Comparing Group and Individual Choices under Risk and Ambiguity:
an Experimental Study•
Marielle Brunette♣, Laure Cabantous♦, Stéphane Coutureℵ
April, 2010
Abstract
In this paper, we investigate attitudes towards risk and attitudes towards ambiguity of groups and
individuals in order to compare them. We are also interested in the impact of collective rule
(unanimity versus majority) on the group decision. Concerning individual choices, we find that, on an
aggregate level, there is difference in individual choice over gain amounts vis-à-vis loss amounts;
most of the subjects are risk averse over gains and risk-seeking over losses with risky prospects; they
are, for the majority, ambiguity averse over gains and ambiguity seeking over losses with ambiguous
prospects. We also find that correlations between attitudes towards risk and ambiguity are positive in
the domain of gains but null in the domain of losses. Concerning group choices, we find that subjects
are risk averse and ambiguity averse, whatever the collective decision rule, but there is no correlation
between both risk and ambiguity attitudes. We also obtain, that, subjects are more ambiguity averse in
the case of the majority rule than in the case of the unanimity one. Finally, our results suggest that
both risk and ambiguity attitudes of individuals are identical of one of groups, whatever the collective
decision rule. Therefore, there is no evidence of differences in the choices of individuals and groups.
Implications of these results are discussed.
Key words: collective decision, unanimity, majority, risk preferences, ambiguity.
JEL classification: C91, C92.

We are grateful to Jean-Marc Rousselle for programming the experiment presented in this paper, and to
Jacques Laye for help in organising experiments. We also thank the financial support of RDT program 2006
“iCrisis” coordinated by Eric Langlais.

INRA, UMR 356 Economie Forestière, F-54000 Nancy, France. AgroParisTech, Engref, Laboratoire
d’Economie Forestière (LEF), 14 rue Girardet, F-54000 Nancy, France.
Tel: +33(0)3 83 39 68 54. Mail: [email protected].

Nottingham University Business School. [email protected].

Corresponding author: INRA, UR 875, Unité de Biométrie et Intelligence Artificielle (UBIA).
[email protected]. Chemin de Borde rouge. BP 52627. 31326 Castanet Tolosan Cedex
1
1. Introduction
Risk and ambiguity1 are important features of many real-world situations. In such a context,
many important economic decisions are made by groups rather than by isolated individuals.
These decisions depend on the risk and ambiguity preferences of all decision-maker members.
In general, individuals have some divergent preferences. More, collective decisions are taken
according to different rules, principally, unanimity or majority. As a result, many questions
emerge: are the group preferences different than the individual ones? How the risk and
ambiguity preferences of different group members are shared to generate the group decision?
What is the importance of the decision rule? Are there differences between decisions taken in
an ambiguous context and ones made under risk? Few previous works have been concentrated
on the comparison between group and individual decision-making under risk, and, to our
knowledge, such a comparison does not exist in an ambiguous context.
In this paper, we contribute to this literature by reporting on an experimental investigation
into individual and collective decision-making under risk or ambiguity. The main objectives
of our experiment are to test the level of risk and ambiguity aversion for individual and group
treatment, and to study the impact of collective decision rule by comparing unanimity-rule
group choice and majority-rule group choice under risk but also under ambiguity.
Decisions taken in a risky environment may diverge as they are made individually or
collectively because of risk preferences. Group members have in general some divergent risk
preferences, therefore collective decision requires agreement and may depend on
compromises by individual members. Many works focus on the possible consequences of
having a group rather than an individual who make a decision. More precisely, some papers
concentrate on the analysis of individual versus group decisions involving Expected Utility
Theory (EUT). Three studies (Bateman and Munro, 2005; Bone et al., 1999; Rockenbach et
al., 2007) examine whether group decisions are more in accordance with EUT, and find no
evidence that EUT describes group decisions better than individual ones. The risk preferences
of individual and group may vary. Four others studies (Harrison et al., 2007; Baker et al.,
2008; Shupp and Williams, 2008; Masclet et al., 2009) explicitly concentrate on individual
versus group risk preferences measurement defined by the number of safe choices using the
lottery-choice experiment developed by Holt and Laury (2002). The experiment results of
these studies report no consensus on significant risk preference differences between
individuals and groups.
In our experiment, the risk preferences of groups and individuals are compared by
implementing two treatments over four independent sessions. We analyse differences in risk
aversion between individuals and group. The originality of our research lies in the fact that we
then introduced ambiguity in lottery-choice experiment for the same participants. In an
ambiguous context, individual decisions are determined by the ambiguity preferences of the
decision-maker. Results of past experiment studies suggest ambiguity aversion in the domain
of gains. Also, most experiments suggest that attitude towards risk and attitude towards
ambiguity are by and large uncorrelated2. As they may exist differences in individual choices
under risk vis-à-vis choices under ambiguity due to some individual divergent risk and
ambiguity preferences, we investigate if it appears some distinctions between individual
1
An uncertain context is called ‘risky’ if the sets of outcomes and of probabilities are known with certainty,
while it is called ‘ambiguous’ if either of these sets are unknown or only partially known.
2
For a general overview of the literature concerning attitudes to risk and uncertainty, we refer to the survey by
Camerer and Weber (1992).
2
decisions and group ones in a risky or ambiguous context, and also dissimilarity between
individual preferences and group ones.
Another originality of our research is to compare group decisions taken into two collective
decision rules: unanimity and majority. There is an important experimental literature that has
explored differences between majority and unanimity with respect to different types of
decisions (see for example Miller, 1985). To our knowledge, no experimental research
explicitly examines the comparative effects of collective decision rules – majority and
unanimity rules – on the kind of choices that interest us, risk and ambiguous choices. Only a
few experimental works have compared the risk preferences of groups and individuals fixing
one formal decision rule3, majority-rule (Harrison et al., 2007) or unanimity-rule (Rockenbach
et al., 2007; Shupp and Williams, 2008; Baker et al., 2008; Masclet et al., 2009). Work with
groups deciding under majority rule concludes no differences in risk aversion between
individual and groups. The general conclusion of the experimental studies with unanimity
decision rule is that groups tend to be more risk averse than individuals. Therefore, collective
decision rule seems to determine the comparison between the risk preferences of individuals
and rule-depending groups, composed of the same subjects.
