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Last updated on:

28-Jun-2007

 

 

 

Ambiguity aversion is the name given by Ellsberg (1961) to a decisional pattern by which clear probabilities are preferred to vague ones for lotteries that would be considered equivalent under subjective expected utility (SEU, Savage 1954). In its simplest formulation the problem goes as follows (Ellsberg two-color problem). Imagine you are given a chance of winning €100 by drawing a ball from one of two urns. The first urn contains 100 balls, 50 of which are black and 50 of which are red. The second urn also contains 100 balls that are either black or red, but the proportion of the colors is unknown. You now have to choose an urn and to announce a color. If the ball you draw from your chosen urn is of the color you announced, you win €100; if it is of the other color you win nothing. Which urn do you choose? Would you change your choice if the prize to be won from the urn with the known color proportion were €99 against a prize of €100 from the urn with the unknown color proportion?

 

In experiments carried out to test this proposition, a majority of subjects is consistently found to prefer the urn with the known proportion of colors (risky urn) over the urn with the unknown proportion of colors (ambiguous urn). Per se there is nothing wrong with such a choice pattern, given that people are generally bad at randomizing and that a null hypothesis of equally many people choosing the one and the other urn may thus be unwarranted. Things however change when the risky urn contains only 49 balls of the winning color and 51 of the losing color, or when the prize to be won with the ambiguous urn is €100 while the prize to be won with the risky urn is €99. In that case, it does indeed become irrational to choose the risky urn over the ambiguous urn, since the latter is then objectively worse. If you are not convinced of the equivalence of the two urns, consider the following argument by Raiffa (1961): if you simply throw a (fair) coin to determine the color you want to bet on, then you should see that you are back at a 50% chance of winning, whatever the actual distribution of colors in the urn may be.

 

The choice pattern described above is also referred to as the Ellsberg paradox. To see what is so paradoxical about this behavior however, we need to put the problem in a different way and add a few facts. Imagine the choice is put to you in the following way. First you are offered to draw from the risky urn and you are asked which color you would rather bet on. Most people declare to be indifferent between the two colors. Then you are offered to draw from the ambiguous urn and are asked which color you would like to bet on. Again most people are indifferent between the two colors. Since the probabilities of the separate events (a black ball is drawn; a red ball is drawn) must add up to one in each urn, this means that   and , where p_b and p_r represent the probability of drawing a black ball and the probability of drawing a red ball respectively, and the superscripts R and A represent the risky and ambiguous urn.

 

Now imagine you are offered the chance to bet on black and given a choice between the two urns. Most people express a strict preference for the risky urn, which tells us that their subjective probability of winning with black in the risky urn is strictly bigger than their subjective probability of winning with black in the ambiguous urn, i.e. . Since we know that , this means that the subjective probability of winning with a black ball from the ambiguous urn must be smaller than ½. Taken together with the equalities in the previous paragraph, this means also that the probability of winning with a red ball from the ambiguous urn must be bigger than ½. If asked to choose between betting on red in the risky urn and red in the ambiguous urn, however, people again declare to strictly prefer to bet on red in the risky urn. This in turn implies that , and hence that . Probabilities in the ambiguous urn can thus be shown to add up to less than one assuming such choice behavior, and hence the paradox.

 

 

What causes ambiguity aversion?

Nobody would arguably state explicitly that the total probability of winning in the risky urn is less than one. What then explains the observed behavior? Before attempting to answer this question let us take a closer look at the nature of the problem at hand. The first thing we will notice is the somewhat unnatural nature of the task. There are two urns, and in one of them some information seems to be missing. Indeed, the absence of information that appears as salient for the decision making process (Frisch and Baron 1988) for the ambiguous urn would seem to play a central role in this issue. This absence of information that might be salient for the decision making process (but is not according to SEU) is especially highlighted by the direct comparison of the ambiguous urn with an urn for which all conceivable information is known.

 

The latter point may be worth some closer consideration. Since the classical Ellsberg task involves a choice between two lotteries, a direct confrontation of the two is inevitable. Ambiguity aversion can however also be studied in other ways, for instance by asking subjects how much they would be willing to pay to play the lottery in question. Making use of exactly this method, Fox and Tversky (1995) elicited subjects' willingness to pay (wtp) for the two urns jointly (in a comparative setting) and separately (in a non-comparative setting, between subject). Whereas in joint evaluation people give a much higher wtp for the risky urn than for the ambiguous one and are thus ambiguity averse, when the two urns are evaluated in isolation the resulting wtp for the two urns is is not significantly different - people are no longer ambiguity averse. The comparative nature of the task, be it in choice or joint evaluation, seems thus to be a necessary condition for ambiguity aversion to occur.

