3
Effects of Risk Preferences in Repeated Social
Dilemmas:
A Game-theoretical Analysis and Evidence
from Two Experiments
Van Assen, M.A.L.M., & Snijders, C. (2001, accepted for publication). Effects of risk
preferences in social dilemmas: a game-theoretical analysis and evidence from two
experiments. In Ramzi, Suleiman, Budescu, Fischer & Messick (Eds.), Contemporary
Psychological Research on Social Dilemmas.
38
Chapter 3
Acknowledgements
We would like to thank Werner Raub for his close collaboration and substantial input, and
Jeroen Weesie for many helpful discussions.
Effects of Risk Preferences in Repeated Social Dilemmas
3.1
39
Introduction
Consider a situation where a group of workers bargain with their employer about their
working conditions or wages. Suppose that the negotiations are at an impasse and the workers
decide that a strike is called for. This situation corresponds to a social dilemma: a worker can
either invest effort to join the strike and improve the situation for the group as a whole, or
free ride on the contributions of other workers who invest. Common intuition suggests that
groups threatened with a loss (worse working conditions or wages) will be more likely to
protest than groups confronting the uncertainty that a potential gain (improvement in working
conditions or wages) may not materialize. More generally, cooperative behavior of the
collective action kind will be attained more easily to counteract a threat when the threat is an
impending loss than when it is the non-appearance of a gain. This intuition has been likewise
put forward by several social scientists working in the domain of collective action (Walder,
1994; Berejikian, 1992). However, this intuitively plausible claim – that losses promote
collective action – is at odds with insights from game theory and individual decision making
with respect to risk preferences, as we have argued earlier (Raub and Snijders, 1997, van
Assen, 1998).1
Individual behavior is said to be “risk averse” when the individual prefers the
expected value of a gamble over the gamble itself. For example, a risk averse individual
would prefer an outcome of 50 over a 50% chance of obtaining 100. An individual is “risk
seeking” when preferring the gamble over its expected value. Risk seeking individuals prefer
a 50% chance of obtaining 100 over obtaining a certain 50. Once again, we might intuitively
expect that if the workers in our example are risk seeking, they are more willing to run the
risk of being the only one, or among the few workers, to invest effort in improving the
conditions for the workers. When we perceive the collective action problem as a repeated
Prisoner's Dilemma (PD), then it is straightforward to show that this intuitive reasoning is
wrong. In fact, risk averse workers are expected to contribute to the collective good more
readily than risk seeking workers. This can be shown mathematically (we provide a brief
outline below; the technical details can be found in Raub and Snijders, 1997), but the
essential argument is that in a repeated PD, actors choose their preferred behavior on the basis
of conditionally cooperative strategies. Risk seeking actors are more willing to run the risk of
an accidental non-cooperative move being sanctioned by future defection of the other actors.
Given that risk aversion actually favors cooperation in a (repeated) collective action
context, we can go one step further. Research into individual decision making shows that
most individuals behave in accordance with “S-shaped utility”, that is, individuals are risk
seeking for losses and risk averse for gains. If this is really the case, and risk aversion favors
cooperative behavior, cooperative behavior should be attained more easily for gains than for
losses, and not vice versa. In our example this would mean that workers are more likely to
cooperate when they can improve their working conditions or wages than when they have to
prevent a worsening of working conditions or wage rate. The relevance of research into the
relation between risk preferences and cooperative behavior in social dilemmas can be backed
up in a similar way by making use of another insight into individual decision making. It is
known from the literature that people evaluate gains and losses from a reference point whose
position can easily be influenced by seemingly irrelevant variations in presentations of the
1 Chapter 4 corresponds to van Assen (1998).
40
Chapter 3
situation (Kahneman and Tversky, 1979; Tversky and Kahneman, 1981). That is, a situation
can be stated or described such that it is perceived as a loss situation or as a gain situation.
Applied to our example, a conflict between employer and employee about working conditions
or wages can be formulated as a loss situation, in which the employees have the idea that if
they do not cooperate they will have worse conditions than their status quo. Alternatively, it
can be formulated as a gain situation where they have the idea that if they cooperate then
optimal working conditions can be obtained rather than the presently less than optimal
conditions. In other words, if risk preferences have an effect on cooperation in repeated social
dilemmas and if the reference point of the employees can be induced, as previous research
suggests, then through shrewd framing one can influence the probability that cooperation will
emerge.
Earlier, we tested the relation between risk preferences and cooperation
experimentally by making use of two-person repeated PDs (Raub and Snijders, 1997). In
accordance with our hypothesis, there indeed existed a positive relation, although weak,
between risk aversion and cooperation in the repeated PDs. Here we report on a second
experiment, designed to replicate and extend the one described by Raub and Snijders (1997),
using the improvements in utility measurement suggested by van Assen (1998).2 The most
important extension that we used was a different way of measuring subjects’ risk preferences.
The first experiment made use of a rather straightforward method of measuring risk
preferences, which at that time seemed sufficient for our purpose. We replicated the
experiment, making a more refined measurement of utility. Firstly, the same simple method
was used more extensively, to obtain a more accurate and reliable estimate of a subject’s risk
preferences. This simple method assumes that subjects evaluate probabilities linearly, which
research in individual decision making argues is typically not correct: subjects overestimate
small probabilities and underestimate intermediate and large probabilities (Tversky and
Kahneman, 1992; Gonzalez and Wu ,1999). Therefore, we included an additional utility
assessment method, known as the tradeoff method (Wakker and Deneffe, 1996; Fennema and
van Assen, 1999)3, which does not depend on the assumption that subjects evaluate
probabilities linearly. By making use of both the more extensive original method and the
robust tradeoff method we hoped to obtain a more reliable and precise estimate of subjects’
risk preferences, and hence a more accurate measure of the relation between risk preferences
and behavior in repeated social dilemmas.
The structure of the text is as follows. Firstly, we briefly describe the theoretical
background behind the effect of risk preferences on cooperation in repeated social dilemmas,
using insights from game theory and individual decision making. We then briefly discuss the
experiment of Raub and Snijders (1997) and its results, before elaborating in some detail on
why a more precise measurement of utility is necessary. The results of the new, second
experiment, including a more detailed and precise measurement of risk preferences, are
discussed in the next section. The results of both experiments are compared and evaluated in
the conclusion and discussion section.
2 In fact, the experiment on which we report is not a replication in the strict sense of the word since it is
different in some non-negligible respects (see below).
3 Chapter 2 corresponds to Fennema and van Assen (1999).
Effects of Risk Preferences in Repeated Social Dilemmas
3.2
41
Effects of risk preferences on cooperation in repeated social dilemmas:
theoretical background
We firstly derive hypotheses concerning our research question using game theory and results
from theories of individual decision making. Before doing so we briefly describe related
predictions found in the literature which are not derived from game theory but are based on
other arguments.
Walder (1994, p.9) argues “that loss is a more powerful motivator than gain, or that
groups threatened with loss will be more likely to protest than groups that seek proactively to
achieve a gain”. Applied to our example described in the introduction, Walder believes that
workers are more likely to cooperate in a conflict with their employee when they have to
prevent a worsening of working conditions than when they can act collectively to improve
their working conditions. Walder derives this hypothesis from historical evidence and from
the empirical regularity in individual decision making that actors in general are risk seeking
in situations where the outcomes are losses, and risk averse in situations where the outcomes
are gains. Berejikian (1992) arrives at a similar conclusion, arguing that in a loss situation
actors are risk seeking and therefore more willing to cooperate or revolt to limit the loss,
thereby risking that other actors will not participate in the revolt and that actor’s own
contribution does not pay off. In a gain situation, Berejikian (1992) argues that actors are risk
averse and are less likely to be willing to run the risk of useless effort, preferring to opt for
the riskless status quo.
Social dilemmas can be characterized as collective action problems where rational
self-interested actors do not obtain the best collective (Pareto-optimal) outcome (Taylor,
1987). We derive our hypothesis using the well-known two-person repeated PD. Although we
are aware that not all collective action problems are repeated PDs, we feel, like many others,
that a relevant set of real life collective action problems can be approximated by a repeated
PD reasonably well (e.g. Taylor, 1987; Hardin, 1982). Repeated PDs are believed to be more
valid approximations of social dilemma situations than one-shot PDs because a dilemma can
often be considered as a chain of events in which all actors involved have to decide whether
or not to contribute to the public good. To derive and test our predictions of the effect of risk
preferences on cooperation in social dilemmas, for reasons of simplicity we model social
dilemmas as two-person PDs instead of N-person PDs. The derived results discussed below
can be generalized to N-person PDs.
In the two-person PD, actors move simultaneously and each actor has two pure
strategies, denoted by ‘cooperation’ (C) and ‘defection’ (D). It is customary to denote the four
payoffs corresponding to the four different outcomes in the following way: T(emptation)
when ego defects and alter cooperates, R(eward) when both alter and ego cooperate,
P(unishment) when both alter and ego defect, and S(ucker) when ego cooperates and alter
defects. The PD requires that T > R > P > S. We assume that the PD is repeated in a following
round with continuation probability w.4 Therefore, w is the constant probability that round
t+1 will be played after round t has been played, while 1-w is the probability that the game
terminates after round t has been played. It is assumed that both actors know the value of w.
