Abstract
This paper reports the results of experiments testing prevalence of non-neutral ambiguity attitudes and how these attitudes change as a result of interpersonal interactions. To address the first question we conducted experiments involving individual choice between betting on ambiguous and unambiguous events of the subject’s choice. We found that a large majority of subjects display ambiguity neutral attitudes, many others display ambiguity incoherent attitudes, and few subjects display either ambiguity-averse attitudes or ambiguity-seeking attitudes. To address the second question we designed a new experiment with a built-in incentive to persuade. We found that interpersonal interactions without incentives to persuade have no effect on behavior. However, when incentives were introduced, the ambiguity neutral subjects were better able to persuade ambiguity seeking and ambiguity incoherent subjects to adopt ambiguity neutral choice behavior and, to a lesser extent, also ambiguity averse subjects.
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Notes
See Becker and Brownson (1964), MacCrimmon (1968), MacCrimmon and Larsson (1979), Fox and Tversky (1995), Chow and Sarin (2001), Halevy (2007), Liu and Colman (2009), Ahn et al. (2011), Bossaerts et al. (2010), Keck et al. (2011), Keller et al. (2009), Binmore et al. (2012). Ambiguity aversion seems to also extend to non-Western, non-student populations. Akay et al. (2011) find ambiguity with Ethiopian farmers in the two-color problem.
This type of suspicion may be engraved into our psyche or learned. In “The Idyll of Miss Sarah Brown”, Damon Ranyon describes the advice given to young Sky Masterson as he is about to leave home, by his father. “Son”, the old guy says, “no matter how far you travel, or how smart you get, always remember this: Some day, somewhere”, he says, “a guy is going to come to you and show you a nice brand-new deck of cards on which the seal is never broken, and this guy is going to offer to bet you that the jack of spades will jump out of this deck and squirt cider in your ear. But son”, the old guy says, “do not bet him, for as sure as you do you are going to get an ear full of cider.”
These problems are referred to in the literature as “truth wins” problems (see Cooper and Kagel 2005). In such problems, if one member of a group gets the right answer, she is able to persuade the other members on the correct course of action.
Other than the number of red slips, the composition of the remaining blue and green slips was unknown. The subjects were told that the co-authors picked the composition in each envelope (this was true) and that the experimenter didn’t know how this was generated. Also, the subjects were invited to check the envelopes at the end of the experiment.
Implicit in this classification is the presumption that subjects do not firmly believe that, in each envelope, one of the colors, say green, is present in number larger than red. If this were the case then the subject should choose this color thoughout and will be classified as ambiguity seeking even though he may well be ambiguity neutral.
The subjects were not informed of the second stage of the experiment while engaged in the first stage, so there was no concern that an ambiguity neutral subject might employ a strategy of displaying ambiguity averse attitudes in order to be matched with other ambiguity neutral subjects in the second stage. And of course they were also unaware of the matching process.
But if the subject is risk averse then \(u\left ( y\right ) <u\left ( x\right ) \frac {y}{x},\) which reinforces the argument.
Complete individual choice profiles for both versions of the experiment can be found in Appendix B.
This figure should be interpreted with some care. When there are fewer than 12 red slips the probability of red is smaller than 0.31, and when there are more than 12 red slips, the corresponding probability is at least 0.36. A subject that was classified as ambiguity neutral assigns a probability higher than 0.31 to one of the other colors in the former case and a probability lower than 0.36 in the latter case. When there are 12 red slips, so that the probability of red is 1/3, there is room for a subject to entertain beliefs that one of the other colors has a probability higher than 1/3 and yet chooses to bet on red. Consequently, our set of ambiguity neutral subjects might include some (slightly) ambiguity averse types. Further investigation of this issue, by refining the partition (e.g., using 90 slips) or using our design with a total of 37 instead of 36 slips is a subject for future research. We thank Luca Rigotti for raising this issue.
The full set of decisions for each individual can be found in Appendix B to this paper.
