Prospect Theory: Gains, Losses, and “Reflection Effects”

A second part of prospect theory pertains to the notion of a reference point from which gains and losses are evaluated. In experiments with money payments, the most obvious candidate for the reference point is the current level of wealth, which includes earnings up to the present. The reference point is the basis for the notion of “loss aversion,” which implies that losses are given more weight in

choices with outcomes that involve both gains and losses. A second property of the reference point is known as the reflection effect, which pertains to cases where positive payoffs are multiplied by minus one in a manner that “reflects” them around 0. The reflection effect postulates that the risk aversion exhibited by choices when all outcomes are gains will be transformed into a preference for risk when all outcomes are losses.

A comparison of behavior in the gain and loss domains is difficult in a laboratory experiment for a number of reasons. First, the notion of a reference point is not precisely defined. For example, would “paper” earnings recorded up to the present point (but not actually paid) in cash be factored into the current wealth position? A second problem is that human subjects committees do not allow researchers to collect losses from subjects in an experiment, who cannot walk out with less money that they started with. One solution is to give people an initial stake of cash before they face losses, but it is not clear that this process will really change a person’s reference point. The notion that an initial stake is treated differently than hard-earned cash is called the “house-money effect.” There is some evidence that gifts (e.g. candy) tend to make people more willing to take risks in some contexts and less willing in others (Arkes et al. 1988, 1994). It is at least possible that the warm glow of a house-money effect may cause people to appear risk seeking for losses when this may not ordinarily be the case with earned cash. One solution is to give people identical stakes before both gain and loss treatments, which holds the house-money effect constant. And making people earn the initial stake through a series of experimental tasks is probably more likely to change the reference point.

Kahneman and Tversky (1979) presented strong experimental evidence for a reflection effect. The design involved taking all gains in a choice pair like those in Table 7.1 and reflecting them around zero to get losses, as in Table 7.2. Now the “safe” lottery, S*, involves a sure loss, whereas the risky lottery, R*, may yield a worse loss or no loss at all. The choice pattern in Table 7.1, with 80% safe choices, is reversed in Table 7.2, with only 8% safe choices.

Table 7.2. A Reflection Effect Experiment (Kahneman and Tversky, 1979)

Lottery S* (selected by 8%) Lottery R* (selected by 92%) minus 3,000 with probability 1.0 minus 4,000 with probability 0.8

0 with probability 0.2

Holt and Laury (2002) evaluate the extent of the hypothetical bias in a reflection experiment. They took a menu of paired lottery choices similar to that in Table 4.1 and reflected all payoffs around 0. Recall that this menu has safe lotteries on one side and risky lotteries on the other, and that the probability of the higher payoff number increases as one moves down the menu. Risk aversion is inferred by looking at the number of safe choices relative to the number of safe choices that would be made by a risk-neutral person (in this case 5). All participants made decisions in both the gains menu and in the losses menu, with the order of menu presentation alternated in half of the sessions. In their hypothetical payoff treatment, subjects were paid a fixed amount $45 in exchange for participating in a series of tasks (search, public goods) in a different experiment, and afterwards they were asked to indicate their decisions for the lottery choice menus with the understanding that all gains and losses would be hypothetical. When all payoffs were hypothetical gains, about half of the subjects were risk averse, and slightly more than 50% of those who showed risk aversion for gains were risk seeking for losses. The modal pattern in this treatment was reflection, although other patterns (e.g. risk aversion for gains and losses) were also observed with some frequency. The choice frequencies for the hypothetical choices are shown in the left panel of Figure 7.1. The modal pattern of reflection is represented by the tall spike in the back-right corner of the left panel.

The real-incentive treatments for gains and losses were run in a parallel manner with the same choice menus. Participants were allowed to build up earnings of about $45 in a different experiment using the same tasks used under the hypothetical treatment. In contrast with earlier results, the modal pattern of behavior with real incentives did not involve reflection. The most common pattern was for people to exhibit risk aversion for both gains and losses. The real-payoff choice frequencies are shown in the right panel of Figure 7.1. The modal pattern of risk aversion in both cases is represented by the spike in the back-left side of this panel. There is a little more risk aversion with real payoffs than with hypothetical payoffs; sixty percent of the subjects exhibit risk aversion in the gain condition, and of these only about a fifth are risk seeking for losses. The rate of reflection with real payoffs is less than half of the reflection rate observed with hypothetical payoffs.

Despite the absence of a clear reflection effect, there is some evidence that gains and losses are treated differently. On average, people tended to be essentially risk neutral in the loss domain, but they were generally risk averse in the gain domain. This result provides some support for the notion of a reference point, around which gains and losses are evaluated, which suggests that laboratory data should be analyzed using utility as a function of earnings (gains and losses), not final wealth. In other words, if the net worth of a person’s assets is w, and if a decision may produce earnings or losses of x, then the analysis of expected utility

(with or without the probability weighting of prospect theory) should be expressed in terms of U(x), not U(w+x).

