Group Decision Making under Vagueness

Group Decision Making under Vagueness

_______________

Steffen KECK Enrico DIECIDUE David BUDESCU 2010/79/DS

Group Decision Making under Vagueness

Steffen Keck*

Enrico Diecidue**

David Budescu***

31 July 2010

* PhD Candidate in Decision Sciences at INSEAD, Boulevard de Constance 77305Fontainebleau Cedex Ph: 33 (0)1 60 72 91 17 Email: [email protected]

** Associate Professor of Decision Sciences at INSEAD, Boulevard de Constance 77305

Fontainebleau Cedex Ph: 33 (0)1 60 72 44 47 Email: [email protected] *** Anne Anastasi Professor of Psychometrics and Quantitative Psychology at Department of

Psychology, Fordham University, Dealy Hall, Bronx, New York 10458, USA Ph: (1) 718 817 3786 Email: [email protected]

A Working Paper is the author’s intellectual property. It is intended as a means to promote research to interested readers. Its content should not be copied or hosted on any server without written permissionfrom [email protected] Click here to access the INSEAD Working Paper collection

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Abstract

We report results of an experiment in which participants provided certainty equivalents for 15 risky or vague (with imprecise probabilities) two-outcome gambles. Participants made their decisions in three different settings: a) individually without prior social interactions, b) individually after discussing decisions with other participants and c) in groups of three. We also manipulated the degree of payoff communality between participants: Either all group members received the same payoff resulting from a decision or payoffs were allowed to differ depending on the outcomes of the gambles. Our results do not show a significant influence of payoff communality on either attitudes towards risk or vagueness. However, we find a significant effect of discussions with others and group decision making. Groups are more likely to make vagueness neutral decisions than individuals and individuals make more vagueness neutral decisions after discussing the decisions with others. We conclude that vagueness neutrality is a persuasive argument in group discussions which significantly affects vagueness attitudes of groups and individuals. Keywords: Risk; Vagueness; Ambiguity; Group Decision Making.

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1. Introduction

Managers in organizations can rarely anticipate the exact consequences of

their actions. Often they might not even be able to make precise judgments

concerning the probabilities with which various outcomes will occur. Consider for

example an executive who needs to decide whether to launch a new product to the

market, or whether to invest money in a research project exploring an innovative but

untested technology. In both cases the Decision Maker (DM) cannot assign precise

probability estimates to the likelihood of failure or success of these ventures. To cope

with such deep uncertainties managers usually seek advice from experts and consult

with peers before deciding on a course of action. Often critical decisions are delegated

to groups of decision makers (for example committees, juries and boards of directors).

Motivated by these issues, our study explores the effects of discussing decisions with

others and the need to aggregate individual preferences into a group decision in the

presence of either risk (probabilities of outcomes are precisely defined) or vagueness

(probabilities of outcomes are imprecise)1.

Starting with Ellsberg (1961) numerous studies have shown that when

probabilities of outcomes are only vaguely specified individuals’ decisions cannot be

reconciled with classical Subjective Expected Utility (SEU) theory (Savage, 1954).

The most common finding in the literature is that individuals act as if they are averse

to vagueness. As a consequence, when evaluating vague gambles or investment

decisions they demand an additional “vagueness premium” on top of the normal risk

premium (for an overview of experimental findings see Camerer & Weber, 1992;

Etner, Jeleva & Talon, 2009). Several studies found that individuals’ attitudes towards

1 While we prefer the term vagueness, following Ellsberg (1961) such situations are also often referred to as ambiguous.

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vagueness and attitudes towards risk are not closely related (Cohen, Jaffray & Said,

1985; Curley, Yates & Abrams, 1986; Hogarth & Einhorn, 1990; Kuhn & Budescu,

1996). Recent findings also suggest the existence of a separate neural brain systems to

evaluate different levels of risk and vagueness (Hsu et al., 2005).

Although aversion to vagueness remains the most common finding, previous

studies have reported diversity in vagueness attitudes. Keren & Gerritsen (1999)

report that for small probabilities of gains and large probabilities of losses most

individuals exhibit vagueness seeking rather than vagueness averse attitudes. Budescu

et al. (2002), who analyze certainty equivalents for gambles with imprecise

probabilities and imprecise outcomes, find no modal attitude towards vagueness in

probabilities and even vagueness seeking for imprecision in outcomes. Du & Budescu

(2005) demonstrate that attitudes towards vagueness in general are malleable and

depend strongly on factors such as its source (probabilities or outcomes), the choice

domain (gains vs. losses) and response modes (pricing or choice).

While vagueness attitudes in individual decisions have been widely studied,

only a very small number of studies have investigated the effects of social and

organizational context on decisions under vagueness. Curley, Yates & Abrams (1986)

found that individuals who were observed by uninvolved others during their decisions

exhibited significantly more vagueness aversion than DMs who made their decisions

alone. They attribute this finding to the participants’ fear of being evaluated

negatively in case the chosen vague alternative leads to undesirable outcomes.

Trautmann, Vieider & Wakker (2008) report results of an experiment in which the

participants’ preferences over outcomes where unknown to the experimenters, so they

could completely rule out the possibility of a negative evaluation by others. This

manipulation significantly decreased vagueness aversion compared to a situation in

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which preferences were known by the experimenter, supporting the interpretation

proposed by Curley et al. (1986). Although both studies involved individual decision

making, their results suggest that the interaction with others either before an

individual decision or as part of group decision making procedure could influence

DMs’ attitudes to vagueness.

Starting with Stoner (1961) a large number of studies have explored the effects

of group discussions and aggregation of individual preferences on attitudes towards

risk (for a comprehensive overview see Isenberg, 1986). Although, attitudes towards

risk and vagueness are not necessarily related, the theoretical frameworks developed

to analyze differences between individuals and groups with respect to risk-taking

offer some guidance in identifying factors likely to influence attitudes towards

vagueness. In this paper we consider two of these factors which we believe to be of

particular importance for the context of vagueness:

Diffusion of responsibility: Wallach, Kogan & Bem (1964) show that “group

decisions bring about a diffusion of responsibility” among group members which

pushes decisions towards more risk-taking. Given that most individuals are averse to

vagueness, one would expect a similar effect, i.e., less vagueness aversion, when

responsibility for decisions is shared with others.

Persuasive Arguments: Several studies have shown that group members

change their individual preferences when confronted with convincing arguments

which persuade them to do so (Bishop & Myers, 1974; Burnstein, 1982; Vinokur &

Burnstein, 1978). This result is in line with findings form a number of experimental

results on strategic interaction situations which indicate that group decisions are more

consistent with economic rationality than individual decisions. (Bornstein & Yaniv,

1998; Cooper & Kagel, 2005; Kocher & Sutter, 2005).

