Social decisions under risk. Evidence from the Probabilistic Dictator Game.
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Social decisions under risk.
Evidence from the Probabilistic Dictator
Game.∗
Michal Krawczyk† and Fabrice Le Lec‡
May 13, 2008
Abstract: This paper reports the results of a ’probabilistic dictator game’
experiment in which subjects had to allocate chances to win a prize between
themselves and a dummy player. Using a within-subject design we manipu-
lated two aspects of the decision: the relative values of the prizes–being equal
for both players, higher for the dictator or higher for the dummy–and the
nature of the lottery determining the earnings–independent draws for the two
players (‘noncompetitive’ condition) vs. a single draw (’competitive’ condi-
tion). We also asked for decisions in a standard, non-probabilistic, setting.
The main results can be summarized as follows: first, a substantial fraction
of subjects do share chances to win, also in the competitive treatments, thus
showing concern for the other player that cannot be explained by outcome-
based inequality aversion or quasi-maximin models. Second, this concern
hardly ever leads to equalizing expected payoffs. Third, subjects share less
in the probabilistic treatments than in the deterministic control treatment,
possibly because in the former even a generous choice cannot remove outcome
∗
We thank Gary Charness, Martin Kocher, Frans van Winden, and participants in
seminars and conferences in Warsaw, Zurich, Munich and Amsterdam. The financial sup-
port from the European Commission via the Marie Curie Research and Training Network
ENABLE is gratefully acknowledged.
†
CREED, Universiteit van Amsterdam. Roeterstraat, 11. 1018WB Amsterdam, The
Netherlands, m.w.krawczyk@uva.nl, tel +31 20 5 25 4383, fax + 31 205255283
‡
CREED, Universiteit van Amsterdam. Roeterstraat, 11. 1018WB Amsterdam, The
Netherlands, lelec@uva.nl, tel +31 205255651, fax + 31 205255283
1
inequalities. Fourth, subjects appear to be somewhat efficiency-oriented, as
they share more when partner’s prize is relatively high.
Keywords: social preference, other-regarding utility, risk attitude, inequal-
ity aversion, social concern, social preference under risk
JEL classification: A13, B49, C62, C65, D63.
1 Introduction
Many everyday situations involve choices between options with uncertain
consequences for the decision maker and some other individuals. A particu-
larly simple example is a decision whether or not to share chances to obtain
some reward. A student may help a less-able friend before an exam or an em-
ployee may–or may not–play fair against a competitor applying for the same
position in the firm. In both cases, the active agent may choose to sacrifice
some chances of success, passing an exam or being promoted respectively,
to the benefit of the other agent due to some moral or social considerations.
The subtle difference between these two examples is that while in the first
case equality of outcomes may be obtained, e.g. both students pass, in the
other one chances may be equal, but eventually only one of the candidates
will enjoy the promotion. Intuition and anecdotal evidence suggests that at
least some individuals may consider the notion of fair chances sufficiently
compelling to offer some chance to competitors even in the latter case.
In these examples, non-selfish behavior can hardly be explained by models
incorporating social concerns defined on outcomes only. This is particularly
evident in the second example, where the final allocation is necessarily ex-
tremely unequal anyhow. Several such models emerged in the past decades.
Being able to account for non-egoistic behavior observed in experimental
studies, they now enjoy substantial popularity. The main difference with
respect to standard assumptions on economic agents is that these models
assume that individuals’ preferences take into account others’ situations in
some way. One well known class of models is based on aversion to inequality
of payoffs (Fehr and Schmidt 1999, Bolton and Ockenfels 2000). The ba-
2
sic idea is that subjects experience disutility from having outcomes different
from others’ – being behind (”envy” in Fehr and Schmidt’s terminology) and
ahead (”guilt” or ”shame”). Using a different notion of social concerns, a
more recent model was developed by Charness and Rabin (2002) on the idea
of maximization of both total payoff (social utilitarianism) and of the payoff
of the worst off agent (maximin). Several other ways to model non-selfish
behaviors have been suggested. The question to know which model captures
subjects’ preferences over allocations more accurately is still open and un-
der intense scrutiny (Engelmann and Strobel 2004, Fehr, Naef, and Schmidt
2006, Bolton and Ockenfels. 2006, Engelmann and Strobel 2006).
The important point about these models is that they all restrict the defini-
tion of social concerns to payoffs or consequences, although intuition suggests
that fairness goes beyond equality of outcomes: equality of chances may mat-
ter at least as much as equality of outcomes. In other words, people may
want to share chances to win some prize fairly, in the same way they appear
to share actual outcomes fairly. Yet, the literature, both at the empirical
and theoretical level, has very little to say about how social concerns operate
in the presence of risk. This contrasts strongly with the overwhelming evi-
dence suggesting concerns for ’procedural fairness’ in various realms: many
institutions use random procedures to ensure fairness (Broome 1984); court
decisions provide some justification for such a practice, as in the well-known
case US vs. Holms1 ; the concept of equality of opportunity is widespread
across political and philosophical literature (Rawls 1971); and last but not
least, it is a common habit to draw lots when non-divisible goods or costs are
to be shared, as in the Machina’s mom example (Machina 1989). These cases
suggest that social concerns cannot be assessed solely by taking into account
consequences. Rather, their definition should consider the distribution of
risk, e.g. by comparing expected payoffs.
