Prospect Theory

Yesterday, I talked about expected utility theory (EUT). Today I will write about one on the major departures from EUT: Prospect Theory. This theory was developed by Nobel laureate Daniel Kahneman and Amos Tversky (Econometrica 1979). The four key characteristics of prospect theory are:

1. Individuals use decision weights, π(p), rather than probabilities, p, when making decisions.
2. The value function is defined as deviations from a reference point. Thus, earning $100,000 this year is perceived differently for an individual earning $50,000 last year compared to one making $1 million last year.
3. Individuals are risk averse with respect to gains but risk loving with respect to losses. This implies that the value function is concave for gains, but convex for losses.
4. The value function is steeper for losses than for gains . This means that losses of $1000, hurt more than gains of $1000.

Why is there a need for prospect theory?

Much experimental evidence has shown that EUT does not only hold. Consider the Allais paradox. Which of the following lotteries would you choose:

• A: ($1m, 1) vs. B: ($1m, .89; 0, .01; $5m .10)
• C: (0, .89; $1m, .11) vs. D: (0,.90; $5m, .10)

Most people choose A and D. Yet it can be shown that under the EUT, these lotteries are mathematically equivalent and this leads to a preference reversal.

Another example is the following:

• A: (6000, .45; 0, .55) vs. B: (3000, .90; 0, .10)
• C: (6000, .001; 0, .999) vs. D: (3000,.02; 0, .998)

The majority of people surveyed here chose B and C. Again these A and C are two are equivalent lotteries as are B and D. The probability of winning in the latter pair is simply dived by 450. Thus, we see a preference reversal according to traditional EUT theory.

Also we see that people treat losses and gains differently.

• A: (6000, .25; 0, .75) vs. B: (4000, .25; 2000, .25; 0, .5)
• C: (-6000, .25; 0, .75) vs. D: (-4000, .25; -2000, .25; 0, .5)

Kahneman and Tversky find that 82% of people choose B over A, but 70% of people choose C over D. This implies that individuals are risk averse with respect to gains and risk loving with respect to losses.

Editing

Before utility functions are evaluated, Kahneman and Tversky say that choices are “edited.” The reason for this is 1) it helps to prevent obvious contradictions in which would occur if editing was not included, and 2) the editing process may more accurately reflect the process by which individuals make choices. Here are some of the ways which individuals edit:

• Coding: Outcomes are perceived as gains and losses with respect to a reference point. The reference point may be the status quo or it may be an expectation. For instance, if my monthly before-tax earnings are $2000 but my after-tax earnings are $1500, I may perceive this as a $500 loss from my expected income rather than a simple $1500 gain.
• Combination: Identical outcomes are simplified so that (100, .25; 100, .25; 0 .50) = (100, .50; 0 .50).
• Segregation: Risky components are separated from non-risky components. For instance, (500, .7; 100, .3) is decomposed into a sure gain of $100 and a lottery of (400, .7; 0, .3).
• Cancellation. This implies that when lotteries are compared, common outcomes are eliminated. “For example, the choice between (200, .20; 100, .50; -50, .30) and (200, .20; 150, .50; -100, .30) can be reduced by cancellation to a choice between (100, .50;-50, .30) and (150, .50; -100, .30).”

Evaluation

After editing, individuals make decisions according to the following utility function:

• Σ π(p)v(x)

Tversky and Kahneman (J Risk Uncert 1992) show that empirically, the function gives an inverted-S shape (see graph). “…for both positive and negative prospects, people overweight low probabilities and underweight moderate and high probabilities. As a consequence, people are relatively insensitive to probability difference in the middle of the range.” As mentioned earlier, the value x is evaluated with respect to a reference point. It is either a gain or a loss. For gains, v”<0 but for losses v”>0. Also, because the utility function is steeper for losses than gains, v(x)<-v(-x). For a graphical display of a prospect theory value function, see a picture from the U of RI economics website.

Another lottery experiment to support prospect theory is the following:

• A: (5000, .001; 0, .999) vs. B: 5
• C: (-5000, .001; 0, .999) vs. D: -5

Of those surveyed, 72% of people choose A over B, but 83% of people chose C over D. This would seem to imply that individuals are risk loving for gains and risk averse to losses. However, the bulk of the evidence has shown that this is not the case. As mentioned above, it seems more likely that people tend to overweight low probability events and underweight high probability events.

While Prospect Theory still is not as popular in mainstream economics as Expected Utility Theory. This is likely due to the added data needed regarding how an individual is editing, and what the individual’s reference point would be. Further, one wonders whether or not individuals become more ‘rational’ in the expected utility sense if they receive feedback from repeated games. Nevertheless, Prospect Theory seems to very accurately explain many of the findings in experimental economics and more work in this area is needed.

• Daniel Kahneman; Amos Tversky (1979) “Prospect Theory: An Analysis of Decision under Risk” Econometrica, Vol. 47, No. 2. Mar., pp. 263-292.
• Amos Tversky; Daniel Kahneman; (1992) ” Advances in prospect theory: Cumulative representation of uncertainty” Journal of Risk and Uncertainty, Volume 5, Number 4 / October, 1992, pp. 297-323.