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In the context of our study we highlight two, possibly complimentary,

persuasive arguments. First, maximizing expected value might be perceived as a

normative (“correct”) way of making decisions by participants and thus be persuasive

in a group discussion. If this is the case, groups should be both more risk- and

vagueness- neutral than individuals. Furthermore, we suggest that arguments in favor

of vagueness-neutrality might be persuasive even if risk-attitudes are non-neutral.

Unlike risk preferences which are to a large extent influenced by preferences over

possible outcomes, vagueness attitude is concerned with the precision of probabilities.

It is natural and intuitive to assume (although this is typically not spelled out) that all

values in the range are equally likely (e.g., Fox & Rottenstreich, 2003; Seale,

Rapoport & Budescu, 2005) and we expect that DMs will perceive averaging over all

possible values in the range as a compelling way to resolve the imprecision and reach

a decision. A number of studies has tested the effect of persuasive arguments for

vagueness neutrality on individuals’ vagueness attitudes (MacCrimmon, 1968; Slovic

& Tversky, 1974; Curley et al. 1986). Contrary to our hypothesis their results showed

that in spite of exposure to rational arguments for vagueness neutrality vagueness

attitudes remained mainly unchanged. However, in all these studies the arguments

were put forward by the experimenter and not by other participants during a group

discussion which might be have a considerably stronger effect than persuasion

attempts by an experimenter.

Keller, Sarin & Sounderpandian, (2009) is, to our knowledge, the only study

of group decision making under vagueness. They compared the willingness of

individuals and dyads to pay for risky and vague gambles. Dyads tended to be more

risk averse than their individual members, but there was no difference with respect to

vagueness attitudes. Their interesting study is limited in a number of important ways.

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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 vagueness-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 an experiment in which participants made binary choices

between sure amounts of money and different risky and vague gambles. Each gamble

offered the possibility of winning $20 (and receiving $0 otherwise) with varying

probabilities (p = 0.20, 0.35, 0.50, 0.65, 0.80) and different levels of vagueness

operationalized by symmetric spreads around these probabilities (∆ = 0, ±0.05, ±0.10,

±0.20, ±0.30, ±0.50).

We distinguish between individual decisions, individual decisions made after

exchanging information with others, and group decisions. This distinction allows us to

disentangle two effects which are typically confounded in studies of group decisions.

The first is the influence of the group discussion and exchange of arguments in favor

or against a certain decision (see section on “Persuasive Argument Theory”). This

factor affects both, individual decisions after a group discussion and standard group

decisions. The second effect is the process of aggregation of individual preferences

into a group decision, which is present only in the group decisions.

We also examine another aspect of group decisions which has not been studied

systematically – the payoff sharing arrangement among the group members. In some

instances the outcome of a group decision affects all individuals involved identically,

i.e., all participants share the benefits (or the costs) equally. We refer to this situation

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as the group sharing a “common fate”. In other situations the outcome of a group

decision varies across the individual members. In some extreme cases there could be

positive outcomes for some individuals and negative for others. Wallach & Kogan

(1964) argue that an essential component for the emergence of a “shared

responsibility” for decisions among group members is that consequences of the group

decisions are shared by all members equally and no member can escape the

consequences of a bad decision. Following this line of reasoning we hypothesize that

the nature of the payoff sharing arrangement – equally or differentially – will

influence the degree of responsibility individuals feel for the group decision and,

according to the “Diffusion of Responsibility Theory” will affect risk and vagueness

attitudes. Sutter (2009) found that individuals invested significant higher amounts of

money in a risky investment if they shared their payoffs with other participants,

providing support for this hypothesis for the case of risk.

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 in a law-firm 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.

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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.

Our experimental design is similar to studies by Shupp & Williams (2008) and

Baker, Laury & Williams (2008) both of which compared risk attitudes of groups of

three with attitudes of individuals. Both studies used two-outcome gambles which

offered the possibility of winning a fixed monetary prize with varying probabilities.

Group decisions were made after an unstructured face-to-face discussion between all

members and participants’ choices had real financial consequences. We use the same

approach in out study.

Shupp & Williams (2008) elicited certainty equivalents (CEs) for gambles

from both groups and individuals and found that the CEs of the groups were

significantly lower that those of individuals’ for gambles with low probabilities of

winning, significantly higher than the individuals’ CEs for gambles with high winning

probabilities, and not significantly different for gambles with medium winning

probabilities. Baker et al. (2008) used a risk-taking measure adapted from Holt &

Laury (2002) and in a within-subject comparison found results consistent with Shupp

& Williams (2008). Baker et al. (2008) also asked participants to make another round

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of individual decisions after the group decisions, and found that individual risk

attitudes shifted significantly in the direction of the risk attitude exhibited in the group

decision.

We go beyond these studies by examining decisions made under vagueness, in

additions to decision under risk. This allows us to extend previous research on

differences between individuals and groups with respect to risk-taking to the setting of

vagueness, which is often encountered in practice. Furthermore, we are able to

compare results for individual decisions from the literature on vagueness attitudes,

with decisions made in groups and by individuals after being exposed to the opinions

and preferences of others.

2. Experimental Method

2.1 Experimental Tasks

DMs in our experiment made binary choices between sure amounts of money

and 15 two-outcome gambles that were either risky (probabilities of outcomes defined

precisely) or vague (probabilities of outcomes presented as a range of possible

values). Each gamble offered participants the possibility of winning $20 with a

probability, p (or a range of probabilities, p ± ∆), or receiving $0. Gambles differed

from each other with respect to the probability of winning (p = 0.20, 0.35, 0.50, 0.65

and 0.80) and the level of imprecision ∆ (∆ = 0.05, 0.10, 0.20, 0.30, and 0.50) of the

probabilities. Not all possible combinations of p and ∆ are feasible. For example we

had 5 different gambles with the p = 0.50 that offered participants 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. But probabilities p = 0.20 and

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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.

All choices were presented to participants in the form of a “decision sheet.”

Sheets 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 choice a DM should always prefer the gamble and

for the last choice always prefer the sure amount of money. The 15 decision sheets

were presented in randomized order. As a consistency check we presented one of the

decision sheets (p = 0.50, ∆ = 0) twice. Table 1 summarizes the characteristics of the

15 gambles and the decision sheets they were presented on. Examples of decision

sheet for risky and vague gambles can be found in the appendix2.

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Insert Table 1 about here

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We inferred CEs for each gamble from the switching point on each decision

sheet, which is the point at which DMs (individuals or groups) switched from

preferring the gamble to preferring the sure amounts. 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 decision sheet a participant

2 In addition to the 15 decision sheets, 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 tasks and a brief d overview of our findings in the appendix.

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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=$8.503,4.