1
Crew members were prosecuted for excluding themselves when drawing lots to decide
who should leave an overcrowded and leaking lifeboat.
3Situations where such concerns may play a role are abundant in everyday
life: any resource that increases the likelihood to obtain some goods and is
shared between different individuals provides a relevant case for such social
preferences under risk. These include cases such as firms choosing not to
take unfair advantage when competing for a contract, traders in financial
markets freely exchanging information, fair-play in sports, sharing of organs
by living donors and many other. In the context of strategic interaction,
it may, among other things, help understand why some inefficient mixed-
strategy Nash equilibria emerge instead of pure efficient ones, e.g. in the
Battle of the Sexes (Cooper, DeJong, Forsythe, and Ross 1989). In public
economics, it may shed light on the reasons why redistribution policies enjoy
weaker support in countries where citizens believe to a greater extent in the
existence of equality of opportunity, e.g US vs Europe (Alesina and Angeletos
2005).
Still, except for a few path-breaking studies (Bolton, Brandts, and Ock-
enfels 2005, Brennan, Güth, Gonzalez, and Levati 2008, Güth, Levati, and
Ploner 2008, Karni, Salmon, and Sopher 2008), very little experimental evi-
dence is readily available on how risks and outcomes interact in the context
of social preferences. The purpose of this paper is to explore how subjects
behave in a simple distributive decision situation, where choices are likely to
be affected by attitudes toward risks as well as concerns for others. A partic-
ularly relevant issue given the current state of the literature on the subject
seems to be the extent to which choices can be explained by social concerns
on outcomes only. Put differently, the question tackled here is to know how
strong social concerns are in terms of probabilities: do subjects equalize ex-
pected payoffs or chances to win some prize, or are they merely inequality
averse with regards to consequences? When outcomes are necessarily asym-
metric, are subjects willing to use risk allocation to counterbalance payoff
inequality? Related issues include the importance of individual risk attitude
and whether the situation is ‘competitive’ or not, i.e. whether participants
compete for the same reward or if independent rewards are available.
4These questions are addressed through a series of simple decision tasks
where subjects are asked to share tokens which represent monetary units or
probabilities to win some prize. The results show that social concerns are
indeed affected by risk. More specifically, a substantial fraction of subjects
chooses non-selfishly when given an option to share chances to win mutually-
exclusive rewards. However, choices in this condition are somewhat less
generous than in case of sharing money, thus indicating that both aversion to
inequality of outcomes and aversion to inequality of opportunities operate,
the former being perhaps less potent. The data suggests to extend current
models (by taking into account chances to see some consequence occurring),
since all of them seem to wrongly predict subjects’ behavior in some of the
treatments at least. We also find little evidence of willingness to equalize
expected or actual payoffs. It rather seems that subjects think it fair to give
something to a less fortunate opponent. Eventually, our subjects seem to be
rather efficiency- than equality-oriented, as they share more when each token
is worth more to the partner than to themselves.
The remainder of the paper is organized as follows. The next section re-
calls the related existing experimental studies and their findings. Section
3 describes the design and procedures. Section 4 summarizes the predic-
tions of various models of other-regarding preferences. Section 5 exposes the
most important findings of our experiment and eventually the last section
concludes with some suggestions for possible further research.
2 Previous Experimental Studies
Despite its major implications, only a few studies have addressed experi-
mentally the issue of other-regarding concerns in the context of risk. These
include Bolton, Brandts, and Ockenfels (2005), Karni, Salmon, and Sopher
(2008) Brennan, Güth, Gonzalez, and Levati (2008) and Güth, Levati, and
Ploner (2008).2
2
We do not consider here some experiments that used lottery tickets as a convenient way
to reward subjects and hence have de facto tackled the issue of social preference under risk,
5The first paper, a path-breaking study in the field, provides evidence of
other-regarding preference in the presence of risk, as subjects played Ultima-
tum Game with chances to win some prize. The authors report that in such a
‘probabilistic’ Ultimatum Game second players tended to reject when offered
overly low chances to win. However, the interactive context makes it hard to
disentangle preferences on outcomes and procedures per se from reciprocity
and strategizing. Nevertheless, it suggests that subjects do consider chances
to win some amount as equivalent to some certain monetary rewards in some
contexts.
The second study directly tests Karni and Safra’s (2002) model and thus
addresses questions only tangential to our purpose. The authors chiefly focus
on whether an agent may want to give up (a bit of) her (expected) payoff
to achieve greater procedural fairness between two other agents: tradeoff
between maximizing own expected payoff and claiming only a fair chance
for oneself is essentially not addressed. Furthermore, they do not consider
a control treatment with divisible goods, and some of their results are only
based on hypothetical, i.e. non-incentivized, questions. Besides, we tend to
think it is instructive to start with the simplest, two-person case. Yet, Karni,
Salmon, and Sopher (2008) find that individuals are willing to sacrifice some
chances to win in order to restore equality of chances for the other two players.
