“Reflections on gains and losses: Bernoulli vs. Tversky and Kahneman,” by Antoni Bosch-Domènech and Joaquim Silvestre

“Theories of choice are at best approximate and incomplete... When faced with a complex problem, people employ a variety of heuristic procedures in order to simplify the representation and evaluation of prospects. (T&K, 1992, p.317) Basic point of agreement with TK

According to T&K, the pattern of risk attitudes can be organized in a remarkable simple manner, on the bases of two dichotomous variables: Gains/losses Low/high probability We believe, instead, that the pattern of risk attitudes can be parsimoniously described by one variable, the amount of money at risk (Bosch-D. & Silvestre, 1999). Basic point of contention with TK

according to TK

K&T proposed a formal theory to explain the four-fold pattern, Prospect Theory (PT). Its major components are a value function and a probability weighting function.

The probability weighting function indicates how the probability levels contribute to evaluate a gamble. Their probability weighting function overestimates small probabilities and underestimates large ones. The value function is defined over changes of wealth rather than final wealth levels. The function is concave for gains and convex for losses and exhibits “loss aversion”, i. e., the function is steeper for losses than for gains.

“the method of hypothetical choices emerges as the simplest procedure by which a large number of theoretical questions can be investigated. The use of the method relies on the assumption that people often know how they would behave in actual situations of choice, and on the further assumption that the subjects have no special reason to disguise their true preferences.” (K&T) On what experimental method the empirics of the four-fold pattern is based?

Today we are going to show that in one particular instance, i. e., when participants face losses, the use of a different method, namely, real-money choices, yields results that PT cannot possibly accommodate. In addition, these observations fit nicely with our previous statement that a parsimonius description of risk attitudes can be based on the amount of money at risk.

PT claims that people display risk attraction in choices involving high- probability losses. This assertion follows from two basic postulates of PT, namely: a)The value function is strictly convex for losses. b) People underweight high probabilities. Specifically :

But our experimental participants tend to avoid fair risks for large (up to €90), high- probability (80%) losses.

The experiment Real losses Avoiding Windfall-gain bias Avoiding House-money bias

Imagine that you are attending a convention in Rome, and you walk into the casino in Vila Cassinata. While passing the slot machines, you put €1 into one machine and, surprisingly, you win €500. Now what? Will your gambling behavior for the rest of the evening be altered? The question has general relevance since decisions are rarely made in temporal isolation. Current choices are often evaluated with the knowledge of the outcomes which have preceded them.

Consequently, to give punch to the participants’ losses, we organize the experiment in two temporally separated sessions: First session: a quiz that allows us to rank participants in four earning groups (€30, €45, €60 and €90) Second session: (several months later; no precise date): they make their choice.

The first session resulted in 4 groups of participants according to their correct answers to the quiz. Each participant earned the following: 1.If in the first group, each earned €30 2.If in the second group, each earned €45 3.If in the third group, each earned €60 4.If in the fourth group, each earned €90

We have indirect evidence that, by the beginning of the second session, about 70% of participants had integrated in their wealth this previously-earned cash, which had vanished in their everyday flow of expenditures.

In the second session, 1. A participant in Group 1 was randomly assigned to four possible amounts of money to lose (€3, €6, €12, €30) 2. in Group 2, r. a. to five amounts (€3, €6, €12, €30, €45) 3. in Group 3, r. a. to six amounts (€3, €6, €12, €30, €45, €60) 4. in Group 4, r. a. to seven amounts (€3, €6, €12, €30, €45, €60, €90).

But before knowing to which class she would eventually belong, the participant was asked to make a very simple choice between: a)The certain loss of 0.2 times the amount of money in the class b)The uncertain prospect of losing the money amount of the class with probability 0.2 and nothing with prob. 0.8

This treatment was called L. In a second treatment, L’, the 0.2 probability was replaced by a 0.8 probability. Definition: We say that a participant displays risk attraction (resp. risk aversion) in a particular choice if she chooses the uncertain (resp. certain) alternative.

Experimental results The numbers indicate the fraction of participants who display risk attraction (in red when majority)

1.Look at one row at a time.You’ll see the Amount effect 2.Look at columns. You’ll see a probability effect

Comparisons with the literature:

4. The probability effect nearly vanishes in real money treatments (New?) 1. High frequency of risk attraction for small real-money losses. ( New but does not contradict PT ) Summary of comparative results: 2. High frequency of risk aversion for large real-money losses ( New and contradicts PT ) 3. The amount effect vanishes in hypothetical treatments ( Hogarth& Einhorn, 1990; Holt & Laury 2002 ).

Bottom line When we try to understand behavior confronting real losses, the explanatory power of the amount of money dominates that of the probability. Whereas the probabilities provide the best organizing principle for the pattern observed in hypothetical-money experiments.

Robustness check with our previous treatments with gains (G and G’)

Conclusion: Our real-loss results are not out of place in the previous table that includes gains. There is a common pattern in all the results, i.e.: The four-fold pattern is hinted (losses vs. gains; overest. low prob. vs. underest. high prob) but is overwhelmed by the amount effect.

Our results for small money amounts GainsRisk-seeking increasing Risk-aversion LossesRisk-seeking increasing Risk-seeking Four-fold pattern Our results for small large amounts GainsRisk-aversion increasing risk-aversion LossesRisk-aversiondiminishing risk-aversion

Conclusion

“The use of the method of hypothetical payments relies on the assumption that people often know how they would behave in actual situations of choice.” (K&T, Econometrica 79) 1.“...subjects facing hypothetical choices cannot imagine how they would actually behave under high-incentive conditions. Moreover these differences are not symmetric: subjects typically underestimate the extent to which they will avoid risk.” H&L, AER Indications by others that something was amiss:

2. If emotions play a significant role in decision making (Loewenstein et al. (2001) “Risk as feelings”, or Slovic, P. et al. (2002) “The affect heuristic”), one should expect different behavior when choices involve hypothetical gains and losses, or when the choices are real gains and losses. BTW, it is not surprising that economists, trained to value incentives, only reluctantly have come to accept that hypothetical choices may provide insights into real decision making. What appears to be surprising is that psychologists, trained to value emotions, have gingerly accepted hypothetical choices as valuable to understand decision making, when it is hard to imagine a more emotion-barren choice than a hypothetical one.

When it matters, most people are risk averse. In our words (Bosch-Domènech & Silvestre 1999, 2002, 2006a, 2006b): When it does not matter, the four-fold pattern works fine as a description of tendencies, but not as a description of proportions of subjects.

“...Everyone who bets any part of his fortune, however small, on a mathematically fair game of chance acts irrationally... The imprudence of a gambler will be the greater, the larger part of his fortune which he exposes to a game of chance.” Daniel Bernoulli (1738) believed that risk aversion was universal:

Bernoulli redux