Decisions under unpredictable losses: An examination of the restated diversification principle

Judgment and Decision Making, Vol. 2, No. 5, October 2007, pp. 312–316
Decisions under unpredictable losses: An examination of the
restated diversi?cation principle
Ali M. Ahmed?
School of Management and Economics
Växjö University
Abstract
An experimental test of the descriptive adequacy of the restated diversi?cation principle is presented. The principle
postulates that risk-averse utility maximizers will pool risks for their mutual bene?t, even if information is missing about
the probabilities of losses. It is enough for people to assume that they face equal risks when they pool risks. The results
of the experiment support the principle.
Keywords: Group behavior, loss sharing, unpredictable losses, experiment, risk pooling.
1 Introduction
of losses. Without such information, the pricing will be-
come arbitrary and the negotiation cost may be large.
In a nutshell, the diversi?cation principle says that, if
Insurance under apparent ambiguity, when information
risk-averse utility maximizers can choose between two
about the probabilities of losses is lacking, may not be
assets with identical but random returns, they will prefer
possible at all (Hogarth and Kunreuther, 1989). Loss-
to invest half of their endowment in each asset (Roth-
sharing, however, can be undertaken without pricing the
schild & Stiglitz, 1971). The principle can be para-
potential loss.
phrased in the following means: If two risk-averse utility
The restated diversi?cation principle was ?rst pre-
maximizers with assets of the same value face the same
sented by Skogh (1999) in the case of two identical pool
distributions of potential losses, they will gain by sharing
members. Skogh and Wu (2005) generalized the princi-
the potential losses equally (Skogh, 1999).
ple to the case where individuals’ losses differ in amount
The restated diversi?cation principle holds for any dis-
or in probability and to the case where individuals’ atti-
tribution of outcomes, as long as the distribution is the
tudes toward risks differ. This paper tests the descriptive
same for both people. It does not matter whether the
adequacy of the restated diversi?cation principle with an
probability of a loss is small or large; in both cases, the
experiment. In particular we test the following two hy-
people gain by sharing the loss. This implies that sharing
potheses:
loss is also mutually favorable if information is missing
about the probabilities of losses. It is not even necessary
H1 : Under apparent ambiguity, people will share po-
that the distribution of outcomes is the same for both peo-
tential losses.
ple; it is enough that they both accept that there is no rea-
son to assume that their probabilities of losses differ (pre-
H2: If the distribution of potential losses is different
sumption of equality). Possible differences can therefore
across people but unknown, people will still share poten-
be neglected if there is no knowledge about them.
tial losses.
When information about the probabilities of losses is
missing, mutual sharing of losses may be superior to in-
The experimental design, results, and discussion are
surance since an insurance premium is normally based
presented in Sections 2, 3, and 4, respectively.
on technical or actuarial information on the probability
?I owe gratitude to Göran Skogh who introduced me to this line
2 Experimental design
of research. Also, thanks to Jonathan Baron and a reviewer for their
suggested improvements. This project was ?nancially supported by
Each participant received SEK20 as a show-up fee. Par-
Jan Wallanders and the Tom Hedelius Foundation. Address: School
of Management and Economics, Växjö University, SE–351 95 Växjö,
ticipants were then divided into groups of four. They
Sweden. Email: ali.ahmed@vxu.se
were told that they would go through three rounds, and,
312

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Judgment and Decision Making, Vol. 2, No. 5, October 2007
Risk pooling in groups
313
at the beginning of each round, they would receive an en-
dowment of 80 tokens, half of which may be lost by the
Table 1: Number (Percentage) of players adopting each
end of the round. In each round, the participants had to
action.
pick a ball from a bucket containing 100 colored balls. If
Round 1 Round 2 Round 3
a black ball was picked from the bucket, the participant
lost 40 of the 80 tokens.
No action
5 (6.25)
2 (2.5)
6 (7.5)
The participants could carry the potential loss individ-
Equal loss sharing
72 (90)
72 (90)
68 (85)
ually, insure themselves against the loss, or share it with
Partial loss sharing
3 (3.75)
6 (7.5)
6 (7.5)
other players in their group. They could cooperate only
Sell insurance
0 (0)
0 (0)
0 (0)
with other members of their group. Participants could
freely communicate within the group, and there was no
Buy insurance
0 (0)
0 (0)
0 (0)
time limit set for each round. Participants had the follow-
N
80 (100) 80 (100) 80 (100)
ing alternatives in each round:
1. No action. The round played alone, with the poten-
2. Participants were given 80 tokens. In the second
tial loss of 40 tokens.