From the experimental data we draw the main conclusions. For the individual decisions, we
find that, preferences to risk and to ambiguity are domain-dependent: most of the participants
are risk averse/ambiguity averse in the domain of gains and risk lover/ambiguity lover in the
domain of losses. We also obtain that correlations between attitudes towards risk and
ambiguity are positive in the domain of gains but null in the domain of losses. Concerning the
group decisions, we find that subjects are risk averse and ambiguity averse, whatever the
collective decision rule, but there is no correlation between both risk and ambiguity attitudes.
We also obtain, that, subjects are more ambiguity averse in the case of the majority rule than
in the case of the unanimity rule. Finally, our results suggest that both risk and ambiguity
attitudes of individuals are identical of one of groups, whatever the collective decision rule.
There is no evidence of differences in the choices of individuals and groups.
The rest of the article is structured as follows. In Section 2, we summarize the relevant
previous works, focusing on our main questions mentioned above. The experimental design is
detailed in Section 3, and the results examined in Section 4. Finally in Section 5, we provide a
summary of the results, and some concluding remarks.
2. Related literature and our empirical questions
Many experimental studies deal with analysis of individual behaviours under risk and
ambiguity. More precisely, some works concentrate on the quantification of both risk and
ambiguity attitudes. Although there exists many experimental approaches that can be used to
estimate risk aversion, there is fewer studies aimed at quantifying ambiguity aversion. Indeed,
as noted by Harrison and Rutström (2008), five general elicitation procedures have been used
to estimate risk attitudes of individuals, and one of the major experimental procedures, and
the most common approach, used to elicit risk attitudes is the Multiple Price List (MPL)
method. The MPL approach was principally made easy to implement by Holt and Laury (HL)
(2002). The MPL is a relatively simple procedure for eliciting risk attitudes
where the subject faces an ordered array of choices between two lotteries
or a lottery and a guaranteed payoff. The results of such choices allows to
directly quantifying the risk aversion parameter of the subject4. Such MPL
approach has been extensively used in many experiments (Bruner, 2007;
3
For the same topic, other papers (Bone et al., 1999; Bateman and Munro, 2005) examine two-person group
decisions based on informal discussion.
3
Andersen et al., 2006; Harrison et al., 2005; Holt and Laury, 2005; Eckel
and Wilson, 2004; Goeree et al., 2003; Eckel and Grossman, 2008).
Concerning the evaluation of the ambiguity preferences, few attempts to
quantify ambiguity attitudes have been proposed, based on simple
experimental protocols in a model-free way (Cohen et al., 2009) or in a
theoretical-model way (Halevy, 2007; Chakravarty and Roy, 2009). The
experimental protocols used to elicit individuals’ attitudes towards
ambiguity are principally an extension of Ellsberg two-urn experiment. The
Chakravarty and Roy’s experiment is the first attempt to use the MPL
method to elicit ambiguity attitudes. Based on the theoretical foundations
provided in Klibanoff et al. (2005), the experiment results allow the
authors to quantify ambiguity preferences.
Some experimental works analyse behaviours towards risk and ambiguity function of the domain (gains/losses) and probabilities of occurrence (high/low). Some authors (Kahneman and
Tversky, 1979; Curley and Yates, 1985, 1989; Cohen et al., 1987; Wehrung, 1989; Tversky
and Kahneman, 1992; Di Mauro and Maffioletti, 2004; Chakravarty and Roy, 2009) conclude
to the existence of a “Reflection effect”. Indeed, the authors show that when probabilities are
high, then risk averse behaviours appear in gain domain and risk-loving ones in loss domain,
while opposite behaviours emerge when probabilities are low. Di Mauro and Maffioletti
(2004) generalize this result in ambiguous context of uncertainty. At the same time, some authors deepen the analysis of uncertain decisions by concentrating on the link between risk and
ambiguity preferences. There is no experimental consensus on the correlation between risk
preferences and ambiguity ones. Cohen et al. (1987), Hogarth and Einhorn (1990) and Di
Mauro and Maffioletti (2004) find that attitudes towards risk and towards ambiguity are not
closely linked while Lauriola and Levin (2001) report a positive correlation. Chakravarty and
Roy (2009) find that correlation is domain dependent. They obtain a positive and significant
correlation over the domain of gains but no such association over the domain of losses.
Numerous
empirical researches based on group and individual decisions can be found. Some
works aim at comparing group and individual decisions (Bone et al., 1999; Blinder and
Morgan, 2005; Cox and Hayne, 2006; Rockenbach et al., 2007). A number of experimental
results indicate that group decisions are more consistent with rationality than individual
decisions. Other works concentrate on the analysis of individual versus group decisions
involving Expected Utility Theory (EUT). Bateman and Munro, 2005; Bone et al., 1999;
Rockenbach et al., 2007 examine whether group decisions are more in accordance with EUT,
and find no evidence that EUT describes group decisions better than individual ones. More
interestingly, other works compare attitude to risk in group and individual decisions. For
instance, Harrison et al. (2007) show that, on average, the collective decisions and the
individual ones are similar with a majority-rule. In other words, the individual’s risk
preferences are identical to the collective ones. On the contrary, Baker et al. (2008) and
Masclet et al. (2009) prove that, on average, groups are more risk averse than individuals,
with an unanimity rule.
Given these works, it seems justified to wonder about the differences between individual and
group preferences in a risky or ambiguous context but also about the impact of decision rule
on collective choices under risk or ambiguity. Concerning individual decisions, we
concentrate on the existence of a “Reflection effect” in risky and ambiguous context and to
4
In such approaches, the expected lottery payout is increased as the subject proceeds through the series of the
dichotomous choices so as to induce the subject to switch from the less risky to the more risky choice. The
decision at which the subject switches produces an interval estimate of the subject’s risk attitude.
4
the link between risk and ambiguity attitudes, in order to test the existing results. We
implement an experimental design to try to answer to these questions.
