Decisions under unpredictable losses: An examination of the
restated diversification principle
Ali M. Ahmed1
School of Management and Economics
Växjö University
Judgment and Decision Making, vol. 2, no. 5, October 2007, pp. 312-316
Abstract
An experimental test of the descriptive adequacy of the restated
diversification principle is presented. The principle postulates that
risk-averse utility maximizers will pool risks for their mutual
benefit, 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
In a nutshell, the diversification principle says that, if
risk-averse utility maximizers can choose between two assets with
identical but random returns, they will prefer to invest half of their
endowment in each asset (Rothschild & Stiglitz, 1971). The principle
can be paraphrased in the following means: If two risk-averse utility
maximizers with assets of the same value face the same distributions of
potential losses, they will gain by sharing the potential losses
equally (Skogh, 1999).
The restated diversification principle holds for any
distribution of outcomes, as long as the distribution is the same for
both people. It does not matter whether the probability of a loss is
small or large; in both cases, the people gain by sharing the loss.
This implies that sharing loss is also mutually favorable if
information is missing about the probabilities of losses. It is not
even necessary that the distribution of outcomes is the same for both
people; it is enough that they both accept that there is no reason to
assume that their probabilities of losses differ (presumption of
equality). Possible differences can therefore be neglected if there is
no knowledge about them.
When information about the probabilities of losses is missing, mutual
sharing of losses may be superior to insurance since an insurance
premium is normally based on technical or actuarial information on the
probability of losses. Without such information, the pricing will
become arbitrary and the negotiation cost may be large. Insurance under
apparent ambiguity, when information about the probabilities of losses
is lacking, may not be possible at all (Hogarth and Kunreuther, 1989).
Loss-sharing, however, can be undertaken without pricing the potential
loss.
The restated diversification principle was first presented by Skogh
(1999) in the case of two identical pool members. Skogh and Wu (2005)
generalized the principle to the case where
individuals' losses differ in amount or in probability
and to the case where individuals' attitudes toward
risks differ. This paper tests the descriptive adequacy of the restated
diversification principle with an experiment. In particular we test the
following two hypotheses:
H1 : Under apparent ambiguity, people will share potential losses.
H2: If the distribution of potential losses is different across people
but unknown, people will still share potential losses.
The experimental design, results, and discussion are presented in
Sections 2, 3, and 4, respectively.
2 Experimental design
Each participant received SEK20 as a show-up fee. Participants were then divided
into groups of four. They were told that they would go
through three rounds, and, at the beginning of each round, they would
receive an endowment of 80 tokens, half of which may be lost by the end
of the round. In each round, the participants had to pick a ball from a
bucket containing 100 colored balls. If a black ball was picked from
the bucket, the participant lost 40 of the 80 tokens.
The participants could carry the potential loss individually, insure
themselves against the loss, or share it with other players in their
group. They could cooperate only with other members of their group.
Participants could freely communicate within the group, and there was
no time limit set for each round. Participants had the following
alternatives in each round:
- No action. The round played alone, with the potential
loss of 40 tokens.
- Equal loss sharing. Each of the four group members covers
one-fourth of the losses of the group. For instance, if one of the
group members drew a black ball, each participant in that group would
pay 10 tokens to cover the loss. If two participants received a black
ball, each group member would pay 20 tokens, and so on.
- Partial loss sharing. Fewer than four group members share
their losses. This was the case when one or two participants in a group
chose to take another action instead of sharing the losses.
- Sell insurance, A player insures other players by
charging a premium in advance. Each player was allowed to insure losses
up to 40 tokens in each round. This was to avoid insolvent insurers.
- Buy insurance. The player buys insurance and pays one or
more other members in the group to cover the potential loss.
Note that participant could agree on any type of insurance solution, as
long as they did not insure for losses of more than 40 tokens. The
insurance premium had to be set by participants themselves. Thus,
insurance involved negotiating between buyers and sellers over the
insurance premium. Let us now describe the differences among the three
rounds.
- Participants were given 80 tokens. In the first round,
participants received the information that the bucket contains 100
balls of various colors and that if a black ball were picked, they would
lose half their endowment. There was no information on the number of
black balls or even if there were any in the bucket at all. All
participants picked a ball from the same bucket.