round, participants received the information that
there were four different buckets and that each con-
2. Equal loss sharing. Each of the four group members
tained 100 balls of various colors. Each participant
covers one-fourth of the losses of the group. For in-
was randomly assigned one bucket. If a black ball
stance, if one of the group members drew a black
were picked, 40 tokens were lost.
ball, each participant in that group would pay 10 to-
kens to cover the loss. If two participants received a
3. Participants were given 80 tokens.
In the third
black ball, each group member would pay 20 tokens,
round, participants were told that there were four
and so on.
buckets, each of which contained 100 balls of var-
ious colors. They were informed that one bucket
3. Partial loss sharing. Fewer than four group mem-
had 70 black balls, one bucket had 50 black balls,
bers share their losses. This was the case when one
one bucket had 30 black balls, and one bucket had
or two participants in a group chose to take another
10 black balls. Each participant was randomly as-
action instead of sharing the losses.
signed to one of the buckets but never knew which
bucket. He or she knew only that the buckets had
4. Sell insurance, A player insures other players by
different numbers of black balls but did not know
charging a premium in advance. Each player was al-
if his or her bucket contained the lowest or highest
lowed to insure losses up to 40 tokens in each round.
number of black balls. Again, if a black ball were
This was to avoid insolvent insurers.
picked, 40 tokens were lost.
5. Buy insurance. The player buys insurance and pays
The decision of each participant was written on a form,
one or more other members in the group to cover the
one for each round. After each round, losses were cal-
potential loss.
culated and payments were made. The experiment took
about 30 minutes. For each token left, the participants
Note that participant could agree on any type of insur-
received SEK0.5. The experiment took place at different
ance solution, as long as they did not insure for losses of
occasions in 2006 at Växjö University. A total of 80 par-
more than 40 tokens. The insurance premium had to be
ticipants — 20 groups of four — participated in the ex-
set by participants themselves. Thus, insurance involved
periment. The average age of participants was 22 years,
negotiating between buyers and sellers over the insurance
and 40 percent of them were women.
premium. Let us now describe the differences among the
The Appendix shows the instructions.
three rounds.
1. Participants were given 80 tokens. In the ?rst round,
3 Results
participants received the information that the bucket
contains 100 balls of various colors and that if a
The results of the experiment are presented in Table 1. In
black ball were picked, they would lose half their
the ?rst round, participants picked a ball from the same
endowment. There was no information on the num-
bucket. If a black ball was picked, they lost 40 tokens.
ber of black balls or even if there were any in the
A ?rst option for the participants was to take no action.
bucket at all. All participants picked a ball from the
Only ?ve participants made this choice; four of these be-
same bucket.
longed to the same group. A second option was to share

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Judgment and Decision Making, Vol. 2, No. 5, October 2007
Risk pooling in groups
314
losses equally among all group members. Seventy-two
of a loss; in the ?nal round, there was full information
participants, or 18 groups, chose this action. Three partic-
of the probability. The results showed that participants
ipants partially shared their losses since one in that group
shared losses when information about the probabilities of
chose not to do anything.
losses was missing, in a way that supports the restated
The pattern in the second round, with four different
principle. However, a large percentage of the participants
buckets with unknown distribution of losses, was similar
chose to do nothing. Participants may have believed that
to the ?rst round. The number of individuals or groups
cooperation did not pay: In each round, half of the en-
that chose to share their losses among all group members
dowment could be lost. If they believed probabilities of
did not change. Two individuals from different groups
losses would be relatively large, bankruptcy could be the
chose to take no action; therefore, the remaining group
expected outcome of the ?ve rounds and gambling, in that
members chose partial loss-sharing.
case, may be a way to survive by luck. In this paper, we
In the third round, participants knew the number of
present results from an experiment in which we overcome
black balls in different buckets, but they did not know
the problem of bankruptcy by giving our participants a
which bucket they would pick from. The results were
new set of tokens in each round.
still similar to the previous two rounds. The majority —
The second difference from Ahmed and Skogh (2006)
68 participants or 17 groups — chose to share their losses
is that we also tested the second hypothesis stated above
equally among all group members. Six participants chose
by using different urns for different group members in the
not to take any action, and another six participants shared
last two rounds.
their losses partially.
