Group Decision Making Under Ambiguity Steffen Keck INSEAD, Boulevard de Constance, 77305 Fontainebleau Cedex, France [email protected]

Enrico Diecidue INSEAD, Boulevard de Constance, 77305 Fontainebleau Cedex, France [email protected]

David Budescu Department of Psychology, Fordham University, Dealy Hall, Bronx, New York 10458, USA [email protected]

Friday, August 12, 2011

1

Abstract We report results of an experimental elicitation of certainty equivalents for 15 risky or ambiguous two-outcome gambles. Decision makers (DMs) made decisions in three different settings: (a) individually, (b) individually after discussing decisions with others, and (c) in interacting groups of three. We also manipulated the degree of payoff-communality between subjects. We show that groups are more likely to make ambiguity neutral decisions than individuals, and that individuals also make more ambiguity neutral decisions after discussing the decisions with others. This shift towards higher ambiguity neutrality in groups and after a group discussion is caused by a reduction in both ambiguity aversion and ambiguity seeking. Surprisingly, our results do not show a significant influence of payoff-communality on attitudes towards risk and ambiguity. We attribute the results to the effective and persuasive communication that takes place in groups.

Keywords: Ambiguity, Group Decision Making, Imprecision, Risk, Vagueness JEL classification: C92, D71, D81

2

1. Introduction Keynes (1921) and Knight (1921) emphasized that most economic decisions involve vague probabilities (ambiguity1) as opposed to probabilities which are precisely defined (risk). In many cases, these decisions are delegated to groups of DMs -for example committees, management teams and boards of directors. Previous work has studied the differential risk attitudes of individuals and groups, but it is unclear whether groups also differ from individuals with respect to their attitudes towards ambiguity. We address this question in a laboratory experiment in which individuals and groups make decisions under risk or under ambiguity. One rationale for delegating important decisions to groups is that they outperform individuals in a large variety of tasks (Kerr & Tindale, 2004). For example, groups have been shown to learn faster and achieve higher outcomes in strategic settings such as beauty contest (Kocher & Sutter, 2005), signaling- (Cooper & Kagel, 2005) and coordination games (Feri, Irlenbusch & Sutter, 2010). In decisions under risk the evidence about group performance is mixed. Groups show fewer violations of stochastic dominance and Bayesian updating rules (Charness, Karni & Levin, 2007) and make better investment decisions (Sutter, 2007; Rockenbach, Sadrieh & Mathauschek, 2007). Bone, Hey and Suckling (1999) report that decisions under risk made by individuals and groups show similar rates of expected utility violations such as the common-ratio effect. Rockenback et al. (2007) confirm this result and also report no difference between individuals and groups with respect to the rates of preference reversals.

1 Following Ellsberg (1961) vague probabilities are commonly referred to as ambiguous. We follow this convention in our paper. However, see Budescu, Kuhn, Kramer and Johnson (2002) for a discussion pointing out the inadequacy of ambiguity as a descriptive term and advocating the use of imprecision or vagueness instead. We use both terms -ambiguity and vagueness- in this paper.

3

Group decisions often involve the aggregation of group members’ attitudes. Previous studies comparing risk attitudes of individuals and groups have found somewhat conflicting results. Zhang and Casari (2009) found groups to be less risk averse than individuals. In contrast, Harrison, Igel-Lau, Rutström and TarazonaGómez (2005), Ambrus, Greiner and Pathak (2009) and Deck, Lee, Reyes and Rosen (2010) did not find differences between the risk attitudes of individuals and groups. Several studies suggest that differences in risk attitudes between individuals and groups depend on the risk level of the decision. Shupp and Williams (2008) showed that groups are less risk averse in low-risk situations and more risk averse when decisions involve high levels of risk. Baker, Laury and Williams (2008) as well as Masclet, Colombier, Denant-Boemont and Lohéac (2009) found a similar pattern. All of these studies have focused on the comparison of risk attitudes between individuals and groups but did not explore potential differences in attitudes towards imprecision. Starting with Ellsberg (1961) numerous studies have shown that in settings which involve ambiguity, individuals’ decisions cannot be reconciled with classical Subjective Expected Utility (SEU) (Savage, 1954). The most common (though, by no means, universal) experimental finding in the literature is ambiguity aversion. Consequently, when evaluating ambiguous gambles or investments, individuals demand an additional “ambiguity premium” on top of the normal risk premium (for an overview of experimental findings see Camerer & Weber, 1992; Etner, Jeleva & Tallon, 2009). Although ambiguity aversion remains the most common finding, diversity in ambiguity attitudes is often found. Ambiguity seeking is commonly observed in the domain of losses (Mangelsdorff & Weber, 1994) and for small probabilities of gains (Einhorn & Hogarth, 1986; Kahn & Sarin, 1988). Budescu, Kuhn, Kramer and

4

Johnson (2002), who analyze certainty equivalents for gambles with vague probabilities and vague outcomes, find no strong modal attitude towards ambiguity in probabilities and predominantly ambiguity seeking (aversion) for vague gains (losses). Du and Budescu (2005) demonstrate that attitudes towards ambiguity in general are malleable and depend on factors such as the locus of the ambiguity (probabilities or outcomes), the choice domain (gains vs. losses) and response modes (pricing or choice). Only a very small number of studies have investigated the effects of social factors on decisions under ambiguity. Curley, Yates & Abrams (1986) found that individuals, who were observed by uninvolved others during their decisions and the resolution of the uncertainty, exhibited significantly more ambiguity aversion than DMs who made their decisions alone. They attributed this finding to the subjects’ fear of being evaluated negatively in case the chosen ambiguous alternative leads to undesirable outcomes. Muthukrishnan, Wathieu and Xu (2009) obtain a similar result. In their study, subjects who anticipated to be informed in the presence of others about the true probability of winning a prize in an ambiguous gamble, were more ambiguity averse, than subjects who anticipated to be informed in private. Trautmann, Vieider and Wakker (2008) report results of an experiment in which the subjects’ preferences where kept unknown to the experimenters, so the possibility of a negative evaluation by others could be completely ruled out. This treatment significantly decreased ambiguity aversion, supporting further the interpretation proposed by Curley et al. (1986). Although all three studies involved individual decisions their results suggest that the interaction with others, either before an individual decision or as part of group decision making procedure, could affect the DMs’ attitudes to ambiguity.

5

Keller, Sarin and Sounderpandian (2009) compared the willingness of individuals and dyads to pay for risky and ambiguous gambles. Dyads tended to be more risk averse than their individual members, but there was no difference with respect to ambiguity attitudes. Their interesting study is limited in a number of important ways. First, they only studied dyads. More importantly, their design did not allow them to distinguish between the effects of information sharing and the need to aggregate individual preferences into a group decision. Finally, they did not study systematically how prior individual ambiguity-preferences (like aversion, neutrality or seeking) are aggregated into a group decision and how these attitudes change in this process.

1.1 The Present Study We conducted a laboratory experiment in which DMs made binary choices between sure amounts of money and different risky and ambiguous gambles. We distinguish between individual decisions, individual decisions made after exchanging information with others, and group decisions. This distinction allows us to disentangle two effects often confounded in studies of group decisions. The first is the influence of discussing decisions with others and exchanging information and opinions (see Trautmann and Vieider (2011) for a comprehensive discussion of social influence processes during group interactions). This factor affects both group decisions and individual decisions after a group discussion. The second is the process of aggregation of individual preferences into a group decision and it affects only the group decision. We also examine the payoff sharing arrangement among the group members. In some treatments the outcome of a group decision affected all individuals identically i.e., the group shared a “common fate”. In other situations the outcome of a group

6

decision varied across the individual members. Wallach and Kogan (1964) argued that an essential component for the emergence of a “shared responsibility” is that consequences of the group decisions are equally shared by all members and no member can escape the consequences of a bad decision. Following this line of reasoning, we hypothesized that the nature of the payoff sharing arrangement will influence the degree of responsibility individuals feel for the group decision and thus affect risk and ambiguity attitudes. Sutter (2009) found that individuals invested higher amounts of money in a risky investment if they shared their payoffs with other subjects, providing support for this hypothesis for the case of risk. Further evidence is provided by Charness and Jackson (2009), who showed that responsibility for the payoff of another person significantly influenced subjects’ decisions to take strategic risks in a coordination game. Accordingly, we distinguish between four different decision settings each of which has real-life counterparts: a) Group decisions with shared consequences: The decision is made by a group of individuals and all group members experience the same consequences. Consider for example a group of partners deciding whether to expand their business overseas or not. The decision is made by all partners and they all bear its financial consequences whether they are positive or negative. b) Group decisions with individual consequences: The decision is made by a group of individuals but this decision leads to different consequences for each member. For example, a group of franchisees can decide jointly on a common marketing strategy but, due to the particular circumstances of each franchisee, they may end up with different outcomes.