3. The experimental design
We employ an experimental Multiple Price List (MPL) procedure eliciting both attitudes
towards risk and towards ambiguity, based on that of Holt and Laury (2002) and on the
modified version proposed by Chakravarty and Roy (2009) to consider risky and ambiguous
contexts in order to recover not just attitudes to risk but also attitudes to ambiguity. The
experiment was computerized and the scripts were programmed using the z-tree platform
(Fischbacher, 2007).
All subjects participated to a lottery choice experiment with two treatments. In the first
treatment, called the individual treatment, subjects were provided with a series of binary
choices for four tasks. The first two tasks involved sequential choices between two risky
lotteries, a sequence of 10 choices in the domain of gains and another sequence of 10 choices
in the domain of losses, whereas the second two tasks deal with the choices between
ambiguous lotteries, 10 sequential choices in the domain of gains and 10 other sequential
choices in the domain of losses.
In the second treatment, called the group treatment, subjects were anonymously selected to
compose three-person groups, and were presented with two tasks of binary choices that were
the same as in the individual treatment in the domain of gains. These tasks were composed of
a sequence of 10 binary risky choices and another sequence of 10 binary ambiguous choices.
In the group treatment, all the payoffs of the lottery choices were in the domain of gains. After
each choice, the groups were randomly reshuffled.
Finally, the subjects were also required to complete a questionnaire concerning basic
demographic variables like age, sex, and professional activity or not.
At the end of the experiment, two decisions were randomly selected by the computer, one in
the individual treatment and the other in the collective treatment. For each selected decision,
the computer determines the payoff earned by the subject according to her chosen option and
the payoff-probability lottery. The sum of the two payoffs determines the final payment of the
respondent.
3.1 Risk and ambiguous tasks in the individual treatment
Each subject is presented with four tables of 10 sequential choices between two lotteries.
Tables 1 and 2 (see Appendix A) present the 10 decisions with risky prospects in the domain
of gains and losses, respectively. For each decision the subject chooses lottery A (safe option)
or lottery B (risky option). A risk-neutral subject should switch from an option to other
option when the expected value of each is about the same, so, in the domain of gains, a riskneutral subject would choose A for the first four decisions and B thereafter and in the domain
of losses, a risk-neutral subject would choose B for the first four decisions and A thereafter.
The objective of this task is to identify the number of safe choices made by each subject for
both payoff domains that allows us to quantify the risk preferences of the subject depending
on the specific domain.
Tables 3 and 4 (see Appendix A) describe the 10 sequential decisions with ambiguous
prospects for the gain payoffs and the loss payoffs respectively. Each decision consists of the
choice between a non-ambiguous option that changed across all decisions (option A) and an
5
ambiguous option that remained constant (option B). For each sequence, we obtain the
number of non-ambiguous options selected by the subject that allows us, using the same
approach as Chakravarty and Roy (2009), to evaluate the ambiguity preferences of the subject
for both gain and loss domains. An ambiguity-neutral subject is indifferent between the two
options at the decision 5 for the gain and loss domains. Thus an ambiguity-neutral subject
should switch from the non-ambiguous option to the ambiguous option at the 5th or the 6th
decision in the gain or loss task.
3.2 Risk and ambiguous tasks in the group treatment
In the experiment, the participants must make 60 decisions, 40 were individuals while 20
were collectives. For these 20 collective decisions, they were gathered by groups of three
persons. These groups changed after each decision. Within these groups, the decision rule was
either unanimity rule or majority one, so that the decision rule is a “between-subject” factor in
this experiment. For the majority rule, the choice emerges simultaneously because there were
two options and three participants. Inversely, for the unanimity rule, an iterative process was
implemented. For each decision, the group members had five possibilities to try to coordinate.
If they did not manage, then a message “disagreement” appeared in the screen and, the groups
were formed again for the following decision.
The 20 collective decisions are composed of a sequence of 10 choices between two risky
lotteries and a sequence of 10 choices between a non-ambiguous lottery and an ambiguous
lottery, with the same probabilities and payoffs as in the individual treatment. All the payoffs
of the lotteries are in the domain of gains.
3.3 Order effects
So as to control potential order effects, several measures were applied. First, some
experimental sessions began with individual decisions while other started with collective
ones. Second, in the individual treatment, the order of appearance of the four tasks (implying
each ten decisions) was random. Nevertheless, for matching constraints, in the collective
treatment, we cannot randomize the order of presentation of the two tasks, so that each subject
began with risky prospects and then followed with ambiguous prospects.
3.4 Participants
The experiment was conducted at the Laboratory of Forest Economics, in Nancy (France). 60
students were recruited to realise this experience from different study programmes. Four
experimental sessions were conducted during around two hours. 31 students were male while
29 were female. The mean age was 21.55 years. The payments of subjects vary between 0 and
26 Euros with an average of 11 Euros.
4. The experiment results
We first present some preliminary information concerning the analysis of order effects and the
statistic approach used to study the experiment results before presenting the main results of
our experiment.
4.1 Preliminaries
In this experiment, we have one between-subject variable, collective decision rule, with two
components (unanimity versus majority), and three within-subject variables, each with two
components: treatment (individual versus collective), context of uncertainty (risk versus
6
ambiguity), and payoff domain (gain versus loss). Therefore, we have 60 observations for
each within-subject variables and only 30 observations for the between-subject variable.
In the statistical analysis, we have eliminated some inconsistent preferences (multiple
switching behaviour: MSB); the proportions of such observations vary across case between
5% and 15%. Such proportions are generally observed in the previous studies that have
employed a MPL risk elicitation mechanism. Holt and Laury (2002) report 13.2% of their
subjects exhibit MSB in hypothetical choices. In Eckel and Wilson (2004), such proportion is
12.9%. Andersen et al. (2006) find that 5.8% of their subjects switch multiple times when
allowing for indifference option (taken by 24.3% of subjects). Bruner et al. (2008) observe
that the proportion of MSB is 20%, and more recently, Jacobson and Petrie (2009) report 55%
of their subjects switch multiple times. When the proportion of MSB choices is small, the
typical solution for solving these mistakes has been to discard these observations. This is the
solution we select to limit the decisions inconsistent with economic theory.
