economist P. A. Samuelson in his 1963 article “showed that expected-utility theory implies that if […] a person turns down a particular gamble, then he or she should also turn down an offer to play n > 1 of those gambles” (Rabin, 2000, p. 206). Rabin demonstrates that this is not empirically true. Sure, you can turn down a single 50-50 lose-$100-or-gain-$200 bet (paragraph (3) below explains why you may actually do that), but you would be “legally insane” (as Rabin puts it), if you rejected an aggregated gamble consisting of 100 consecutive 50-50 lose-$100-or-gain-$200 bets. This aggregated gamble has expected yield of $5000 (50 times $200 minus 50 times $100), and there is only a 1 : 2300 chance of possibly losing any money in the end (Rabin, 2000).
It is, in fact, true that people sometimes, and often quite systematically, fail to aggregate possible gains and especially possible losses (in exactly the same way Samuelson did in his 1963 paper). Kahneman & Lovallo (1993) show this in their article. They analyze the costs of isolation (i. e. of not aggregating) vs. the profit of adopting the broader frame and aggregating losses over series of gambles (or decisions in general). “Overly cautious attitudes to risk result from a failure to appreciate the effects of statistical aggregation in mitigating relative risk” (Kahneman & Lovallo, 1993, p. 393). Still, we can see there is a psychological difference between accepting Samuelson’s bet on each of 100 days and accepting them all at the same time (Rabin, 2000, p. 207) – the longer the time, the more difficult psychologically for investors/decision makers to aggregate.
(b) Benartzi & Thaler (1995) managed to explain the puzzle of excessively high equity premium by investors’ myopic loss aversion (the higher the risk of loss in an investment – loans, stocks, bonds – the higher the equity premium). This myopic loss aversion arises partly from investors’ failure to aggregate risk, again. Investors evaluate the possible loss of a given asset too often, say, on a yearly basis, even though they are not selling and do not intend to sell their assets in such a short time. If the risk-evaluation period of a given investment was longer, it would mean the possible losses were aggregated, and that would mean the premium would be lower. The authors calculated the equity premiums for five-, ten-, and twenty-year evaluation periods. The numbers obtained were 3%, 2% and 1,4% respectively (the actual equity premium is 6,5% – roughly the same for the past 200 years).
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Prospect theory was introduced by Daniel Kahneman and Amos Tversky in the late 70’s. They and other researchers that joined this approach (and extended it) came with several empirically demonstrable characteristics of human decision making. Prospect theory, to put it shortly, demonstrated that people do not behave as classical expected-utility theory maximizers. I am presenting below a few examples of input-sensitive decision making from prospect theory research: sensitivity to degrees of likelihood of events (paragraphs (1) and (2) below) and to certainty (4), to pending gains/losses (paragraphs (2) and (3)), and to assumed predictability by decision makers’ own expertise (5).
(1) Decision makers tend to overestimate low probabilities and underestimate high probabilities (see Graph 4.1 - adapted from Kahneman & Tversky, 1979, p. 37). In other words, the input of low probability events (like winning in a lottery) bear upon decision makers more strongly than would be “rational” (according to classical expected-utility theory, i. e. with respect to realistically expected gains/losses), and vice versa for inputs of high probability. See for example Kahneman & Tversky, 1979; Tversky & Kahneman, 1992; Fox & Tversky, 1998. Slovic et al. (1980) presents empirical data demonstrating that people overestimate probability of rare causes of death, such as botulism or venomous bite, and underestimate the probability of common causes of death, such as diabetes or stroke – although their estimate might be partly biased by relatively greater subjective availability (see section 5.2) of rare and dramatic causes of death due to differential newspaper/TV coverage.

Graph 4.1: Function f of decision weight dependent on probability of outcomes

We can understand overestimating low probabilities (and underestimating high probabilities) as a simple frequency-dependent heuristic, that results in people’s seeking of rare, unique, hard-to-get things. And we can conceive of an evolutionary argument why this heuristic evolved. Low frequency things are by definition harder to get than high frequency things of equivalent utility value, so natural selection should select against seeking low frequency things. But sexual selection could act against the pressure of natural selection (see Fisher, 2000; Trivers, 1972; Miller, 2001).