7

c) Individual decisions after social interaction with shared consequences: The decision is made by an individual, only after consulting with others. Consider, for example, a senior executive who has the authority to make decisions independently but consults with other members of his organization before deciding. Although the final decision is made by an individual its consequences affect everyone in the company. d) Individual decisions after social interaction with individual consequences: The decision is made by an individual DM who solicits advice from external consultants. Thus, although the decision has benefited from sharing information the ultimate decision made has no effect on the individuals giving advice.

2. Experimental Method 2.1 Experimental Tasks DMs in our experiment (individuals or groups) made choices between sure amounts of money and 15 two-outcome gambles. Each risky (ambiguous) gamble offered the possibility of winning $20 with probability p (range of probabilities p ± ∆), or receiving $0. Gambles differed from each other with respect to the probability of winning p (p = 0.20, 0.35, 0.50, 0.65 and 0.80) and the level of imprecision ∆ (∆ = 0, 0.05, 0.10, 0.20, 0.30, and 0.50) of the probabilities. Not all possible combinations of p and ∆ were implemented: We had 5 different gambles designed around p = 0.50 that offered subjects the possibility of winning $20 with probabilities 0.50 (∆ = 0), 0.45 - 0.55 (∆ = 0.05), 0.40 - 0.60 (∆ = 0.10), 0.20 - 0.80 (∆ = 0.30) and 0 - 1.0 (∆ =0.50), respectively. Probabilities p = 0.20 and 0.80 were only paired with 4 values of ∆ (0, 0.05, 0.10, 0.20), and probabilities p = 0.35 and 0.65 were only paired with ∆ = 0.

8

All choices were presented in the form of a “price list”. Each price list consisted of a number of binary choices (15 or 19 depending on the gamble) between the gamble and increasing sure amounts of money (ranging from either $0.50-$7.50; $1.00 - $19.00 or $12.50 - $19.50 and increasing either in $0.50 or $1.00 increments) such that for the first decision a DM should always prefer the gamble and for the last decision always prefer the sure amount of money. The 15 gambles (and their respective price lists) were presented in random order. We presented one of the price lists (p = 0.50, ∆ = 0) twice to test for consistency. Table 1 summarizes the 15 gambles and the price lists used as stimuli. Examples of price list for risky and ambiguous gambles can be found in appendix A. In addition to the 15 price lists, we also included the two classical Ellsberg tasks in the study. We did not find a significant difference between individuals and groups for either of the two tasks. We provide descriptions of the Ellsberg tasks and a brief overview of our findings in appendix B. Insert Table 1 about here We inferred certainty equivalents (CEs) for each gamble from the switching point from preferring the gamble to preferring the sure amounts on each price list. A DM with monotonic preferences for money should have a unique switching point between the two alternatives. We defined the CE for a particular gamble as the midpoint of the switching interval. For example if on a particular price list a DM preferred the gamble over all sure amounts of money ≤ $8 and the sure amount over the gamble for all amounts ≥ $9 we assume that the CE is ($8.00 + $9.00) / 2 = $8.502,3.

2

In 16 (out of the 6800 price lists = 0.2%) cases subjects had multiple switches. In these cases we use the first (lower) switching points on the price list to calculate the CEs. 15 of these 16 cases were caused by three individuals who had multiple switches on several price lists. To test for the robustness of our

9

To facilitate the decision process subjects were offered the option of using computer-assistance in making their decisions. The computer assistance automatically filled in all choices located on the price list above the choice for which a subject preferred the sure amount of money over the gamble. Analogously, the computer assistance filled in the choices below the choice for which a subject preferred the gamble over the sure amount of money. For example, if a subject indicated a preference for $10 over the gamble the computer-assistance automatically determined that she would also prefer all sure amounts > $10. Similarly if a subject indicated a preference for the gamble over $9 the computer assistance automatically determined that she also prefers the gamble over all sure amounts < $9. Used effectively the computer assistance allowed subjects to complete a particular price list with only two clicks by indicating her switching point. Subjects could deactivate the computer assistance at any time they wished.

2.2 Subjects We recruited a total of 240 undergraduate students from a large east coast university (90 male, 150 female) by an e-mail announcement. Students varied widely with respect to their majors. Average age of the subjects was 20.7 years. All sessions were run at a behavioral laboratory in April 2009. Subjects were paid a $5 show-up

results we also run our analysis excluding those subjects and their groups. This has no influence on the significance of any of our results. 3 In 253 (out of the 6800 price lists = 3.7%) cases decision makers preferred either the gamble over all sure amounts (176 cases = 2.6%) or all sure amounts over the gamble (77 cases = 1.1%) and never switched between the two. To calculate a CE in these cases, we assume that the decision maker would have switched at the next item that could have been listed on the price list. For example if a decision maker preferred a gamble to win $20 with p=0.20 over all amounts between $0 and $7.50 (the upper end on this price list) we assume that the decision maker would have switched for the sure amount of $8.00 and infer the CE to be $7.75 (the midpoint between $7.50 and $8.00). To test for the robustness of our results we also run our analysis excluding those subjects and their groups. This has no influence on the significance of any of our results.

10

fee plus what they won (individually or in groups) during the experiment and earned on average $23.2.

2.3 Experimental Design Subject were randomly assigned to one of the 5 experimental treatments summarized in table 2. The experiment consisted of two stages (individual and group decision making) in treatments “GD(shared)”, “GD(shared; counterbalanced)” and “GD(separate)” and of three stages (individual decisions, group discussion and second round of individual decisions) in treatments “IDIN(shared)” and “IDIN(separate)”. Insert Table 2 about here At each stage subjects made decisions for all 16 (15+1 repeated) price lists. In all but one treatment subjects started the experiment with the individual decision making stage. This stage was followed by a group stage and in treatments “IDIN(shared)” and “IDIN(separate)” by another round of individual decision making. To control for possible order-effects, we ran a treatment “GD(shared; counterbalanced)” where we reversed this order and asked subjects to make their decisions first as a group and then individually. To determine final payment for subjects we employed the random incentive system (Starmer & Sugden, 1991; Hey & Lee, 2005). One of the choices made at each stage (except the group discussions in treatments “IDIN shared” and “IDIN separate”) was randomly selected at the end of the experiment and subjects were paid according to their decisions. Depending on their stated preference they either received the sure amount of money, or played the chosen gamble (by a random draw from a real urn filled with red and black chips). For all gambles, drawing a red chip resulted in winning the $20 and drawing a black chip in winning nothing. As explained to

11

subjects in the instructions, all urns contained a total of 100 red and black chips in a proportion corresponding to the characteristics of the chosen gamble. For example, a gamble with p = 0.50 and ∆ = 0.10 was represented by a draw from an urn containing 100 chips in total where 40 - 60 were red and the rest black. We chose this procedure in order to make the characteristics of the gambles and the resolution of uncertainty as vivid as possible for subjects4. Whereas the individual decision making stage was identical in all treatments, the group stage varied across treatments in the following way: Treatment “GD(shared)” (Group decisions with shared outcomes): After finishing the first round of individual decisions, subjects were randomly assigned to groups of three and completed the same tasks in the groups. Group members had to make a joint decision about how to fill in the price lists. They were allowed to discuss their choices in a face to face interaction as long as they wished before making one joint decision. Disagreements were resolved by discussing the problem and, if necessary, by a majority vote. Subjects were informed that their payoffs for the group stage (independent of the payoff for their individual decisions) would be determined according to one randomly selected choice the group made. All group members were paid according to the group decision. If the group chose the sure amount each member received this amount. If the group chose the gamble, the gamble would be played once and each member received either the winning prize of $20 or nothing. Treatment “GD(shared; counterbalanced)” (Group decisions with shared outcomes and reversed order): The treatment was identical to “GD(shared)” but the

4

The procedure had the potential disadvantage of introducing concerns about a biased urn and thus possibly increase aversion towards ambiguity. To alleviate this concern, subjects were informed that the proportion of red and black chips was -within the parameters of the gambles- determined purely by chance and they had the possibility of inspecting the urns after the payment had taken place (see also Curley et al., 1986 who did not find a significant effect of ruling out such potential concerns).