 

Once this condition is fulfilled it is however still not exactly clear why people would shy away from the urn about which they think to know less. Many different explanations have been proposed over the years. Curley, Yates, and Abrams (1986) test six such explanations and find what they call other-evaluation - the fact that others observe their decision and may judge them for it - to be the only significant one. In the same line, Trautmann, Vieider, and Wakker (2008) find that eliminating the possibility that decision maker be judged by others by making her preferences unknown to potential onlookers is enough for ambiguity aversion to disappear. They also find that people with a higher score on the Leary (1983) scale measuring fear of negative evaluation (Watson and Friend 1969) are more likely to choose the risky urn when exposed to the potential evaluation of others.

 

In sum it seems thus that ambiguity aversion is mainly due to accountability pressures. When called upon to make a decision which they might be required to justify in front of an audience who's views on the matter are unknown (Lerner and Tetlock 1999), people usually take the decision which they deem most easily justifiable (Shafir et al. 1993). When it comes to deciding which of the two urns to choose in the Ellsberg task, there is a common presumption that choosing the risky urn is more justifiable given that all information is known for it. In other words, a choice of the ambiguous urn seems hard to justify when an otherwise perfectly equivalent urn with more information is available.

 

There is a further methodological point to be made here. As we have seen, ambiguity aversion may be investigated both through choice tasks and through valuation tasks. Trautmann, Vieider, and Wakker (2009) however find preference reversals to occur between choice and matching tasks. In particular, they find that subjects who choose the ambiguous gamble very often give a higher wtp for the risky gamble. Combining these findings with previous findings in the literature one could now construct a ‘scale’ of ambiguity aversion whereby comparative evaluation tasks produce the strongest ambiguity aversion; certainty equivalents elicited though choice produce the second strongest ambiguity aversion due to the asymmetric strength of valuation (but less than WTP because of the absence of preference reversals); choice produces somewhat less pronounced, but still strong ambiguity aversion (depending on the strenght of other-evaluation); and finally separate evaluations do not produce any ambiguity aversion.

 

 

References

 

Curley, Shawn P., J. Frank Yates and Richard A. Abrams (1986). Psychological Sources of Ambiguity Avoidance. Organizational Behavior and Human Decision Processes 38, 230-256.

Ellsberg, Daniel (1961). Risk, Ambiguity and the Savage Axioms. The Quarterly Journal of Economics 75(4), 643-669

 

Fox, Craig R. and Amos Tversky (1995). Ambiguity Aversion and Comparative Ignorance. The Quarterly Journal of Economics.

Frisch, Deborah and Jonathan Baron (1988). Ambiguity and Rationality. Journal of Behavioral Decision Making 1, 149-157

Leary, Mark R. (1983). A Brief Version of the Fear of Negative Evaluation Scale. Personality and Social Psychology Bulletin 9(3), 371-375

Lerner, Jennifer S. and Philip E. Tetlock (1999). Accounting for  the Effects of Accountability. Psychological Bulletin 125, 255-275.

Raiffa, Howard (1961). Risk, Ambiguity and the Savage Axioms: Comment. The Quarterly Journal of Economics, vol. 75, no.. 4, pp. 690-694.

Savage, L. J. (1954). The Foundations of Statistics. Wiley, New York.

 

Shafir, Eldar, Itamar Simonson and Amos Tversky (1993). Reason-based Choice. Cognition 49, 11-36.

Trautmann, Stefan T., Ferdinand M. Vieider, and Peter P. Wakker (2008). Causes of Ambiguity Aversion: Known versus Unknown Preferences. Journal of Risk and Uncertainty 36, 225-243.

Trautmann, Stefan T., Ferdinand M. Vieider, and Peter P. Wakker (2009). Preference Reversal under Ambiguity. Working Paper.

Watson, David and Ronald Friend (1969). Measurement of Social-Evaluative Anxiety. Journal of Consulting and Clinical Psychology 33 (4), 448-457

Ambiguity Aversion