0.5 × v− 1000 + 0.5 × v0 > v− 500, implying risk-loving behavior. Though the monetary outcomes of the gamble are the same, framing the choice as a gain or a loss leads people to consider the outcomes in very different ways, leading to loss aversion.

Prospect Theory

Kahneman and Tversky proposed prospect theory as a potential solution to the reflection effect puzzle. The value function neatly explains how risk preferences could flip so dramatically over very similar gambles. However, simply using the value function does not address many of the other violations of expected utility theory we have observed in the previous chapter. To address each of these various effects, Kahneman and Tversky proposed a three-component version of prospect theory to deal with choice under risk. The three components are editing, probability weighting, and the value function.

Editing refers to a phase of the decision-making process in which the decision maker prepares to evaluate the decision. In this phase, the decision maker seeks to simplify her decision, thus making the evaluation of the potential prospects easier. The decision maker reorganizes the information in each choice and sometimes alters the information slightly to make the decision easier. The editing phase is intended to represent the actual motivations and deliberative process of the decision maker. Because the need to edit is based on the limited cognitive ability of the decision maker, this can be thought of as a procedurally rational model. The editing phase consists of six types of activities:

i.Coding: The decision maker determines a reference point (often her current level of wealth or some default outcome). Outcomes are then coded as either gains or losses with respect to the reference point.

ii.Combination: Probabilities associated with identical outcomes are combined. Thus, if a decision maker is told that a fair die will be rolled and $300 will be

given to her if an odd number results, she would combine the probability of rolling a 1, 3, and 5 or 16 + 16 + 16 = 12.

iii.Segregation: Certain outcomes are segregated from risky outcomes. Thus, a gamble that yields $40 with probability 0.5 and yields $65 with probability 0.5 is thought of as a sure gain of $40, with a 0.5 probability of gaining $25 and a 0.5 probability of gaining $0.

iv.Cancellation: When the choices being considered all have common components, those components are eliminated when making the choice. Suppose someone were choosing between Choice A, 0.25 probability of gaining $40, a 0.25 probability of gaining $60, and a 0.50 probability of gaining $0, and Choice B, 0.25 probability of gaining $40 and 0.75 probability of gaining $65. The first

outcome is cancelled because they are similar, yielding Choice A, 0.25 probability of gaining $60 and 0.50 probability of gaining $0, and Choice B, 0.75 probability of gaining $65.

v.Simplification: Probabilities and outcomes are rounded. Thus a probability of 0.49 would be thought of as 0.50. As well, an amount equal to $1,001 would be thought of as $1,000.

vi.Detection of dominance: The gambles are inspected to determine if one of the gambles first-order stochastically dominates the other. If one clearly dominates the other, then that gamble is selected.

Activities i, iii, iv, and v lead to specific violations of expected utility theory. For example, i enables the reflection effect by allowing the value function to be employed. Activity iv creates the isolation effect described earlier. In Example 10.2, applying iv would mean the common first-stage lottery would be eliminated from consideration, leading to the observed preference reversal. Because these editing activities represent ways actual decision makers reason through risky decisions, they are a potentially powerful tool.

There are two strong drawbacks to this model of prospect editing. First, the order in which the different activities are applied will affect the edited gambles to be evaluated. Applying i through vi in order might imply a different decision than if one were to apply them in the reverse order or in some other random order. Thus, as a model, editing does not make very specific predictions. For example, consider the choice between

Notice that if we first applied v and just rounded the probabilities, we would replace the gambles with

At that point we might apply vi, and recognize that Gamble G dominates Gamble H and choose G. Alternatively, if we first rounded just the amounts, we would arrive at

In this case we might notice that H dominates G and choose H. Similarly, applying each of the phases of editing in different orders can produce different outcomes. Moreover, it is clear from some of the anomalies in Chapter 9 that people do not always detect dominance. Hence it is possible to find convoluted gambles that can lead people to select dominated gambles.

A second drawback is that editing is difficult to apply to many real-world risky decisions. For example, if we were to consider stock investments, people are not told probabilities of specific outcomes, and the choices might not involve two or three potential gambles. Rather, outcomes could be anything in a continuous range of values, and the number of decisions could also be continuous (as in the number of shares to purchase). In this case, many of the editing activities become meaningless. Because of this, many economic applications of prospect theory have ignored all components of the editing phase aside from coding.

Once the editing phase has occurred, the decision maker then evaluates the gamble through the use of a probability-weighting function and a value function. Kahneman and Tversky originally proposed using a subadditive probability-weighting function as introduced in Chapter 9. Thus, the decision maker will maximize

where pi is the probability of outcome i and xi is outcome i. The use of the probabilityweighting function allows many of the same anomalies described in Chapter 9, including the certainty effect, the common-outcome effect, and Allais’ paradox. Alternatively, to eliminate the intransitive preferences implied by the subadditive probability-weighting function, it has become common to replace the probability-weighting function with a rank-dependent weighting function as described in Chapter 9. We refer to the use of prospect theory with rank-dependent weights as cumulative prospect theory.

Prospect Theory and Indifference Curves

Deriving indifference curves in the Marschak–Machina triangle is a little more difficult when dealing with probability weights. If we consider all possible gambles with outcomes of $100, $50, and $0, we are interested in all probability values that satisfy (see equation 9.3)

where x, y, and 1 − x − y are the probabilities of receiving $100, $0, and $50, respectively. For the sake of example, let us set v0 = 0 (this assumption can be made without losing any generality). Without specifying the probability weighting function, we cannot determine the shape of the implied curve in the triangle. However, we can approximate the shape locally by totally differentiating equation 10.3 with respect to x and y to obtain

where π′(.) is the derivative (or slope) of the probability weighting function. Thus, the shape of the indifference curve depends on the probabilities of each of the outcomes.