12

order in which individual and group decisions were made was reversed. Subjects were assigned to groups of three and completed tasks as a group first, followed by the same tasks individually. Treatment “GD(separate)” (Group decisions with separate outcomes): This treatment differs from treatment “GD(shared)” only in the way final payoffs for the group stage was determined. At the end of the experiment one choice made during the group stage was selected randomly and the group members paid according to their group decision. Either they all received the sure amount corresponding to this choice, or the gamble was played separately for each group member. Thus, the final payoffs of the members could be different, as some won the gamble while others did not. Treatment “IDIN(separate)” (Individual decisions after group interaction with separate outcomes): After finishing their first round of individual decisions, subjects were assigned to groups of three. Group members worked through the same tasks as at the individual stage, but the group decisions did not affect their final payoffs. Subjects were told that the group stage only served to expose them to the other group members’ opinions which they could take into account when making individual decisions later on. Subjects were allowed to discuss their decisions freely, and as long as they wished. After the group stage, each group member completed, again, all the tasks individually with the understanding that this second round of individual decisions count towards their final payoff (on top of earnings bases on the first round). The final payoffs for this stage were determined by choosing one decision for each group member separately and either playing out the gamble or paying the sure amount of money depending on the subject’s decision. Treatment “IDIN(shared)” (Individual decisions after group interaction with shared outcomes): This treatment differs from treatment “IDIN(separate)” only in the

13

way final payoffs for the third stage was determined. One choice of a randomly determined group member was randomly selected for the whole group. Depending on the decisions the group member made in the round of individual decisions for this choice problem, the corresponding gamble was played out and each group member received either $20 or nothing, or the sure amount was credited to all group members. Thus, payoffs were identical for all members of the group.

2.4 Procedure Subjects were welcomed to the lab, instructed about the general procedure of the experiment and assigned to an individual computer. All instructions were presented to subjects on their computer screens, and a hard copy of the instructions was available for reference5. The software included an introduction of the tasks subjects were asked to complete, and an explanation of how to use the software to make decisions. To ensure that all subjects fully understood the instructions, they were required to pass four quiz questions before being allowed to start the experiment. Subjects were also encouraged to ask questions at any time they wished. After all subjects completed the price lists individually, the computer instructed them to move to their “group” computers which were located in the same room (to avoid inter-group communication the group computers were positioned at different corners of the room). All three group members were seated in front of one monitor and one of them was assigned to enter the group decisions. In treatments “GD(shared), GD(shared; counterbalanced) and GD(separate)” the experiment ended after the second stage and all group members were paid and debriefed. In treatment “GD(shared; counterbalanced)” all 3 subjects started the experiment at their group

5

A copy of the instructions can be found can be found in the supplementary material.

14

computers, and then moved to the individual computers. In treatments “IDID(shared)” and “IDID(separate)” subjects returned to their individual computers after the group stage, and repeated the same tasks individually. Upon completion of the task, subjects were debriefed and paid.

3. Results 3.1 Reliability To measure the reliability and consistency of the subjects’ decisions we calculated the Mean Absolute Difference (MAD) between the two CEs obtained from the replicated gamble and its price list (p = 0.50, ∆ = 0). Groups made more reliable decisions than individuals (MAD = 0.34 for groups compared with MAD = 0.91 for individuals; aggregated over all three group treatments). We also found that individual decisions made after group discussions are more reliable (MAD = 0.70) than those made before (MAD = 1.00). In the subsequent analysis we averaged the two CEs obtained from the repeated gamble leaving us 15 observations for each individual/group at each stage of the experiment. 3.2 Monotonicity in probabilities Violations of monotonicity are very common (Birnbaum, 1992; Birnbaum, Coffey, Mellers & Weiss, 1992; Birnbaum & Sutton, 1992; Charness et al., 2007). Birnbaum (1992) used a procedure similar to ours to elicit CEs and found that “70% of the subjects showed at least one violation of monotonicity and 50% of the subjects violated monotonicity more often than they satisfied it” (p.312). We find that for 173 of the 240 individual subjects (72%) in all five treatments (excluding stage III decisions) and 39 of the 45 groups (87%) in the three group treatments, monotonicity

15

in

p

was

fully

satisfied

and

we

observed:

CE p = 0.2 ≤ CE p = 0.35 ≤ CE p = 0.5 ≤ CE p = 0.65 ≤ CE p = 0.8 .6 Consistent with the results of Charness, Karni and Levin (2007) groups showed less violations of monotonicity than individuals. 3.3 Group vs. Individual Decisions Table 3 shows the mean individual and group CEs for the 15 gambles (risky and ambiguous) in treatments “GD(shared), GD(shared; counterbalanced) and GD(separate)”. Insert Table 3 about here 3.3.1 Risk attitudes For all probability levels p>0.20, the means of individual- and group CEs of risky gambles were lower than the gambles’ expected values which indicates risk aversion. For p = 0.20, there was no systematic difference between the mean of group and individual CEs and the gamble’s expected value. Group CEs were consistently higher than individual CEs indicating less risk aversion in groups. A 5 (probability level) X 2 (individuals vs. groups7) X 3 (treatment) mixed ANOVA (N=45) revealed that the difference in CEs for risky gambles between individuals and groups was significant, (F[1,42]=11.78, p=0.001). It also showed a significant effect of probability levels on CEs, (F[4,168]=1397.34, p<0.001). CEs did not differ significantly between the three treatments (F[2,42]=0.46, p=0.63) and there was no significant interaction effect between any of the variables (p>0.10 for all interactions).

6 Most violations of monotonicity are caused by comparisons between CEs obtained from gambles with similar probabilities. If we exclude gambles with p = 0.35 and p= 0.65 from the analysis, almost all CEs (239 individuals and 45 groups) are monotonic in p. 7 Individuals are represented by the mean CE of the three group members’ individual decisions.

16

3.3.2 Ambiguity attitudes We computed ambiguity premiums (APs) for each of the 10 ambiguous gambles by subtracting the CE of the ambiguous gamble from the CE of the risky gamble with the same probability level8. Ambiguity premiums greater than (equal to, smaller than) zero indicate ambiguity aversion (neutrality, seeking). We computed the proportion of individual and group decisions which exhibited a particular ambiguity attitude (seeking, neutral, averse) based on this measure, and we classified each DM according to the ambiguity attitude he/she exhibited for the majority of the 10 ambiguous gambles. A DM was classified as ambiguity seeking (neutral, averse) when 6 or more of the DM’s decisions were ambiguity seeking (neutral, averse). About one third (31% of the groups and 35% of the individual DMs) were classified as ambiguity averse; 20% of the groups were ambiguity neutral compared to 11% of the individuals and 8% of the individuals, but none of the groups, were ambiguity seeking; 49% of the groups and 46% of the individuals did not exhibit a dominant ambiguity attitude. Table 4 presents mean ambiguity premiums and the distribution of ambiguity attitudes for each of the 10 ambiguous gambles for individuals and groups across all three conditions. Insert Table 4 about here With only three exceptions, mean individual and group ambiguity premiums were strictly positive indicating ambiguity aversion. A 10 (gambles) X 2 (individuals9 vs. groups) X 3 (treatments) mixed ANOVA (N=45) with ambiguity premiums as the 8

For all 10 ambiguous gambles, ambiguity premiums were positively correlated with risk premiums (mean r=0.50 for groups and mean r=0.42 for individuals). Previous studies have found mixed result concerning the correlation of risk and ambiguity attitudes. Cohen, Jaffray, & Said, 1985, Curley et al., 1986 and Kuhn, & Budescu 1996 report no correlation whereas Lauriola and Levin (2001), Lauriola, Levin and Hart (2007) and Kocher and Trautmann (2010) find risk and ambiguity attitudes to be positively correlated. 9 Individuals are represented by the mean AP of the three group members’ individual decisions.