4 The meaning of w as the continuation probability of the repeated PD is equivalent to interpreting the repeated
game as an infinitely repeated PD with exponential discounting of constituent game payoffs with discount
parameter w. Both interpretations are used in the literature. The continuation probability interpretation is chosen
here because it corresponds to the experimental setting in both experiments.
42
Chapter 3
In round t, each actor has been informed about the behavior of the other actor in all previous
rounds 1,2, … t-1. We assume that actors evaluate outcomes using their utility function U. As
is well known, analyzing the one-shot PD results in the conclusion that a rational selfinterested actor will choose to defect, because both T > R and P > S. However, the situation
changes in the repeated PD. Assume that actors either choose defection in all rounds or play a
“trigger” strategy. An actor playing a trigger strategy cooperates until the other actor defects,
and as soon as that happens, the actor defects in all rounds thereafter. Hence a trigger strategy
represents conditional cooperation to the extreme.
Mutual cooperation, resulting in R for both actors instead of the worse P, is feasible
for rational self-interested actors if it is supported by a subgame perfect equilibrium. If we
assume that actors either play defection in all rounds or play a trigger strategy, the repeated
two-person PD has a (subgame perfect) equilibrium such that both actors cooperate on the
equilibrium path if, and only if
w≥
U (T ) − U ( R )
U (T ) − U ( P )
(1)
Equation (1) reveals that the shadow of the future as reflected by the continuation
probability w, must be larger than a certain value to make cooperation possible in
equilibrium. If w is smaller than the ratio in (1) there is no equilibrium that yields only
cooperative choices, because the trigger strategy, the strongest possible punishment for a
defection, does not provide a strong enough punishment (P instead of R as long as the game
lasts) for a first defection in comparison to the gain from this defection (T - R). Note that the
ratio in the equation does not contain the PD outcomes itself but their utilities. In the
derivation it is assumed that both actors evaluate outcomes similarly, that is, they have equal
utility functions.5 A derivation of the above result and an elaborate discussion of related
results can be found in Chapter 4 of Taylor (1987).
To see the effect of the utility function on the conditions for conditional cooperation,
consider Figure 3.1. Figure 3.1 depicts a utility function, known as an S-shaped utility,
commonly found and hypothesized in the individual decision making literature (Tversky and
Kahneman, 1992; Fennema and van Assen, 1999). The x-axis depicts the outcome, for
example a monetary value, and the y-axis depicts its evaluation or utility. It is assumed that
the function is strictly increasing, that is, the larger the outcome the more desirable it is for
the actor. If we consider linear utility functions then the ratio in (1) is equal to the ratio of
payoffs (T-R)/(T-P). This is not the case if we consider a part of a utility function with
decreasing marginal utility (negative second derivative), as in the gain domain of the function
in Figure 3.1. In the case of decreasing marginal or concave utility the middle outcome R is
evaluated more highly relative to outcome T than in the case of linear utility, as can be seen
by a higher positioning of U(R) on the y-axis then R’s position in the case of linear utility.
The fact that R is evaluated more highly leads to a smaller ratio in (1) because the numerator
contains the reciprocal difference U(T)–U(R). Therefore, conditions for cooperation in the
5 In fact, to derive (1) we only require that the actors’ ratios of utilities in (1) are equal and that this is common
knowledge (they know it, and both know that the other actor also knows it, etc.). This does not imply equal
utility functions, because utilities for outcomes different from T, R, and P might be different for both actors. We
do need to assume that in calculating the equilibria, subjects do not under- or overestimate probabilities.
However, in our theoretical argumentation we explicitly take the possibility into account that actors do
transform probabilities nonlinearly. We return to this issue in the discussion section.
Effects of Risk Preferences in Repeated Social Dilemmas
Figure 3.1:
43
S-shaped utility function. Ratio in (1) is highest if utility is concave (in
gain domain with outcomes P, R, and T), intermediate if utility is linear,
and lowest if utility is convex (in loss domain with outcomes P', R', and
T').
repeated PD would be better for decreasing marginal utility functions because a smaller
continuation probability would suffice for mutual cooperation to be in equilibrium. On the
other hand, we can also imagine that utility functions have increasing marginal or convex
utility, as in the loss domain of the function in Figure 3.1. Here we have the opposite relation.
The middle outcome R is evaluated less (more negatively) in comparison to outcome T than
in the case of linear utility, resulting in less favorable conditions for mutual cooperation; a
larger continuation probability is needed for mutual cooperation to be in equilibrium.
Therefore, our analysis suggests that if actors are rational and comply with behavioral
assumptions of game theory then conditions for cooperation are better for concave utility
functions and worse for convex utility functions. Our main hypothesis is then
H1:
The cooperation rate in the repeated PD is larger for people with concave
utility functions than for people with convex utility functions.
We already mentioned that people are considered to have S-shaped utility and that this would
imply that actors’ cooperation rates are larger when outcomes represent gains than when
outcomes represent losses. Note that this prediction is exactly opposite to the predictions of
Walder and Berejikian discussed earlier. We believe that the main reason for the difference in
predictions is the interpretation of the concept “risk”. Note that in our introduction we discuss
our hypotheses in terms of “risk preferences”, but in this paragraph we do not use the concept
“risk” in deriving our hypotheses. The reason is that the risk concept is ambiguous in that it
loses its obvious interpretation in interdependent situations, as we will explain below.
“Risk”, and the related concept “risk preferences”, are commonly used in the field of
individual decision making. If an actor makes a decision under conditions of risk, he chooses
between alternatives of which at least one consists of probabilistic outcomes, and where the
44
Chapter 3
values of the probabilities are known to him and are not equal to 0 or 1 (see Luce and Raiffa,
1957, for a discussion on terminology). The concepts risk aversion and risk seeking refer to
the behavior of actors under conditions of risk. An actor’s behavior in a situation is called (i)
risk seeking when he prefers to gamble over obtaining a certain outcome equal to the
expected value of the gamble, (ii) risk neutral when he is indifferent between the gamble and
its expected value, and (iii) risk averse when he prefers the expected value over the gamble.
For example, if an actor prefers $10 to a gamble ($20, ½, $0), with a 50% chance of winning
$20 and a 50% chance of playing even, then this behavior is known as risk averse.
Risk preferences are directly related to the shape of the utility function if we assume
that actors are rational in the sense of Expected Utility theory as formulated by von Neumann
and Morgenstern (1944).6 Risk aversion then implies and is implied by a concave utility
function, risk neutral behavior implies and is implied by a linear utility function, and risk
seeking behavior implies and is implied by a convex utility function. The utility of a gamble
is then equal to a linear combination of the utilities of the outcomes of the gamble, with
weights equal to the probabilities. For example, in the loss domain of Figure 3.1, the utility of
a gamble ($T, p, $P) is equal to the point pU($T) + (1-p)U($P), which is on the line
connecting U(T) and U(P). The utility U[p$T+(1-p)$P] of the expected value of the gamble is
equal to the point on the utility curve at this value on the x-axis. Note that the whole utility
curve in the loss domain is below the straight line, implying that an actor with a convex
utility function would prefer the gamble above its expected value, which is risk seeking
behavior. Therefore, under Expected Utility theory we could substitute our hypothesis H1
with
H1’
The cooperation rate in the repeated PD is larger for people who are risk
averse in gambles containing the outcomes of the PD than for people who are
risk seeking in these gambles.
It is important to emphasize that because there are reasons not to believe in Expected Utility
as a good descriptive theory of human behavior, as we will argue in our section on utility
measurement, we will refer to H1 as our main hypothesis.
A natural question to ask is whether actors’ risk preferences depend on the range of
the outcomes used in gambles. For economists it was a truism that the utility of money is
marginally decreasing, and therefore that utility is concave (e.g. Bentham 1789). Prospect
theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992) suggested that utility is
indeed concave for gains, but convex for losses. Empirically it is well established that utility
for gains is concave for most actors (see e.g. Fennema and van Assen, 1999). However, the
evidence for the shape of the utility function for losses was inconclusive because of
variability between individuals and particular bias in the methods used to assess utilities.
Fennema and van Assen (1999) elicited utility for a pool of subjects using a utility
assessment method that avoids this biases. On the basis of this method, they found that utility
for losses was convex for most subjects, supporting Prospect Theory and not the traditional
economic literature. Using the additional assumption of S-shaped utility, hypothesis H1’ can
be restated as
H1’’
The cooperation rate in the repeated PD is larger when outcomes in the PD are
6 In the section on utility measurement we will see that risk preferences and utility are not directly related
because actors tend to transform probabilities.
Effects of Risk Preferences in Repeated Social Dilemmas
45
gains then when the outcomes in the PD are losses.
Walder (1994) and Berejikian (1992) proposed a hypothesis that is the opposite of
H1’’. We think that the reason for this difference is the ambiguity of the concept “risk”. In the
reasoning of Walder and Berejikian the concept of risk is not limited to choice behavior in
gambles in comparison to the expected value of the gamble. In their work, “risk” refers to the
fact that some outcomes are uncertain even after having chosen a particular course of action
(e.g., defection or cooperation). However, in this interpretation there are multiple risks
involved in the decision. In a repeated PD, if an actor cooperates (s)he runs the risk of being
exploited by the other actor and hence obtaining the sucker's payoff S. If the actor chooses not
to cooperate (s)he runs the risk that his or her choice will lead to future retaliation by the
other actor. Because both cooperative and non-cooperative behavior is in some sense risky,
according to the latter use of the concept risk, it is unclear how a risk averse actor might
proceed. We believe that Walder and Berejikian relate their reasoning only to the risk of
being exploited by the other actor when cooperating, and if this is done, it leads to the
complementary hypothesis of H1’’. To avoid ambiguities and to be consistent with the
individual decision making literature, we use the term “risk averse” for an actor who prefers
the expected value of a gamble over the gamble.