Sexual selection may favor seeking low frequency things. With what result? With the result of higher male mortality (Trivers, 1972), because it is typically the males (or more generally the sex with lower parental investment) that are selected by sexual selection of the females (or the sex with higher parental investment), and hence also suffer the repercussions of natural selection pressure. And why would, then, sexual selection act on this trait (seeking low probability things)? Exactly because it is costly and so it can serve as a fitness indicator of potential mate’s genes: If potential mates could survive using hard-to-get things (low frequency things), they are probably fitter then other potential mates struggling along with easy-to get things (high frequency things of equivalent practical utility). This is also called handicap principle (Zahavi, 1975; see also Ridley, 1993). (Interestingly, altruism – an important factor in cooperative behavior, see section 6.3 paragraphs (1), (2), (3) – could possibly represent a handicap, as understood by Zahavi, see Roberts, 1998.)
The critical question is, how would you induce males to actually seek low frequency hard-to-get things? You can (a) make them value low frequency things more (and we can no doubt find evidence for that), or you can (b) make them overestimate low probabilities of success and underestimate high probabilities of success (which would bias their behavior into seeking low probability things), or you can (c) put in both these heuristics. Of course these are just hypotheses that would need refining and empirical testing.
You might for example think that in getting people to seek low frequency things it suffices to make them value low frequency things relatively higher (a). Now, look at the decision between “low-frequency” and “high-frequency” strategy through the eyes of a male, who at a certain moment only knows this – that hard-to-get things are valuable (a) – and who also knows that they are low frequency and by definition hard-to-get, and who might therefore calculate that it has no sense trying (i. e. trying the “low-frequency” strategy), since the expected utility of hard-to-get things obtained through energy output O is at best the same (due to their higher value and lower availability) as the expected utility of easy-to-get things obtained through the same energy output O.
Now, if you add heuristic (b) into this man’s mental repertory, he will judge that the net expected utility of hard-to-get things obtained through energy output O is greater (due to their higher value and fair availability) than the expected utility of easy-to-get things obtained through energy output O. This subhypothesis predicts that in case (a), but not in case (b) lower availability offsets higher value. That of course needs empirical testing.
(2) Decision makers are risk seeking for gains and risk aversive for losses of low probability, and risk aversive for gains and risk seeking for losses of high probability. In one study (Kahneman & Tversky, 1979, pp. 20-22; see also Tversky & Kahneman, 1992) 86% of decision makers preferred a 90% probability gain of 3000 over a 45% probability gain of 6000 (= risk aversion for gains of high probability; the outcomes of the two bets are mathematically identical), while 92% of decision makers preferred a 45% probability loss of 6000 over a 90% probability gain of 3000 (= risk seeking for losses of high probability; note that the prospects of this bet are identical and represent a mirror image of the prospects in the first bet). And 70% of subjects preferred 0,1% chance to gain 6000 over 0,2% chance to gain 3000 (= risk seeking for gains of low probability), while 70% of subjects chose a 0,2% probability loss of 3000 over 0,1% probability loss of 6000 (= risk aversion for losses of low probability).
Some studies (e. g. Fagley & Miller, 1997) showed that people are even more risk seeking when dealing with human life or death situations when losses of high probability are pending. Wang (1996a, b); and Wang et al. (2001) found that this effect is even more pronounced in small, and/or kin-related groups. The differential risk seeking tendency for losses of high probability can not be explained by prospect theory itself, as Fagley & Miller (1997) noted – but relatively higher risk seeking for losses in small kin-related groups is consistent with (and possibly explained by) evolutionary theory that predicts that altruistic action towards kin-related organisms might increase inclusive fitness of the altruist, even though it lowers his own individual fitness (as any risky action by definition does) (see also section 6.3, paragraph (1)). In other words, additional risk taken by the “altruist” in kin-related group is off-set by an increase of his inclusive fitness.
(3) The value function is concave for gains and convex and steeper for losses (see Graph 4.2 – adapted from Kahneman & Tversky, 1979, p. 34). That means that losses weigh more than gains – this is called loss aversion (loss aversion does not contradict risk seeking, actually risk seeking for losses of high probability is direct product of loss aversion). See also Tversky & Kahneman, 1991; Kahneman et al., 1991; Johnson et al., 1993; Quattrone & Tversky, 1988, with interesting examples – pp. 453-461.
Graph 4.2: A hypothetical utility function f concave for gains and convex for losses

The concavity/convexity of utility function applies, other things being equal. The situation, however, may be different in some special cases: “The utility function of an individual who needs $60.000 to purchase a house may reveal an exceptionally steep rise near the critical value”, Kahneman & Tversky, 1979, p. 33. In this case there are circumstances that transform an uncompleted sum of parts into a complete whole, which is formally analogical to the “certainty effect” discussed in the next paragraph.

(4) People value (overvalue) certainty. This “certainty effect” causes preference for a sure gain over a larger gain that is merely probable. For example in one study 80% people chose a sure gain of 3000 and only 20% people chose a 80% probable gain of 4000 (Kahneman & Tversky, 1979, p. 22). The defining quality of certainty effect is that the change of probability from uncertainty to certainty (0% or 100% probability) has greater impact on decision makers (represents a disproportionately stronger input) than the same change of probability that only makes a possibility more or less probable.
(5) People’s decision making (willingness to bet) when it comes to betting on uncertain events depends not only on the degree of uncertainty of the events but also on its source. Fox & Tversky (1998) showed that experts in a field (people with interest in basketball) preferred betting on basketball to betting on another chance event with matched probability of outcomes (the probability of the outcomes of basketball matches was first estimated by the subjects). The classical theory predicts that there will be no preference for either type of bet, since both yield identical outcomes.