17

dependent variable did not show a significant difference between individuals and groups (F[1,42]=0.23, p=0.64). Ambiguity premiums were also not significantly different across the three treatments (F[2,42]=0.70, p=0.50). Naturally the ambiguity premium varied significantly across gambles (F[9,378]=14.2, p<0.001). None of the interactions reached significance (p>0.09 for all interactions). The classification of decisions as either ambiguity seeking, ambiguity neutral or ambiguity averse allows for a more detailed analysis. It is remarkable that for all levels of p and of ∆ the proportion of ambiguity neutral decisions was higher for groups than for individuals (table 4). This difference between individuals and groups was significant according to a Wilcoxon Signed-ranks Test (WST), N=45, z=4.41, p<0.00110. This shift towards ambiguity neutrality was caused by a reduction in both ambiguity averse and ambiguity seeking group decisions: 46% of individuals decisions were ambiguity averse but only 36% of all groups decisions showed ambiguity aversion, WST: N=45, z=-2.84, p<0.01; 29% of all individual decisions were ambiguity seeking, compared to only 18% of group decisions, WST: N=45, z= 3.29, p<0.01. Table 5 presents the proportion of ambiguity neutral individual and group decisions for each treatment. The middle column (labeled “median model”) shows the proportion of ambiguity neutral choices which would be predicted if each group had implemented systematically the median individual ambiguity attitude of its members. For example, if for a particular decision one group member made an ambiguity seeking, one an ambiguity neutral and one an ambiguity averse choice the median model prediction for this decision would be ambiguity neutrality. A Wilcoxon Signedranks Test revealed that the proportion of ambiguity neutral decisions was

10

All Wilcoxon- Signed-ranks tests are based on proportions aggregated over all levels of p and ∆

18

significantly higher in groups compared to individuals (WST: N=45, z=4.41, p<0.001) and the median individual DM model (WST: N=45, z=2.47, p=0.02). Insert Table 5 about here This result indicates that the increase in ambiguity neutrality in group decisions cannot be explained by the simple aggregation of group members’ individual ambiguity attitudes. Further evidence for this is provided by an analysis of individual ambiguity attitudes and the corresponding group decision. In 94 gambles at least 2 of the 3 players decided individually in a vagueness seeking fashion, but the modal choice of the group in these cases (51%) was for vagueness neutrality. The most common pattern was to have a majority of vagueness avoiders (200 gambles), yet over one third (39%) of these groups provided ambiguity neutral CEs. In treatments “GD(shared)” and “GD(separate)” individual decisions were followed by the group decisions.

In these two conditions we found that group

decisions were significantly more likely to be ambiguity neutral than individual decisions (46% vs. 26%), WST: z=2.98, p<0.01. It is possible that the shift in group decisions reflect a carry-over or learning.

However, in treatment “GD(shared;

counterbalanced)” the decision order was reversed and we found the same pattern as in the other two treatments. 47% of all group decisions were ambiguity neutral compared to only 22% of individual decisions, WST: z=3.24, p<0.01 (see also table 3). In fact, a Mann–Whitney U test reveals that group decisions made at stage I of the experiment in condition “GD(shared; counterbalanced)” were significantly more likely to be ambiguity neutral than individual decisions made at stage I in conditions “GD(shared)” and “GD(separate)” (N=45; z=-3.31, p<0.01). Thus, the results are not due to learning.

19

We did not find any significant influence of payoff schemes on behavior. Group decisions were more likely to be ambiguity neutral than individual decisions under both shared (45% vs. 25%; WST: N=31 z= 2.35, p=0.02 ) and separate payoffs (46% vs. 28%; WST: N=14: z= 1.82, p=0.07). To smooth out irregularities, the data were also analyzed under specific parametric assumptions. The results are consistent with the non-parametric analysis. A description of the methodology and the detailed results can be found in appendix C.

3.4 Individual decisions before and after group discussion We now turn to comparing risk and ambiguity attitudes before and after the group discussion stage in the two treatments with repeated individual decisions: “IDIN(shared)” and “IDIN(separate)”. Table 6 shows the mean CEs before and after the group discussion for each gamble and in each treatment. Insert Table 6 about here To test for the effect of group discussions on risk attitudes we employed a 5 (probability level) X 2 (before vs. after group discussion11) X 2 (treatment) mixed ANOVA with the CEs of the risky gambles as the dependent variable. There was no significant difference between decisions before and after the group discussion (F[1,33]=2.15, p=0.15). Moreover, there was no significant effect of the treatment (F[1,33]=1.53, p=0.23). Naturally, the probability level had a significant effect on CEs (F[4,132]=1063.56, p<0.001). None of the interactions was significant (p>0.54 for all interactions). We next analyze ambiguity attitudes. We computed ambiguity premiums and classified ambiguity attitudes as either seeking, neutral or averse. A 10 (gambles) X 2 11

Due to the group discussion individual decisions are no longer independent from each other. Therefore, we compare the mean values of the three individual group members’ decisions before and after the group discussion.

20

(before and after group discussion12) X 2 (treatment) mixed ANOVA (N=35) did not show difference in ambiguity premiums before and after the group discussion (F[1,33]=2.42, p=0.13); There was no significant effect of the treatment (F[1,33]=1.96, p=0.17) and no significant interaction between any of the variables (p>0.47). Only the gamble had a significant influence on APs (F[9,297]=15.98, p<0.001). Although APs were not significantly different, we found that after the group discussion individual decisions – just like the group decisions in the group decision making treatments- were likely to shift towards ambiguity neutrality (36% aggregated over all levels of p and ∆ and both treatments) compared to individual decisions before the discussion (26% aggregated over all levels of p and ∆ and both treatments). This difference was significant (WST: N=35, z=2.88, p<0.01). This result seems to stand in contrast to our previous finding that the timing of the group decisions (before or after the individual decisions) did not affect ambiguity attitudes. However, an important difference between the three group decision making treatments and the treatments with repeated individual decision is that in the latter subjects were particularly instructed to take into account the opinions and attitudes of group members when making their subsequent individual decisions. This was not the case in the group decision making treatments in which individual decisions were introduced to subjects as completely independent from the group decisions. We found no effect of the different payoff schemes. In treatment “IDIN(shared)” as well as in treatment “IDIN(separate)” more individual decisions were ambiguity neutral after the discussion with group than before (WST: N=18, z=2.20, p=0.02 and N=17, z=1.76, p=0.08 respectively).

12

See footnote 10).