We now briefly restate the results of a first experimental test of hypotheses H1’ and
H1’’, as reported by Raub and Snijders (1997).
3.3
Experiment 1 (Raub and Snijders, 1997)
3.3.1 Method
The experiment was run over eight sessions using different subjects. In total, 190 subjects
participated. To test our hypotheses each subject played two repeated PDs, one PD with
outcomes in the loss domain and one PD with outcomes in the gain domain. Outcomes T+, P+,
and R+ were equal to 16, 7 and 4 Dutch guilders respectively.7 Outcome S+ was varied over
the sessions between -1, 0, and 1 guilders. Outcomes T-, R-, P-, and S- in the loss domain were
obtained by subtracting 16 guilders from the outcomes of the PD in the gain domain. The
value of the continuation probability w in the experiment was 0.75.
All participants completed three tasks. The assessment of risk preferences of subjects
for both value domains was conducted in the first part of the experiment. To test the relation
between subjects' behavior in repeated PDs and their risk preferences, the outcomes of the
gambles used to elicit risk preferences were equal to those used in the repeated PDs. Subjects
were required to make a choice between (T+, 1-w, P+) and the certain outcome R+. Probability
w was equal to the continuation probability also used in the repeated PDs. Additionally,
subjects made a choice between (T-, 1-w, P-) and the certain outcome R-. The order in which
both choices had to be made was varied across different sessions.
In the second task subjects completed a comprehension check and questionnaire,
which contained several questions about possible relevant background variables and in which
their comprehension of the repeated PD was tested. The most time consuming third and last
task of the experiment consisted of subjects playing the repeated PD in both the loss and gain
domain. Sessions were run in groups with a maximum of 30 subjects. Participants were
7 At the time we conducted the first experiment, 1 US dollar was equivalent to approximately 1.7 Dutch
guilders.
46
Chapter 3
evenly divided over two semi-separated sides of a room such that they could not see each
other. Pairs of players playing a repeated PD together were separately determined at random
for each repeated PD. The order in which both repeated PDs were played was
counterbalanced between sessions. Subjects were not allowed to communicate during the
experiment and were told that they would not find out the identity of the players they were
connected with. The procedure of the repeated PD was roughly in line with the procedure
used in Murnighan and Roth (1983). Choices of subjects were written down on paper and the
experimenters passed the choices on to their partners on the other side of the room. A roulette
wheel with 36 slots was then spun. If the ball fell in one of 27 slots (with probability equal to
0.75) mentioned on the card of a pair of players then this pair of players played another round
of the repeated PD.
At the start of the experiment subjects received 20 Dutch guilders. We hoped to
minimize the gambling with the house money effect (Thaler and Johnson, 1990) by explicitly
mentioning that the money was meant to cover their time and effort, and by letting
participants put the money in their wallet immediately. The gambling with the house money
effect might make participants more risk seeking than they would have been if they had not
received 20 guilders at the start of the experiment.
3.3.2 Results
As expected, the majority of subjects were risk seeking in the loss domain (66.3%).
Unexpectedly, only 44.7% of the subjects were risk averse in the gain domain, and across risk
patterns for both domains, of 190 subjects only 43 (23%) had risk preferences in agreement
with Prospect Theory. One explanation for the absence of risk aversion in the gain domain is
that outcomes of the gambles are low in comparison to the outcomes used in other
experiments. It is known from other studies that subjects in general are more risk averse when
the range of outcomes of the gambles is larger (Krzysztofowicz and Duckstein, 1980).
Another explanation is provided in the section on utility measurement.
The main hypothesis to be tested was whether there is a positive relation between risk
aversion and cooperation in the repeated PD. Cooperation was measured by the first response
in the repeated PD. An important reason for neglecting the responses in round 2 and beyond
is that the responses in later rounds are very likely be influenced by the responses of their
opponent in previous rounds. Comparing the proportions of cooperation for subjects who
were risk seeking with those subjects who were risk averse confirmed our hypotheses. In the
gains domain, 38% of the risk averse subjects cooperated, as opposed to 26% of the risk
seeking subjects (p = 0.054, Fischer exact test). In the losses domain the observed difference
was larger; of the risk averse subjects 48% cooperated, as against only 26% of the risk
seeking subjects (p = 0.002, Fischer exact test).
The connection between risk preferences and cooperation was also tested on the basis
of a probit analysis of the probability of cooperation, using as independent variables a
subject’s risk aversion, whether the game was one involving losses or gains, and some other
background or control variables such as age and sex. The effect of risk preferences remained
significant (p < 0.05) in the probit analyses including all variables. The effects of all other
variables except one were not significant. The one variable for which the effect was
significant indicated that subjects who participated in the experiment as a course requirement
were more inclined to cooperate than others (p < 0.01). Summarizing the results, our
Effects of Risk Preferences in Repeated Social Dilemmas
47
hypothesis that risk preferences have an effect on cooperation in repeated social dilemmas
was confirmed.
3.4
Utility measurement
Although other experiments suggest that most subjects have S-shaped utility, in our
Experiment 1 we found that only 23% of all subjects were risk seeking for losses and risk
averse for gains. Although risk seeking behavior was observed for losses, the majority of
subjects were also risk seeking for gains. One explanation of this unexpected finding of a
majority of risk seekers for gains is that our design did not sufficiently account for the
gambling with the house money effect. There are two other complementary explanations for
this unexpected finding. Firstly, there can be an experimental bias. The subject’s payoff in the
experiment was not fixed but depended on the subject’s responses, the responses of other
subjects, and on chance. In the worst case scenario a subject would earn 3 Dutch guilders and
in the best possible scenario a subject would earn 36 guilders. The expected utility of
participating in the gamble for a person who is risk averse for gains would be lower than for a
person who is risk seeking for gains. Therefore, the experiment might attract a relatively large
proportion of subjects who are risk seeking for gains. Other experiments investigating risk
preferences used either fixed payoffs (e.g., Tversky and Kahneman, 1992; Fennema and van
Assen, 1999) or a variance in the payoffs that was substantially lower than in our experiment.
Therefore Experiment 1 might indeed have had a (larger) bias in selecting subjects who are
risk seeking for gains.
Another explanation for the relative abundance of risk seeking behavior involves
probability weighting by subjects (this argument was made earlier by van Assen, 1998). In
Section 3.2 we explained that, assuming Expected Utility theory, subjects with concave
utility prefer the expected value of a gamble over the gamble itself because only then pU[T] +
(1-p)U[P] < U[pT + (1-p)P] = U[R]. However, if subjects transform not only utilities but also
probabilities then Expected Utility no longer holds and risk aversion is no longer equivalent
to behaving in accordance with concave utility, just as risk seeking behavior is no longer
equivalent to behaving in accordance with convex utility. We will show below that the
abundance of risk seeking for gains can at least partially be explained by concave utility for
gains together with probability weighting as assumed in (Cumulative) Prospect Theory of
Tversky and Kahneman (1992).
For example, assume that a subject’s utility function is concave and can be
0.88
represented by the power function u[x] = (x) . Furthermore, assume that the subject
overweighs a probability of 0.25 relative to a probability of 0.75, such that the weight of
probability 0.25 is π[0.25] = 0.27 and π[0.75] = 1 - π[0.25] = 0.73. Then, although utility is
concave, the subject prefers the gamble (T+, 0.25, P+) over its expected value R+: 0.27 * (16)0.88
+ 0.73 * (4)0.88 > (7)0.88. Evidence for subjects transforming probabilities in this way –
overweighing small probabilities and underweighing moderate to large probabilities – is
overwhelming in individual decision making research. Most researchers in the field of
individual decision making accept Expected Utility theory as an adequate normative theory or
as a way of making the underlying mechanisms of decision situations apparent. Most of them
also agree that the theory is not an adequate descriptive theory of how people actually make
decisions. Kahneman and Tversky (1979) describe sets of choice situations involving
gambles that elicit responses from subjects that cannot be accommodated by Expected Utility
theory. They could convincingly explain the deviations from Expected Utility theory by
48
Chapter 3
Probability weight
1
.8
.6
.4
.2
0
0
Figure 3.2:
.2
.4
.6
Objective probability
.8
1
A typical probability weighting function, reflecting overweighing of a
small probability and underweighing of moderate to large probabilities.
assuming that subjects transform probabilities in a particular way. Their theory of how actors
transform probabilities, known as Prospect Theory, was developed in 1979, and was extended
by Tversky and Kahneman (1992), being relabeled Cumulative Prospect Theory. In
Cumulative Prospect Theory different weight functions π for probabilities belonging to losses
and gains are assumed. A property of the functions π is that for gambles with two outcomes
in the same domain (losses or gains) the functions assign weight π(p) to the probability p
corresponding to the largest absolute outcome. The smallest absolute outcome is assigned
decision weight 1- π(p). The decision weight functions π can be different for losses and gains.