The third and fourth studies are also closely related and seek to establish
whether subjects care about the amount of risk assumed by another individ-
ual. To this end, they elicit willingness to pay or willingness to accept of
social lotteries using the Becker-DeGroot-Marschak procedure. Their main
finding is that although some subjects care about the allocation of expected
payoffs, they do not seem to treat risk for themselves and others in the same
way. More specifically, they are willing to pay to reduce variance for their
own payoff but not for others’. Yet, as the authors point out themselves, the
yet had quite different goals and used methods questioned by experimental economics, e.g.
involved deception or extremely low stakes. The general picture seems to be that replacing
money with lottery tickets makes little difference, though this observation has to be treated
with great caution, as these studies did not offer suitable control treatments.
6preference elicitation method used may be cognitively demanding for sub-
jects,3 who may as a result have restricted attention to their own prospect.
In addition to this possible explanation, the design used by the two studies
asked the subjects to compare a monetary amount for the decision maker for
sure with risky prospects for another subject. It might be that the effect in
this case is too small to reach conventional significance levels, given that the
following two effects combine: on average certain outcomes are preferred to
risky ones and own payoffs matters more than others’.
As a conclusion, it seems that some evidence, albeit fragmentary, ex-
ists that subjects may take into account distribution of risks when socially-
oriented. Yet, none of these studies directly tackles the question how chances
to obtain some prize are dealt with when they are to be allocated in the most
basic case, namely among two subjects. They also offer little data on how
willingness to share differs between the probabilistic and the deterministic
context and how relative size of the pie matters. We think that these impor-
tant issues deserve further attention.
3 Design, procedures and methodological is-
sues
As the focus of the study is on social preferences under risk, we chose a
very simple design, that rules out phenomena such as reciprocity or signaling
of intentions. It thus involved one-time distributive decisions in a series of
(generalized) ”Dictator Games” (Bolton, Zwick, and Katok 1998), hereafter
DG. This method allows for comparison with the numerous experiments on
allocation tasks run with monetary amounts (Engelmann and Strobel 2007).
While the DG design has important weaknesses (Bardsley forthcoming), it
is natural and very simple to understand for the subjects.
3
The fact that they also tested in the same session how delays in receiving a prize were
assessed by socially-oriented people could also have rendered the whole setting even more
complicated for subjects.
7At the beginning of the experiment, each subject was assigned the role of
player A (’dictator’) or player B (’dummy’). This was fixed through the ex-
periment, to prevent subjects from justifying selfish behavior on the grounds
that the ‘victim’ will have a chance to be dictator herself in the following
round. For the same reason we decided not to use the strategy method, fa-
vored in many related experiments (Andreoni and Miller 2002, Güth, Levati,
and Ploner 2008), which may promote selfish behaviors with the procedu-
rally fair role allocation being taken as an excuse. For the same reason, it
was important that subjects do not perceive the allocation of roles as just
random, as this, again could affect the perceived ex ante fairness of the sit-
uation. But, relying on some criterion possibly interpreted as legitimate by
subjects (e.g. score at a previous task or a “first come, first served” pro-
cedure) could lead to some “entitlement effect” (Gächter and Riedl 2005).
Eventually, using some “obviously unfair” criterion (e.g. racial) would be
disputable from an ethical viewpoint and could lead to a strong emotional
reaction overshadowing whatever distributional preferences subjects could
have in the first place. Thus, role assignment was based on arrival time but
not in a monotonic way,4 and subject were not led to believe that the latter
was the case. In the post-experiment questionnaire many subjects indicated
that it would have been fair to allocate the roles randomly rather than ”just
like that”, suggesting that our manipulation worked well.
For all treatments, each dictator was matched to a new dummy and had
to distribute 10 tokens between the two of them. What each token repre-
sented exactly varied during the experiment on two dimensions. The ’prizes’
at stake, that is, the maximum amount that each player could earn in a
given round, and the lottery used to allocated these prizes, were systemati-
cally manipulated. The prizes, in euro, were (10; 30), (20; 20) or (30; 10) for
the dictator and dummy respectively. In other words, on top of the typical
symmetric case, we considered an asymmetric dictator-favoring distribution
of prizes and an asymmetric dummy-favoring one. This prize variation also
4
Precisely, odd-numbered subjects took the roles of dictators while even-numbered ones
played dummies.
8guaranteed that at least in some treatments, some choices could be expected
to be more socially oriented than in the case of the symmetric prizes. For
instance, for subjects being concerned about efficiency as defined by Char-
ness and Rabin, prizes (10; 30) should yield additional deviations from the
selfish choice. This allows to have a certain level of socially-oriented deci-
sions, which are the only ones plausibly affected by the probabilistic payment
scheme treatments. This is achieved without departing from the standard
DG structure. Indeed, other possibilities to reduce the fraction of totally
selfish choices have been evoked, such as having specific recipients instead
of other subjects, e.g. charity organizations, or to induce some “entitlement
effect”, for instance by setting that the pie to be divided could depend on the
effort of both players in a previous task. Although such methods are indeed
known to raise the level of amounts given to the dummy player, they may
have non-controlled influences on the effect that is primarily tested here, that
is, how subjects take risks into account within social concerns.
The second, more fundamental, dimension is the type of the game (here-
after game type). In the Deterministic condition, each token simply corre-
sponds to one tenth of the prize, such that keeping all the tokens means
winning the full prize. For example, if prizes are (30; 10), keeping 7 tokens
and passing 3 to the other player results in earnings for the dictator and
dummy being equal to 21 and 3 euro respectively.5 In the other two condi-
tions, tokens represent a probabilistic analogon of a fraction of the prize: in
the Competitive condition each token stands for a 10% chance of winning the
prize, with the events of winning for both players being mutually exclusive.