21

4. General Discussion 4.1 Summary of main findings We compared individual and group decisions made under risk, ambiguity and various payoffs sharing arrangements. In addition, we explored the effects of exposure to other individuals’ opinions and attitudes on subsequent individual decisions. Differences between individuals and groups in their attitudes towards ambiguity have not been addresses systematically in the past and we present three new empirical results: (1) For all probability levels and all levels of ambiguity groups made ambiguity neutral decisions more often than individuals, and more often than predicted by the median preferences in the group; (2) This shift towards ambiguity neutrality is not unidirectional but caused by a reduction in ambiguity seeking and ambiguity aversion in groups; (3) Decisions made after non binding group discussions with other DMs are more ambiguity neutral than decisions before the interaction. Besides our main findings on ambiguity attitudes we confirmed that groups are less risk averse than individuals (see also Zhang & Casari, 2009). However, we did not find strong evidence for an interaction between attitude to risk and the riskiness of the gambles as reported by Baker et al. (2008), Masclet et al. (2009) and Shupp and Williams (2008). 4.2 Explaining the key findings Our three main results for ambiguity attitudes are consistent with two possible explanations. The first is that making a joint group decision or at least having the possibility of interacting with others before the decision reduces the fear of negative evaluation. Curley et al. (1986) as well as Trautmann et al. (2009) have shown that the fear of negative evaluation is an important driver of ambiguity aversion. Although this

22

explanation is consistent with the reduction in ambiguity aversion in groups it cannot explain finding that groups are also significantly less likely to make ambiguity seeking decisions. The second explanation is that ambiguity attitudes are influenced by persuasive arguments in favor of ambiguity neutrality during the group interaction. Persuasive arguments have a strong effect on group decisions and group members’ subsequent individual decisions (Burnstein & Vinokur, 1977). Previous research has shown that ambiguity attitudes are robust to persuasion attempts by the experimenter (MacCrimmon, 1968; Slovic & Tversky, 1974; Curley et al. 1986). However, ambiguity attitudes might be more malleable when persuasive arguments are put forward by group members. If ambiguity neutrality is a persuasive argument during a group discussion this account can explain all three findings: A shift towards ambiguity neutrality in group decisions, reduction in ambiguity averse and ambiguity seeking decisions, and the shift towards higher neutrality in individual decisions after a group interaction. An alternative, simpler, explanation for any group effects is that the members “compromise” and converge to the central and least controversial position. Two results suggest that this is not case in our study: First, we have shown that our groups are systematically and significantly more neutral than the median opinion (which would be expected by a pure compromise model). Secondly, we found that vagueness neutrality is a prominent outcome even when a majority of the participants hold views that are inconsistent with this pattern: Obviously, the group interaction does more than simply aggregating the individual opinions. 4.3 Future directions Future research should explore further to what degree and under what circumstances attitudes towards ambiguity are malleable to persuasion by group

23

members, or other sources, by analyzing the content of the group discussion to determine which arguments individuals put forwards during the group interaction and to what extent they are accepted or rejected by others (see Cooper & Kagel, 2005).. We found no support for our hypothesis that the common fate and shared responsibility (induced by common payoffs) would affect the groups’ attitudes towards ambiguity. However, Sutter (2009) found that payoff commonality had a strong effect on individual decisions and significantly increased risk-taking. Sutter (2009) compared decisions made by individuals with payoff-commonalities to those made by individuals alone. In contrast, in our setting subjects were either members of a group or had prior interactions with others. Thus, the effect of payoff-commonality might have been swamped by the stronger effect arising from group discussions and exposure to other opinions and we did not detect the influence of the different payoff sharing arrangements. Future research should explore further under what circumstances payoff-communality and the perception of common fate they induce in the group affects decisions under risk and vagueness, as well as the possible impact of other social factors such as apprehension of evaluation (Curley et al., 1986; Trautmann et al., 2009) on such decisions. Recently, Abdellaoui et al. (2010) have shown that attitudes towards natural sources of uncertainty such as future temperatures or stock prices can be analyzed in a tractable way while providing new insights into the nature of ambiguity attitudes. Running further studies with individuals and groups considering natural sources of uncertainty would be another possible extension.

24

4.4 Practical implications We pointed out in the introduction that many important decisions are delegated to groups (panels, committees, juries, etc). The common wisdom in the literature is that teams are better in solving intellective tasks (e.g., Laughlin, 1980; Maciejowsky & Budescu, 2007), play more strategically and rationally (closer to the equilibrium) in a variety of games (e.g., Bornstein & Yaniv, 1998; Cooper & Kagel, 2005; Kocher, & Sutter, 2005; Kugler, Kocher, Sutter & Bornstein, 2007). Part of the teams’ success can be attributed to their ability to engage in effective communication and persuasion (e.g., Cooper & Kagel, 2005) that is relevant to the task at hand. Our results suggest those teams could also induce better outcomes in tasks that require proper estimation and use of vague probabilities. Such situations are quite common in a variety of domains such as planning in the face of long – term environmental and climatologically uncertainties, military intelligence, etc. Probably the most familiar domain is insurance that requires the proper estimation of various risks and potential losses and setting proper premiums. In some cases (life insurance for healthy individuals or car insurance) all relevant parameters are well understood and reliably quantified, so insurance underwriters face decisions under risk. However in other cases, such as insurance against unintended side-effects of newly developed technologies, materials, drugs, treatments, etc., the insurers face decisions with vague probabilities (and in some cases, outcomes).

Several studies (Cabantous, 2007;

Cabantous, Hilton, Kunreuther, & Erwann-Kerjan, 2011; Hogarth & Kunreuther, 1989; Kunreuther, Meszaros, Hogarth, & Spranca, 1995; Viscusi & Cheson, 1999) documented the sensitivity of insurance professionals to the source and the perceived (im)precision of the probabilities. The most consistent result is that for low probability events (especially with high consequences) their actions are consistent

25

with vagueness avoidance. The intriguing possibility suggested by our laboratory experiment is that this tendency could be reduced, or maybe totally eliminated, if such decisions were assigned to small groups rather than individuals.

26

Acknowledgments We thank Professor Andrew Schotter for access to the facilities of the Center for Experimental Social Sciences at New York University to run the study, and James Marcus for assistance in data collection.

Funding This research was funded by the INSEAD R&D committee and the INSEAD alumni fund.

27

References Ambrus, A. Greiner, B., Pathak. P. (2009). Group versus individual decision-making: Is there a shift? Working Paper, Harvard University. Abdellaoui, M., Baillon, A., Placido, L., & Wakker, P. (2010). The rich domain of uncertainty: Source functions and their experimental implementation. American Economic Review, forthcoming. Baker, R.J., Laury, S.K., & Williams, A. W. (2008). Comparing group and individual behavior in lottery-choice experiments. Southern Economic Journal, 75, 367– 382. Birnbaum, M. H. (1992). Violations of monotonicity and contextual effects in choicebased certainty equivalents. Psychological Science, 3, 310-314. Bimbaum, M. H., & Sutton, S. E. (1992). Scale convergence and utility measurement. Organizational Behavior and Human Decision Processes, 52, 183-215. Bimbaum, M. H., Coffey G., Mellers, B. A., & Weiss, R. (1992). Utility measurement, configural-weight theory and the judges point of view. Journal of Experimental Psychology Human Perception and Performance, 8, 331346. Bornstein, G., & Yaniv, I. (1998). Individual and group behavior in the ultimatum game: Are groups more “rational” players? Experimental Economics, 1, 101108. Budescu, D. V., Kuhn, K. M., Kramer, K. M., & Johnson, T. (2002). Modeling certainty equivalents for imprecise gambles. Organizational Behavior and Human Decision Processes, 88, 748–768. Burnstein, E.. & Vinokur, A. (1977). Persuasive argumentation and social comparison as determinants of attitude polarization. Journal of Experimental Social Psychology. 13. 315-332. Cabantous, L.(2007). Ambiguity aversion in the field of insurance: Insurers' attitudes to imprecise and conflicting probability estimates, Theory and Decision, 62, 219-240. Cabantous, L.; Hilton, D.; Kunreuther, H.; Erwann-Kerjan, M.(2011). Is imprecise knowledge better than conflicting expertise? Evidence from insurers' decisions in the United States, Journal of Risk and Uncertainty, forthcoming.