If the outcomes of the two-outcome gamble are from different value domains with probability
q that the positive outcome occurs and with probability 1-q that the negative outcome occurs,
then the decision weights assigned to the outcomes are π+(q) and π-(1-q) respectively. Note
that, as opposed to the decision weights of two-outcome gambles with outcomes from the
same value domain, the sum of decision weights does not need to be unity. Intuitively, the
probability transformations can be understood as a higher sensitivity for changes in
probability near certainty than in the middle range of probabilities. This tendency is
corroborated by several empirical studies (Kahneman and Tversky, 1979; Tversky and
Kahneman, 1992; Camerer, 1995; Camerer and Ho, 1994; Gonzalez and Wu, 1999). A
typical probability weighting function found in these studies is depicted in Figure 3.2.
The argument from Cumulative Prospect Theory used to explain the relative
abundance of risk seeking behavior in the gains domain can be summarized as follows. A
typical subject has concave utility for gains and overweighs the probability of 0.25 for the
largest outcome T+. The overweighing of the largest outcome makes the gamble more
attractive. This increase in attractiveness overcomes the effect of the concavity of the utility
function that makes the gamble less attractive. Both effects taken together result in a
preference for the gamble over the certain outcome R+.
Allowing for probability weighting has considerable consequences for our reasoning
outlined above and for our experimental test. We stated that under Expected Utility the
Effects of Risk Preferences in Repeated Social Dilemmas
49
?
X
1/3
1/3
2/3
2/3
-5
Figure 3.3:
-10
Structure of the gambles used in the tradeoff method to assess utilities
in the gain domain. The gain X in the left gamble is either X1 = 0 , X2 =
5, X3 = 10, or X4 = 20. The subject specifies the amount ? such that (s)he
is indifferent between the two gambles.
20
1/3
1/3
2/3
2/3
X
Figure 3.4:
50
?
Structure of the gambles used in the tradeoff method to assess utilities
in the loss domain. The loss X in the left gamble is either X1 = 0 , X2 = 5, X3 = -10, or X4 = -20. The subject specifies the amount ? such that
(s)he is indifferent between the two gambles:
hypotheses H1 and H1’are equivalent. If probability weighting is taken into account, this is
no longer true. With our first experiment we tested assumption H1’, but we were mainly
interested in hypothesis H1 which describes the relation between the shape of the utility
function and cooperation, as derived from game theory. Moreover, the utility assessment
method used in Experiment 1 is not robust with respect to probability distortions and
therefore provides only indirect information on subjects’ utility functions. For a stricter test
of hypothesis H1 we need a utility assessment method which is robust with respect to
probability distortions. However, all traditional utility assessment methods (see Farquhar,
1984, for a general overview) assume that subjects do not transform probabilities. Wakker
50
Chapter 3
and Deneffe (1996) developed a method that can measure utility functions independently to
utility transformations. Their method, called the “tradeoff method”, will be used to assess
utilities in Experiment 2.
The tradeoff utility assessment method is robust with respect to probability
transformations. It can be used to assess utilities under different theories of individual
decision making and under risk assuming probability transformations, including Expected
Utility theory, Prospect Theory, Rank Dependent Utility theory, and Cumulative Prospect
Theory.8 We will give a self-contained explanation of the tradeoff method here, assuming
that subjects have different decision weight functions for gains and losses, as in Cumulative
Prospect Theory. The subject’s task in the tradeoff method is to choose between two gambles
with two outcomes. The lowest outcomes, and therefore also the largest outcomes, occur with
the same probability for both gambles. Moreover, the outcomes with equal probabilities are
of the same value domain, that is, both are losses or both are gains. In the gambles used in the
tradeoff method here, the lowest outcome is from the loss domain and occurs with probability
2/3, and the largest outcome is from the gain domain and occurs with probability 1/3. The
structures of the choice situations used in the second experiment to elicit utilities in the gain
and loss domains, are shown in Figure 3.3 and Figure 3.4 respectively.
Consider first the structure of the choice situation in the gain domain. The losses from
the gambles are fixed at a loss of 5 guilders in the left hand gamble and a loss of 10 guilders
in the right hand gamble.9 The gamble on the left hand side contains a gain X. The subject’s
task in the long run is to specify the value of the gain such that (s)he is indifferent between
choosing the gambles on the left and right hand sides. Consider two choice situations that
differ in gain X in the left hand gamble, e.g. X3 = 10 and X4 = 20. The responses of the subject
in these situations are denoted by ?3 and ?4 respectively. Assuming Cumulative Prospect
Theory, the response ?3 to the first situation implies that π-(2/3)u(-5) + π+(1/3)u(10) = π(2/3)u(-10) + π+(1/3)u(?3). Rearranging the terms in the equality leads to
U (? 3 ) − U (10) = [U ( −5) − U ( −10)]
π − (2 / 3)
π + (1 / 3)
(2)
Repeating this exercise for the second choice situation leads to the same equation but with the
left hand side equal to u(?4) - u(20). Thus, the two responses in the choice situations together
imply that the subject’s utility differences u(?4) - u(20) and u(?3) - u(10) are equal. Note that
possible probability transformations are canceled out because they have identical effects in
both choice situations. The only assumption we make using the tradeoff method is that both
probability transformations and utility for losses of 5 guilders and 10 guilders, are invariant
over the choice situations. In a similar fashion, utilities can be assessed for the losses domain.
Wakker and Deneffe (1996) and also Fennema and van Assen (1999) used a so-called
“chained outward” tradeoff method to elicit the utilities. The term “outward” is used because
8 Another merit of the tradeoff method as opposed to traditional utility assessment methods is that it can be used
under conditions of uncertainty, that is, when outcomes are obtained from events of which the probabilities are
unknown. Explanations of how the tradeoff method works under conditions of uncertainty and what
considerations should be taken into account when using the tradeoff method assuming one of the different
theories of individual decision making under risk, can be found in Wakker and Deneffe (1996) and Fennema
and van Assen (1999).
9 At the time we conducted the second experiment 1 US dollar was worth approximately 2 Dutch guilders.
Effects of Risk Preferences in Repeated Social Dilemmas
51
the absolute value of the subject’s response ? is larger than the absolute value of the outcome
X. Note that likewise the outward tradeoff method is used in the choice situations of Figure
3.3 and Figure 3.4. The method is said to be “chained” because the subject’s response in a
given situation is substituted for the X outcome in the next choice situation. As a result, a
sequence of connected equal utility intervals is obtained for each subject. In our experiment
we are interested in the shape of utility for a narrow domain of outcomes used in the repeated
PDs. The outcomes used in the PDs are in the interval ranging from a loss of 25 guilders to a
gain of 25 guilders. For our purposes an unchained tradeoff method seemed more convenient,
yielding a more precise measurement of the interval of relevant outcomes. In our new
experiment we substituted for X in Figure 3.3 the values of 0, 5, 10, and 20 guilders, and 0, 5, -10, and -20 guilders for the losses in Figure 3.4.
In our Experiment 1, risk preferences were assessed using single questions relating the
outcome R to gambles (T,1-w,P) that have an expected value equal to R. We also chose to use
this simple method in Experiment 2, although elaborated with two additional questions,
alongside the tradeoff method. By including both measurements in the experiment we are
able to compare the results of both experiments, and are able to compare the results of the
tradeoff method with the results of the traditional method. Empirical studies (Wakker and
Deneffe, 1996; Fennema and van Assen, 1999) have found that the tradeoff method results in
less concave utility than a method comparable to that used in Experiment 1. However, it was
not evident that this result would also be found in the second experiment. In the two
experimental studies cited, utility was assessed for the whole gain domain. Previous research
has shown that when utility is assessed using the simple method in gambles with outcomes
lying far apart utility is more biased to concavity (Krzysztofowicz and Duckstein, 1980).
More details of the precise utility assessment methods used in the second experiment
are provided below, where the design and results of the second experiment are discussed in
detail.
3.5
Experiment 2
To test the hypothesis that there is a relation between cooperation in a repeated PD and the
shape of utility in the domain of the payoffs used in the PDs, an experiment consisting of two
parts was performed. In the first part, utility of subjects was assessed, while in the second part
subjects played a number of repeated PDs. The method and results of the experiment are
reported below.
3.5.1 Method
3.5.1.1 Subjects
Subjects were first recruited by flyers and by advertising in the university newspaper.
Because only a small number of students was reached using these methods we sent an email
to approximately 13,000 students of the University of Groningen in which we advertised our
experiment. A sufficient number of potential subjects reacted positively to our email. In total
227 subjects participated in the experiment, of whom 216 finished both sessions. Most
subjects were undergraduates from the University of Groningen. The age of subjects was
between 17 and 29, and 50.5% of them were male.
52
Chapter 3
3.5.1.2 Procedure
A computer program was written for each of the two parts of the experiment, which were run
on different days.10 We attempted to ensure that the amount of time between participating in
the first and second part of the experiment was at least 24 hours and at most a week. The
experimental sessions took place in a large room with 15 computers distributed in five rows
of three. In the experimental sessions both subjects participating in the first part and subjects
participating in the second could be present. Although the configuration of the room was such
that it was barely possible to observe other subjects' responses, the subjects participating in
the second part were always positioned in the back of the room to prevent the other subjects
suspecting what might be involved in the second part. Subjects were explicitly asked to be
silent during the experiment. One experimenter was always present in the room to answer any
questions of the subjects.
