Prospect Theory and Disappointment Aversion
The Allais paradox is a problem designed by French economist Maurice Allais in 1953 to show common inconsistencies between the predictions of expected utility theory and actual observed choices of individuals. Many theories have been proposed to solve the Allais paradox, among which the Prospect Theory and the Theory of Disappointment Aversion, and both will be described in this essay.
Prospect Theory
Guthrie, C. (2003) describes how the two original proponents of the Prospect Theory, Daniel Kahneman and Amos Tversky, would meet every mid-day for a long time, and spend their time inventing interesting sets of gambles and observing our their intuitive preferences. If they decided on an equivalent choice, they assumed that it was a characteristic of humankind and went on to investigate the theoretical ramifications. According to them, as reported by Guthrie, C. (2003), in some months they raced through more than twenty different theoretical formulations. In 1979, Kahneman and Tversky officially introduced their theory in their article “Prospect Theory: An Analysis of Decision Under Risk.” According to them, people make high-risk or uncertain decisions within the following four methods:
First, individuals make decisions relative to some reference point, usually the status quo. When selecting options that appear to be gains compared to that benchmark, a person makes risk-averse choices; faced against the appearance of losses, people have a tendency to make risk-seeking choices. For instance, individuals will pick a definite $1,000 prize over a 50% chance at receiving a $2,000 prize, but will choose to face a 50% chance of paying out a $2,000 fine over paying out a definite $1,000 fine. This outcome is contradictory with the premise of risk neutrality or risk aversion in the face of gains and losses. (Guthrie, C. 2003)
Second, individuals' risk preferences have a tendency to reverse when faced against low-probability outcomes. Individuals tend to make risk-seeking decisions when selecting between options that appear to be low-probability gains and risk-averse choices when facing perceived low-probability losses. When choosing between a certain $50 award and a 5% chance at winning a $1,000 prize, individuals make the risk-seeking option and opt for the gamble. When selecting between paying for a certain $50 good and dealing with a 5% possibility of paying out a $1,000 fine, individuals tend to make the risk-averse choice and opt to make the sure payment. Again, this empirical finding conflicts with rational choice theory. (Guthrie, C. 2003)
Third, individuals tend to overvalue losses against gains of the same magnitude. In other words, the pain of a given feels tougher than the satisfaction of an equivalent gain. A possible $1,000 loss will have higher weight on a decision maker than a prospective equivalent gain. (Guthrie, C. 2003)
Fourth, Guthrie, C. (2003) describes that individuals show a tendency to overvalue certainty. For example, most individuals prefer a definite award of a one-week paid vacation over a 50% possibility of winning a three-week longer vacation. However, when given an option between a 5% chance of winning the three-week vacation against a 10% chance at the week-long option, they choose the 5% gamble.
Other authors, such as Betts, S. and Zinaida, T. (2006), have looked for empirical evidence for the prospect theory predictions that customers are risk seeking when the dependability of a specific brand of a vehicle is below average, and risk averse if its reliability is above average. Their article concludes with evidence supporting the original theory of Kahneman and Tversky.
Prospect theory is not the only alternative explanation for risk and preferences. Schmidt, U. (1999) describes other models that explain apparently contradictory preference behavior, such as the dual expected utility, the semi-weighted utility, and, more importantly, the theory of disappointment aversion, our next topic.
Disappointment Aversion
Cheung, K., Chong. W., and S Yam, C. (2015) wrote a complete summary of the evolution of the Theory of Disappointment Aversion. They initially describe the 1985 work of Bell that can explain variants of the Allais Paradox. Bell, in fact, proposed a ‘modified expected utility’ to quantify human satisfaction under dissatisfactory conditions, with an increasing and concave function for the expected utility, and increasing reverse S-shaped function for dissatisfaction. With this breakthrough design, Bell explained the violations of independence axiom of the original utility theory.
Cheung, K., Chong. W., and S Yam, C. (2015) further explain Gul’s Disappointment Aversion Model, in which the author suggested decomposing lotteries into two factors, ‘elation’ and ‘disappointment.’ He then suggests a value function with a modified expected utility that introduced a ‘disappointment aversion coefficient' for the decision maker and a ‘disappointment averse certainty equivalent.’ The authors describe the main challenge of applying his theory in practice as the implicit nature for the definition of the certainty equivalent, making the applied mathematics difficult to handle.
Lastly, the authors describe a more recent (from 2006) development in which the researchers Cillo and Delquié suggested a new design, without previous expectation. Instead of assuming that the decision maker set a guide point before resolving the lottery, Cillo and Delquié proposed a generalized model in which any outcome could be a conceivable reference point. They backed up their reasons with three main arguments. First, it is hard to judge whether a single previous ‘expectation’ is more appropriate than none. Second, the reference points generally do not match actual results acquired from the lottery. Third, recent empirical research had shown that disappointment feelings could be triggered by comparing different outcomes. Cillo and Delquié then proposed a modified expected utility function based on expected values, disappointment and independent lottery results.
As we can see from the work of Cheung, K., Chong. W., and S Yam, C. (2015), Disappointment Aversion Theories attempt to handle the Allais paradox under a different light: even positive outcomes could have yielded a better result, and the mere possibility of these ‘could-have-beens’ changes people’s behaviors and risk aversion practices. A common example is the investor that has unexpected gains in the stock market, and is joyful up and until he finds out that he could have sold his stock some time later to an even better result. What was then a source of joy becomes painful, and the learned feeling of disappointment changes the investor behavior: he is bound to hold his stocks longer next time, contradicting his original risk preferences.
There is yet much to be learned from possible answers to the Allais Paradox. As we can see from both Prospect Theory and Disappointment Aversion, the issues are not simple and have to be tested extensively until a more robust answer to the paradox presents itself with clarity. There is mounting evidence, however, that traditional expected utility is not a sufficient tool to encompass the idiosyncrasies and subtleties of human preference.
Bibliography
Betts, S. C., and Zinaida, T., 2006. A Test of Prospect Theory in the Used Car Market: The
Non-Linear Effects of Age and Reliability on Price. Academy of Marketing Studies
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Cheung, K.C., Chong,W.F., and S Yam, C.P., 2015. The Optimal Insurance under
Disappointment Theories. Insurance: Mathematics and Economics 64: 77-90.
Available through ScienceDirect < http://www.sciencedirect.com>. [Accessed 9 May
2016].
Guthrie, C., 2003. Prospect Theory, Risk Preference, and the Law. Northwestern University
Law Review 97.3. Available through Questia <http://www.questia.com>. [Accessed 9
May 2016].
Schmidt, U., 1999. Efficient Risk-Sharing and the Dual Theory of Choice under Risk.
<http://www.questia.com>. [Accessed 9 May 2016].