The observations clearly show that participants share
Note that not a single participant chose to insure
losses when information about the probabilities of losses
someone else and, consequently, not a single participant
is missing in a way that supports the descriptive adequacy
bought insurance in any of the rounds. In the ?rst two
of the restated diversi?cation principle. That is, sharing
rounds, the reason for that choice could be that it is dif-
is chosen exclusively because it eliminates the problem of
?cult to set a premium ex ante because of missing infor-
pricing when losses are unpredictable. Our observations
mation about the probabilities of losses. However, par-
also show that participants share losses even if they know
ticipants did not buy or sell insurance in the ?nal round
that the distribution of losses is different across group
either, when the distribution of balls was known. We will
members — but they do not know in what way because
discuss this further in the next section.
of the lack of information that discriminates among group
The results here are clear cut; however, for the record,
members. Participants share potential losses by applying
the proportion of participants that choose equal sharing
the presumption of equality.
was signi?cantly larger than the proportions of partici-
A noteworthy result that needs to be commented is
pants choosing alternative actions (Chi-square test, p <
that none of the participants bought or sold insurance
0.0001). Also, the change in the proportion of partici-
in the experiment. In the ?rst two rounds, the reason is
pants choosing equal sharing compared to other actions
probably that an insurance premium could not be set, as
across rounds was not signi?cant (McNemar test).
information about the probabilities of losses was lack-
ing. However, in the last round, this information was
known, so the question remains as to why not a single
4 Discussion
insurance contract was established. One reason could be
that participants found insurance to be a more compli-
cated choice than loss-sharing since it involved negotia-
One previous study, Ahmed and Skogh (2006), examined
tion between the buyer and the seller over the insurance
the restated diversi?cation principle, but the present paper
premium. Another explanation could be that mutual in-
differs from the previous study in two important ways:
surance, where participants insure each other at the same
First, in Ahmed and Skogh (2006), participants re-
premium, is equivalent to loss-sharing. Hence, if every-
ceived 1,000 tokens as an endowment that could be lost in
one in the group wanted to buy and sell insurance, then
?ve risky rounds. Participants were divided into groups
they might as well just share losses instead. An inter-
of four and informed of an urn that contained 100 balls
esting extension of the experiment would be to include a
of various colors. In each round, each player randomly
?fth person in the group that acts only as an insurer. A
picked one ball. A black ball would cost the partici-
possible explanation could also be that cooperating with
pant 500 tokens. In each round, participants could also
other group members in previous two rounds establishes
take actions to reduce the risk of large losses. One of
a bond among participants that makes it is hard to devi-
the actions was to share losses with other participants.
ate from the group behavior to individual behavior. In-
The information on the risk varied across rounds: In the
troducing anonymity among group members might limit
?rst round, there was no information on the probability
this possibility. Replications are requested.

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Judgment and Decision Making, Vol. 2, No. 5, October 2007
Risk pooling in groups
315
The paper relates to the vast literature on decision un-
Theory 3, 66–84.
der risk. Savage (1954) put forward the subjective prob-
Savage, L. J. (1954). The foundations of statistics. New
ability theory, where the distinction between known and
York: Wiley.
unknown probabilities is meaningless because subjective
Skogh, G.. (1999). Risk-sharing institutions for unpre-
probabilities are never unknown. Empirical evidence,
dictable losses. The Journal of Institutional and Theo-
however, has shown that people do, in fact, make such
retical Economics, 155, 505–515.
a distinction (Ellsberg, 1961). A justi?cation of the sub-
Skogh, G. & Wu, H. (2005). The diversi?cation theo-
jective probability theory is that people cannot make deci-
rem restated: Risk-pooling without assignment of loss
sions without assignment of probabilities. Yet, the results
probabilities. Journal of Risk and Uncertainty 31, 35–
of the present experiment show that this is not completely
51.
correct: Loss-sharing among people takes place without
assignment of probabilities.
The paper also relates to the prospect theory of Kah-
Appendix: Translation of written in-
neman and Tversky (1979). Prospect theory predicts that
structions
people are risk-averse in the domain of gains and risk-
seeking in the domain of losses. The results in this pa-
General instructions
per might then look inconsistent with the prediction of
prospect theory, as participants did not appear to be risk-
Welcome! Thank you for participating in this experi-
seeking when facing potential losses. Alternatively, it
ment. The experiment will take a maximum of one hour.
may be the case that participants actually were in the do-
The purpose of the study is to gain greater insight into
main of gains when making their decision since they re-
economic decision-making. To express our gratitude and
ceived a show-up fee that could not be lost in the exper-
to compensate you for your time, you have already been
iment; in addition, they gained 40 tokens guaranteed in
given a show-up fee of SEK 20. In addition to this, you
each round. Putting it in this way, participants should
also have the opportunity to earn more money during the
be risk-averse, and the experiment may have not im-
experiment.
plemented potential losses. On the other hand, a simi-
As you already have noticed, this experiment will
lar experiment by Kühberger, Schulte-Mecklenbeck, and
be conducted in groups of four, and you have already
Perner (2002) shows that participants who have lost half
been matched with three other participants whom you
of their given endowment have a tendency of being risk-
are seated with. In this experiment, you and your group
seeking. It is dif?cult to arrive at de?nite conclusions by
members will go through three rounds, and in each round
hypothesizing from known facts and observations from
you and your group members have to make a decision. In
other studies.
each round, you and your group members will receive 80
tokens each. The total gains to you from the experiment
depend on the tokens remaining when the three rounds
References
are over. For each token you have left, you will be paid
SEK0.5.