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Risk Averse

Risk Seeking

Risk Averse Risk Seeking

Hypothetical Losses

Hypothetcal Gains

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Risk Averse

Risk Seeking

Risk Averse Risk Seeking

Real Losses

Real Gains

Figure 7.1. Inferred Risk Aversion for Hypothetical Payoffs (Left) and Real Payoffs (Right) Source: Holt and Laury (2002)

The absence of a clear reflection effect in the Holt and Laury data is a little surprising given the results of several other studies that found reflection with real money incentives (Camerer, 1989; Battalio et al. 1990). One difference is that instead of holding initial wealth constant in both treatments (at a level high enough to cover losses), these studies provided a high initial stake in the loss treatment, so the final wealth position is constant across treatments. For example, a lottery over gains of 4,000 and 0 could be replaced with an initial payoff of 4,000 and a choice involving losses of 4,000 and 0. Each presentation or “frame”

provides the same possible final wealth positions (0 or 4,000), but the framing is in terms of gains in one treatment and in terms of losses in the other. A setup like this is exactly what is needed to document a “framing effect.” Such an effect is present since both studies report a tendency for subjects to be risk averse in the gain frame and risk seeking in the loss frame. Whether these results indicate a reflection effect is less clear, since the higher stake provided in the loss treatment may itself have induced more risk seeking behavior, just as gifts of candy and money tend to increase risk seeking in experiments reported by psychologists.

sense that people can see past gains and losses and focus on the variable that determines consumption opportunities (final wealth). The Camerer (1989) and Battalio et al. (1989) experiments provide strong evidence that decisions are framed in terms of gains and losses, and that people do not “integrate” gains and losses into a final asset position. Indeed, there is little if any experimental evidence for such “asset integration.” Rabin (2000) and Rabin and Thaler (2001) also provide a theoretical argument against the use of expected utility as a function of final wealth; the argument being that the risk aversion needed to explain choices involving small amounts of money implies absurd levels of risk aversion for choices involving large amounts of money. Most analyses of risk aversion in laboratory experiments are, in fact, already done in terms of gains and losses (Binswanger, 1980; Kachelmeyer and Shehata, 1992; Goeree, Holt, and Palfrey, 2002, 2003).

Even if expected utility is modeled in terms of gains and losses, there is the issue of whether to incorporate other elements like nonlinear probability weighting that represents systematic misperceptions of probabilities. As noted above, the evidence on this method is mixed, as is the evidence for the reflection effect. Some, like Camerer (1995), have urged economists to give up on expected utility theory in favor of prospect theory and other alternatives. More recently, Rabin and Thaler (2001) have expressed the hope that they have written the final paper that discusses the expected utility hypothesis, referring to it as the “hypothesis” with the same tone that is sometimes used in talking about an ex-spouse. Other economists like Hey (1995) maintain that the expected-utility model outperforms the alternatives, especially when decision errors are explicitly modeled in the process of estimation. In spite of this controversy, expected utility continues to be widely used, either implicitly by assuming risk neutrality or explicitly by modeling risk aversion in terms of either gains and losses or in terms of final wealth.

Some may find the mixed evidence on some of these issues to be worrisome, but to an experimentalist it provides an exciting area for further research. For example, there remains a lot of work to be done in terms of finding out how people behave in high-stakes situations. One way to run such experiments is to go to countries where using high incentives would not be so expensive. For example, Binswanger (1980) studied the choices of farmers in Bangladesh when the prize amounts sometimes involved more than a month’s salary. (He observed considerable risk aversion, which was more pronounced with very high stakes.) Similarly, Kachelmeier and Shehata (1992) performed lottery choice experiments in rural China. They found that the method of asking the question has a large impact on the way people value lotteries. For example, if you ask for a selling price (the least amount of money you would accept to sell the lottery), people tend to give a high answer, which would seem to indicate a high

value for the risky lottery, and hence a preference for risk. But if you ask them for the most they would be willing to pay for a risky lottery, they tend to give a much lower number, which would seem to indicate risk aversion. The incentive structure was such that the optimal decision was to provide a “true” money value in both treatments (much as the instructions for the previous chapter provided an incentive for people to tell the truth about their probabilities in a Bayes’ rule experiment). It seems that people go into a bargaining mode when presented with a pricing task, demanding high selling prices and offering low buying prices. The nature of this “willingness-to-pay/willingness-to-accept bias” is not well known, at least beyond the simple bargaining mode intuition provided here. Nevertheless, it is important for policy makers to be aware of the WTP/WTA bias, since studies of non-market goods (like air and water quality) may have estimates of environmental benefits that vary by 100% depending on how the question is asked. Given the strong nature of this WTP/WTA bias, it is usually advisable to avoid using pricing tasks to elicit valuations. For more discussion of this topic, see Shogren, et al. (1994).

A number of additional biases have been documented in the psychology literature on judgment and decision making. For example, there may be a tendency for people to be overconfident about their judgments in some contexts.

Some of the systematic types of judgmental errors will be discussed at length in later chapters, such as the “winner’s curse” in auctions for prizes of unknown value. For further discussion of these and other anomalies, see Camerer (1995).

Questions

1. Show that a risk-neutral person would prefer a 0.8 chance of winning 4,000 to a sure payment of 3,000, and that the same person would prefer a 0.2 chance of winning 4,000 to a 0.25 chance of winning 3,000.

2. If the probability of the 4,000 payoff for lottery R in Table 7.1 is replaced by 0.7, show that a risk neutral person would prefer Lottery S.