In the first part of the experiment, subjects' utility functions were assessed in different
ways (see below). In the second part, subjects played five different repeated PDs. Outcomes P
for mutual defection in these PDs were 5, 0, -5, -10, -20 Dutch guilders respectively. The
continuation probability w of the PDs was equal to (T-R)/(T-P) = 0.5. In the PDs, T = P + 20,
R = P + 10, and S = P - 5, resulting in a domain of PD outcomes ranging from -25 to 25. In
the first part of the experiment utility was obviously assessed for precisely this domain.
3.5.1.3 First part of the experiment
The first part consisted of five tasks. The subjects started with an extension of the utility
assessment procedure used by Raub and Snijders (1997). For the second task, subjects were
required to indicate their indifference point in comparisons of a gamble with a certain
outcome. Subsequently, the TO method was used to elicit subjects' utilities. The fourth task
was a questionnaire. All subjects had to complete the first four tasks, and if they were
completed within 45 minutes the subject continued with a test-retest reliability task. Here the
subject was given exactly the same choice situations as in the first three tasks. A more
detailed description of each of the tasks is provided below.
Task 1: Extension of utility assessment procedure as used by Raub and Snijders
Raub and Snijders assessed risk preferences for the domain [P,T] in a PD with these
outcomes and a continuation probability w equal to 0.75, by requiring subjects to make a
preference comparison between gamble (T, 1-w, P) and the expected value R of the gamble.
Subjects preferring the gamble are "risk seeking", while subjects preferring the certain
outcome R are "risk averse". Subjects also made made preference comparisons between (T, 1w, P) and R, but this time with w equal to 0.5. We refer to this way of measuring the risk
preference as the "traditional method". To extend the measurement two additional preference
comparisons were constructed: a comparison between (T, 1/3, P) and R, and one between (T,
2/3, P) and R. Using the answers to these three preference comparisons, four different types
of risk preferences can be distinguished: "very risk averse" if a subject does not prefer any of
the gambles above the certain outcome, "risk averse" if the subject only prefers the gamble
when the probability of obtaining the best outcome T from the gamble is equal to 2/3, "risk
seeking" if the subject prefers the gambles for probabilities 2/3 and 0.5, and "very risk
seeking" if the subject prefers all gambles above R. In the analyses, we refer to this way of
10 The computer programs are written in Turbo Pascal 7.0 and are available from the first author upon request.
Effects of Risk Preferences in Repeated Social Dilemmas
53
measurement as the "extended traditional method."
A brief instruction explained the task to the subjects, who were told to imagine that
they would choose between the gamble and the certain outcome only once and that the
outcomes of the choice situation were their own money. To practice the procedure, subjects
made three preference comparisons between (60, p, -20) and 20 with p equal to 2/3, 0.5, and
1/3 respectively.
The task contained four sets of three preference comparisons, with P equal to 0, -5, 10, or -20. The order of the sets was varied among subjects with a Latin Square design
(Edwards, 1968) consisting of four orderings. Within a set, subjects always first stated their
preference for the comparison with p equal to 0.5. After subjects had stated their preference
they were asked to confirm their preference. They were also given the possibility of changing
answers they gave earlier.
Task 2: Extra choice situations
Five additional choice situations involving a gamble and a certain outcome were put before to
the subjects. In theory, the answers of the subjects in these situations allow us to estimate
parametric utility functions. Because we do not report on parametric utility functions here, we
also do not report on the details of the second task.
Task 3: The tradeoff method
Utility was assessed for gains and losses using the unchained tradeoff method as described
above. Four choice situations were used in each value domain. The choice situations for
assessing utility for gains and losses are shown in Figure 3.3 and Figure 3.4 respectively. For
gains, outcomes 0, 5, 10 and 20 are substituted for X, and for losses outcomes 0, -5, -10, and
–20 are substituted.
To obtain the subject's indifference point "?" in the right hand gamble of a particular
choice situation, the computer program first substitutes outcome X + Y for "?". The value of Y
is 5, 10, 15, or 20 in the case of gains, and -5, -10, -15, or -20 in the case of losses. The
probability of each of the values of Y is 0.25. In general the subject then has four options: he
can either be indifferent between the two gambles, he can prefer the right hand gamble or the
left hand gamble, or he can recognize that he has made a “mistake” earlier in the procedure
and choose to return to an earlier choice situation to change a response. If the subject is
indifferent between the gambles, the indifference point is determined and the computer
program provides a choice situation with another value of X (and "?"). If the subject prefers
one of the gambles to the other, the computer program generates another value of "?" in the
right hand gamble of the same choice situation. Consider, for example, that the subject
prefers the left hand gamble to the right hand gamble. In the case of losses, this implies that
the indifference point of the subject is between X and X + Y. The computer program then
generates the following choice situation where the right hand gamble is made more attractive
than in the first situation. The new value of "?" will then be equal to a loss of X + 0.5Y,
bisecting the interval which contains the indifference point. If the subject had preferred the
right hand gamble, the loss of the right hand gamble in the previous choice situation is
increased by an additional loss Y, again chosen randomly from the distribution specified
above. This procedure, choosing "?" with bisection or an additional random loss, is continued
54
Chapter 3
until the subject indicates he is indifferent between the two gambles.11 In the case of gains the
procedure is identical, choosing "?" with bisection or an additional random gain.
Half of the subjects started with the tradeoff method for gains, while the other half
started with the TO method for losses. The order of choice situations within each of the two
value domains was such that a subject first determined his or her indifference point for the
two most extreme values of X (0 and 20, or 0 and -20), followed by that for the two other
choice situations (X equal to 5 and 10, or -5 and -10). The 24 possible orderings were varied
among subjects using a Latin Square design.
The instructions for the tradeoff method given to subjects were extensive because
previous research with the method indicated that at least initially, subjects have some
problems in grasping the complexity of the choice situation. The procedure described above
was explained and applied to find the subject's indifference point "?" such that he is
indifferent between (40, 1/3, -10) and (?, 1/3, -20). To assist the subject in finding the value
of "?", he was asked to think of how much more the gain "?" (in the right hand gamble)
should be than the 40 guilders (in the left hand gamble) in order to compensate for the
additional loss of 10 guilders (compared to the left hand gamble), which is twice as likely as
the additional gain. The instruction also contained a second choice situation in which subjects
had to specify a negative indifference point. The subjects were again told to imagine that they
would choose between the gambles only once and that the outcomes of the choice situation
involved their own money.
Task 4: Questionnaire
The participants were asked about their age, sex, number of siblings, whether they studied or
not, whether they had a (part-time) job, subjects they followed in secondary school, the
political party they voted for at the last elections, and whether they had knowledge of game
theory. Subjects' responses to the questions were used as controls in our analyses. The
questionnaire was concluded with a measure of the social orientation of the participant
(Messick and McClintock, 1968; McClintock and Liebrand, 1988). Social orientation
acknowledges that for one's utility in a choice situation, not only one’s own payoff is
important but also the payoff of the other actors involved in the choice situation. That is, in
the PD, ego's utility of the payoff can also depend on the payoff obtained by alter. The higher
ego's social orientation, the more important alter's payoff is for one’s utility relative to one’s
own payoff. A high social orientation can transform a PD game into another game in which
defection is no longer the dominant choice. Therefore, social orientation is a relevant
background variable for our analyses in the results section. Social orientation is measured in
the questionnaire by requiring the subjects to order pairs of outcomes (53,53), (64,39),
(74,30), and (58,104). The first outcome is that of ego and the second outcome is that
outcome for alter. Assuming that the actors' utility of outcome (x, y) is equal to x + θy, where
θ is the social orientation parameter, then six possible preference orders of the four outcome
pairs are obtained from different values of θ, and hence six different classes of participants
with varying social orientation.
11 The procedure was also stopped when the length of the interval, which contained the indifference point, was
equal to 1. The subject was then required to state the value which was closest to his "real" indifference point.
This resulted in integers as indifference points.
Effects of Risk Preferences in Repeated Social Dilemmas
55
Task 5: Test-retest reliability
The choice situations in tasks 1, 2, and 3 were repeated once until 45 minutes had passed
since the start of the experimental session. The order of choice situations in Task 1 and Task
2 was randomized. The choice situations of Task 3 with extreme values of X were repeated
first, followed by the four other choice situations.
3.5.1.4 Second part of the experiment
The second part of the experiment started with extensive instruction towards the subjects. The
PDs were presented to the subjects using 2x2 tables. The relation between the choices made
by the players and their outcomes was explained by letting participants play a repeated PD
twice, with outcomes T = 60, R = 20, P = -20, S = -40, and w equal to 0.5 as in the games of
interest.
Subjects were told that they did not play against a player present in the lab but against
a randomly selected player from a previous experiment, drawn with replacement before each
repeated PD. The experiment from which players were drawn was actually that of Raub and
Snijders (1997). In Appendix 3.1 we explain how we were able to model the strategies of
human players accurately and realistically by making use of the responses of subjects in the
1997 experiment. Advantages of the procedure we used are also enumerated in Appendix 3.1.
A lot of effort was invested in the instruction to make participants in our second experiment
realize that (i) they played against a human player, although the computer generated the
responses, and (ii) that they were not deceived in any way (which was true).
After participants had played the two repeated PDs it was explained to them how their
payoff would be determined at the end of the experiment. The procedure for rewarding the
subjects is described later. Finally, before playing the repeated PDs of interest, subjects were
told to imagine that outcomes of the PDs involved their own money.