28

Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of Risk and Uncertainty, 5, 325-370. Charness, G., Karni, E., & Levin, D. (2007). Individual and group decision making under risk: An experimental study of Bayesian updating and violations of firstorder stochastic dominance. Journal of Risk and Uncertainty, 35, 129–148. Charness, G., & Jackson, M.O. (2009). The role of responsibility in strategic risktaking. Journal of Economic Behavior and Organization, 69, 241-247. Cohen, M., Jaffray, J. Y., & Said, T. (1985). Individual behavior under risk and under uncertainty: An experimental study. Theory and Decision, 18, 203–228. Cooper, D. J., & Kagel, J. H. (2005). Are two heads better than one? Team versus individual play in signaling games. American Economic Review, 95, 477-509. Curley, S. P., Yates, J. F., & Abrams., R. A. (1986). Psychological sources of ambiguity avoidance. Organizational Behavior and

Human Decision

Processes, 38, 230-256. Deck, C., Lee, J.,

Reyes, J., & Rosen, C. (2010). Risk taking behavior: An

experimental analysis of individuals and pairs. Working Paper, University of Arkansas Du, N., & Budescu, D.V. (2005). The effects of imprecise probabilities and outcomes in evaluating investment options. Management Science, 51, 1791-1803. Einhorn, H. J., & Hogarth, R. M. (1986). Decision making under ambiguity. Journal of Business, 59, 225–250. Ellsberg, D. (1961). Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics, 75, 643-669. Etner, J., Jeleva, M., & Tallon, J. M. (2009). Decision theory under uncertainty. Unpublished Manuscript. Feri, F., Irlenbusch, B., Sutter, M. (2010). Efficiency gains from team-based coordination – Large-scale experimental evidence. American Economic Review, 100, 1892–1912. Harrison, G. W., Igel Lau, M., Rutström, E. & Tarazona-Gómez, M. (2005). Preferences over social risk. Unpublished Manuscript. Hogarth, R.M., & Kunreuther, H. (1989). Risk, ambiguity, and insurance. Journal of Risk and Uncertainty, 2, 5-35. Hey, J. D., & Lee, J. (2005). Do subjects separate (or are they sophisticated)? Experimental Economics, 8, 233-265. 29

Kahn, B. E., & Sarin, R. K. (1988). Modeling ambiguity in decision under uncertainty. Journal of Consumer Research, 15, 265–272. Keller, L. R.,

Rakesh K., & Sounderpandian, S. J. (2009). An examination of

ambiguity aversion: Are two heads better than one? Judgment and Decision Making, 2, 390–397. Kerr, N. L., & Tindale, R. S. (2004). Group performance and decision making. Annual Review of Psychology, 55, 623–655. Keynes, J. M. (1921). A Treatise on Probability. London: Macmillan. Knight, F. (1921). Risk, Uncertainty, and Profit. New York: Houghton Mifflin. Kocher, M. G., & Sutter, M. (2005). The decision maker matters: Individual versus group behaviour in experimental beauty-contest games, Economic Journal, 115, 200-223. Kocher, M. G., & Trautmann, S. T. (2010), Selection into auctions for risky and ambiguous prospects. Economic Inquiry, forthcoming. Kugler, T., Kocher, M., Sutter, M., & Bornstein, G. (2007). Trust between individuals and groups: Groups are less trusting than individuals but just as trustworthy. Journal of Economic Psychology, 28, 646-657. Kuhn, K. M., & Budescu, D. V. (1996). The relative importance of probabilities, outcomes, and vagueness in hazard risk decisions. Organizational Behavior and Human Decision Processes, 68, 301–317. Kunreuther, H., Meszaros, J, Hogarth, R.M., & Spranca, M. (1995). Ambiguity and underwriter decision processes, Journal of Economic Behavior and Organization, 26, 337-352. Laughlin, P. R. (1980). Social combination processes of cooperative, problem-solving groups on verbal intellective tasks. In M. Fishbein (Ed.). Progress in Social Psychology. Vol. 1. (pp. 127-155). Hillsdale, NJ: Lawrence-Erlbaum. Lauriola, M., & Levin, I. P. (2001). Relating individual differences in attitude toward ambiguity to risky choices. Journal of Behavioral Decision-Making, 14, 107– 122. Lauriola, M., Levin, I. P., & Hart, S. S. (2007). Common and distinct Factors in decision making under ambiguity and risk: A psychometric study of individual differences. Organizational Behavior and Human Decision Processes, 104, 130−149.

30

Maciejovsky, B., & Budescu, D. V. (2007). Collective induction without cooperation? Learning and knowledge transfer in cooperative groups and competitive auctions. Journal of Personality and Social Psychology, 92, 854-870. Mangelsdorff, L., & Weber, M. (1994). Testing Choquet expected utility. Journal of Economic Behavior Organization, 25, 437–457. MacCrimmon, K. R. (1968). Descriptive and normative implications of the decisiontheory postulates. In Borch, K. & Mossin, J. (Eds.). Risk and Uncertainty. London, MacMillian Masclet D., Colombier N., Denant-Boemont L., & Loheac Y. (2009). Group and individual risk preferences: A lottery-choice experiment with self-employed and salaried workers. Journal of Economic Behavior and Organization, 70, 470-484. Muthukrishnan, A. V., Wathieu, L., & Xu, A. J. (2009). Ambiguity aversion and preference for established brands. Management Science, 55, 1933−1941. Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behaviour and Organization, 3, 323–43. Rockenbach, B., Sadrieh, A. & Matauschek, B. (2007). Teams take the better risks. Journal of Economic Behavior and Organization, 63, 412–422. Savage, L. (1954). The foundations of statistics. New York: Wiley. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57, 571–87. Shupp, R. S., & Williams, A. W. (2008). Risk preference differentials of small groups and individuals. Economic Journal, 118, 258–283. Slovic, P.,l., & Tversky, A. (1974.) Who accepts savage's axiom? Behavioral Science, 19, 368-373. Starmer, C., & Sudgen, R. (1991). Does the random-lottery incentive system elicit true preferences? An experimental investigation. American Economic Review, 81, 971-978. Sutter, M. (2007). Are teams prone to myopic loss aversion? An experimental study on individual versus team investment behavior. Economics Letters, 97, 128132. Sutter, M. (2009). Individual behavior and group membership: Comment. American Economic Review, 99, 2247–225.

31

Trautmann, S. T., Vieider, F. M., & Wakker, P. P. (2008). Causes of ambiguity aversion: Known versus unknown preferences. Journal of Risk and Uncertainty, 36, 225-243. Trautmann, S. T.,

& Vieider F. M. (2011). Social influences on risk attitudes:

Applications in economics. In: Roeser, S. (ed.), Handbook of risk theory, Springer, Chapter 30, forthcoming Viscusi, K.W., & Chesson, H.W. (1999). Hopes and fears: The conflicting effects of risk ambiguity. Theory and Decision, 47, 153-178. Wallach, M. A., Kogan, N., & Bem, D. J. (1964). Diffusion of responsibility and level of risk taking in groups. Journal of Abnormal Social Psychology, 68, 263-274. Zhang, J. & Casari, M. (2009). How groups reach agreement in risky choices. Economic Inquiry, forthcoming.

32

Tables Table 1: Overview of price lists

p 0.20 0.20 0.20 0.20

∆ 0.00 0.05 0.10 0.20

Range of sure amounts of money lowest highest increments $0.50 $7.50 $0.50 $0.50 $7.50 $0.50 $0.50 $7.50 $0.50 $0.50 $7.50 $0.50

0.35

0.00

$0.50

$13.50

$1.00 and $0.50

15

0.50* 0.50 0.50 0.50 0.50

0.00 0.05 0.10 0.30 0.50

$1.00 $1.00 $1.00 $1.00 $1.00

$19.00 $19.00 $19.00 $19.00 $19.00

$1.00 $1.00 $1.00 $1.00 $1.00

19 19 19 19 19

0.65

0.00

$6.50

$19.50

$1.00 and $0.50

15

0.80 0.80 0.80 0.80

0.00 0.05 0.10 0.20

$12.50 $12.50 $12.50 $12.50

$19.50 $19.50 $19.50 $19.50

$0.50 $0.50 $0.50 $0.50

15 15 15 15

No. of choices 15 15 15 15

Notes: *presented twice

33

Table 2: Experimental treatments

Condition GD(shared)

Payoffs shared

Stage I Individual Decisions

Stage II Group Decisions

GD(shared; shared counterbalanced)

Group Decisions

Individual Decisions

GD(separate)

separate

Individual Decisions

Group Decisions

IDIN(shared)

shared

Individual Decisions

Group Discussion

IDIN(separate)

separate

Individual Decisions

Group Discussion

No. of Groups 16

No. of participants 48

15

45

14

42

Individual Decisions

18

54

Individual Decisions

17

51

Stage III

34

Table 3: Certainty equivalents (CEs) for the three group treatments `

p 0.20 (EV=4) 0.20 0.20 0.20

∆ 0.00 0.05 0.10 0.20

GD(shared) Individuals Groups (N=48) (N=16) 3.79 (0.70) 4.06 (0.81) 3.90 (0.62) 4.16 (0.82) 3.54 (0.72) 4.22 (0.74) 3.64 (0.66) 4.00 (0.80)