As previously mentioned, an order effect can appear between individual and group treatments,
and this is for that reason that we have realised two types of session. On the aggregate level,
the order effect concerning treatment is not significant. Indeed, in risky context, the
comparison of variance in individual and collective treatments gives a F statistic of 0.681 with
a p value of 0.413. In the same way, in ambiguous context F is equal to 1.217 with a p value
of 0.275. Consequently, we accept the null hypothesis of equality of variances, allowing us to
conclude with an absence of order effect.
To statistically analyse the experiment results, we run three different Analysis of Variance
(ANOVA) on the dependent variable that is the choice variable. The first ANOVA aims at
testing the impact of payoff domain on both risk and ambiguity attitudes of individuals. The
second one is running in order to analyse the effect of the collective decision rule on the
attitudes toward risk and the attitudes towards ambiguity of groups. Finally, the third
ANOVA deals with the comparison of individual and group choices under risk and ambiguity.
Sometimes, these ANOVA are completed with other tests.
4.2 The analysis of individual decisions
4.2.1 Risk and ambiguity preferences
Table 5 provides information on the lottery choice frequencies for both gain and loss domains.
Consistent with Holt and Laury’s approach5, it is also possible to calibrate the bounds for the
risk aversion parameter and the proportion of safe choices for each bound. We use a popular
Constant Relative Risk Aversion (CRRA) characterization of risk attitudes, with U(x) = (x1−r )/
(1-r), for x > 0, where r ≠ 1 is the CRRA coefficient. Hence a value of 0 denotes risk
neutrality, negative or positive values indicate risk-loving, and positive or negative values
indicate risk aversion, in the domain of gains or losses respectively.
Table 5: Risk preferences classification in the domain of gains and losses
Number
of safe
choices
0 and 1
2
3
4
Bounds for relative
risk aversion
U(x)=x1-r/1-r
r < -0.95
-0,95 < r < -0.49
-0.49 < r < -0.15
-0.15 < r < 0.15
Gain domain
Risk preferences
classification
Highly risk loving
Very risk loving
Risk loving
Risk neutral
Proportion
of choices
7
15.8
Bounds for relative
risk aversion
U(x)=-(-x)1-r/1-r
r >1.37
0.97 < r < 1.37
0.68 < r < 0.97
0.41 < r < 0.68
Loss domain
Risk preferences
classification
Proportion
of choices
Extremely risk loving
Highly risk loving
Very risk loving
Risk loving
2
21.6
29.4
5
The risk tasks along with the CRRA EU representation (U(x) = x1-r/(1-r)) allow us to calibrate the bounds of the
risk aversion parameter r.
7
5
0.15 < r < 0.41
Slightly risk averse
6
0.41 < r < 0.68
Risk averse
7
0.68 < r < 0.97
Very risk averse
8
0.97 < r < 1.37
Highly risk averse
9 and 10
1.37 < r
Stay in bed
Proportion of risk averse subjects (r > 0.15)
21.1
26.3
19.3
7
3.6
77.3
0.15 < r < 0.41
Slightly risk loving
-0.15 < r < 0.15
Risk neutral
-0.49 < r < -0.15
Risk averse
-0.95 < r < -0.49
Very risk averse
r < -0.95
Stay in bed
Proportion of risk averse subjects (r > 0.15)
39.2
7.8
0
In the domain of gains, 90% of the subjects choose between 4 and 7 safe options before to opt
for a risky one, and 77.3% of them are risk averse. In the domain of losses, the average
number of safe choices varies between 3 and 6, and no one present risk aversion (92.2% are
risk lover). It seems that, on an aggregate level, most of subjects are risk averse in the domain
of gains and are risk lover in the domain of losses.
Figure 1 shows the proportion of safe choices for each of the 10 decisions listed in Tables 1
and 2. The dashed lines correspond to predicted behaviour under risk neutrality, for the
domain of gains and losses, respectively. For the two domains, the percentage choosing the
safe option falls as the probability of the higher payoff increases. Thus, in the domain of
gains, the subject chooses the option A for the first four decisions and then switches for the
option B. We observe that the curve is above the dashed line, confirming the risk aversion of
the majority of subjects in the domain of gains. In the domain of losses, the subject chooses
the option B for the first four decisions and then switches for the option A. In this domain, the
curve is always inferior to the dashed line of neutrality, indicating that most of the subject is
risk lover. The Figure 1 proves that risk preferences are not constant and vary across payoff
domains.
Figure 1: The proportion of individual safe choices in each decision
in the domain of gains and losses
1
0,9
0,8
Percentage
0,7
0,6
Gain
0,5
Loss
0,4
0,3
0,2
0,1
0
1
2
3
4
5
6
7
8
9
10
Decision num ber
Table 6 presents ambiguity preferences classification in the domain of gains and losses. Using
the same approach as Chakravarty and Roy (2009), the number of non-ambiguous lottery
choices allows us to determine the bounds for the ambiguity aversion parameter s using the
KMM (Klibanoff, Marinacci and Mukerji, 2005) representation of risk and ambiguity
attitudes. The KMM representation (KMM(x) = E φ(EU(x))) allows to disentangle pure risk
attitudes from pure ambiguity attitudes; the function U characterizes the subject’s attitude
towards risk whereas the function φ characterizes the subject’s attitude towards pure
ambiguity.