After the instruction the subject played four repeated PDs in a random order. The
value of P in the PD was -20, -10, -5, or 0. The subjects were told that their responses in these
games could have an effect on their reward at the end of the session. If 40 minutes had not
yet passed since the start of the experimental session, the subject continued to play these four
repeated PDs in a random order. The responses in these additional games had no effect on the
subject's reward. The subjects were, however, told that their behavior in these games was
important for the experiment and they were asked to behave as if they would experience the
outcomes as real payoffs. When either the subject had played each repeated PD twenty times
or 40 minutes had passed, the subject played a fifth repeated PD with P equal to 5. This game
was also important with regard to the subject's reward. The computer program then showed
the possible rewards for participating in the experiment.
3.5.1.5 Reward for participation in the experiment
After completion of the first part of the experiment, subjects were given a reward of 35
guilders (about US$ 17.5). Rewards were not dependent on the subject's responses in the
preference comparisons. We were not concerned that using hypothetical payoffs in the choice
situations would bias our results because previous studies on individual decision making
under risk using real incentives yielded results similar to studies using hypothetical payoffs
(e.g., Tversky and Kahneman, 1992; Camerer, 1995; Beattie and Loomes, 1997).
56
Chapter 3
Table 3.1:
Risk preferences for the gains and losses domain using the traditional
method with a single choice per subject.
Risk averse for
gains
No
Yes
Risk averse for losses
No
Yes
114
37
45
20
Total
159
57
Total
151
65
216
To make the repeated PDs in the second part of the experiment as realistic as possible
and because payoffs dependent on behavior were used in Experiment 1, we chose to make the
subject's reward dependent on his or her behavior (and on the behavior of the other player).
Since it was possible for the subject to money in some of the repeated PDs, and hence in the
second part of the experiment, the reward for participation in the first part was high.
Separating the two parts had the advantage that the "gambling with the house money effect"
(Thaler and Johnson, 1990) was, in all likelihood, minimized.
The subject's reward in the second part of the experiment was determined by the
random selection method. The outcome of the last round of each of the five different repeated
PDs played for the first time was selected as a possible payoff. This was explained to the
subjects during the instruction at the start of the second part of the experiment. As shown by
Raub and Snijders (1997), selecting only the last round excludes wealth effects and does not
affect the strategic structure of the repeated PD.
We were mainly interested in the subjects' behavior in the four repeated PDs with P =
0, -5, -10, and -20. In the present study, only the subjects’ response to the two repeated PDs
with P equal to 0 (positive PD) and -20 (negative PD) are analyzed; responses to the two
other (mixed) PDs are analyzed in a future study. Because most outcomes in the last round of
a repeated PD were determined by mutual defection, the expected value of the subjects'
reward with the random selection method would have been negative. Therefore, the fifth
repeated PD with P = 5 was included at the end of the experiment. At the end of the
experiment the subjects drew one card from a pile of six. To further increase the expected
value of the procedure, two jesters were included which were associated with the fifth
repeated PD; the other repeated PDs were associated with clubs, diamonds, hearts, or spades.
By the simulation of both actor strategies we predicted that the expected value of the second
part of the experiment was around zero guilders. Our prediction was approximately correct,
the average reward given to subjects in the second part being -0.4 guilders. The minimum
reward for participation in both parts of the experiment was 10 guilders (35-25) and the
maximum reward was equal to 60 guilders (35+25). Because participation in the second part
was not lucrative for the subjects, we were concerned that a number of subjects would not
turn up. Therefore, we asked them to sign a contract with their name, address, telephone
number, and a promise that they would arrive for the second part. Only 8 subjects did not turn
up for the second part of the experiment.
Effects of Risk Preferences in Repeated Social Dilemmas
Table 3.2:
Risk
averse for
gains
57
Risk preferences for the gains and losses domains using the traditional
method with three choices per subject. The values ‘0’, ‘1’, ‘2’, ‘3’, represent
the number of gambles out of three that the subject prefers above the certain
outcome. Therefore, a larger value corresponds to greater risk aversion.
0
1
2
3
Total
0
11
17
7
1
36
Risk averse for losses
1
2
19
8
67
25
34
15
3
2
123
50
3
2
2
1
2
7
Total
40
111
57
8
216
3.5.2 Results
3.5.2.1 Distribution of risk preferences
We first report on the data obtained on subjects’ risk preferences based on the traditional
method employed by Raub and Snijders (1997). That is, to assess risk preferences for gains,
we only used the subject’s preference for the gamble (T+, 1-w, P+) and the certain outcome R+.
The choice between (T-, 1-w, P-) and the certain outcome R- determines the risk preference of
the subject for losses. We found that only 30.1% (65/216) of the subjects are risk averse for
gains, and that 73.6% (159/216) are risk seeking for losses. Compared to Raub and Snijders
(1997) we find a relatively large percentage of risk seeking choices, especially in the gains
domain. Only 20.8% (45/216) of the subjects are categorized as risk averse for gains and risk
seeking for losses. Table 3.1 summarizes these results. In Table 3.2 we report the frequencies
of the degree of risk aversion based on three choices per domain.
One crucial difference between the experiment reported by Raub and Snijders (1997)
and this one is the use of the more sophisticated measurement of risk preferences, the tradeoff
method. Based on this method, subjects can be categorized in the same way as with the
traditional method. As outlined previously in the section on utility measurement, the tradeoff
method generates four responses ?1, ?2, ?3, ?4 per value domain (gain or loss). These four
responses provide three independent pieces of evidence to determine the shape of utility in
this domain (see also Fennema and van Assen, 1999). For example, in the case of gains, if the
difference ?1 – 0 is smaller than ?2 – 5 then this is evidence for concavity of the utility
function (more money needs to be added to 5 than to 0 to bridge the same utility difference).
Similarly, subsequent differences can be compared and evidence can be listed. If two or more
pieces of evidence point to concave, linear, or convex utility, then the subject’s pattern of
responses is coded in Table 3.3 as concave, linear, or convex respectively. If a response
pattern reveals exactly one piece of evidence for each of convex, linear, and concave utility
then the subject is categorized as “missing”. We have 45 “missings” in the losses domain and
34 in the gains domain. We observe that of the 142 subjects who are assigned a risk
preference, the largest proportion (54 subjects) have concave utility for gains and convex
utility for losses, in accordance with S-shaped utility. Therefore, the tradeoff method does
generate response patterns that are more in line with standard results on risk preferences.
58
Table 3.3:
Utility for
gains
Chapter 3
Utility for the gains and losses domain using the tradeoff method.
“Missing” is denoted by ‘*’.
Convex
Linear
Concave
*
Total
Convex
9
4
54
6
73
Utility for losses
Linear
Concave
0
11
18
5
7
34
7
13
32
63
*
4
3
30
8
45
Total
24
30
125
34
213
The tradeoff method is also more stable than the traditional method. The average testretest reliability correlation between the responses before and after the questionnaire in a
given choice situation equals 0.67 for the tradeoff method, whereas this average correlation
equals 0.50 for the traditional method. Surprisingly, the correlation between the results from
the extended traditional method and the tradeoff method is low (0.10, p=0.16). An
explanation and discussion of this weak correlation is postponed until the conclusion and
discussion section.
3.5.2.2 Risk preferences and cooperation
Hypothesis H1’’ stipulated that the cooperation rate is higher for the gains domain and our
results support this hypothesis. The percentage of cooperation in the first repeated PD for
gains is 36.5, whereas this percentage is only 26.9 for the losses domain (paired t-test, t=2.16,
p=0.02). The finding of a higher cooperation rate for gains together with evidence of Sshaped utility as assessed by the tradeoff method suggests that our hypotheses H1’ and H1
will also be supported. However, this is not the case, as we demonstrate below.
Hypothesis H1’ stated that there is a relation between risk preferences, as measured by
the traditional method, and the cooperation rate in the repeated PD, such that a higher
cooperation rate corresponds to greater risk aversion. If we compare the choices of subjects in
the repeated PDs, we find small differences between risk averse and risk seeking subjects
according to the traditional method. In the first round of the first gains dilemma that subjects
play, 38.5% of the risk averse subjects cooperate, as opposed to 35.1% of the risk seeking
subjects (1-sided Fisher’s exact=0.37). If we use the extended version of the traditional
method, we find that the percentage of cooperation decreases for only the most risk seeking
subjects: the most risk averse subjects score 37.5%, and the more risk seeking categories of
subjects score 38.6%, 36.9%, and 30%. In the first round of the first losses dilemma that
subjects play, 31.6% of the risk averse subjects cooperate, as opposed to 25.2% of the risk
seeking subjects (1-sided Fisher’s exact=0.22). The extended version of the traditional
method yields a percentage of cooperation that decreases as subjects become more risk
seeking, from 42.9% to 30.0% and 26.8% to 19.4% for the most risk seeking subjects. In
summary, the differences in cooperation rates between risk averse and risk seeking subjects
are small, but show the correct trends for the traditional measurements.
We next analyze whether a relation between risk preferences and behavior in repeated
PD’s can be established. For this aim we use subjects’ risk preferences as mentioned above,
Effects of Risk Preferences in Repeated Social Dilemmas
Table 3.4:
Logistic regression on the choice in round 1 of the repeated PD
excluding the control variables. To account for the multiple cases per
subject, standard errors are calculated on the basis of Huber (1967).