GD(shared; counterbal.) Individuals Groups (N=45) (N=15) 3.65 (1.27) 4.18 (1.52) 3.33 (1.20) 3.67 (1.77) 3.45 (1.45) 3.43 (1.98) 3.73 (1.11) 3.58 (1.84)

GD(separate) Individuals Groups (N=42) (N=14) 4.33 (0.80) 4.07 (1.34) 4.33 (0.62) 4.25 (1.02) 4.29 (1.01) 4.00 (1.01) 4.55 (0.85) 3.86 (0.9)

0.35 (EV=7)

0.00

5.67 (1.48)

5.94 (1.71)

5.14 (1.53)

5.70 (2.31)

5.95 (1.08)

5.71 (1.89)

0.50 (EV=10) 0.50 0.50 0.50 0.50

0.00 0.05 0.10 0.30 0.50

8.94 8.83 8.65 8.54 7.33

9.63 9.38 9.44 8.75 8.28

9.18 8.81 8.69 7.99 7.57

9.73 9.77 9.53 9.30 8.53

9.31 8.95 8.92 8.70 8.12

9.75 9.93 9.57 8.71 8.57

(1.41) (1.56) (1.46) (1.56) (2.27)

(1.09) (1.20) (1.39) (1.65) (1.76)

(1.48) (1.85) (1.49) (2.02) (2.51)

(1.76) (1.78) (1.87) (2.08) (2.89)

(1.56) (1.57) (1.65) (2.04) (1.83)

(1.24) (1.28) (1.73) (2.01) (2.43)

0.65 (EV=13) 0.00

10.80 (1.51)

11.75 (2.24)

11.42 (1.48)

12.12 (2.40)

11.32 (1.59)

12.48 (1.44)

0.80 (EV=16) 0.80 0.80 0.80

15.00 14.80 15.24 14.50

15.25 15.38 15.34 14.88

15.74 15.59 16.02 14.83

16.08 16.05 16.42 15.40

15.68 15.45 15.75 14.96

15.93 15.68 15.57 15.39

0.00 0.05 0.10 0.20

(0.97) (0.88) (0.79) (0.70)

(0.52) (0.76) (0.88) (0.87)

(1.29) (1.36) (1.09) (1.02)

(1.46) (1.35) (1.22) (1.71)

(1.36) (0.81) (1.06) (0.94)

(0.95) (0.94) (0.91) (0.91)

Notes: Mean values; standard deviation in parentheses.

35

Table 4: Ambiguity premiums and attitudes aggregated over all three group treatments

Individuals (N=135)

Groups (N=45)

Am. Attitudes* p ∆ 0.20 0.05 0.20 0.10 0.20 0.20 Total p=0.20

A 0.36 0.43 0.36 0.38

N 0.31 0.30 0.28 0.30

S 0.33 0.27 0.36 0.32

0.50 0.05 0.50 0.10 0.50 0.30 0.50 0.50 Total p=0.50

0.39 0.45 0.56 0.60 0.50

0.30 0.24 0.16 0.21 0.22

0.31 0.31 0.28 0.19 0.27

0.80 0.05 0.80 0.10 0.80 0.20 Total p=0.80

0.42 0.38 0.61 0.47

0.30 0.23 0.18 0.24

0.27 0.39 0.21 0.29

Total overall

0.46 0.25 0.29

Am. Attitudes* AP** 0.07 (0.66) 0.17 (0.80) -0.04 (0.97)

A 0.20 0.22 0.33 0.25

N 0.62 0.62 0.44 0.56

S 0.18 0.16 0.22 0.19

AP** 0.08 (0.67) 0.22 (1.16) 0.29 (0.84)

(0.99) (0.95) (1.22) (1.62)

0.20 0.27 0.58 0.64 0.42

0.60 0.49 0.24 0.24 0.39

0.20 0.24 0.18 0.11 0.18

0.02 0.19 0.78 1.24

0.19 (0.73) -0.20 (0.82) 0.71 (0.75)

0.31 0.27 0.56 0.38

0.53 0.44 0.36 0.44

0.16 0.29 0.09 0.18

0.27 0.39 0.73 1.48

(0.56) (0.99) (1.36) (1.79)

0.04 (0.70) -0.03 (0.79) 0.53 (1.16)

0.36 0.46 0.18

Notes: *Proportion of decisions which are ambiguity averse (A), ambiguity neutral (N) or ambiguity seeking (S); **Mean ambiguity premiums; standard deviation in parentheses.

36

Table 5: Proportion of ambiguity neutral decisions by treatment

p ∆ 0.20 0.05 0.20 0.10 0.20 0.20 Total p=0.20

GD(shared) Median Model Ind. Groups (N=48) (N=16) (N=16) 0.29 0.38 0.56 0.29 0.38 0.63 0.31 0.44 0.44 0.30 0.40 0.54

GD(shared; counterbal.) Median Model Ind. Groups (N=45) (N=15) (N=15) 0.31 0.47 0.53 0.24 0.40 0.67 0.24 0.33 0.40 0.27 0.40 0.53

GD(separate) Median Model Ind. Groups (N=42) (N=14) (N=14) 0.33 0.50 0.79 0.36 0.43 0.57 0.29 0.43 0.50 0.33 0.45 0.62

0.50 0.05 0.50 0.10 0.50 0.30 0.50 0.50 Total p=0.50

0.38 0.23 0.15 0.17 0.23

0.50 0.31 0.31 0.13 0.31

0.50 0.44 0.13 0.19 0.31

0.27 0.27 0.09 0.18 0.20

0.13 0.27 0.07 0.20 0.17

0.67 0.47 0.33 0.20 0.42

0.24 0.21 0.24 0.29 0.24

0.43 0.36 0.29 0.29 0.34

0.64 0.57 0.29 0.36 0.46

0.80 0.05 0.80 0.10 0.80 0.20 Total p=0.80

0.27 0.23 0.17 0.22

0.31 0.31 0.25 0.29

0.56 0.50 0.56 0.54

0.27 0.18 0.18 0.21

0.47 0.40 0.27 0.38

0.60 0.53 0.27 0.47

0.38 0.29 0.19 0.29

0.64 0.43 0.36 0.48

0.43 0.29 0.21 0.31

Total overall

0.25

0.33

0.45

0.22

0.30

0.47

0.28

0.41

0.46

37

Table 6: Certainty equivalents for treatments IDIN (shared) and IDIN(separate)

p 0.20 (EV=4) 0.20 0.20 0.20

∆ 0.00 0.05 0.10 0.20

IDIN(shared) (N=54) Bef. After 3.98 (0.97) 4.07 (1.08) 4.23 (1.06) 3.96 (1.28) 3.84 (1.10) 4.02 (1.25) 4.14 (1.03) 4.12 (1.22)

IDIN(separate) (N=51) Bef. After 3.91 (0.59) 3.62 (0.64) 3.85 (0.44) 3.65 (0.52) 3.71 (0.57) 3.50 (0.64) 3.80 (0.50) 3.68 (0.65)

0.35 (EV=7)

0.00

5.86 (1.97)

5.95 (2.09)

5.42 (1.29)

5.13 (0.74)

0.50 (EV=10) 0.50 0.50 0.50 0.50

0.00 0.05 0.10 0.30 0.50

9.71 9.19 9.41 8.91 8.70

9.34 9.26 8.98 8.61 8.54

9.34 8.74 8.87 8.21 7.74

8.98 8.62 8.46 7.79 7.81

(1.68) (1.72) (1.81) (1.82) (2.11)

(1.97) (2.06) (2.11) (1.97) (2.26)

(1.35) (1.47) (1.32) (1.61) (1.72)

(1.20) (1.27) (1.25) (1.53) (1.69)

0.65 (EV=13) 0.00

12.11 (1.94)

12.13 (2.20)

11.47 (1.40)

11.20 (2.01)

0.80 (EV=16) 0.80 0.80 0.80

16.13 15.60 15.97 15.04

15.76 15.71 15.87 15.34

15.69 15.36 15.35 14.93

15.58 15.19 15.61 14.89

0.00 0.05 0.10 0.20

(1.11) (1.01) (1.23) (1.08)

(1.39) (1.34) (1.39) (1.31)

(1.25) (1.40) (1.31) (1.14)

(1.43) (1.34) (1.61) (1.19)

Notes: Mean values; standard deviation in parentheses.