To speed up the decision process participants were offered the option of using

computer-assistance in making their decisions. The computer assistance automatically

filled in all choices located on the decision sheet above the choice for which a

participant preferred the sure amount of money over the gamble. Analogously, the

computer assistance filled in the choices below the choice for which a participant

preferred the gamble over the sure amount of money. For example, if a participant

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

participant indicated a preference for the gamble over $9 the computer assistance

automatically determined that she also preferred the gamble over all sure amounts <

$9. Used effectively the computer assistance allowed participants to complete a

particular decision sheet with only two clicks by indicating her switching point.

Participants could deactivate the computer assistance at any time they wished.

2.2 Participants

We recruited a total of 240 undergraduate students from New York University

(90 male, 150 female) by a mass e-mail announcement. Students varied widely with

3 In 16 (out of the 6800 sheets = 0.2%) cases participants had multiple switches. In these cases we use the first (lower) switching points on the decision sheet to calculate the CEs. 15 of these 16 cases were caused by three individuals who had multiple switches on several decision sheets. 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. 4 In 253 (out of the 6800 sheets = 3.7%) cases DMs (individuals or groups) 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 DM would have switched at the next item that could have been listed on the decisions sheet. For example if a DM preferred a gamble to win $20 with p=0.20 over all amounts between $0 and $7.50 (the upper end on this decision sheet) we assume that the DM 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).

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respect to their majors. Average age of the participants was 20.7 years. All sessions

were run at the NYU Center for Experimental Social Science in April 2009. Subjects

were paid a $5 show-up fee plus what they won (individually or in groups) during the

experiment and earned on average $23.2.

2.3 Experimental Design

Each participant was randomly assigned to one of the 5 experimental

conditions summarized in table 2.

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The experiment consisted of two stages (individual and group decision

making) in conditions “GD shared” , “GD shared (reversed)” and “GD separate” and

of three stages (individual decisions, group decisions and second round of individual

decisions) in conditions “IDIN shared” and “IDIN separate”. In most groups

participants started the experiment with the individual decision making stage. At this

stage participants made their choices on all 16 (15+1 repeated) decision sheets. This

stage was followed by a group stage and in conditions “IDIN shared” and “IDIN

separate” by another round of individual decision making. To control for possible

order-effects, we ran a condition “GD shared (reversed)” where we reversed this order

and asked participants to make their decisions first as a group and then individually.

To determine final payment for participants we employed the random

incentive system (Starmer & Sugden, 1991; Hey & Lee, 2005). One of the choices

made at each stage (except the group decisions in conditions “IDIN shared” and

“IDIN separate”) was randomly selected at the end of the experiment and participants

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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 an urn filled with chips). For all gambles drawing a red chip resulted in

winning the $20 and drawing a black chip in winning nothing. As explained to

participants 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.

While the individual decision making stage was identical in all conditions, the

group stage varied across conditions in the following way:

Condition “GD shared” (Group decisions with shared outcomes): After

finishing the first round of individual decisions, participants were randomly assigned

to groups of three. Each group completed all of the decision sheets. Group members

had to make a joint decision about how to fill in the decision sheets. 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. Participants were informed that their

payoffs for the group stage (which was 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.

Condition “GD shared (reversed)” (Group decisions with shared outcomes and

reversed order): The condition was identical to the “GD shared ” but the order in

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which individual and group decisions were made was reversed. Participants were

assigned to groups of three and completed tasks as a group first, followed by the same

tasks individually.

Condition “GD separate” (Group decisions with separate outcomes): This

condition differs from condition “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.

Condition “IDIN separate” (Individual decisions after group interaction with

separate outcomes): After finishing their first round of individual decisions,

participants were assigned to groups of three. Group members completed the same

tasks as at the individual stage, but the group decisions did not affect their final

payoffs. The group stage only served to expose all subjects to the other group

members’ opinions. Participants 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 participant’s decision.

Condition “IDIN shared” (Individual decisions after group interaction with

shared outcomes): This condition differs from condition “GD separate” only in the

way final payoffs for the third stage was determined. One choice of a randomly

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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 reference. 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 participants fully understood the instructions, they

were required to pass a brief quiz before being allowed to start the experiment.

Participants were also encouraged to ask questions at any time they wished.

After all subjects completed the decision sheets 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 conditions “GD

shared, GD shared (reversed) and GD separate” the experiment ended after the second

stage and all group members were paid and debriefed. In condition “GD shared

(reversed)” all 3 participants started the experiment at their group computers, and then

moved to the individual computers. In conditions “IDID shared” and “IDID separate”

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subjects returned to their individual computers after the group stage, and repeated the

same tasks individually. After they had finished subjects were debriefed and paid.

3. Results

3.1 Overview

Reliability

To get a measure of the reliability of participants’ choices we calculate for

each DM the Mean Absolute Difference (MAD) between the two CEs for the

replicated decision sheet (p = 0.50, ∆ = 0). Groups made more reliable decisions than

individuals (MAD = $0.34, SD = $0.07 compared with MAD = $0.91, SD = $1.38),

and individual decisions made after the group discussion are more reliable (MAD =

$0.70, SD = $1.25) than those made before (MAD = $1.00, SD = $1.43). In the

subsequent analysis we average the two CEs obtained for this decision sheet. This

leaves us with 15 observations for each individual/group at each stage of the

experiment.

Monotonicity

To test for monotonicity of participants’ choices we compare the CEs for the

five risky gambles (∆ = 0). For a DM with monotonic preferences we should observe:

%80%65%50%35%20 ===== ≤≤≤≤ ppppp CECECECECE . Violations of monotonicity are

quite common (Birnbaum et al. 1992; Birnbaum & Sutton, 1992; Charness, Karni &

Levin, 2007). For example, 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). Subjects in our experiment were much more consistent: For 173

of the 240 individual participants (72%), 68 of the 80 groups (85%) and 83 of the 105

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individuals in the stage III decisions (79%) monotonicity in p was fully satisfied5.

Groups show less violations of monotonicity confirming findings from Charness et al.

(2007) who compared decisions of individuals and groups in decision settings of

varying complexity. The average Kendal’s τb for individual decisions (excluding stage

III decisions) was τb = 0.90, for groups τb = 0.95 and for third stage decisions after

group discussions τb = 0.93.

Descriptive data overview:

Figure 1 presents the mean CEs of all 15 gambles for all three stages of the

experiment. Several observations stand out:

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Insert Figure 1 about here

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1. For p = 0.50 CEs are monotonically decreasing as ∆ increases, indicating

increasing vagueness aversion, but there is no clear pattern for p = 0.20 or p = 0.80.

2. Most mean CEs of risky gambles (∆ = 0) are lower than their expected values

(EVs) indicating risk aversion. The only exception is for p = 0.20 where CEs are risk-

neutral on average.

3. CEs of individual decisions are systematically lower than group CEs.

4. There are no systematic differences between individual CEs obtained before and

after the group discussion.