Absolute t-values are given in brackets. ‘*’ denotes significance at the
5% level, ‘**’ denotes significance at the 1% level.
Risk seeking
Gains dilemma
Constant
Adj. R2
Observations
Table 3.5:
59
Risk preference assessment
Traditional (1)
Traditional (3)
Tradeoff
-0.22
(0.98)
-0.21
(1.49)
0.12
(0.86)
0.42*
(2.03)
0.43*
(2.05)
0.51*
(2.04)
-0.84**
(3.75)
-0.62*
(2.06)
-1.20**
(3.47)
0.01
0.01
0.01
432
432
347
Logistic regression on the choice in round 1 of the repeated PD
including the control variables. To account for the multiple cases per
subject, standard errors are calculated on the basis of Huber (1967).
Additional independent variables are included, absolute t-values are
given in brackets. ‘*’ denotes significance at the 5% level, ‘**’
denotes significance at the 1% level.
Risk seeking
Gains dilemma
Male
Know game theory
Social orientation
Constant
Adj. R2
Observations
Risk preference assessment
Traditional (1)
Traditional (3)
Tradeoff
-0.25
(1.09)
-0.22
(1.49)
0.12
(0.84)
0.46*
(2.17)
0.46*
(2.18)
0.53*
(2.08)
0.34
(1.52)
0.33
(1.52)
0.36
(1.49)
0.59**
(2.59)
0.58**
(2.60)
0.65**
(2.65)
-0.03
(0.36)
-0.02
(0.42)
-0.03
(0.32)
-1.15**
(4.18)
-0.93*
(2.68)
-1.57**
(4.02)
0.01
0.03
0.03
432
432
347
and we correlate these with the choices of subjects in the first round of the first gains PD (P =
0) that they played, and in the first round of the first losses PD (P = -20) that they played.
Therefore, we consider two cases per subject simultaneously. Only the choices in the first
rounds were used in the analysis, because choices in subsequent rounds are affected by the
choices of the other player. Table 3.4 reports the results using a repeated measures design
(logistic regression with two cases per subject). Table 3.5 reports the same results, including
several extra independent variables. In contrast to our first experiment we find no significant
relation between risk aversion as measured by the traditional method and the probability of
cooperation. We observe that only the traditional method in its extended form (using three
choices instead of a single choice) comes close to being significant at the p=0.10 (tvalue=1.65) level. The size of the effect on the probability of cooperation of a single
difference in category equals 0.05 for both traditional methods of utility measurement.
Separate analyses performed on only the gains or only the losses dilemma give similar
results. The added independent variables in Table 3.5 show some interesting results. Men are
60
Chapter 3
Table 3.6:
Logistic regression on the choice in rounds 1 and 2 of the repeated PDs
for gains, and for rounds 1 and 2 of the repeated PDs for losses,
excluding the control variables. To account for the multiple cases per
subject, standard errors are calculated on the basis of Huber (1967).
Absolute t-values are given in brackets. ‘*’ denotes significance at the
5% level, ‘**’ denotes significance at the 1% level.
Cooperation in
the gains
dilemma
Cooperation in
the losses
dilemma
Risk seeking
Adj. R2
Observations
Risk seeking
Adj. R2
Observations
Risk preference assessment
Traditional (1)
Traditional (3)
Tradeoff
0.07 (0.26)
-0.07 (0.47)
0.10 (0.49)
0.00
0.00
0.00
431
425
358
-0.20 (0.80)
-0.32* (2.07)
0.38* (2.51)
0.01
0.01
0.02
431
431
347
not more likely to cooperate, and those who claim to have heard of “game theory” do
cooperate more often (an effect size of about 0.12 on the probability of cooperation).
Additionally, contrary to most previous results on the effects of social orientation in social
dilemma situations, subjects with high social orientation values are also not more likely to
cooperate in the repeated PD. Moreover, we find that cooperation occurs more often in round
1 of the gains dilemma (an effect size of about 0.10 on the probability to cooperate).
Therefore, the argument that “losses loom larger than gains and/or people are more risk
seeking for losses, thus subjects are more willing to risk being the sucker in a repeated PD for
losses, and are hence more likely to cooperate in a repeated PD for losses” is not only a
mistake theoretically, as we outlined in the previous sections, but also is not corroborated
empirically.
Hypotheses H1 stated that there is a relation between the shape of the utility function
as measured by the tradeoff method and the cooperation rate in the repeated PD, such that a
larger cooperation rate corresponds to greater concavity of the utility function. In the case of
the first round of the first gains dilemma, the tradeoff method yields no difference in
cooperation rates. Subjects with concave utility cooperate in 37.6% of the cases, and the
percentages of those who have linear utility or convex utility are 36.7% and 37.5%
respectively. For the first round of the losses dilemma, the trend of the percentages of
cooperation given by the tradeoff method is in the opposite direction. Subjects with concave
utility cooperate in 23.8% of the cases, which increases to 25.0% and 32.9% for those who
have linear and convex utility respectively. The tradeoff method thus gives results that clearly
contradict our hypothesis: we find no difference for the gains dilemma, but a slight increase
in cooperation for subjects who have more convex utility.
In similar fashion to our analyses of the results of the traditional methods, we analyze
whether there is a relation between concavity of the utility function and subjects' behavior in
repeated PDs. Results of the analyses are reported in Table 3.4 and Table 3.5. We observe
that in both the analysis with control variables and without, the relation between shape of the
utility function and cooperation is not significant. The effect size of concavity of utility on the
probability of cooperation of a single difference in category equals 0.017.
Effects of Risk Preferences in Repeated Social Dilemmas
61
Evidently, the effects of both risk aversion and the shape of the utility function on
cooperation as put forward in H1 and H1' receive little support. However, if we extend our
analyses and include the choice in the second round of the repeated PD, we do find some
significant effects. Table 3.6 reports these results.
Table 3.6 shows that risk preferences may have an effect in the losses dilemma, more
sothan in the gains dilemma. For the losses dilemma, the extended traditional method shows
a significant effect in the hypothesized direction (effect size = 0.07), whereas the tradeoff
method reveals an effect of the same size in the opposite direction. If we run additional
analyses, including extra rounds of the losses dilemma, we find that the negative effect of the
extended traditional method remains significant at p=0.10 with an average effect size of
approximately 0.10 for a difference of two categories. The positive effect of the tradeoff
method disappears immediately when the third rounds of the losses dilemma are included,
and does not return when subsequent rounds are included.
3.6
Conclusion and discussion
In Raub and Snijders (1997) a theoretical analysis suggested that a connection exists between
the shape of the utility function and cooperation rates in a repeated PD. The theoretical
argument there is that concavity should favor cooperation. Raub and Snijders (1997) assumed
Expected Utility theory and in that case risk preferences and the shape of the utility function
coincide. An experimental test supported their argument that risk aversion (concavity) favors
cooperation. Moreover, cooperation was on average less prevalent in situations where it was
necessary to avoid a loss. Van Assen (1998) argued, among other things, that the utility
assessment method used by Raub and Snijders (1997) needed to be improved. We reported
here on an experiment meant to replicate, combine, and extend the work in these two papers.
We conducted an experiment consisting of two separate sessions, one to measure a
subject’s risk preferences and utility function, and a second to measure cooperation in
different repeated PDs. Three hypotheses were put to the test. One hypothesis (H1’’) stated
that cooperation should be more easily achieved in a situation where mutual cooperation is
necessary to achieve a gain than in a situation where mutual cooperation is necessary to avoid
a loss. Our results clearly support this hypothesis. Once again, the intuitively appealing
argument that people are more likely to contribute to the collective good in a situation where
cooperation is necessary to avoid a loss, as argued, for instance, in Berejikian (1992), is
refuted. Moreover, the acceptance of hypothesis H’’ together with the fact that framing
effects exist leads to the conclusion that a party interested in the development of cooperation
can influence the probability that cooperation will emerge by framing the situation as a gains
dilemma.
The second hypothesis (H1’) stated that the cooperation rate in the repeated PD is
greater for people who are risk averse in gambles containing the outcomes of the PD than for
people who are risk seeking in these gambles. Although significant (but small) positive
effects of risk aversion were found by Raub and Snijders (1997), the evidence in our
additional experiment is less convincing. Using the same method of measuring risk
preferences as in the earlier experiment, we find significant effects of risk aversion only when
we extend the measurement from a single gamble to three gambles. Risk aversion favors
cooperation only in the repeated PD for losses, with an effect size on the probability of
cooperation of about 0.10. The third hypothesis (H1) stated that the cooperation rate in the
62
Chapter 3
repeated PD is greater for people with concave utility functions than for people with convex
utility functions. To test this hypothesis, we measured utility functions using the tradeoff
method (Wakker and Deneffe, 1996; Fennema and van Assen, 1999), a method that does not
depend on the assumption that subjects evaluate probabilities linearly. The tradeoff method
does elicit risk preferences that are more in line with earlier results in the literature on risk
preferences. Whereas the traditional method as used by Raub and Snijders (1997), and
replicated here, resulted in a relatively large number of risk seeking choices in the gains
domain, the tradeoff method shows more convex utility for losses and much more concave
utility for gains. However, there is no relation between utility as measured using the tradeoff
method and cooperation in the repeated PDs
Surprisingly, the shape of utility as measured using the tradeoff method hardly
correlates with risk preferences as measured by the extensive traditional method. Several
complementary reasons can be provided for the low correlation between the results of using
the two methods. Firstly, the bias in the traditional method resulting from probability
weighting can vary between individuals, which decreases the correlation. Secondly, it is
likely that the correlation is also obscured because we make use of rather crude nonparametric measures of utility and risk preferences. We expect the correlation between the
two measures to be higher when estimating risk preferences and utility parametrically. In
principle it is possible to use the data to estimate the utility function in a parametric way, but
we did not do so because our main goal was to test the relation between the shape of the
utility function and cooperation in the repeated PDs. Other disturbing factors that could have
been involved in reducing the correlation are random and independent errors of subjects in
both methods. The moderate test-retest reliability correlations indicate that within-subject
variation is indeed substantial. Although there are reasons to expect only a low to moderate
correlation between the two methods, the finding that it is near zero is worrying and implies
that the question should be asked of whether utility as a subject characteristic exists at all
(Shoemaker, 1982). A recent study by Isaac and James (2000) also questions the view that
individuals have a stable and consistent risk attitude. They measured utility of individuals
using two methods and found a negative correlation between the two methods, although the
correlation was not significantly different to zero. Weber and Milliman (1997) hypothesize
that risk preference might be a stable personality trait after all, but that differences in the
choices made in different situations or methods may be the result of changes in risk
perception. That is, different choices made by the same individual in different circumstances
can be the result of different perceptions of risk, while risk preferences remain stable.