38

Appendix A: Examples of price lists Picture 1: Price List (p=0.5, ∆=0)

Picture 2: Price List (p=0.5, ∆=0.5)

39

Appendix B: Results from two and three-color Ellsberg tasks In the two-color Ellsberg task participants were offered a gamble that paid $20 upon drawing a red chip from a particular urn. Participants had to choose between draw from an urn with 50 red and 50 black chips, or from an urn containing 100 red and black chips in unknown proportion. In the three-color task participants were asked to make two binary choices between two different gambles described below. All gambles paid $10 upon drawing a ball of a certain color from an urn containing 90 chips in total out of which 30 were red. The remaining chips were either black or yellow in unknown proportion: Choice 1: Participants made a binary choice between: Gamble A: Pays $10 upon drawing a black chip Gamble B: Pays $10 upon drawing a red chip Choice 2: Participants made a binary choice between: Gamble C: Pays $10 upon drawing either a red or a yellow chip Gamble D: Pays $10 upon drawing either a black or a yellow chip Participants made two choices between two different urns and thus could exhibit four different choice patterns: two of them are consistent with subjective expected utility (SEU) and two of them not. In order to facilitate our analysis we focus on differences in SEU consistency between groups and individuals rather than analyzing differences in the four different choice patterns separately. In the two-color problem, 4 out of 45 groups (9%) chose the ambiguous urn, compared to 19 out of 135 individual decision makers (14%). There is no significant difference between group decisions and the number of choices of the ambiguous urn predicted by the majority preference within the group (4 out of 45 [9%]), Sign test: N=45, p=1.00. In the repeated individual decision conditions, 20 out of 105

40

individuals (19%) chose the ambiguous urn before the group discussion and 16 out of 105 after the discussion (15%), Sign test: N=35, p= 0.43. For the three-color task groups (26 out of 45 [58%]) make SEU consistent decisions slightly more often compared to individuals (66 out of 135 [49%]). This difference is not significant, Sign test: N=45 p=0.74. The proportion of individuals making SEU inconsistent choices is slightly higher after compared to before the group discussion (54 out of 105 [51%] after the discussion vs. 44 out of 105 [42%] before the discussion) but this difference is not significant, Sign test: N=35 p=0.08. In the two-color Ellsberg tasks participants were forced to choose between a risky and an ambiguous option. There was no possibility for participants to express ambiguity neutrality (in this case indifference). Similarly in the three-color tasks participants made forced choices between either a risky and an ambiguous gamble or between two ambiguous gambles. Again there was no way for participants to express ambiguity neutrality by indicating indifference. Thus it is not surprising that we did not find the same shift away from ambiguity aversion and ambiguity seeking towards ambiguity neutrality as demonstrated in the main body of the paper.

41

Appendix C: Parametric analysis of group and individual decisions We adopted the most popular model of choice for risk and ambiguity, i.e. the rankdependent models (Quiggin, 1982; Schmeidler, 1989). According to these models the evaluation of the risky gamble L = ( x , p; 0) (receiving x with probability p, zero otherwise) is given by: W ( p ) ⋅ u ( x ) + (1 − W ( p )) ⋅ u (0)

(1)

where W(.) is a probability weighting function, and u(.) a utility function. For ambiguous gambles L = ( x, p ± ∆; 0) (receiving x with a probability interval of [p+∆, p-∆], 0, otherwise) we assumed that gambles are evaluated according to: W ( f ( p ± ∆ )) ⋅ u ( x ) + (1 − W ( f ( p ± ∆ ))) ⋅ u (0) (2)

where f ( p ± ∆ ) captures DMs’ attitude towards ambiguity. Thus the difference between f ( p ± ∆ ) and p gives a measure of a DMs’ ambiguity attitude. In particular: f ( p ± ∆ ) = p
the decision is ambiguity neutral f ( p ± ∆ ) > p
the decision is ambiguity seeking f ( p ± ∆ ) < p
the decision is ambiguity averse

For utility we assumed a power function: u ( x) = x α α > 0. For the probability weighting we assumed the one parameter function suggested by Prelec (1998) w( p) = exp[−(− ln( p)) γ ] . With x=20 and u(0)=0, the evaluation is: exp[ −( − ln( p i )) γ ] ⋅ 20 α

(3) for risky gambles

and exp[−(− ln( f ( pi∆ )))γ ] ⋅ 20α

(4) for ambiguous gambles

For each risky gamble we have: 42

CE iα = exp[ −( − ln( pi ))) γ ] ⋅ 20α (5)

Rearranging and taking logs repeatedly this becomes: ln( − ln(

CE i )) = ln(1 / α ) + γ ( − ln( p i )) (6) 20

Using the CEs for the 5 risky gambles we estimated values for α and γ in equation 6 with a standard OLS regression. Based on these estimates we then used the CEs of the ambiguous gambles to directly calculate the values of f ( p ± ∆ ) for each of the ambiguous gambles. We also computed the absolute distance between p and f ( p ± ∆ ) This distance measures how sensitive DMs are to ambiguity. Consistent with our results in section 3.3.1 the mean value of α for groups and individuals was significantly smaller than 1 for individuals (Mα=0.91; t(44), p<0.01), but not for groups (Mα=0.95, t(44), p=0.08). Thus, the utility function was slightly less concave for groups than for individuals, WST: N=45, z=1.89, p=0.06. We did not find a significant difference between individuals and groups in the shape of the probability weighting function (Mγ=1.00 for groups and Mγ=0.99 for individuals; WST: N=45, z=0.01, p=0.99). Table 7 shows the mean values of f ( p ± ∆ ) and |p - f ( p ± ∆ ) | for individuals and groups. For all levels of p and ∆, the values of f ( p ± ∆ ) for groups were closer to p than the values of f ( p ± ∆ ) for individuals. We counted the number of decisions for which |( f ( p ± ∆ ) Group - pi)| < [( f ( p ± ∆ ) Gr..Member 1 - pi ) + ( f ( p ± ∆ ) Gr..Member 2 - pi )+ ( f ( p ± ∆ ) Gr..Member 3 - pi )] / 3 . and found that for 70.4% of all decisions (across all values of p and ∆) the value of f ( p ± ∆ ) for group decisions was closer to p than for the group members’ individual decisions. A Wilcoxon Signed-rank Test confirmed the significance of this result (N=45, z=4.78, p<0.001).

43

Table 7: Mean values of f ( p ± ∆ ) and | p − f ( p ± ∆ ) | aggregated over the three group treatments.

Individuals (N=135)

Groups (N=45)

∆ p 0.20 0.05 0.20 0.10 0.20 0.20 Total p=0.20

f ( p ) | pi − f ( p ) |

f ( p ) | p i − f ( p i∆ ) |

0.23 0.22 0.23 0.23

0.06 0.06 0.07 0.06

0.22 0.21 0.21 0.21

0.04 0.05 0.05 0.05

0.50 0.05 0.50 0.10 0.50 0.30 0.50 0.50 Total p=0.50

0.49 0.49 0.47 0.43 0.47

0.06 0.06 0.09 0.11 0.08

0.51 0.50 0.47 0.45 0.48

0.04 0.05 0.05 0.08 0.05

0.80 0.05 0.80 0.10 0.80 0.20 Total p=0.80

0.80 0.82 0.78 0.80

0.04 0.05 0.05 0.05

0.80 0.81 0.78 0.80

0.03 0.03 0.04 0.03

Total overall

---

0.07

---

0.05

∆ i

∆ i

∆ i

44