8
Table 6: Ambiguity preferences classification in the domain of gains and losses
Number of
nonambiguous
choices
Gain domain
Bounds for ambiguity
Ambiguity
aversion
preferences
KMM(x)=Ej(EU(x))
classification
with j(x)=xs
0
1
2
3
4
5
6
7
8
9
Proportion
of choices
Bounds for ambiguity
aversion
KMM(x)=Ej(EU(x))
with j(x)=-(-x)s if x<0
Extremely
ambiguity loving
Highly
ambiguity loving
Very
ambiguity loving
s> 1.92
1.92 ≤ s < 1.59
1.59 ≤ s < 1.35
1.35 ≤ s < 1.15
0.43 ≥ s > 0.30
Ambiguity neutral
Slightly
ambiguity averse
0.86 ≤ s < 0.75
0.75 ≤ s < 0.66
Ambiguity averse
Very
ambiguity averse
Highly
ambiguity averse
Extremely
ambiguity averse
0.66 ≤ s < 0.43
0.43 ≤ s < 0.30
10
s≥ 0.30
Proportion of ambiguity averse subjects (s < 1)
5.3
0.66 ≥ s > 0.43
0.75 ≥ s > 0.66
17.5
31.6
0.86 ≥ s > 0.75
1 ≥ s > 0.75
14
12.3
1.15 ≥ s > 1
1.35 ≥ s > 1.15
14
1.59 ≥ s > 1.35
5.3
1.92 ≥s > 1.59
Proportion
of choices
Extremely
ambiguity loving
Highly
ambiguity loving
Very
ambiguity loving
0.3 ≥ s
Ambiguity loving
Slightly
Ambiguity loving
1.15 ≤ s < 1
1 ≤ s < 0.86
Loss domain
Ambiguity
preferences
classification
5.4
5.4
Ambiguity loving
Slightly
Ambiguity loving
39.3
26.8
7.1
Ambiguity neutral
Slightly
ambiguity averse
8.9
7.1
Ambiguity averse
Very
ambiguity averse
Highly
ambiguity averse
Extremely
ambiguity averse
s > 1.92
Proportion of ambiguity averse subjects (s > 1)
45.6
23.1
In the domain of gains, we observe that the majority of the participants opts for 4 to 7 nonambiguous options. Note that 45.6% of the participants are ambiguity averse in the domain of
gains and 31.6% are risk neutral. In the domain of losses, the number of non-ambiguous
lottery choices is between 2 and 8, with only 23.1% of the participants which is ambiguity
averse and 50.1% which are ambiguity lover. As in risky context, it seems that most of the
subjects are ambiguity averse in the domain of gains and ambiguity lover in the domain of
losses, as illustrated by Figure 2. The dashed line represents the predicted behaviour under
ambiguity neutrality. This figure shows that preferences to ambiguity are not constant and
vary across payoff domains.
Figure 2: The proportion of individual non-ambiguous choices
in the domain of gains and losses
1
0,9
0,8
Percentage
0,7
0,6
Gain
0,5
Loss
0,4
0,3
0,2
0,1
0
1
2
3
4
5
6
7
8
9
10
Decision number
9
A significant fact appears from the experiment results. It seems that most of the subjects are
risk averse in the domain of gains and risk lover in the domain of losses one. In the same way,
the majority of subjects are ambiguity averse in the domain of gains and ambiguity seeking in
the domain of losses. Therefore both attitudes towards risk and ambiguity are domainspecific.
In the following Figure 3, we can easily observe that higher percentages of participants are
associated to aversion to risk and ambiguity in the domain of gains. Inversely, such
preferences are in the minority in the domain of losses because, in this domain risk seeking
and ambiguity seeking behaviours are observe in majority.
Figure 3: Individual preferences to ambiguity:
domain of gains versus domain of losses
100
90
80
Percentage
70
Risk averse
60
Risk neutral
Risk lover
50
Ambiguity averse
40
Ambiguity neutral
Ambiguity lover
30
20
10
0
Gain
Los s
In order to test this potential “Reflection effect”, we realize an ANOVA with two withinsubject factors: context of uncertainty (risky/ambiguous) and domain (gain/loss). The
assumption of normality distribution being a prerequisite of ANOVA, we indicate that for
individual decisions (whatever the domain and the context of uncertainty), Skewness
coefficients belong to [-0.586, 0.673] and Kurtosis coefficients to [-0.042, 0.647], so that the
distribution of safe choices in individual decisions follows a Normal distribution. The
ANOVA shows that the context of uncertainty is not significant (F(1;41) = 1.868; p value =
0.179). This result means that in risky and ambiguous context the preferences of the subjects
are identical. In other words, aversion is the dominant behaviour in the two contexts of
uncertainty. We also prove that the domain is significant at the level of 1% (F(1;41) = 25.752; p
value = 0.000). This means that the preferences to risk and ambiguity of the participants are
different function of the domain, gains versus losses. This result seems to confirm the
previous intuition indicating that the subjects are, for the majority of them, risk averse and
ambiguity averse in the domain of gains and risk lover and ambiguity lover in the domain of
losses. The cross variable “Domain*Context” is not significant (F(1;41) = 0.733; p value =
0.397). Consequently, it appears that, for a majority of the subjects, the standard “Reflection
effect” appears. We summarize this result:
Result 1: preferences to risk and to ambiguity are domain-dependent: most of the participants
are risk averse/ambiguity averse in the domain of gains and risk lover/ambiguity lover in the
domain of losses.
1
This first result is agreed with the conclusion of Di Mauro and Maffioletti (2004) and
Chakravarty and Roy (2009).
4.2.2 Relation between risk and ambiguity attitudes
Chakravarty and Roy (2009) assert that the result, concerning relation between risk and
ambiguity attitudes, is domain dependent. Consequently, we first analyse the domain of gains
and second, the losses one.
To test a potential correlation between risk and ambiguity behaviours in the domain of gains,
we compare the number of safe decisions with the number of non-ambiguous ones. The null
hypothesis stipulates the absence of relationship between these two numbers. We obtain a
Pearson correlation coefficient equals to 0.398 with a p value of 0.003. Consequently, we
reject the null hypothesis and we show the existence of a positive and significant correlation
(at the level of 1%) between individual attitudes towards risk and ambiguity, in the domain of
gains.
In the domain of losses, we also compare the number of safe choices with the number of nonambiguous ones. The null hypothesis is the same as previously mentioned. In this case, we
obtain a Pearson correlation coefficient of -0.072 with a p value of 0.627. In other words, we
accept the null hypothesis, meaning that there is no relationship between attitudes towards
risk and ambiguity, in the domain of losses.