Consider an ambiguity averse subject who chooses one of the “other colors” when the number of reds is 1–9 and switches to red when the number of reds is 10 and up. Her choice indicated that
$ \left ( x,\frac {10}{36}\right ) \succ \left ( x,\frac {9}{36}\right ) \succ \left ( x,\frac {k(36-9)}{36}\right ) , $where \(x\) is the money payoff and \(\frac {10}{36}\) and \(\frac {9}{36}\) are the objective probabilities of red in line with 10 and 9 red balls, respectively, and \(\frac {k(36-9)}{36},\) is the probability of the “other color”. Thus,
$ \frac {9}{36}>\frac {k(36-9)}{36}. $Thus
$ k<\frac {9}{27} $Let \(y>x\) be the payoff if there is an agreement and suppose that the subjects are risk averse. Normalize the utility function so that \(u(0)=0,\) and let \(k=9/25,\) then \(y\) must satisfy the following:
$ \left ( x,\frac {10}{36}\right ) \succ \left ( y,\frac {9(36-10)}{27\times 36} \right ) . $Under expected utility this implies
$ u\left ( x\right ) \frac {10}{36}>\frac {y}{x}u\left ( x\right ) \frac {9\times 26}{ 27\times 36}>u\left ( y\right ) \frac {9\times 26}{27\times 36}. $Hence, choosing \(y\) such that
$ \frac {y}{x}<\frac {10\times 27\times 36}{36\times 9\times 26}=\frac {270}{234} =1.154 $is not going to make the AA type change her switch point.
However, there may be some evidence of ambiguity aversion when the number of red slips in the envelope is 12, so that this is costless; however, this is a matter of interpretation regarding what one considers the base rate. We find that 50% of choices (82 of 164) in this case are to bet are on red. If everyone is ambiguity neutral, they might randomize amongst the three colors (although it could be argued that red is more focal, in terms of there being red and non-red categories) who are indifferent in this case. However, there are many ambiguity averse people and even with the tiniest degree of ambiguity aversion an individual will choose red. Of course, if there is more than the tiniest amount of ambiguity aversion we should expect to see this when the number of red slips in the envelope is 11.
The simple binomial test indicates no significant overall tendency for the subsamples with and without premium. For the full sample, Z \(=\) 0.23, with no premium Z \(=\) 0.00 and with premium Z \(=\) 0.26.
For the full sample the simple binomial test shows that the tendency to switch from ambiguity seeking to ambiguity neutrality as opposed to switching in the other direction is significant (Z \(=\) 2.33, p \(=\) 0.010, one-tailed test, p \(=\) 0.019, two-tailed test). The same test for the subsample with no premium, Z \(=\) 1.50, p \(=\) 0.067, one-tailed test, p \(=\) 0.134, two-tailed test. For the subsample with a premium, Z \(=\) 1.81, p \(=\) 0.035, one-tailed test, p \(=\) 0.070, two-tailed test.
For the full sample the simple binomial test shows that the tendency to switch from ambiguity incoherence to ambiguity neutrality as opposed to switching in the other direction is significant. For the full sample (Z \(=\) 1.73, p \(=\) 0.042, one-tailed test, p \(=\) 0.083, two-tailed test). For the subsample with no premium, Z \(=\) 1.41, p \(=\) 0.079, one-tailed test, p \(=\) 0.157, two-tailed test. For the subsample with a premium, Z \(=\) 1.10, p \(=\) 0.136, one-tailed test, p \(=\) 0.272, two-tailed test.
The binomial test gives \(Z=3.50\), \(p=0.000\), one-tail or two-tail test.
Low rates of ambiguity averse attitudes were recently reported in Binmore et al. (2012) which led them to conclude that “ambiguity aversion is not always as powerful and robust a phenomenon as it is sometimes said to be”. (p. 21). Wakker (2010) provides supporting evidence to this conclusion in the context of cumulative prospect theory. According to Wakker, there is more ambiguity seeking than ambiguity aversion for losses. Similarly, there is more ambiguity seeking than ambiguity aversion for gains and unlikely events.
Note that if a subject thought that there were many more blue than green slips he would choose blue when blue is one of the choices and green if green is the alternative to red. Such a subject would be classified as ambiguity incoherent while, in fact he might be ambiguity neutral. If such subjects were present, this would partially explain the very high (35.6%) rate of ambiguity incoherent attitudes observed in version 2.
These ambiguity premiums are much higher than those observed in other studies. Recall, however, that Becker and Brownson (1964) only allowed subjects who were ambiguity averse to participate in their experiment.
Camerer and Weber (1992) do not report which studies controlled for what we consider a possible explanation for the preference of risky over ambiguous bets, namely subjects’ suspicion of the ambiguous urns being rigged.
Note that this result is in line with our results suggesting the presence of ambiguity aversion in the knife-edge case when all actions give the same expected value.
They refer to ambiguity as vagueness.
Keck et al. also studied group decision making in which decisions were taken by majority vote of the group and the payoff shared among the membership. This is an issue that is outside the scope of our work.
One possible explanation for this is that they offered their subjects the option of letting the computer assist in the decision making process. This assistance allows the computer to automatically fill in all the choices located on the decision sheet above the choice for which a participant preferred the sure amount of money over the gamble. Using this procedure may rule out what would have been inconsistent choices.