Nevertheless, in the context of the present study, the low reliability and small correlations
between methods imply that if a correlation is expected between assessed utility and
cooperation in the PD games, then it is very likely to be small.
While the low reliability and small correlations between methods at least partly
explain the lack of empirical support for hypotheses H1’ and H1’’, several other disturbing
factors may be involved as well. Firstly, it is theoretically possible that subjects in our study
were trying to reach an equilibrium, where behavior on the equilibrium path alternated
between cooperation and defection, yielding payoffs (S, T) and (T, S). However, this is
unlikely for two reasons. Firstly, outcome R was larger than the average of outcomes S and T.
Subjects could therefore observe that alternating defection and cooperation does not lead to a
higher payoff than adhering to mutual cooperation. Secondly, alternating response patterns
Effects of Risk Preferences in Repeated Social Dilemmas
63
were observed only infrequently observed in both experiments. Somewhat related to this
argument is the argument that (T-R)/(T-P) may not be the correct index to study in the
repeated PD (although it is the right index to study if game theoretical rationality is assumed).
However, Murnighan and Roth (1983) found that this index was one of the best predictors of
cooperation rates in repeated PDs with different payoffs.
The experiment that we conducted was different to the original experiment of Raub
and Snijders (1997) in an important way other than just using a more extensive measurement
of risk preferences and utility. Instead of letting subjects play repeated PDs against each
other, we let them play against the computer, telling them so. The computer was programmed
to mimic the behavior of a subject in the earlier experiment. This allowed subjects to wok
through the experiment at their own pace, without delaying others, thereby speeding up
matters dramatically. Although subjects were explicitly and repeatedly told during the
instruction that the computer program played like a human player randomly selected from the
Raub and Snijders (1997) experiment, this procedure may have resulted in a certain lack of
reality for the subjects (“there was no real person on the other side”). However, after
completing the experiment, only a few (less than five) subjects asked for a detailed typed
explanation of the computer program they played against. The explanation was an elaborated
version of Appendix 3.1. The fact that only a few subjects asked for the explanation suggests
that participants believed that they had not been tricked, and that they played against a
computer that faithfully mimicked a human player.
The data from the experiment allow other tests of the effects of risk preferences on
cooperation in repeated PDs to be studied. As van Assen (1998) showed, one can expect more
dramatic effects of risk preferences for repeated PDs where the payoffs are mixed (that is,
partly positive and partly negative). Although we do not report it here, we did also let
subjects play such mixed dilemmas. We plan a future study on the effect of loss aversion on
cooperation in repeated mixed dilemmas.
Theoretically, two improvements of our current models seem particularly useful. Thus
far, we have assumed that ego considers the utility function of alter to be the same as his own.
A logical extension would be to include incomplete information about the utility function of
alter, reflecting in one way the uncertainty of ego when considering the move of alter.
Additionally, we could improve the theory by looking more closely at the effects of
evaluating probabilities nonlinearly on the properties of the equilibrium condition. In
measuring the utility functions we took care of this, but likewise, there are arguments that the
equilibrium condition will be influenced if we base our theory on Cumulative Prospect
Theory. For example, part of the derivation of the equilibrium condition involves calculating
(infinite) discounted sums. These discounting weights are typically probabilities, calling for
adaptations of the derivation that accurately reflect the nonlinear evaluation of probabilities.
64
Appendix 3.1
Chapter 3
Simulation of other actor's strategy in repeated PDs
The optimal social dilemma experiment would allow the experimenters to gather a maximum
number of responses per subject per time unit, requiring as little time of as few experimenters
as possible, without losing the reality of actual interaction. We use a way of letting subjects
interact that is not very common: they interact with a computer program that pretends to
behave like a human and we inform the subjects about this. In Experiment 1 a procedure was
used where subjects played repeated PDs against other players in the same room. This
procedure has several disadvantages. First of all, it is rather slow and quite demanding for the
experimenters. The experimenters - sometimes five in number per session - passed the
responses of subjects to their opponents. Only after all subjects had made their choice and
had received the choice of their opponent was a roulette wheel spun to determine the pairs
that could continue to the next round of the repeated PD. This procedure is quite timeconsuming and in fact a lot of the subjects’ time is spent waiting (until other subjects had
made their choice, or until the experimenters had delivered the choices of the opponents, or
until the roulette wheel was spun). This is not efficient and moreover, it tempts subjects to
start looking around or talking to others. We considered speeding up the procedure by using a
computer network connecting pairs of players, which also requires less effort on the part of
the experimenters during the experiment. However, the slower one of the two subjects still
determines the speed of the procedure, and subjects may not believe that they are actually
playing against another subject. Therefore, we chose to try a procedure that has the
advantages of being fast and not very demanding, and avoids the disadvantage that subjects
are not sure whether they are being tricked into believing that they are not interacting with
someone else. We told subjects that they were interacting with a computer program that
mimics the behavior of a person in previous experiments. This implies that subjects play at
their own pace, any number of subjects can play at the same time, and the experimenters'
efforts during the experiment are minimized. Naturally, it might be the case that "playing
against the computer" is not as realistic to the subjects as playing against a real opponent
would be.
A statistical model of actors' strategies was constructed and fitted to an actor's
behavior in Experiment 1. The model is a logistic regression model
P (C i ) =
1
1 + exp( − X i )
where P(Ci) denotes the probability that the actor cooperates in round i. Function Xi, denoting
the propensity of actor z to cooperate in round i, is dependent on four parameters:
 i −1

 i −1





X i = β 0 + β D  ∑ 2( j − i + 1)(1 − y jA)  + β C  ∑ 2( j − i + 1) y jA +
 j =1

 j =1


 i −1

 i −1

β M  β D  ∑ 2( j − i + 1)(1 − y jZ)  + β C  ∑ 2( j − i + 1) y jZ  




 j =1

 j =1


Variable YjZ denotes actor z's choice in round j, YjA denotes the other actor's choice in round j,
where y is 1 if the choice is cooperation and is 0 if the choice is defection. The sums in the
equation imply that the responses in round i-1-k are 2-k times as relevant for the propensity to
Effects of Risk Preferences in Repeated Social Dilemmas
65
cooperate than the responses in round i-1. Moreover, using a discount factor equal to 0.5
implies that the responses in the previous round are never less relevant than the whole history
of the game before the previous round. The parameters have the following interpretation:
β0
Propensity to cooperate in the first round of the repeated PD
βD
Effect of defection in the previous round on the propensity to cooperate in the
present round
βC
Effect of cooperation in the previous round on the propensity to cooperate in
the present round
βM
Relative importance of the opponent's choice in comparison to the own choice
in the previous round for the propensity to cooperate in the present round
The statistical model was fitted to the data obtained from Experiment 1. More precisely, the
observed frequencies of histories in round 1, round 2, round 3, and the proportions of mutual
defection or mutual cooperation up to round 5 were predicted by the logistic model. The
model was able to predict the 2 ('C' and 'D') + 4 ('DD', 'DC', 'CD', 'CC') + 8 ('DDD', 'DDC',
etc.) + 2 ('DDDDD' and 'CCCCC') proportions very accurately. Estimates of the parameters
β0, βD, βC, and βM were equal to –0.7369, -1.1, 1.3, and 0.85 respectively. These estimates
indicate that cooperation in the previous round had a larger effect on the present response
than defection, and that the subject’s own response had a larger effect on the present response
than the response of the opponent. The chi-square statistic comparing expected and observed
frequencies was equal to 3.24. It is difficult to devise a statistical test because the frequencies
are clearly not independent. However, even for 1 degree of freedom the obtained chi-square
value is not significant, which indeed indicates that our statistical model accurately describes
actor's behavior in the first experiment.
To use the statistical model in Experiment 2 with the fitted values of the four
parameters, the computer program generates random numbers in the unit interval [0,1]. The
program cooperates if the number is in interval [0,P], where the value of P is determined by
the history of the game up to the previous round, otherwise it defects.