Measures of risk and vagueness attitude:

Certainty equivalents are a direct measure of participants’ evaluations of each

of the gambles and reflect both risk and vagueness attitudes. They provide a natural 5 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 79 groups) are monotonic in p.

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starting point for our analysis of the effects of the different experimental conditions.

However, analyzing CEs does not allow us to disentangle risk and vagueness attitudes

and to determine their respective directions (aversion, neutrality or seeking). To

overcome this limitation we consider two additional measures assessing risk-attitudes

and two measures assessing vagueness attitude (see for example Curley, Yates and

Abraham, 1986 who used the same measures).

1. For the 5 precise gambles (∆=0) we compute risk premiums (RPs) by

subtracting CEs from the expected values (EVs) of the gamble such that a risk

premium greater than (equal to, smaller than) than 0 indicates risk aversion

(neutrality, seeking).

2. Based on the RPs for each of the 5 precise gambles we compute the

proportion of decisions which exhibit a particular risk attitude (seeking, neutral,

averse)6. We consider both the proportions for each gamble separately as well as

aggregated over all levels of p.

3. We compute vagueness premiums (VPs) for each of the 10 vague gambles

(∆ > 0) by subtracting the CE of the vague gamble from the CE of the risky gamble

with the same p. VPs greater than (equal to, smaller than) 0 indicates vagueness

aversion (neutrality, seeking).

4. Based on the VPs for each of the 10 imprecise gambles we compute the

proportion of decisions which exhibit a particular vagueness attitude (seeking, neutral,

averse). For each probability level we aggregate over all levels of ∆ to obtain one

measure of vagueness-attitude for each probability level. We also consider the

proportion of vagueness-neutral choices aggregated over all levels of p.

6 Due to our method of calculating CEs to be the midpoint between two choices, RPs are by definition never exactly=0. To obtain a measure of risk-neutrality we count a decision to be risk neutral if |RP|≤0.5.

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3.2 Group decision making

3.2.1 Effect of payoff-communality

We conducted a 5 (conditions) X 15 (decision sheets) mixed ANOVA (N=80)

on group CEs to test for differences between conditions. As expected there is a highly

significant effect of the decisions sheets on certainty equivalents (F7.36,552.33 =1135.82,

p<0.01) 7. There was no significant effect of the conditions (F4,75=0.80, p=0.53), nor

significant interaction effects with the decision sheets (F29.56,552.33=1.10, p=0.30). We

tested a number of pre-planned contrasts:

1. The contrast between conditions “GD shared” and “GD shared (reversed)” was not

significant (F1,75=0.003, p=0.96) confirming that the order in which decisions are

made does not have an effect on the average CEs.

2. The contrast between conditions “GD shared” and “GD separate” was not

significant (F1,75=0.07, p=0.79), rejecting the notion that the payoff sharing among

members affects group decisions.

3. The contrast comparing group decisions that counted towards final payoff (“GD

shared”, “GD shared (reversed)” and “GD separate”) and those that did not (“IDID

shared” and “IDID separate”) was not significant (F1,75=0.25, p=0.62).

The first result from the contrasts indicates that group choices are not

influenced by whether individuals are already familiar with the decisions from the

prior round of individual decisions. However, the results from conditions “IDID

shared” and “IDID separate” (which we discuss in section 3.3) show that a prior

group discussion has a strong influence on subsequent individual decisions. Therefore

in the following comparison of individual and group decisions we exclude data from

condition “GD shared (reversed).” Given that we found no difference between group

7All degrees of freedom and p-values of this and the following repeated-measure ANOVA’s are computed with Greenhouse-Geisser corrections for violations of sphericity.

20

decisions in the other four conditions we pool the data from the remaining 65 groups

for our analysis.8

3.2.2 Aggregation models

We proceed by comparing several models for the aggregation of individual

CEs into group CEs. We consider four different aggregation rules:

1. Min: The group CE is equal to the minimum of the three individual stage one CEs.

2. Mean: The group CE is equal to the mean of the three individual stage one CEs

3. Median: The group CE is equal to the median of the three individual stage one CEs

4. Max: The group CE is equal to the max of the three individual stage one CEs

Table 3 shows the mean absolute deviation (MAD) and mean signed deviation

(MSD) between each model’s predictions and the actual group CE. The right-hand

side of the table shows the total number of group decisions which violate internality,

which refers to cases where CEs are either below the minimum of individual CEs

(GroupCE < min) or above their maximum (GroupCE > max).

-----------------------------------------


-----------------------------------------

The models with the lowest MADs are median and mean and there are no

systematic differences in model performance for the cases of risk and vagueness. All

models, except Max, systematically under-predict group CEs which indicates a shift

in group attitudes towards higher CEs. There is no systematic difference in the level

of under-predictions for risk and vagueness. We find a relatively low number of

8 We also tested for differences in RPs, VPs, the number of (risk) ambiguity seeking, neutral and averse choices across conditions. None of these differences is significant.

21

internality violations. Only 53 (5.5%) group CEs are below the minimum of

individual CEs and 82 (8.4%) are above the maximum. Consistent with our finding

that group CEs shift upwards compared to average individual CEs, the number of

upper internality violation (GroupCEs>max) is higher than the rate of lower violations

(GroupCEs<min).

3.2.3 Risk- and vagueness attitudes of groups and individuals

To test for differences in risk attitudes between individuals and groups we

conducted a 5 (probability levels) X 2 (individual vs. group decision) repeated-

measure ANOVA (N=65) on RPs where the individuals are represented by the mean

RP of the three group members. The ANOVA shows significant main effects of both

factors (F3.25,207.99=26.71, p<0.01 for probability levels; F1,64=8.87, p<0.01 for mean

individual vs. groups) and a significant interaction between the two (F2.89,185.25=3.05,

p=0.03). To explore this interaction we employ a Wilcoxon matched-pairs tests to

compare individual and group decisions for each level of p separately. The results

show that differences between mean group member decisions and groups are only

significant for medium probabilities, i.e., p=0.5 (z=-2.65, p<0.01) and p = 0.65 (z=-

3.21, p<0.01). For all other levels of p, the differences between mean group member

RPs and group RPs are not significant although the results point qualitatively in the

same direction.

We also find that for all levels of p, groups make risk neutral decisions more

often than individuals (46% of all group decisions compared to 29%). This difference

is significant by a sign test for the proportions aggregated over all levels of p (N=65,

p<0.01), and confirms our hypothesis that risk neutrality serves as a persuasive

22

argument during a group discussion. A detailed descriptive overview of our findings

for risk-attitudes can be found in the appendix.

Table 4 summarizes vagueness premiums and the proportion of choices which

exhibit either vagueness averse, vagueness neutral or vagueness seeking attitudes for

different levels of p and ∆.