- Participants were given 80 tokens. In the second round,
participants received the information that there were four different
buckets and that each contained 100 balls of various colors. Each
participant was randomly assigned one bucket. If a black ball were
picked, 40 tokens were lost.
- Participants were given 80 tokens. In the third round,
participants were told that there were four buckets, each of which
contained 100 balls of various colors. They were informed that one
bucket had 70 black balls, one bucket had 50 black balls, one bucket
had 30 black balls, and one bucket had 10 black balls. Each participant
was randomly assigned to one of the buckets but never knew which
bucket. He or she knew only that the buckets had different numbers of black
balls but did not know if his or her bucket contained the lowest or
highest number of black balls. Again, if a black ball were picked, 40
tokens were lost.
The decision of each participant was written on a form, one for each
round. After each round, losses were calculated and payments were made.
The experiment took about 30 minutes. For each token left, the
participants received SEK0.5. The experiment took place at different
occasions in 2006 at Växjö University. A total of 80
participants - 20 groups of four - participated in the experiment. The
average age of participants was 22 years, and 40 percent of them were
women.
The Appendix shows the instructions.
Table 1: Number (Percentage) of players adopting each action.
| Round 1 | Round 2 | Round 3 |
|
No action | 5 (6.25) | 2 (2.5) | 6 (7.5) |
| Equal loss sharing | 72 (90) | 72 (90) | 68 (85) |
| Partial loss sharing | 3 (3.75) | 6 (7.5) | 6 (7.5) |
| Sell insurance | 0 (0) | 0 (0) | 0 (0) |
| Buy insurance | 0 (0) | 0 (0) | 0 (0) |
| N | 80 (100) | 80 (100) | 80 (100) |
|
|
3 Results
The results of the experiment are presented in Table 1. In the first
round, participants picked a ball from the same bucket. If a black ball
was picked, they lost 40 tokens. A first option for the participants
was to take no action. Only five participants made this choice; four of
these belonged to the same group. A second option was to share losses
equally among all group members. Seventy-two participants, or 18
groups, chose this action. Three participants partially shared their
losses since one in that group chose not to do anything.
The pattern in the second round, with four different buckets with
unknown distribution of losses, was similar to the first round. The
number of individuals or groups that chose to share their losses among
all group members did not change. Two individuals from different groups
chose to take no action; therefore, the remaining group members chose
partial loss-sharing.
In the third round, participants knew the number of black balls in
different buckets, but they did not know which bucket they would pick
from. The results were still similar to the previous two rounds. The
majority - 68 participants or 17 groups - chose to share their losses
equally among all group members. Six participants chose not to take any
action, and another six participants shared their losses partially.
Note that not a single participant chose to insure someone else and,
consequently, not a single participant bought insurance in any of the
rounds. In the first two rounds, the reason for that choice could be
that it is difficult to set a premium ex ante because of missing
information about the probabilities of losses. However, participants
did not buy or sell insurance in the final round either, when the
distribution of balls was known. We will discuss this further in the
next section.
The results here are clear cut; however, for the record, the proportion
of participants that choose equal sharing was significantly larger than
the proportions of participants choosing alternative actions
(Chi-square test, p < 0.0001). Also, the change in
the proportion of participants choosing equal sharing compared to other
actions across rounds was not significant (McNemar test).
4 Discussion
One previous study, Ahmed and Skogh (2006), examined the restated
diversification principle, but the present paper differs from the
previous study in two important ways:
First, in Ahmed and Skogh (2006), participants received 1,000 tokens as
an endowment that could be lost in five risky rounds. Participants were
divided into groups of four and informed of an urn that contained 100
balls of various colors. In each round, each player randomly picked one
ball. A black ball would cost the participant 500 tokens. In each
round, participants could also take actions to reduce the risk of large
losses. One of the actions was to share losses with other participants.