Recall that a steeper slope of the indifference curve reflects greater risk aversion. In this case, the slope is steeper when πy is small relative to π1 − x − y. Recall from Chapter 9 that the probability-weighting function is believed to be steeper for small and large probabilities than for those closer to 0.5 (see Figure 9.8). Thus, equation 10.4 tells us that people tend to be more risk averse when probabilities for the largest outcome are near 0.5 relative to the probabilities for either the middle or lower outcome. Alternatively, when the probability of the largest outcome is extreme relative to that of the middle outcome, the person behaves more risk loving. This risk-loving behavior when the probabilities of gains are relatively small is the primary contribution of the proba- bility-weighting function.

If we consider the shape of the indifference curves over losses of $100, $50, or $0, we must remember to reorder the axes. Now the y axis, which represents the probability of the superior outcome, is associated with a loss of $0, and the x axis, representing the probability of the most inferior outcome, is associated with a $100 loss. In this case, equation 10.4 can be rewritten as

In comparing equations 10.5 and 10.4, it is notable that the equations are almost exactly multiplicative inverses of one another (the difference being the negative values of the value function arguments). The gambler will display greater risk aversion if the probability of the lowest outcome is extreme relative to the probability of the middle outcome. Thus, the reference point of prospect theory implies that indifference curves over losses should be close to a reflection around the 45-degree line, as has been observed in laboratory experiments and as we displayed in figures 9.6 and 9.7. This provides some further evidence that people make decisions with respect to reference points and that they display loss-averse preferences.

EXAMPLE 10.3 Prospect Theory, Risk Aversion, Risk Loving, and

Observed Behavior

Using data from laboratory experiments with 25 participants, Tversky and Kahneman attempted to find parameters for cumulative prospect theory that could explain the results of several risky choices. Each participant was asked to choose among several gambles, each with a pair of possible outcomes. Though the data are limited, they were able to find parameters that provide a relatively close fit to the observed behavior. The estimated model is of the form

All parameter values are assumed to be positive. The rank-dependent weighting function in equation 10.7 allows the weighting to differ depending on whether the outcome associated with the probability is a gain or a loss. The parameters α and β in the value function (equation 10.6) determine the curvature of the value function (concavity or convexity). The parameter λ determines the degree to which the marginal utility of losses differs from that of gains at the reference point, with smaller positive values for λ corresponding to a wider difference in slopes. They find

α = 0.88

β= 0.88 λ = 2.25 γ = 0.61 δ = 0.69

Thus, the curvature of the value function for gains and losses is similar, though the difference in slope is severe (greater than a factor of 2).

Figure 10.5 pictures the estimated value function, having the familiar concave shape for gains, and convex for losses. The dominant feature, however, is the stark difference in slopes over gains and losses. Figure 10.6 displays the estimated probability-weighting

p(100) = 1

Probability of $100

Probability of $0

appears also to depend on the types of risky choices being presented. For example, when facing tradeoffs between gains and losses, prospect theory performs particularly well. Alternatively, when dealing with only gains and correlated gambles, regret theory appears to perform much better.

where $1.640.09 = $18.22 is the payoff if the bet is won with 9 percent making similar bets. Note that this is a riskier prospect. The probability of success is lower, and the

expected payoff is lower. However, there is a potentially higher payoff ($18.22 vs. $13.67) if the second-favored horse wins. The bettor, being risk averse over gains, would rather bet on the favored horse than on the second-favored horse.

Now suppose the bettor lost her first eight bets and was considering betting on the ninth and final race. Now instead of the $500 she arrived with, she only has $484. If she remembers her loss and does not adjust her reference point, she must win at least $16 more than she bet to return to her reference point of $500. A win of $11.67 results in a reduction of loss for a total loss of only $4.33. An additional loss of $2 results in a total loss of $18. The current valuation is v− 16 = − 25.81, and the bettor will be willing to take any bet that puts her in a better position. Thus, calculating as before we find

Now the bettor prefers betting on the riskier horse because it might allow the larger return and the possibility of returning to her reference point of wealth despite the added risk. Because the bettor now codes all outcomes in the loss domain, she behaves risk loving. As losses rack up, people become more and more willing to risk lower-probability wins for higher potential payouts. This pattern of race track betting becoming more and more risky throughout the day appears to be a relatively robust phenomenon, and it often results in lower returns on risky horses in the last races each day as more patrons start betting down the ticket.

EXAMPLE 10.5 Loss-Averse Contract Labor

Labor contracts are often written in terms of a base level of pay plus some bonus for good performance and potentially minus some penalties for poor performance. Such contracts seem to be a natural application of prospect theory.

Consider a contract between a firm that sells pizza and a worker. The worker can either choose to put in high effort or low effort. With high effort the probability of producing a high-quality pizza is equal to 1. With low effort, the probability of producing a high-quality pizza is equal to 0.5. The firm can sell high-quality pizza for a higher price than it can sell low-quality pizza. But suppose the firm cannot observe the level of effort the laborer gives, only the quality of the pizza. Further, suppose high effort costs the laborer the equivalent of 2 utils in effort, and low effort costs the laborer only 1 util. The firm is trying to determine how to structure the contract.