-----------------------------------------


-----------------------------------------

To test for differences in vagueness premiums we conducted a 10 (decision

sheet) X 2 (individual vs. group decision) repeated-measure ANOVA (N=65) on VPs

where individuals are represented by the mean VP of the three group members. The

ANOVA shows a significant main effect of decision sheets (F3.86,247.16= 27.51,

p<0.01) but the results do not suggest the existence of a significant difference in

vagueness premiums between individuals and groups (F1,64=0.01, p=0.94) or a

significant interaction effect with decision sheets (F5.20,332.80=1.27, p=0.28). However,

as the data in table 4 show the variance in VPs is considerably lower in groups than in

individual decisions.

We also find that groups make more vagueness neutral choices than

individuals at all levels of p and ∆. This difference is statistically significant (N=65,

p=0.03) by a sign-test for the proportions aggregated over all levels of p and ∆. This

pattern is consistent with our hypothesis that vagueness neutrality is a particular

persuasive argument in group discussion.

To explore this shift towards vagueness neutrality further we compare the

attitudes to vagueness exhibited by the majority of group members with the attitude

exhibited by the corresponding group decision.

23

-----------------------------------------


-----------------------------------------

The first three sections of table 5 show the total number and the proportion of

cases for which a majority of either vagueness seeking, neutral or averse attitudes

resulted in a particular group attitude The last section gives the number and

proportion of group attitudes for the case where each of the individual group members

exhibited exactly one of the possible vagueness attitudes. For neutral and averse

vagueness attitudes approximately half (46% and 51% respectively) of the group

decisions preserve the vagueness attitude of the majority of individual group

members. For vagueness seeking attitudes this proportion is only 21% which suggests

that a group discussion has a particularly strong effect when a majority of individuals

are vagueness seeking. As the results clearly show, there is a strong shift from

vagueness aversion in individual attitudes to vagueness neutrality in group decisions

(93 decisions shift this way compared with only 38 decisions for which there is a

switch from neutrality to aversion). This provides a partial explanation for our finding

of increased vagueness neutrality in group decisions. Finally for the 127 choices for

which no majority exists at the individual level, the modal group decision is

vagueness neutral (43%) followed by vagueness aversion (34%) and vagueness

seeking (23% of all decisions).

24

3.3 Risk- and vagueness attitudes after group interactions

In this section we examine the effect of group interactions on individuals’

subsequent decisions as observed in the two IDID conditions. Because the participants

interacted with each other in groups before their second round of individual decisions

we can no longer treat these decisions as independent. Therefore, we analyze the data

at the group level (with N=35 groups) by comparing the mean attitudes of group

members before and after the group interaction stage.

The first step in our analysis is to test for differences between conditions

“IDID shared” and “IDID separate”. A 15X2 mixed ANOVA (N=35) on the mean

CEs obtained from stage III decisions shows a significant main effect of the decision

sheets (F4.17,137.50=795.24, p<0.01) but does not indicate a significant main effect of

the payoff sharing arrangements (F1,33=1.76, p=0.19) nor a significant interaction

effect with the decision sheets (F4.17, 137.50=0.45, p=0.78)9. Thus, in the following

anlyses we do not distinguish between the two conditions.

We focus first on the case of risk (∆ = 0). The results presented in Figure 1 did

not suggest a systematic difference in risk premiums between stage I and stage III

decisions. This is confirmed by a 5 (probability level) X 2 (before vs. after group

interaction) repeated-measures ANOVA (N=35) on RPs with no significant effect of

group interaction (F1,34=2.15, p=0.15) nor a significant interaction effect with

probability levels (F4,31=0.43, p=0.74). We detect a significant increase in the

proportion of risk neutral choices made after the discussion (34%) compared to before

(27%) (sign test: N=35, p=0.04, aggregated over all levels of p ). This is consistent

with our results from the comparison of individual and group decisions and, as such,

confirms our hypothesis that risk-neutrality is a persuasive argument during a group 9 We also test for differences in RPs, VPs, the number of (risk) vagueness seeking, neutral and averse choices across conditions. None of these differences is significant.

25

discussion which causes individuals to revise their attitudes in this direction. A

detailed descriptive overview of the results can be found in the appendix.

Table 6 presents our findings for vagueness attitudes before and after group

interaction.

-----------------------------------------


-----------------------------------------

A 2 (before and after group discussion) X 10 (decision sheet) repeated-

measures ANOVA (N=35) on mean VPs results in F9,26=10.1, p<0.01 for the main

effect of decision sheets, F1,34=2.58, p=0.12 for the main effect of group discussion

and F9,26=0.09, p=0.54 for the interaction effect. We do not find a significant

difference between the pre- and post- group interaction mean VPs, but the variance of

VPs is lower after the group discussion stage for all decision sheets.

Table 6 also shows that participants made significantly more vagueness

neutral decisions after the group discussion at all levels of p and ∆. The difference in

the proportion of vagueness neutral choices is significant according to a sign test

(N=35, p<0.01, aggregated over all levels of p). This confirms our hypothesis that

vagueness-neutrality acts as a persuasive argument during group discussion, which

influences group decisions (see previous section) as well as individual decisions after

the group interaction.

To analyze further to what extent individuals change their vagueness attitudes

as a consequence of the outcome of the prior group discussion we compare the

marginal distribution of attitudes to vagueness in each of the 3 stages10. These are

presented in Table 7. The last row in the table is a simple average of the distribution 10 Another way to investigate this question is a regression analysis which models changes between stage I and stage III decisions as a function of stage II group decisions. The results are similar to those presented here. Details can be found in the appendix.

26

of initial individual attitudes (stage I) and group attitudes (stage II). This average

predicts qualitatively the overall pattern of the final individual attitudes in Stage III

and can be viewed as a compromise between the attitudes in stages I and II. While

the mean predicts accurately the proportion of vagueness seeking decisions, it over

(under) estimates the instances of vagueness neutrality (avoidance), which can be

attributed to the persuasive arguments in favor of neutrality.

-----------------------------------------


-----------------------------------------

4. Discussion

We compared individual and group decisions made under risk and vagueness

and various payoffs sharing arrangements, and we explored the effects of exposure to

other individuals’ opinions and attitudes on subsequent individual decisions. We

found that groups are, on average, less risk averse than individuals at all probability

levels and make risk-neutral decisions more often. Furthermore, our results showed

that the shift towards risk neutrality persists in subsequent individual decisions.

Differences between individuals and groups in their attitudes towards vagueness have

not been addresses systematically in the past, and we present three novel results. We

found that 1) groups make vagueness neutral decisions more often than individuals for

all probability levels and degrees of vagueness; 2) This shift towards vagueness

neutrality persists in subsequent individual decisions; 3) Individuals’ vagueness

attitudes shift systematically towards the outcome of the prior group interaction.

27

Surprisingly, we did not find a significant effect of pay-off sharing arrangement on

either risk attitudes or vagueness attitudes.