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We are grateful to Stefan Trautmann, Peter Duersch, Luca Rigotti and an anonymous referee for their useful comments. We also benefited from comments and suggestions of the participants of the conference on “Ambiguity: Theory and Experiments,” at the Center for the Economic Analysis of Risk, Georgia State University, September 20–21, 2012.
Appendices
Appendix A
1.1 A.1 Instructions 1
Thank you for participating in our experiment. This experiment is conducted as a part of a research project on people’s preferences. The whole session will last between 30 minutes and one hour. After we read the instructions and you understand the task, we shall proceed.
Consider six containers that have 36 chips in each. Each container will have a different (but known) number of Red chips, with the remaining chips in the container being either Blue or Green; you will not be told how many chips are Blue and how many chips are Green.
You’ll be faced with a table with six rows, with each row representing one container. The first row will ask you to make a choice when there are 9 Red chips in the container (container 9), and so 27 Blue or Green chips; the second row to consider has 10 Red chips in the container (container 10), and so 26 Blue or Green chips; and so on up to 14 Red chips (and 22 Blue or Green chips) in the container (container 14). For each row your task is to choose one of the three colors to bet on.
After people make their choices, we will randomly draw a number from 9–14 to determine which line in the table is to be implemented (played). Once a line is selected it determines which container will be used. We will draw one chip from that container and pay $10.00 to each person who picked that color.
1.1.1 A.1.1 Decision sheet
In the table below, please circle R or G or B in each row. If you have any questions, please raise your hand, and we will come to assist you. Please do not speak to any other person.
Red numbers | Blue or Green numbers | Bet on Red [R], bet on Green [G] or bet on Blue [B] |
|---|---|---|
9 | 27 | R G B |
10 | 26 | R G B |
11 | 25 | R G B |
12 | 24 | R G B |
13 | 23 | R G B |
14 | 22 | R G B |
1.2 A.2 Instructions (2) [Premium]
We now proceed to pair each person with another person amongst the participants. Look for the person who has the same ID number as you. You will now have the opportunity to consult directly with this other person regarding the selections made on the second decision sheet that follows. The same rules apply in terms of selecting a color and the amount of the prize.
You may speak (quietly) with the person with whom you are paired and you’ll have 5 minutes to deliberate about the decisions to be made. Afterwards each person is free to make his or her own decision. [These decisions need not be the same; however, if these decisions are indeed the same, then an additional $1.25 will be added to the potential prize upon winning for each person (so that it becomes $10.00 + $1.25 = $11.25, rather than $10.00).] If all of the decisions are not the same, the potential prize for each person remains $10.00.
1.2.1 A.2.1 Decision sheet 2
In the table below, please circle R or G or B in each row. If you have any questions, please raise your hand, and we will come to assist you. Please do not speak to any other person besides the person with whom you have been paired.
Red numbers | Blue or Green numbers | Bet on Red [R], bet on Green [G] or bet on Blue [B] |
|---|---|---|
9 | 27 | R G B |
10 | 26 | R G B |
11 | 25 | R G B |
12 | 24 | R G B |
13 | 23 | R G B |
14 | 22 | R G B |
1.3 A.3 Instructions (2) [No Premium]
We now proceed to pair each person with another person amongst the participants. Look for the person who has the same ID number as you. You will now have the opportunity to consult directly with this other person regarding the selections made on the second decision sheet that follows. The same rules apply in terms of selecting a color and the amount of the prize.
You may speak (quietly) with the person with whom you are paired and you’ll have 5 minutes to deliberate about the decisions to be made. Afterwards each person is free to make his or her own decision.
1.3.1 A.3.1 Decision sheet 2
In the table below, please circle R or G or B in each row. If you have any questions, please raise your hand, and we will come to assist you. Please do not speak to any other person besides the person with whom you have been paired.
Red Numbers | Blue or Green Numbers | Bet on Red [R], bet on Green [G] or bet on Blue [B] |
|---|---|---|
9 | 27 | R G B |
10 | 26 | R G B |
11 | 25 | R G B |
12 | 24 | R G B |
13 | 23 | R G B |
14 | 22 | R G B |
Appendix B
1.1 B.1 Summary of Ellsberg results
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Charness, G., Karni, E. & Levin, D. Ambiguity attitudes and social interactions: An experimental investigation. J Risk Uncertain 46, 1–25 (2013). https://doi.org/10.1007/s11166-012-9157-1
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DOI: https://doi.org/10.1007/s11166-012-9157-1