Ahmed, A. M. & Skogh, G. (2006). Choices at vari-
In each round, you and your group members have to
ous levels of uncertainty: an experimental test of the
pick a ball from a bucket containing 100 colored balls.
restated diversi?cation theorem. Journal of Risk and
If a black ball is picked from the bucket, you will lose
Uncertainty 33, 183–196.
40 of the 80 tokens. Hence, each round may result in
Ellsberg, D. (1961). Risk, ambiguity, and the Savage ax-
a loss of 40 tokens. You may take this risk individually,
ioms. Quarterly Journal of Economics, 75, 643–699.
share losses with others in the group, or you may also buy
Hogarth, R. M., & Kunreuther, H. C. (1989). Risk, ambi-
or sell insurance. The alternative actions are described
guity, and insurance. Journal of Risk and Uncertainty,
below:
2, 5–35.
A. No action. You go through the round yourself with
Kahneman, D., & Tversky, A. (1979). Prospect theory:
the potential loss of 40 tokens.
An analysis of decision under risk. Econometrica, 47,
B. Equal loss-sharing. Each of you in the group covers
263–291.
one-fourth of the total losses of the group. For instance,
Kühberger, A., Schulte-Mecklenbeck, M., & Perner, J.
if one of your group members draws a black ball, each of
(2002). Framing decisions: Hypothetical and real. Or-
you in the group will pay 10 tokens to cover the loss. If
ganizational Behavior and Human Decision Processes
two in your group receive a black ball, then each of you
89, 1162–1175.
will pay 20 tokens, and so on.
Rothschild, M. & Stiglitz, J. E. (1971). Increasing risk
C. Partial loss-sharing. It is also possible that two or
II: Its economic consequences. Journal of Economic
three of you share the potential losses. This may be the

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Judgment and Decision Making, Vol. 2, No. 5, October 2007
Risk pooling in groups
316
case if one or two of your group members chooses to take
Information given in the second round
another action instead of sharing the losses.
You have been given 80 tokens. There are four buckets
D. Sell insurance. You may insure other group mem-
containing 100 balls of various colors. The four buckets
bers by charging a premium in advance. You are allowed
to insure losses up to 40 tokens in each round. For ex-
will be randomly assigned to you and your group mem-
bers: one bucket for each of you. You will then be asked
ample, the premium for coverage of 40 tokens could be
to pick a ball from your bucket. If you pick a black ball,
equal to 5, 10, 20, 30 or so on, depending on what you
you will experience a loss of 40 tokens. You have the
believe about the risk of receiving a black ball. If you
possibility to take any of the actions de?ned earlier. You
insure another person in your group, you will ?rst pick a
will pick a ball after you and your group members have
ball for yourself and then pick a ball again for the person
decided what actions to take.
you insured. Partial cover of the loss may also be applied.
E. Buy insurance. You may buy insurance from other
group members and pay them a premium in advance to
Information given in the third round
cover the potential loss. You and your insurer have to de-
You have been given 80 tokens. There are four buckets
cide what premium to set. Hence, if you pay a premium
containing 100 balls of various colors: one bucket has 70
to another group member to cover all your losses, your
black balls, one bucket has 50 black balls, one bucket has
earnings from that round will be 80 tokens minus the pre-
30 black balls, and one bucket has 10 black balls. The
mium paid.
four buckets will be randomly assigned to you and your
You can discuss freely with other members of your
group members: one bucket for each of you. You will
group. Your choice of action in each round is to be writ-
not, however, know which bucket contains which number
ten down on a decision form that you will receive before
of black balls. You know only that the buckets contain
each round together with speci?c information related to
different numbers of black balls. You will then be asked
each round. The picking of balls will take place after
to pick a ball from your bucket. If you pick a black ball,
your decision has been collected.
you will experience a loss of 40 tokens. You have the
possibility to take any of the actions de?ned earlier. You
Information given in the ?rst round
will pick a ball after you and your group members have
You have been given 80 tokens. You will be asked to pick
decided what actions to take.
a ball from a bucket that contains 100 balls of various
colors. If you pick a black ball, you will experience a
loss of 40 tokens. You have the possibility to take any of
the actions de?ned earlier. You will pick a ball after you
and your group members have decided what actions to
take. You and your group members will pick a ball from
the same bucket.

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