For example, with prizes (30; 10) and the chosen distribution of tokens (7; 3)
as above, a 10-sided die is rolled: if the outcome is in {1, 2, ..., 7}, A wins 30
euro and B nothing; otherwise B wins 10 euro and A nothing. In the Non-
Competitve condition each token still represents a 10% chance of winning
own prize, yet A’s and B’s chances are independent. With numbers given
above, two dice are rolled, one giving A a 70% chance of winning his prize
and another giving B hers with a 30% chance.
5
It is very easy to note that our symmetric Deterministic treatment is the familiar DG.
9The combination of both dimensions generate nine treatments, which are
summarized in Table 1.
(10;30) (20;20) (30;10)
Certain Outcome Deterministic-10 Deterministic-20 Deterministic-30
Non-Competitive Lottery NonCompetitive-10 NonCompetitive-20 NonCompetitive-30
Competitive Lottery Competitive-10 Competitive-20 Competitive-30
Table 1: The nine treatments
In order to correlate behavior in different situations and increase statistical
power for treatment comparisons, we implemented a within-subject design
in the same spirit as Andreoni and Miller (2002): our design can be seen as
an adaptation of theirs with two probabilistic conditions–Competitive and
NonCompetitive–added. Every subject played all nine games, grouped in
three blocks: Deterministic, Competitive and NonCompetitive. Relevant
instructions were distributed at the beginning of each block, followed by
control questions.6 This organization of the experiment was implemented
to ensure that subjects were not confused about specific payment schemes.
Every erroneous answer to a control question was recorded and had to be
corrected by the participant in question in order to proceed. The order of the
blocks was different in different sessions and the order of particular rounds,
i.e. with different prizes within each block, was randomly manipulated at
individual level. While dictators were busy with their decisions, the dummies
were asked completely analogous, yet hypothetical, questions. A similar
procedure was used by Karni, Salmon, and Sopher (2008): it provides some
additional, yet un-incentivized, data at no additional cost and it renders
unlikely the identification of types by subjects during the experiment. At
the end, it was announced which of the nine rounds would be played for real
and the relevant decisions of the dictators were implemented, with one or
two 10-sided dice being rolled publicly whenever appropriate.
6
Subjects knew that the meaning of tokens would change between blocks and could ask
for instructions in advance but never actually did.
10After this part, a series of three individual decisions under risk was run, in
order to establish individual attitudes towards risk. We used the interactive
Multiple Price List design (Andersen, Harrsion, Lau, and Rutstrom 2006),
to elicit certainty equivalent for a 10, 50 and 90 percent chance to get 20
euro, the average stake level considered in the main task. One third of
the subjects were randomly picked for payment in this second part of the
experiment, and if chosen, subjects were only paid for one of the tasks,
once again randomly determined. Subjects were of course explicitly told so,
and this particular payment scheme was implemented in order to make it
comparable with the decisions made in the first stage, where only one out
of nine decisions was actually paid. Our conjecture was that displayed risk
attitude may correlate with decisions in the first stage of the experiment. In
particular, subjects evaluating a 90 percent chance for 20 euro at much less
than 20 euro (certainty effect) can be expected to be more likely to keep all
the tokens in the probabilistic treatments.
Two short questionnaires were distributed during the experiment. Directly
after the first part subjects were asked to rate their satisfaction from own DG
decisions and obtained outcomes and then to describe what they thought a
typical subject did and should have done from the viewpoint of fairness. At
the end of the experiment demographic data was collected and subjects were
paid their earnings in private.
The experiment was run in October and December 2007 in the laboratory
of the Center for Research in Experimental Economics and Political Deci-
sion Making (CREED) in Amsterdam. It was programmed using the z-Tree
software (Fischbacher 2007). In total 128 subjects, mostly undergraduate
students at the University of Amsterdam, participated. Average earnings
for an experiment lasting from 70 to 100 minutes equaled about 22 euro,
including a 7.50 euro show-up fee.
114 Theoretical predictions
The standard prediction, once the assumption that subjects are selfish is
made, is that all players should keep all the tokens to themselves, regardless
of the relative prize, the type of the game or even their own attitude to-
wards risk. Concerning other-regarding preferences, few models available in
the literature, with a notable exception of Trautmann (2006), seem immedi-
ately suitable for generating predictions for our experiment. To use current
models of social preferences, one needs to make the natural assumption that
preferences are represented by the product of the social utility on outcomes
and the corresponding probabilities. Charness and Rabin (2002) for instance
implicitly endorse this view in their definition of a social-welfare equilibrium
and the corresponding mixed strategies, p. 853.7 This apparently innocuous
and natural generalization leads to some clear-cut and somewhat counter-
intuitive predictions.
4.1 Inequality aversion models
As a generalization of inequality aversion models, we consider a model
defined in Appendix, à la Fehr and Schmidt (1999) or Bolton and Ockenfels
(2000). In a two-person game, it generalizes both models since the measure of
inequality is equivalent in both models when only two agents are concerned.
