Download - Group Decision Making under Vagueness
Group Decision Making under Vagueness
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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
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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Insert Table 2 about here
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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).
-----------------------------------------
Insert Table 3 about here
-----------------------------------------
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 ∆.
-----------------------------------------
Insert Table 4 about here
-----------------------------------------
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
-----------------------------------------
Insert Table 5 about here
-----------------------------------------
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.
-----------------------------------------
Insert Table 6 about here
-----------------------------------------
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.
-----------------------------------------
Insert Table 7 about here
-----------------------------------------
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
Individual Decisions
Individual Decisions
Individual Decisions
Individual Decisions
Group Decisions
Individual 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
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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).