Next we discuss how our findings for risk-attitudes compare to results

obtained in related studies. We then discuss our main results for vagueness attitudes in

more details and point out implications for future research.

. Shupp & Williams (2008) reported that groups were more risk averse than

individuals for low probabilities, approximately equally risk averse for medium

probabilities and less risk averse for high probabilities. We find that groups are less

risk-averse at all probability levels with a particular strong effect for medium

probabilities. There is an important methodological difference between the two

studies that could account for this discrepancy. While we elicited CEs from binary

choices, Shupp & Williams (2008) endowed their participants with money and

elicited buying prices for each lottery. This endowing procedure could affect choices

by introducing differential levels of loss aversion for individuals and groups which are

absent in our elicitation procedure. Given the recent interest in the differences

between decisions by individuals and groups, it is important to investigate

systematically this interaction between elicitation procedures and the deciding entity

(individual or groups) in future research.

Baker et al. (2008) also found interactions between winning-probabilities and

the deciding entity (individuals and groups) which they interpret as being consistent

with Shupp &Williams. However, due to their different methodology for measuring

risk attitudes it is difficult to compare our results to theirs in more detail. Similar to

our conditions “IDIN separate” and “IDIN shared” Baker et al. (2008) asked

participants to make another round of individual decisions after the group stage. Like

Baker et al. (2008) we found that subsequent individual decisions are influenced by a

28

prior group discussion, and tend to shift towards the group decision. However, unlike

Baker et al. (2008) who found that individuals are more risk-taking after the group

discussion our results do not show a significant difference in risk premiums. Again

due to the different methodologies in our two papers a more detailed comparison is

difficult.

Finally, consider the effect of payoffs arrangements. Sutter (2009) investigated

the effects of payoff commonality on individual decisions made under risk and found

that payoff commonality has a strong effect on individual decisions and significantly

increased risk-taking. We did not find such differences. One explanation for this

difference is that Sutter compares decisions made by individuals with payoff-

commonalities to those made by individuals alone. In contrast, in our setting

participants were either members of a group or had prior interaction with others. As a

consequence, the effect of payoff-commonality might have been swamped by the

stronger effect arising from group discussions and exposure to other participants’

opinions such that we could not detect a significant effect of the different payoff

sharing arrangements.

We turn our attention to the analysis of vagueness attitudes. We found that

groups made vagueness neutral decisions more often than individuals. Our analysis

shows that the increase in the proportion of vagueness neutral decisions is mainly due

to the large number of decisions in which a majority of vagueness averse individual

group members reach a vagueness neutral decision, when operating as a group. Only a

small proportion of choices show the opposite pattern in which a majority of

vagueness neutral group members makes a vagueness averse decision.

Secondly, we find that individual vagueness attitudes shift systematically in

the direction of the attitudes reflected by the outcome of the prior group discussion.

29

As a consequence of this shift, a significantly higher proportion of post interaction

individual decisions are vagueness neutral compared to pre-interaction decisions. This

demonstrates that the change in attitudes towards vagueness observed in group

decisions carries over and reflects actual shifts in these attitudes, and it is not caused

solely by the social context of the group decision or the need to compromise by

aggregating individual preferences.

Finally, mirroring our result for risk-attitudes we did not find a significant

effect of pay-off sharing arrangement on vagueness attitudes. This finding contradicts

our initial hypothesis that the degree to which common or separate payoffs introduces

a feeling of common fate and shared responsibility in the group will influence

attitudes towards vagueness. It appears that the experience of group interaction is

quite powerful, and its effects are strong enough to cause people to ignore or

disregard differences in the payoff arrangements.

Our results contradict previous studies which showed that vagueness attitudes

are very robust towards persuasion by rational arguments (MacCrimmon, 1968;

Slovic & Tversky, 1974; Curley et al. 1986). In fact, all three findings are in line with

the “persuasive argument hypothesis” which states that exchange of persuasive

arguments (in this case the argument is vagueness neutrality) during a group

discussion can induce systematic changes in behavior.

Future research should explore further to what degree and under what

circumstances attitudes to vagueness are malleable to persuasion by group members,

or other sources. One interesting direction would be to analyze the content of the

group discussion in order to precisely determine which arguments individuals put

forwards during the group interaction and to what extent they are accepted or rejected

by others. Moreover, future studies should explore further to what extent social

30

factors other than payoff-communality, or apprehension of evaluation (as in Curley et

al., 1986 and Trautmann et al., 2009) influence vagueness attitudes. One possible

extension of our research in this direction would be to ask participants to act as group

representatives and make decisions on behalf of others without being able to

communicate with the group (see, for example, work by Charness, Rigotti &

Rustichini (2007) who investigate the behavior or group representatives in social

dilemmas and coordination games).

In a recent paper 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 vagueness

attitudes. Adopting their methodology and running further studies with individuals

and groups considering natural sources of uncertainty would be another very

interesting extension of our current research.

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. The authors

appreciate the insightful comments by seminar participants at the D&U-workshop 2010 in

Paris, FUR 2010 in Newcastle and ESA 2010 in Copenhagen.

Funding

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

31

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34

Appendix A: Tables, Figures and Screenshots Table 1: Overview of Decision Sheet Characteristics

p ∆ lowest highest increments0.20 0.00 $0.50 $7.50 $0.50 150.20 0.05 $0.50 $7.50 $0.50 150.20 0.10 $0.50 $7.50 $0.50 150.20 0.20 $0.50 $7.50 $0.50 15

0.35 0.00 $0.50 $13.50 $1.00 and $0.50 15

0.50* 0.00 $1.00 $19.00 $1.00 190.50 0.05 $1.00 $19.00 $1.00 190.50 0.10 $1.00 $19.00 $1.00 190.50 0.30 $1.00 $19.00 $1.00 190.50 0.50 $1.00 $19.00 $1.00 19

0.65 0.00 $6.50 $19.50 $1.00 and $0.50 15

0.80 0.00 $12.50 $19.50 $0.50 150.80 0.05 $12.50 $19.50 $0.50 150.80 0.10 $12.50 $19.50 $0.50 150.80 0.20 $12.50 $19.50 $0.50 15

*presented twice

Characteristics of gamble Range of sure amounts of money offered No. of choices on decision sheet

Table 2: Summary of Experimental Conditions and Sample Sizes

Condition Outcome Stage I Stage II Stage IIINo. Groups

No. of participants

GD shared shared 16 48

shared 15 45

GD separate separateIndividual Decisions

Group Decisions 14 42

IDIN shared shared 18 54

IDIN separate separate 17 51

GD shared (reversed)