The information on the risk varied across rounds: In the first round,
there was no information on the probability of a loss; in the final
round, there was full information of the probability. The results
showed that participants shared losses when information about the
probabilities of losses was missing, in a way that supports the
restated principle. However, a large percentage of the participants
chose to do nothing. Participants may have believed that cooperation
did not pay: In each round, half of the endowment could be lost. If
they believed probabilities of losses would be relatively large,
bankruptcy could be the expected outcome of the five rounds and
gambling, in that case, may be a way to survive by luck. In this paper,
we present results from an experiment in which we overcome the problem
of bankruptcy by giving our participants a new set of tokens in each
round.
The second difference from Ahmed and Skogh (2006) is that we also
tested the second hypothesis stated above by using different urns for
different group members in the last two rounds.
The observations clearly show that participants share losses when
information about the probabilities of losses is missing in a way that
supports the descriptive adequacy of the restated diversification
principle. That is, sharing is chosen exclusively because it eliminates
the problem of pricing when losses are unpredictable. Our observations
also show that participants share losses even if they know that the
distribution of losses is different across group members - but they do
not know in what way because of the lack of information that
discriminates among group members. Participants share potential losses
by applying the presumption of equality.
A noteworthy result that needs to be commented is that none of the
participants bought or sold insurance in the experiment. In the first
two rounds, the reason is probably that an insurance premium could not
be set, as information about the probabilities of losses was lacking.
However, in the last round, this information was known, so the question
remains as to why not a single insurance contract was established. One
reason could be that participants found insurance to be a more
complicated choice than loss-sharing since it involved negotiation
between the buyer and the seller over the insurance premium. Another
explanation could be that mutual insurance, where participants insure
each other at the same premium, is equivalent to loss-sharing. Hence,
if everyone in the group wanted to buy and sell insurance, then they
might as well just share losses instead. An interesting extension of
the experiment would be to include a fifth person in the group that
acts only as an insurer. A possible explanation could also be that
cooperating with other group members in previous two rounds establishes
a bond among participants that makes it is hard to deviate from the
group behavior to individual behavior. Introducing anonymity among
group members might limit this possibility. Replications are requested.
The paper relates to the vast literature on decision under risk. Savage
(1954) put forward the subjective probability theory, where the
distinction between known and unknown probabilities is meaningless
because subjective probabilities are never unknown. Empirical evidence,
however, has shown that people do, in fact, make such a distinction
(Ellsberg, 1961). A justification of the subjective probability theory
is that people cannot make decisions without assignment of
probabilities. Yet, the results of the present experiment show that
this is not completely correct: Loss-sharing among people takes place
without assignment of probabilities.
The paper also relates to the prospect theory of Kahneman and Tversky
(1979). Prospect theory predicts that people are risk-averse in the
domain of gains and risk-seeking in the domain of losses. The results
in this paper might then look inconsistent with the prediction of
prospect theory, as participants did not appear to be risk-seeking when
facing potential losses. Alternatively, it may be the case that
participants actually were in the domain of gains when making their
decision since they received a show-up fee that could not be lost in
the experiment; in addition, they gained 40 tokens guaranteed in each
round. Putting it in this way, participants should be risk-averse, and
the experiment may have not implemented potential losses. On the other
hand, a similar experiment by Kühberger, Schulte-Mecklenbeck, and
Perner (2002) shows that participants who have lost half of their given
endowment have a tendency of being risk-seeking. It is difficult
to arrive at definite conclusions by hypothesizing from known facts and
observations from other studies.
References
Ahmed, A. M. & Skogh, G. (2006). Choices at various levels of
uncertainty: an experimental test of the restated diversification
theorem. Journal of Risk and Uncertainty 33, 183-196.
Ellsberg, D. (1961). Risk, ambiguity, and the Savage
axioms. Quarterly Journal of Economics, 75, 643-699.
Hogarth, R. M., & Kunreuther, H. C. (1989). Risk, ambiguity,
and insurance. Journal of Risk and Uncertainty, 2,
5-35.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An
analysis of decision under risk. Econometrica, 47,
263-291.
Kühberger, A., Schulte-Mecklenbeck, M., & Perner, J.
(2002). Framing decisions: Hypothetical and real. Organizational
Behavior and Human Decision Processes 89, 1162-1175.
Rothschild, M. & Stiglitz, J. E. (1971). Increasing risk II:
Its economic consequences. Journal of Economic Theory 3, 66-84.