Consequently, our results seem to be in accordance with the conclusion of Chakravarty and
Roy (2009). We summarize the result as follows:
Result 2: correlation between individuals’ attitudes towards risk and ambiguity is domain
dependent; it is positive in the domain of gains and null in the domain of losses.
4.3 The analysis of collective decisions
4.3.1 Risk and ambiguity preferences
Table 7 presents risk aversion classification based on lottery choices function of the decision
rule, majority or unanimity.
Table 7: Collective risk preferences function of the decision rule
Number of
safe choices
Bounds for relative
Risk preferences
risk aversion
classification
U(x)=x1-r/(1-r)
0 and 1
r < -0.95
Highly risk loving
2
-0,95 < r < -0.49
Very risk loving
3
-0.49 < r < -0.15
Risk loving
4
-0.15 < r < 0.15
Risk neutral
5
0.15 < r < 0.41
Slightly risk averse
6
0.41 < r < 0.68
Risk averse
7
0.68 < r < 0.97
Very risk averse
8
0.97 < r < 1.37
Highly risk averse
9 and 10
1.37 < r
Stay in bed
Proportion of risk averse players (r > 0.15)
Proportion of choices
Majority Unanimity
10
26.7
33.3
30
30
10
40
20
90
70
On average, subjects choose from 4 to 7 safe options. More interesting, whatever the decision
rule, no risk seeking behaviour appears. Indeed, we observe that 90% of the subjects are
collectively risk averse (and 10% are risk-neutral) with a majority rule and 70% with an
unanimity rule (and 30% are risk-neutral). However, risk aversion seems to be higher with a
majority rule than with an unanimity one. The same conclusion emerges from Figure 4. The
1
dashed line represents the predicted behaviour under risk neutrality. The Figure 4 shows that
collective risk preferences are not constant and vary with the decision rule.
Figure 4: The proportion of individual safe choices in each decision
function of the decision rule
1
0,9
0,8
Percentage
0,7
0,6
Majority
0,5
Unanimity
0,4
0,3
0,2
0,1
0
1
2
3
4
5
6
7
8
9
10
Decision number
Table 8 is the equivalent of Table 7 but for ambiguous context. The participants choose from
4 to 7 non-ambiguous options. We observe that, with a majority rule, 90% of the subjects are
ambiguity averse while with an unanimity one, they are 70%. Note that no ambiguity seeking
behaviour appears whatever the decision rule. The same intuition emerges from Figure 5.
Table 8: Collective ambiguity preferences function of the decision rule (in %)
Number of
nonambiguous
choices
0
1
2
3
4
5
6
7
8
9
10
Bounds for
Ambiguity preference
ambiguity aversion
classification
coefficient s
KMM(x)=Eϕ(EU(
x)) with ϕ(x)=xs
s > 1.92
Extremely ambiguity loving
1.92 ≤ s < 1.59
Highly ambiguity loving
1.59 ≤ s < 1.35
Very ambiguity loving
1.35 ≤ s < 1.15
Ambiguity loving
1.15 ≤ s < 1
Ambiguity neutral
1 ≤ s < 0.86
Slightly ambiguity averse
0.86 ≤ s < 0.75
Ambiguity averse
0.75 ≤ s < 0.66
Very ambiguity averse
0.66 ≤ s < 0.43
Highly ambiguity averse
0.43 ≤ s < 0.30
Extremely ambiguity averse
s≥ 0.30
Stay in bed
Proportion of ambiguity averse (s < 1)
Proportion of choices
Majority Unanimity
10
43.3
26.7
13.3
6.7
30
23.3
36.7
10
90
70
Figure 5: Collective ambiguous choice: majority versus unanimity
1
1
0,9
0,8
Percentage
0,7
0,6
Majority
0,5
Unanimity
0,4
0,3
0,2
0,1
0
1
2
3
4
5
6
7
8
9
10
Decision number
On Figure 5, the dashed line represents the predicted behaviour under ambiguity neutrality.
This figure presents a strange point at the fourth decision. For this particular decision, 80% of
participants submitted to majority rule adopt non-ambiguous option while for the fifth
decision, they are 90%. But, we can consider that the curve for the majority rule is above the
curve for the unanimity rule, suggesting that ambiguity aversion is higher with a majority rule
than with an unanimity one.
Consequently, one unique message seems to emerge from Tables 7 and 8 and Figures 4 and 5:
most of the participants seems to be collectively risk averse/ambiguity averse whatever the
decision rule, but aversion seems to be higher with a majority rule than with an unanimity
one. Figure 6 highlights this intuition.
Figure 6: Collective preferences to ambiguity: unanimity versus majority
100
90
80
Percentage
70
Risk averse
60
Risk neutral
Risk lover
50
Ambiguity averse
40
Ambiguity neutral
Ambiguity lover
30
20
10
0
Unanimity
Majority
No risk seeking and no ambiguity seeking behaviours appear on Figure 6. Moreover, an
obvious conclusion emerges from Figure 6, risk aversion and ambiguity aversion dominate
whatever the decision rule but higher aversion appears with majority.
In order to test this intuition, we conduct an ANOVA with a within-subject variable, context
of uncertainty (risky/ambiguous) and a between-subject one, the decision rule
(majority/unanimity). Normality of distribution and homogeneity of variance are prerequisites
for ANOVA with a between-subject factor. The normality assumption is fulfilled for
collective decisions (whatever the decision rule and context of uncertainty) because Skewness
coefficients belong to [-0.339, 0.632] and Kurtosis coefficients to [-1.387, -0.045].
1
Concerning homogeneity of variance, we obtain, in risky context: F(1,58) = 1.949 with a p value
= 0.168, and in ambiguous one: F(1,58) = 0.004 and a p value = 0.949, so that the null
hypothesis is not rejected (H0: error variance of the dependent variable is equal across
groups). Given these prerequisites, we show that the context of uncertainty is not significant
(F(1;58) = 1.295; p value = 0.260). In other words, the collective behaviours are almost the same
in the two contexts of uncertainty. We also prove that the decision rule is significant at the
level of 10% (F(1;58) = 3.258; p value = 0.076). This result implies that collective preferences to
risk and ambiguity are different function of the decision rule adopted. In other words,
collective aversion is higher with a majority rule than with an unanimity one. Moreover, the
cross variable “Context of uncertainty * Decision rule” is not significant (F (1;58) = 0.008; p
value = 0.931). We summarize this result as follows:
Result 3: collective risk and ambiguity preferences are different function of the decision rule:
for most of the participants, risk aversion and ambiguity aversion are higher with a majority
rule than with an unanimity one.