Individual Decisions

Group Decisions





Group Decisions


Group Decisions

Group Decisions

35

Table 3: Comparisons of Fit of Four Aggregation Models

Mean Absolute Deviation Mean Signed Deviation

p ∆ Min Median Mean Max Min Median Mean Max GroupCE<Min

GroupCE>max

0.20 0 1.57 0.89 0.81 1.21 1.42 0.04 0.16 -0.97 4 60.05 1.51 0.79 0.74 1.25 1.36 0.01 0.10 -1.09 5 40.10 1.66 0.84 0.82 1.26 1.47 0.09 0.16 -1.09 3 60.20 1.51 0.88 0.79 1.34 1.34 0.01 0.02 -1.30 4 1Mean 1.56 0.85 0.79 1.26 1.40 0.04 0.11 -1.11 Total 16 17

0.35 0 2.40 1.34 1.38 1.99 2.30 0.44 0.43 -1.43 4 10

0.50 0 2.41 1.13 1.16 1.62 2.28 0.31 0.37 -1.47 1 60.05 2.80 1.18 1.32 1.61 2.63 0.45 0.59 -1.31 5 50.10 2.70 1.45 1.43 1.83 2.50 0.43 0.52 -1.38 5 70.30 2.79 1.43 1.41 2.38 2.56 0.50 0.30 -2.18 5 40.50 2.94 1.61 1.50 2.44 2.66 0.21 0.20 -2.26 5 5Mean 2.73 1.36 1.36 1.97 2.53 0.38 0.40 -1.72 Total 21 27

0.65 0 3.16 1.59 1.45 2.03 3.02 0.82 0.76 -1.55 1 6

0.80 0 1.81 1.07 0.98 1.56 1.72 0.12 0.14 -1.41 2 40.05 1.79 0.99 0.93 1.27 1.67 0.31 0.33 -0.99 2 100.10 1.62 1.02 0.95 1.59 1.57 0.09 0.07 -1.44 3 30.20 1.74 0.81 0.81 1.41 1.68 0.26 0.27 -1.13 4 5Mean 1.74 0.97 0.92 1.46 1.66 0.20 0.20 -1.25 Total 11 22

Mean ∆=0 2.27 1.20 1.16 1.68 2.15 0.35 0.37 -1.37 Total ∆=0 12 32Mean ∆>0 2.01 1.06 1.02 1.56 1.86 0.20 0.24 -1.36 Total ∆>0 41 50

2.27 1.20 1.16 1.68 2.15 0.35 0.37 -1.37 12 32

Violations of internality (Total No.)

Total Overall

Mean Overall

N=65 Table 4: Vagueness Attitudes of Group- and Individual Decisions

Proportion of choicesVagueness Premiums* Vag. seeking Vag. neutral Vag. Averse

p ∆ Ind. Groups Ind. Groups Ind. Groups Ind. Groups0,20 0,05 -0.07 (1.08) -0.04 (0.83) 0,34 0,29 0,33 0,46 0,33 0,25

0,10 0.17 (1.26) -0.05 (0.88) 0,25 0,26 0,36 0,45 0,39 0,290,20 -0.02 (1.58) -0.02 (0.73) 0,34 0,37 0,31 0,34 0,35 0,29Mean 0,31 0,31 0,34 0,42 0,36 0,28

0,50 0,05 0.40 (1.61) 0.35 (0.93) 0,29 0,15 0,28 0,48 0,43 0,370,10 0.35 (1.67) 0.39 (1.12) 0,30 0,14 0,27 0,48 0,43 0,380,30 0.74 (2.26) 0.95 (1.40) 0,27 0,15 0,21 0,25 0,52 0,600,50 1.35 (2.63) 1.60 (1.71) 0,19 0,05 0,23 0,26 0,58 0,69Mean 0,26 0,12 0,25 0,37 0,49 0,51

0,80 0,05 0.33 (1.37) 0.05 (0.88) 0,23 0,22 0,32 0,46 0,45 0,320,10 0.06 (1.57) 0.18 (1.12) 0,31 0,22 0,28 0,43 0,41 0,350,20 0.77 (1.58) 0.61 (0.98) 0,20 0,11 0,22 0,34 0,58 0,55Mean 0,25 0,18 0,28 0,41 0,48 0,41

Mean 0,27 0,20 0,28 0,39 0,45 0,41

*Mean values, standard deviations in parentheses; N=195 for individuals and N=65 for groups

36

Table 5: Aggregation of Individual Vagueness Attitudes into Group Attitudes

Majority ofgroup members Group Decisions

Vag. Seeking Vag. Neutral Vag. Averse Total Vag. Seeking Total Number 24 47 46 117

Proportion 21% 40% 39% 100%Vag. Neutral Total Number 33 61 38 132

Proportion 25% 46% 29% 100%Vag. Averse Total Number 41 93 140 274

Proportion 15% 34% 51% 100%No Majority Total Number 29 55 43 127

Proportion 23% 43% 34% 100%

Table 6: Individual Vagueness Attitudes Before and After Group Interaction

Proportion of choicesVagueness Premiums* Vag. seeking Vag. neutral Vag. Averse

p ∆

0,20 0,05 -0.02 (1.09) 0.04 (0.73) 0,37 0,22 0,30 0,48 0,32 0,300,10 0.13 (1.13) 0.09 (0.63) 0,28 0,21 0,38 0,46 0,34 0,330,20 -0.09 (0.99) -0.05 (0.88) 0,38 0,28 0,30 0,43 0,32 0,30Mean 0,34 0,23 0,33 0,45 0,33 0,31

0,50 0,05 0.51 (1.65) 0.22 (1.46) 0,27 0,25 0,20 0,32 0,53 0,430,10 0.43 (1.64) 0.44 (1.56) 0,29 0,19 0,26 0,31 0,46 0,500,30 1.04 (2.25) 0.95 (2.2) 0,22 0,19 0,19 0,30 0,59 0,510,50 1.42 (2.76) 0.98 (2.06) 0,20 0,20 0,20 0,28 0,60 0,52Mean 0,24 0,21 0,21 0,30 0,55 0,49

0,80 0,05 0.35 (1.37) 0.2 (1.34) 0,22 0,20 0,29 0,40 0,50 0,400,10 0.09 (1.54) -0.08 (1.16) 0,29 0,33 0,29 0,38 0,43 0,290,20 0.93 (1.66) 0.55 (1.21) 0,19 0,16 0,24 0,30 0,57 0,53Mean 0,23 0,23 0,27 0,36 0,50 0,41

Mean 0,27 0,22 0,27 0,37 0,46 0,40

Before Inter.

After Inter.

Before Inter.

After Inter.

BeforeInter.

After Inter.

Before Inter.

After Inter.