At least when compared to the Fehr and Schmidt’s linear model it offers some
extra flexibility to account for the data. This procedure has also the critical
advantage that predictions do not crucially depend on parameters. We will
refer to this model as the ”generalized inequality aversion’” model. As it
is well known, such inequality aversion models predict for the Deterministic
case that choices are between a payoff-equalizing division and a selfish one:
the admissible range for the number of tokens given is thus [0,8] for the
dictator-favoring asymmetric case, [0,5] for the symmetric game and [0,3] for
the dummy-favoring. In the general case, it is not possible to say more, the
results depending on the shape of the utility component based on inequality
7
They also defend this point of view in a former working paper version of their publi-
cation, see footnote 46.
12aversion.
In the case of a symmetric probabilistic DG, some interesting features
emerge. First, when the lottery is competitive, it is straightforward that
the dictator should keep all the tokens. Indeed, total inequality will re-
sult no matter who ends up winning the money. Therefore, even if there
is no self-serving bias in the evaluation of fairness, self-interest will make
agents maximize the probability that they win rather than the opponent.
This is also independent from any possible non-linear probability weighing.
The same is obviously true for the dummy-favorable case, Competitive-10.
However, while by and large the same is true for the dictator-favorable treat-
ment Competitive-30, very strong inequality aversion may lead to non-selfish
choices. It is possible that u(30, 0) < u(0, 10) due to inequality aversion and
hence it may be the case that such subjects pass all the tokens to the other
player. Of course, this also implies that some (very few) subjects can be
exactly indifferent between (30, 0) and (0, 10), thus being free to pass any
number of tokens. In the light of previous studies, such cases should be at
best infrequent.
For the case of two independent gambles–the NonCompetitive condition–
the situation is less straightforward. In the symmetric case, the possibility
to end up in the worst possible outcome, unfair and disadvantageous, (0; 20)
may be partly counterbalanced by the possibility to get the preferred out-
come (20; 20). It can hence be the case that the dictator does not keep all the
tokens. Technically, if the utility function is normalized so that u(0; 0) = 0,
the requirement is that u(20; 20) > 2u(20; 0)8. It means that the subject
must be inequality averse, and that this inequality aversion should be strong
enough to imply that in the Deterministic-20 treatment the decision-maker
has not kept all the tokens (see Appendix for details). This also implies that
no more than half the tokens are given. Regarding situations with asymmet-
ric prizes, the prediction is similar. For instance, the range of possibilities
8
See Appendix for details. It is also assumed that u(20; 0) > u(0; 0) which is not as
such implied by the model.
13for the (10, 30) case is equivalent to the one for (20; 20). In the (30; 10) case,
the number of tokens given is positive if u(30, 10) > 2u(30, 0), which implies
an even greater inequality aversion than in the previous case.
4.2 Quasi-maximin models
Previous studies as Andreoni and Miller (2002) and Charness and Rabin
(2002) tend to show that subjects exhibit a strong heterogeneity concern-
ing social preferences/social concerns, and that utilitarian motives have to
be taken into account. Charness and Rabin have proposed a simple model9
based on the idea that the apparent inequality aversion was in fact the will-
ingness to maximize the minimal payoff, i.e. the payoff of the worst-off
individual. The model also includes a total welfare term in the equation
(see Appendix for details). Here again, the model is linear, but a natural
generalization is possible. It is straightforward that in the standard DG,
Deterministic-20, the fraction of the number of tokens given should lie in
[0, 21 ]10 . In the dummy-favorable case, any allocation of tokens can be cho-
sen; a subject only interested in maximizing the total payoff will give away
all the tokens, and a purely selfish will keep them all. For Deterministic-30,
the appropriate range for the number of tokens given is [0, 43 ]: 34 corresponds
to the maximization of the minimal payoff, and 0 to either the selfish or the
utilitarian motive.
When the model is extended (on the basis of expected utility) to risky
outcomes under Competitive condition, it predicts, not surprisingly, that
the selfish choice results in (almost) all cases: unless social concerns are
implausibly important, the decision maker prefers to keep all the money to
giving all away. In the case of the NonCompetitive treatments, things are
less clear-cut. For the symmetric prize, there is a range of parameters for
which the number of tokens given can be positive: as in the case of the
9
For obvious reasons we disregard here the reciprocity part of their model.
10
It is yet possible that some agents are only concerned by total welfare and are hence
completely indifferent between all cases from passing 10 tokens to keeping all of them. To
the best of our knowledge, no such experimental behaviors has ever been observed.
14inequality-based model, it also requires that u(20; 20) > 2u(20; 0). This
corresponds to very strong social concerns: for example it is never the case
in Charness and Rabin’s specific linear model. The requirements in the
(30, 10) case are even more demanding. But, it is possible that the number
of tokens given in NonCompetitive-10 reaches 10, for instance when subjects
are mainly concerned by total payoff. Yet, it seems fair to say that although
not impossible, non-selfih choices should not be frequent here.
4.3 Probabilistic inequality aversion models
The model by Trautmann (2006) is the most tractable model taking into ac-
count risk and social preferences. Being an expected-payoff generalization of
Fehr and Schmidt’s model, it makes identical prediction for all game types:
it proposes that subjects either keep all the tokens or equalize expected pay-
offs. Again, a simple generalization to non linear functional forms allows
for intermediate choices as well, identical to the predictions of the inequality
aversion model for the Deterministic case. Furthermore, it predicts that at
the individual level, subjects’ choices should not be affected by game type.