This result is of particular interest because it underlines the fact that collective preferences to
ambiguity are the same, the participants are collectively ambiguity averse, but also highlights
that decision rule plays a decisive role on the issue of collective decision process. Thus, our
result shows that the issue of a negotiation largely depends on the rule adopted. The
implications of such a result are important. Indeed, the decision rule may be chosen function
of the expected issue, so that new questions emerge concerning the way collective decision
rule is chosen.
4.3.2 Relation between risk and ambiguity attitudes in collective treatment
We first analyse the potential correlation between risk and ambiguity attitudes with an
unanimity rule and second, with a majority one.
To test a potential correlation between risk and ambiguity behaviour with an unanimity rule,
we compare the number of safe decisions with the number of non-ambiguous ones. The null
hypothesis is the absence of relationship between the two numbers. We find a Pearson
correlation coefficient equals to -0.149 with a p value of 0.431. Consequently, we accept the
null hypothesis and the absence of link between the two attitudes.
With a majority rule, we also compare the number of safe choices with the number of nonambiguous ones. The null hypothesis is the same as previously mentioned. We obtain a
Pearson correlation coefficient of 0.202 with a p value of 0.284, so that we can not reject the
null hypothesis.
To conclude, it seems that there is no relationship between collective attitudes to risk and
ambiguity with the two decision rules: unanimity and majority. We summarize this result:
Result 4: there is no correlation between collective attitudes to risk and ambiguity, whatever
the decision rule.
This result is different from the one obtained in individual decisions. Recall that Result 2
indicates that attitudes towards risk and ambiguity of participants are correlated in the domain
of gains. Consequently, a difference appears between collective decisions and individual one
concerning the link between these two attitudes. This interesting result proves that collective
preferences towards risk and ambiguity are not the addition of individual preferences.
4.3.3 Proportions of disagreement in each decision
1
The experimental sessions conducted with unanimity rule can lead to disagreement among the
group members. Indeed, it was possible that, at the end of the five iterations, the subjects were
not agreed concerning the option to choose. In this case, the message “disagreement”
appeared on the screen and the groups were formed again to move on the following decision.
This disagreement between group members can appear only when the unanimity rule is tested.
We present in Tables 9 and 10, the two scenarios implying collective decisions with
unanimity rule in risky context and ambiguity one, respectively.
Percentage of disagree group
Table 9: Collective decisions in risky context with unanimity rule
1
0,9
0,8
0,7
0,6
iteration 1
iteration 2
0,5
0,4
0,3
0,2
0,1
0
iteration 3
iteration 4
iteration 5
1
2
3
4
5
6
7
8
9
10
Decisions
Percentage of disagree group
Table 10: Collective decisions in ambiguous context with unanimity rule
1
0,9
0,8
0,7
iteration 1
0,6
0,5
0,4
iteration 2
0,3
0,2
iteration 5
iteration 3
iteration 4
0,1
0
1
2
3
4
5
6
7
8
9
10
De cisions
Several remarks can be made on the basis of these two tables. First, most of the ten decisions
imply a disagreement between the group members at the first iteration (9 decisions on 10 in
risky context and 8 in ambiguous one). Second, around half of the decisions conclude with a
disagreement for 20% of groups (decisions 5 in risky context and 4, 5, 6 and 8 in ambiguous
one) and for 10% of groups (decisions 6, 7 and 8 in risky context and decision 10 in
ambiguous one). Third, the percentage of disagree group is higher for decisions 5, 6 and 7 in
risky context and decisions 4, 5 and 6 in ambiguous one. On the contrary, decisions 1, 2 and 3
1
are not really concerned by disagreement. This difference is due to the fact that decision 4
implies risk neutrality in Table 9 and decision 5 neutrality to ambiguity in Table 10.
4.4. Individual and group decisions
4.4.1 Comparison between the individual treatment and the group one
In order to compare individual treatment and collective one, we conduct an ANOVA in the
domain of gains with two within-subject factors: context of uncertainty (risky/ambiguous) and
treatment (individual/collective). Normality of distribution is fulfilled in the two treatments
(cf. Skewness coefficients and Kurtosis ones stated in Sections 4.2 and 4.3). Our results show
that these two variables are non significant. For the context of uncertainty, we obtain F (1;53) =
0.175 with a p value of 0.677 while for treatment, we have F (1;53) = 1.738 and a p value of
0.193. Consequently, there is no difference between the risk and ambiguity preferences of
individuals and groups.
More interesting, we deepen the analysis in order to analyse the role of decision rule.
4.4.2 A test of the existing results
Recall, that works with a collective decision rule of unanimity (Rockenbach et al., 2007;
Shupp and Williams, 2008; Baker et al., 2008; Masclet et al., 2009) report that groups tend to
be more risk averse than individuals, while papers considering majority rule (Harrison et al.,
2004) conclude to no difference in risk aversion between individuals and groups. We test
these results.
First, for unanimity rule, we realize a pair comparison between scenario with collective
decision and scenario with individual decision (in the domain of gain and in risky context).
The null hypothesis stipulates the equality of the average number of safe choices made in each
scenario. We obtain a t of student of 0.182 with a p value of 0.857, so that we accept the null
hypothesis. This result is different to the existing ones. A possible explanation lies on the
implementation of unanimity rule. Indeed, several methods were used to analyse collective
choice with unanimity rule. Baker et al. (2008) used “cheap talk” to implement collective
choice, i.e. the participants discuss in order to take a collective decision. We use a
computerized procedure similar to Masclet et al. (2009) (and Harrison et al. (2004) also) with
a noticeable exception. Masclet et al. (2009) consider that, if the subjects are not coordinated
at the end of the five iterations, then, the computer randomly select one of the two possible
options. In our experiment, we assume that, in such a situation, a failure appeared in the
coordination problem and it represents an important result that we want to conserve, so that
the message “disagreement” appeared. Consequently, differences can emerge in terms of
number of safe choices and then, on preferences to risk, due to the procedure employed.