*Mean values, standard deviations in parentheses; N=105

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Table 7: Distribution of Attitudes to Vagueness (in %) in All Stages of the Experiment

Attitude to VaguenessStage Seeking Neutral AverseI (Individual) 27.0 26.4 46.7II (Group) 17.7 39.1 43.1III (Individual) 22.3 36.6 41.1

Prediction of Stage III from Stages I and IIMean of Stages I and II 22.3 32.8 44.9

N=105 Figure 1: Mean CEs of Gambles as a Function of p, ∆, and Experimental Condition

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00

p=0.2; ∆=0

p=0.2; ∆=0.05

p=0.2; ∆=0.1

p=0.2; ∆=0.2

p=0.35; ∆=0

p=0.5; ∆=0

p=0.5; ∆=0.05

p=0.5; ∆=0.1

p=0.5; ∆=0.3

p=0.5; ∆=0.5

p=0.65; ∆=0

p=0.8; ∆=0

p=0.8; ∆=0.05

p=0.8; ∆=0.1

p=0.8; ∆=0.2

Individual Decisions:All conditions pooled (N=240)

Group Decisions:All conditions pooled (N=80)

Individual Decisions after group discussion:Conditions C and D (N=105)

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Screenshot 1: Decision Sheet (p=0.5, ∆=0)

Screenshot 2: Decision Sheet (p=0.5, ∆=0.5)

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Appendix B: Additional tables for risk attitudes Table A1: Risk Attitudes of Group- and Individual Decisions

Risk Premiumsp Ind. Groups Ind. Groups Ind. Groups Ind. Groups0.20 0.01 (1.47) -0.07 (1.21) 0.38 0.35 0.32 0.40 0.29 0.250.35 1.28 (2.44) 0.88 (2.08) 0.13 0.12 0.31 0.37 0.56 0.510.50 0.66 (2.53) 0.19 (1.48) 0.18 0.15 0.39 0.69 0.43 0.150.65 1.55 (2.85) 0.78 (2.00) 0.18 0.14 0.24 0.51 0.58 0.350.80 0.36 (1.93) 0.29 (1.33) 0.33 0.25 0.21 0.31 0.47 0.45Mean 0.24 0.20 0.29 0.46 0.47 0.34

Risk AverseProportion of choices

Risk seeking Risk Neutral

*Mean values; standard deviations in parentheses; N=65

Table A2: Aggregation of Individual Risk Attitudes into Group Attitudes

Risk Seeking Risk Neutral Risk Averse TotalRisk Seeking 0.08 0.05 0.03 0.16Risk Neutral 0.03 0.14 0.04 0.22Risk Averse 0.04 0.17 0.23 0.45No majority 0.05 0.09 0.04 0.18Total 0.20 0.46 0.34 N=325

Majority of individuals

Group

Table A3: Individual Risk Attitudes Before and After Group Interaction

Proportion of choicesRisk Premiums* Risk Seeking Risk neutral Risk Averse

p0.20 0.05 (1.48) 0.15 (1.19) 0.38 0.28 0.30 0.37 0.31 0.350.35 1.35 (2.63) 1.45 (2.02) 0.13 0.17 0.26 0.26 0.61 0.680.50 0.47 (2.42) 0.83 (2.00) 0.22 0.10 0.39 0.53 0.39 0.360.65 1.2 (2.87) 1.32 (1.32) 0.25 0.07 0.19 0.34 0.56 0.490.80 0.08 (1.98) 0.33 (1.78) 0.41 0.37 0.19 0.18 0.40 0.45Total 1.39 0.99 1.33 1.69 2.28 2.32

BeforeInter.

After Inter.

BeforeInter.

After Inter.

BeforeInter.

After Inter.

BeforeInter.

After Inter.

*Mean values; standard deviations in parentheses; N=105

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Appendix C: Results from regression analysis for post-interaction decisions The table below presents results from ordered-logit regressions for each of the three possible vagueness attitudes separately. All regressions include fixed effects for the 10 decision sheets (not reported here) and standard errors are adjusted for clustering in each of the 35 groups. All of the analysis is done at the group level. Our dependent variable is number of choices that exhibit a particular vagueness attitude (seeking, neutral or averse) after the group discussion (at stage III) but did not do so before (at stage I). For example for the case of vagueness neutrality we count the number of choices in each group which were not vagueness neutral before the group discussion (stage I) but changed to being neutral after the discussion (stage III). To measure the influence of the group discussion on changes in individual attitudes we include a dummy variable which captures the attitude exhibited by the group decision at stage II for a particular choice. For example, in the case of vagueness neutrality this variables equals 1 if the stage II group decision was vagueness neutral and 0 otherwise. We also include the number of choices in each group which exhibit a particular attitude before the discussion (at stage I). This captures the fact that the total number of changes towards a particular attitude is necessarily smaller when a larger number of choices showed this attitude before the group discussion.

Number of switches from not vag.seeking to vag. seeking

Number of switches from not vag. neutral to vag. neutral

Number of switches from not vag. averse to vag. averse

Number of vag. seeking choices before group decision -0.790 (0.209)

Group Decision (1 if group decision is vag. seeking; 0 otherwise) 0.056 (0.393)

Number of vag. neutral choices before group decision -0.997 (0.210)**

Group Decision (1 if group decision is vag. neutral; 0 otherwise) 0.524 (0.28)*

Number of vag. averse choices before group decision -1.24 (0.179)**

Group Decision (1 if group decision is vag. averse; 0 otherwise) 0.911 (0.282)**

Observations 350 350 350Pseudo R2 0,055 0,080 0,139

* p<0.05, ** p<0.01; coefficients from ordered-logit regression; SE in parentheses

As expected in all three regressions the number of choices which showed a

particular attitude at stage I has a highly significant negative influence on the number of choices which changed their attitude in this direction. For vagueness aversion and vagueness neutrality the outcome of the stage II group decisions also has a significant influence on the number of choices which changed towards this attitude. This shows that individuals’ changes in attitudes are not simply the consequence of random changes, but are influenced by the group discussions. Interestingly, we do not find a significant influence of the outcome of the group decision on changes towards vagueness seeking. This suggests that arguments in favor of vagueness neutrality and

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aversion are relatively persuasive to individuals when advocated in a group discussion and causes them to revise their opinions, but not so for vagueness seeking.

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Appendix D: 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.

Three color problem: Proportion of choices being SEU consistent

Two color problem: Proportion of choices choices in favor of risky urn

Individuals (N=195) 0.59 0.84Groups (N=65) 0.45 0.95After discussion (N=105) 0.48 0.89

In the two-color problem compared to individuals, groups are more likely to choose the risky urn (Sign test: N=65, p<0.01). There is no significant difference between individual decisions before and after the group discussion (Sign test: N=35, p= 0.87). For the three-color task the data on table 5 indicates that groups seem to be less likely to be SEU consistent compared to individuals. However, this difference is not significant (Sign test: N=65 p=0.16). Similarly the proportion of individuals making SEU consistent choices is slightly lower after compared to before the group discussion although again this difference is not significant (Sign test: N=35 p=0.08).

Group Decision Making under Vagueness

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