4.4 Some extensions and caveats
Using generalized models of expected utility instead of mere expected util-
ity, for instance by applying a probability transformation function to proba-
bilities, does not change many of the predictions, at least in the Competitive
case. In particular, corner solutions (typically meaning holding all the to-
kens) still hold. In other words, most of these predictions are not based on
the assumption of a linear treatment of probabilities. They depend much
more critically on the fact that probability distributions are not taken into
account in the terms summarizing social concerns, be it inequality aversion
or quasi-maximin.
Likewise, most of these predictions do not rely on specific parameter or
functional forms of the models–and that may explain their vagueness in some
cases. Predictions in the case of inequality aversion are based on the mere
15fact that
u(20; 20) > u(20; 0) > u(0; 0) > u(0; 20)
and in the case of quasi-maximin on the following ranking
u(20; 20) > u(20; 0) > u(0; 20) > u(0; 0)
This simply means that, if the models appear to predict wrongly in some
of our treatments, it cannot be argued that it is because of a specific form
of the model chosen. In particular, no generalization of the corresponding
models, as long as social concerns depend only on consequence, are about to
solve the problem.
For a handy reference, we summarize all these predictions in Table 2,
with fractions of tokens given (figures in parentheses represent possible yet
implausible choice ranges).
Treatment/Predictions Inequality-Based Generalized Trautmann Quasi-Maximin ‘Selfish’
Deterministic-10 [0, 14 ] [0, 14 ] [0, 1] 0
Deterministic-20 [0, 12 ] [0, 12 ] 1
[0, 2 ]([0, 1]) 0
Deterministic-30 [0, 34 ] [0, 34 ] [0, 43 ] 0
NonCompetitive-10 [0, 21 ] [0, 14 ] [0, 1] 0
NonCompetitive-20 [0, 21 ] [0, 12 ] [0, 21 ] 0
NonCompetitive-30 [0, 1] [0, 43 ] [0, 1] 0
Competitive-10 0 [0, 41 ] 0 ([0, 1],1) 0
Competitive-20 0 [0, 21 ] 0 ([0, 1]) 0
Competitive-30 0 (1, [0, 1]) [0, 43 ] 0 0
Table 2: Predicted fraction of tokens given
5 Results
The results of the experiment suggest that social concerns are not only
assessed on consequences and outcomes, but take into account the allocation
of risk ex ante. But, we also find challenges for the probabilistic inequality
16aversion approach, especially since subjects typically do not try to equalize
expected payoffs.
5.1 Overview
To begin our analysis we first note that, while subjects were allowed to
distribute less than 10 tokens, they virtually never chose to do so: this hap-
pened in just 5 out of 1152 possible cases. Therefore it makes no difference
whether one focuses on the number of tokens kept by the decision maker or
the number tokens passed to the partner, be it in the real or hypothetical
case. We choose the latter to report treatment effects more clearly. Fig-
ure 1 gives an overview of the distribution of the number of tokens given in
the nine treatments for decision-makers and Figure 2 of analogous decisions
made hypothetically by the dummies. Tables 3 and 4 report means of this
variable in particular treatments.
Deterministic, 10 Deterministic, 20 Deterministic, 30
0 .2 .4 .6 .8
NonComp., 10 NonComp., 20 NonComp., 30
0 .2 .4 .6 .8
Density
Competitive, 10 Competitive, 20 Competitive, 30
0 .2 .4 .6 .8
0 5 10 0 5 10 0 5 10
Number of tokens given
Graphs by game_type and prize
Figure 1: Frequencies of dictators’ choices in all treatments
17Several observations can readily be made. Starting with decisions made
’for real’ we note that, first, a large fraction of choices in each treatment
entails keeping all the tokens. Actually, giving nothing is always the median
choice and may be chosen by as much as three-quarters of individuals, as in
treatments Competitive-30 and Noncompetitive-30. Overall, mean number
of tokens given is 1.13. However, non-selfish choices cannot be regarded
simply as ’mistakes’. First, there are too many of them. Second, the most
typical mistake to be expected seems to be keeping 0 tokens and giving
10 away, which happened utmost rarely and mostly when own prize was
10, where it could be explained by some concern for efficiency. Moreover,
checking for any link between errors in the control questions and behaviors
in the allocation decisions reveals no correlation between both at conventional
levels: the number of errors made in answering control questions for every
block (Deterministic, Noncompetitive and Competitive) is not significantly
linked to the numbers of tokens kept, and this is true for all prize treatments.
Third, another kind of mistake, distributing less than 10 tokens in total, was
extremely rare, as mentioned before.
Game type / Prizes 10 20 30
Deterministic 1.81 1.33 0.83
NonCompetitive 1.52 1.19 0.52
Competitive 1.50 0.83 0.64
Table 3: Average number of tokens passed by the dummies
On a general note, the non-selfish choices are distributed quite evenly be-
tween 1 and 5. Interestingly, the classic DG case (Deterministic-20) seems to
be the only one in which sharing tokens equally (5-5) appears to be somewhat
prominent as in previous studies, though it was actually chosen by only 11
% of subjects. Moreover, it is hard to see a strong tendency to equalize earn-
ings or expected earnings in asymmetric treatments: apart from the standard
DG with 20 euro, there is only a (limited) peak at the equalizing choice in
Deterministic-10.