Second, for majority rule, we compare scenario with collective decision and scenario with
individual decision (in the domain of gain and in risky context). The null hypothesis is similar
as the one previous mentioned. We obtain a t of student of -0.425 with a p value of 0.674,
implying an equality of means between numbers of safe choices adopted in risky collective
treatment with majority rule and in individual treatment. This result confirms the one of
Harrison et al. (2004).
Consequently, these results associated to the conclusion of the section 4.4.1, allow us to
obtain the following result:
1
Result 5: individual and collective preferences towards risk and ambiguity are similar
whatever the decision rule adopted for collective choices.
This result suggests that subjects behave in the same manner when they made an individual
choice and a group one. Result 1 and Result 3 prove that subjects are risk averse and
ambiguity averse in the domain of gains, meaning that their preferences are similar. However,
Result 5 completes these previous results by showing that aversion to risk and ambiguity is
the same whatever the treatment, individual or collective. In other words, participants are
individually and collectively risk averse and ambiguity averse and the aversion is not higher
or lower function of the treatment.
5. Conclusion and discussion
This paper deals with individual and collective decisions in risky and ambiguous situation.
Our concern is about the comparison of risk and ambiguity preferences during individual
decision and group ones. We are also interested in the decision rule adopted during the
collective decision process, unanimity versus majority. We obtain several important results.
On the one hand, the analysis of individual decisions lets appeared that i) most of the subjects
behave in accordance with the so-called “Reflection effect” and that, ii) a positive correlation
between risk and ambiguity attitudes emerges in the domain of gains while no correlation
appears in the domain of losses. On the other hand, the study of collective decisions shows
that i) aversion is higher with a majority rule than with an unanimity one but also that ii) there
is no correlation between collective attitudes to risk and ambiguity, whatever the decision rule
(unanimity/majority). Finally, the comparison between individual and collective decisions
leads to a result having important repercussions, risk and ambiguity preferences of individuals
and groups are similar whatever the decision rule adopted for collective choices.
Some extensions of this paper can be considered. First, for the individual decision, we have
the domain of gains and losses while for collective decision, we only consider the domain of
gains. This choice is due to our interest for decision rule in the collective decision process.
Indeed, as the decision rule is a between-subject factor, if we consider the domain of gains
and losses for each decision rule, lots of participants are needed. Consequently, a new
experiment considering the two domains and only one decision rule could be an extension.
The objective will be to test the existence of a “Collective Reflection Effect”. Second, we
have seen in this paper that several methods exist in order to implement a collective decision,
especially the “cheap talk” used by Baker et al. (2004) and the computerized method of
Harrison and al. (2004), Masclet et al. (2009) and the extension for unanimity rule proposed
in this paper. A new experiment could test the effect of methodology on the collective
preferences to risk and ambiguity.
1
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2
Appendix A
Table 1: Individual decisions with risky prospects in the domain of gains
1
2
3
4
5
6
7
8
9
10
Option A
Prob. P
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Gains
7€
7€
7€
7€
7€
7€
7€
7€
7€
7€
Prob. (1-P)
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Gains
5€
5€
5€
5€
5€
5€
5€
5€
5€
5€
Option B
Prob. P
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Gains
13 €
13 €
13 €
13 €
13 €
13 €
13 €
13 €
13 €
13 €
Prob. (1-P)
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Gains
0€
0€
0€
0€
0€
0€
0€
0€
0€
0€
Choice
A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
Table 2: Individual decisions with risky prospects in the domain of losses
1
2
3
4
5
6
7
8
9
10
Option A
Prob. P
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Gains
-7 €
-7 €
-7 €
-7 €
-7 €
-7 €
-7 €
-7 €
-7 €
-7 €
Prob. (1-P)
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Gains
-5 €
-5 €
-5 €
-5 €
-5 €
-5 €
-5 €
-5 €
-5 €
-5 €
Option B
Prob. P
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Gains
-13 €
-13 €
-13 €
-13 €
-13 €
-13 €
-13 €
-13 €
-13 €
-13 €
Prob. (1-P)
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Gains
0€
0€
0€
0€
0€
0€
0€
0€
0€
0€
Choice
A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
Table 3: Individual decisions with ambiguous prospects in the domain of gains
Choose a colour: BLACK ○
1
2
3
4
5
6
7
8
9
10
Option A: urn A
In urn A, the distribution of balls
is 5 black and 5 white
The chosen color The chosen color
is obtained
is not obtained
Gains
Gains
13 €
0€
12 €
0€
11 €
0€
10 €
0€
9€
0€
8€
0€
7€
0€
6€
0€
4€
0€
2€
0€
WHITE ○
Option B: urn B
In urn B, the possible distributions
of balls is not known
The chosen color
The chosen color
is obtained
is not obtained
Gains
Gains
9€
0€
9€
0€
9€
0€
9€
0€
9€
0€
9€
0€
9€
0€
9€
0€
9€
0€
9€
0€
Choice
A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
2
Table 4: Individual decisions with ambiguous prospects in the domain of losses
Choose a colour: BLACK ○
1
2
3
4
5
6
7
8
9
10
Option A: urn A
In urn A, the distribution of balls
is 5 black and 5 white
The chosen color The chosen color
is obtained
is not obtained
Gains
Gains
-13 €
0€
-12 €
0€
-11 €
0€
-10 €
0€
-9 €
0€
-8 €
0€
-7 €
0€
-6 €
0€
-4 €
0€
-2 €
0€
WHITE ○
Option B: urn B
In urn B, the possible distributions
of balls is not known
The chosen color
The chosen color
is obtained
is not obtained
Gains
Gains
-9 €
0€
-9 €
0€
-9 €
0€
-9 €
0€
-9 €
0€
-9 €
0€
-9 €
0€
-9 €
0€
-9 €
0€
-9 €
0€
Choice
A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
2