Savage, L. J. (1954). The foundations of statistics.
New York: Wiley.
Skogh, G.. (1999). Risk-sharing institutions for unpredictable
losses. The Journal of Institutional and Theoretical Economics, 155,
505-515.
Skogh, G. & Wu, H. (2005). The diversification theorem
restated: Risk-pooling without assignment of loss probabilities.
Journal of Risk and Uncertainty 31, 35-51.
Appendix: Translation of written instructions
General instructions
Welcome! Thank you for participating in this experiment. The experiment
will take a maximum of one hour. The purpose of the study is to gain
greater insight into economic decision-making. To express our gratitude
and to compensate you for your time, you have already been given a
show-up fee of SEK 20. In addition to this, you also have the
opportunity to earn more money during the experiment.
As you already have noticed, this experiment will be conducted in
groups of four, and you have already been matched with three other
participants whom you are seated with. In this experiment, you and your
group members will go through three rounds, and in each round you and
your group members have to make a decision. In 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 are
over. For each token you have left, you will be paid SEK0.5.
In each round, you and your group members have to pick a ball from a
bucket containing 100 colored balls. If a black ball is picked from the
bucket, you will lose 40 of the 80 tokens. Hence, each round may result
in a loss of 40 tokens. You may take this risk individually, share
losses with others in the group, or you may also buy or sell insurance.
The alternative actions are described below:
A. No action. You go through the round yourself with the
potential loss of 40 tokens.
B. Equal loss-sharing. Each of you in the group covers
one-fourth of the total losses of the group. For instance, if one of
your group members draws a black ball, each of you in the group will
pay 10 tokens to cover the loss. If two in your group receive a black
ball, then each of you will pay 20 tokens, and so on.
C. Partial loss-sharing. It is also possible that two or three
of you share the potential losses. This may be the case if one or two
of your group members chooses to take another action instead of sharing
the losses.
D. Sell insurance. You may insure other group members by
charging a premium in advance. You are allowed to insure losses up to
40 tokens in each round. For example, the premium for coverage of 40
tokens could be equal to 5, 10, 20, 30 or so on, depending on what you
believe about the risk of receiving a black ball. If you insure another
person in your group, you will first pick a ball for yourself and then
pick a ball again for the person 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 cover the potential loss.
You and your insurer have to decide what premium to set. Hence, if you
pay a premium to another group member to cover all your losses, your
earnings from that round will be 80 tokens minus the premium paid.
You can discuss freely with other members of your group. Your choice of
action in each round is to be written down on a decision form that you
will receive before each round together with specific information
related to each round. The picking of balls will take place after your
decision has been collected.
Information given in the first round
You have been given 80 tokens. You will be asked to pick 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 defined 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.
Information given in the second round
You have been given 80 tokens. There are four buckets containing 100
balls of various colors. The four buckets will be randomly assigned to
you and your group members: one bucket for each of you. You will then
be asked to pick a ball from your bucket. If you pick a black ball, you
will experience a loss of 40 tokens. You have the possibility to take
any of the actions defined earlier. You will pick a ball after you and
your group members have decided what actions to take.
Information given in the third round
You have been given 80 tokens. There are four buckets containing 100
balls of various colors: one bucket has 70 black balls, one bucket has
50 black balls, one bucket has 30 black balls, and one bucket has 10
black balls. The four buckets will be randomly assigned to you and your
group members: one bucket for each of you. You will not, however, know
which bucket contains which number of black balls. You know only that
the buckets contain different numbers of black balls. You will then be
asked to pick a ball from your bucket. If you pick a black ball, you
will experience a loss of 40 tokens. You have the possibility to take
any of the actions defined earlier. You will pick a ball after you and
your group members have decided what actions to take.
Footnotes:
1I owe gratitude to Göran Skogh who
introduced me to this line of research. Also, thanks to Jonathan
Baron and a reviewer for their suggested improvements. This
project was financially supported by Jan Wallanders and the Tom
Hedelius Foundation. Address: School of Management and Economics,
Växjö University, SE-351 95 Växjö, Sweden. Email:
ali.ahmed@vxu.se
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