18Deterministic, 10 Deterministic, 20 Deterministic, 30
.5
0
NonComp., 10 NonComp., 20 NonComp., 30
.5
Density
0
Competitive, 10 Competitive, 20 Competitive, 30
.5
0
0 5 10 0 5 10 0 5 10
Number of tokens given
Graphs by game_type and prize
Figure 2: Frequencies of dummies’ hypothetical choices in all treatments
Hypothetical choices made by dummies show quite a different picture:
their allocation decisions were much less selfish. While 10 is still the mode in
every treatment, it is never median. Overall mean number of tokens passed
is up 1.3 from the dictators’ case, that is 2.5 instead of 1.2. Dummies are
less selfish under all conditions and these differences are significant at 1%
level in a Wilcoxon rank-sum test in seven out of nine treatments. This
gives support to the idea that equalization of payoffs is viewed as the socially
or morally desirable outcome when its achievement does not compete with
strict personal interest or total payoff maximization. Interestingly, passing
exactly five tokens appears to be more prominent when choices are made
hypothetically, especially in the deterministic case. Also in Deterministic-10,
a substantial fraction of subjects (22%) made a choice (nearly) resulting in
equalization of payoffs (thus gave 2 or 3 tokens).
19Game type / Prizes 10 20 30
Deterministic 2.55 2.87 2.72
NonCompetitive 2.91 2.33 1.89
Competitive 2.86 2.52 1.70
Table 4: Average number of tokens passed by the dummies
Further insight into individual behaviors may be gained by categorizing
them into classes. A k-means clustering procedure based on the all nine
choices tells us that there are essentially two distinct categories of dictators:
42 of them were nearly perfectly selfish–they almost always chose to keep
all the tokens–and 22 were not. The fraction of ’egoistic’ decision makers
is a bit higher than is typically observed, around two thirds rather than
one half, which could be due to our sample, consisting chiefly of economics
students with a lot of lab experience. A Wilcoxon test comparing the two
clustered populations in terms of lab experience (reported number of previous
experiments) is thereby highly significant (p < .01).
In order to allow comparison with previous studies, a prosocial/’selfish’
split based on the choices in the Deterministic block only might be more rel-
evant. One way to do it is to consider as prosocial any dictator that gave at
least one token across the three Deterministic treatments, i.e Deterministic-
10, Deterministic-20, Deterministic-30, while the rest were seen as self-
interested. Of course, we expect that the selfish subjects will, most of the
time, keep all the tokens in other treatments as well. Indeed, 21 out of 22 of
the non-selfish identified using the k-means method gave at least one token
in one of the Deterministic treatments. One advantage of this rather loose
classification criterion is that it gives a more balanced split: 35 prosocial
dictators and 29 selfish ones.
This classification accentuates the differences between treatments, which
in Table 3 appear low, largely because a majority of subjects consistently
behaved in a selfish way. If we consider only prosocial subjects, the variation
seems much more substantive as shown in Table 5 and discussed in the next
20subsection.
Game type / Prizes 10 20 30
Deterministic 3.31 2.43 1.52
NonCompetitive 2.55 1.97 .86
Competitive 2.29 1.34 1.12
Table 5: Pro-Social Dictators’ means
5.2 Treatment effects
Concerning treatment effects, a glance at tables 3 and 4 is sufficient to
see that the higher the own prize, the more selfish the choices. This is true
for the three game types. Using a Wilcoxon signed-rank test on the sums
of all given tokens to the partner across payment schemes for different prize
treatments allows quite robust conclusions: the hypothesis that this sum
is equal for different prizes is strongly rejected, for (10, 30) vs (30, 10) and
(20, 20) vs (30, 10) with p < .001, and with p < .01 for (10, 30) vs (20, 20).
The effects are quite important: although the average total of tokens given
in three treatments with a particular prize range from 1.98 for (30, 10) to
4.83 for (10, 30), equaling 3.34 for (20, 20), it is clear that subjects are not
affected equally depending on their types. Restricting attention to prosocial
decision-maker shows the number of given tokens more than double as own
prize goes down from 30 to 10. Table 6 summarizes this effect. It is worth
Prize Type 10 20 30
Self-interested Dictators .83 .45 .18
Prosocial Dictators 8.15 5.75 3.49
All Dictators 4.83 3.34 1.98
Table 6: Total number of tokens given across groups of three treatments with
identical prize
noting that there also seems to exist a very small effect on selfish dictators
too, although not significant at any level, which could be interpreted in terms
21of the idea that the higher the stakes the less ’errors’ subjects make (Camerer
and Hogarth 1999).
These findings are roughly confirmed when focusing on the Determin-
istic DG only: the number of tokens kept is significantly different for
Deterministic-10 vs Deterministic-30 and Deterministic-20 vs Deterministic-
30 (p < .01 in a Wilcoxon signed ranked test), although the difference be-
tween Deterministic-10 and Deterministic-20 is only significant at the 10 %
level (p = .06).
On a general note, the effects of prizes in this experiment are equiva-
lent to what is observed by Andreoni and Miller in their seminal study. In
Deterministic-10, 13 out of 64 dictators did choose to give more tokens to
the dummy than to themselves, suggesting a concern for efficiency. This
is similar to the distribution of Andreoni and Miller’s ”perfect substitutes”
type, which composes roughly 20 % of subjects in their experiments. Equal-
ization of earnings in this case seemed to matter to 14 subjects who chose
to keep 7 or 8 tokens, approximately 22 %, once again to be compared with
roughly 30 % of Leontieff subjects, as defined by Andreoni and Miller. The
difference can be explained by the seven subjects who chose either 5 or 6 and
represented 10 % of the dictators.
In contrast, in Deterministic-30, only one subject gave more than 5 tokens
to her counterpart, another one gave 5 tokens, and 52 out of 64 dictators kept
9 or 10 tokens for themselves. The picture is similar for the Deterministic-20
treatment where only one dictator gave more than 5 tokens and where the
equal split was chosen only by 7 dictators. Similar patterns, perhaps miti-
gated, seem to characterize the effect of prize in other game type treatments.
As a conclusion on the effect of prize, our results mimic Andreoni and
Miller’s findings, with a mild skew of the distribution towards more selfish
behaviors. In general, the data give little support to a tendency to equal-
ize payoffs (or expected payoffs) by dictators, and show a certain degree of
22efficiency-driven social concerns.
Comparing results across game types, it appears that decisions are less
selfish in the Deterministic case than in the Competitive one. Summing all
tokens given by dictators when the payment scheme is held constant and com-
paring them in the Deterministic, NonCompetitive and Competitive cases,
reveals some differences: according to a Wilcoxon signed-rank test, the to-
tal numbers of tokens kept in Deterministic and Competitive are different
(p = .03); Deterministic and NonCompetitive are weakly different (p < .10),
and no significant difference at all can be found between Competitive and
NonCompetitive.11 When restricting attention to prosocial subjects as cate-
gorized above, these effects are stronger: the difference between Deterministic
and Competitive treatments is highly significant (p = .0016), and the p-value
when comparing NonCompetitive and Determinist is equal to .012. Yet, here
again, Competitive and NonCompetitive do not yield any significantly dif-
ferent results. The effect itself is less strong than the prize effect reported in
the previous paragraph, the total number of tokens given ranging from 2.97
to 3.97, but once again the whole effect seem to chiefly concern ’prosocial’
dictators (Table 7). It is also worth noting that choices made by prosocial
dictators in the Competitive treament are quite far from the extreme pre-
11
This result holds when controlling for block order. Given the odd number of sessions,
there is a indeed a slight asymmetry of block orders: the Competitive treatment is less
often in the first or second positions than the other ones are. In the next subsection,
some block order effect is reported in the regression analysis. So it could be that the fact
subjects keep more tokens in the Competitive treatment is explained by some block-order
effect referred to below. Indeed, subjects tend to give less in later periods, hence since
Competitive treatments are more often at the end of the experiments, it might be that
the whole difference with other treatments is due to block order effect. It seems that
it is not the case. To control for that we ran the same tests as referred to before, but
restricting data points to groups of three sessions in a way such that block orders are
completely equivalent for all three treatments, namely sessions 1,2 and 3, and sessions 3,4,
and 5. The results are similar: for prosocial dictators, the total number of tokens kept
in Deterministic vs. Competitive is significantly different (p = .03 for sessions 1,2 and 3,
and p = .06 for sessions 3,4 and 5). Comparing Deterministic and NonCompetitive along
these lines gives a very weakly significant difference (p = .12 for 1,2 and 3, and p = .06
for 3, 4 and 5). Likewise, the difference between Competitive and NonCompetitve is not
significant at any conventional level even when controlling for block orders. Finally, the
fact that our findings are not driven by the order effect is also apparent from the regression
analysis reported in the next subsection.
23Game type Deterministic NonCompetitive Competitive
Selfish Dictators null 0.62 0.83
Prosocial Dictators Only 7.26 5.37 4.75
All Dictators 3.97 3.22 2.97
Table 7: Total number of tokens given across blocks (groups of three treat-
ments with identical game type)
dictions of the outcome-based models, i.e. that they should keep all the
tokens.
Such effects are less clear cut when focusing on the different prizes sepa-
rately. The control prize of (20; 20) roughly confirms some of these results:
dictators’ behaviors seem to be different in on the one hand the Determinis-
tic and NonCompetitive treatment and on the other hand the Competitive
treatment. A Wilcoxon signed rank test gives p = .02 and p = .05 respec-
tively. The significance of these effects is confirmed when restricting the test
to the prosocial dictators with p < .001 and p = .05. This suggests that for
symmetrical prizes, the Deterministic treatment and the NonCompetitive
treatments are different from the Competitive one. This is not incompatible
with the overall results that reports above a significant difference between
NonCompetitive and Deterministic: it might be linked to the fact that the
effect of game type is not constant across prize treatments.
But indeed, focusing on the other prize treatments reveals a slightly
different effect: there is no significant pair-wise difference according to
the Wilcoxon signed-rank test used previously between Deterministic-10,
NonCompetitive-10 and Competitive-10, although the mean values of to-
kens kept seem to increase with ’risk’. This seems to make sense given that
a significant share of the prosocial dictators seems partly driven by efficiency
motives. For those subjects, the riskier the game type is, the less selfishly
some might behave assuming for instance that they prefer (30, 0) to (0, 10).
Yet, zooming into the data does not reveal such a pattern: on the contrary,
prosocial dictators seem to behave significantly differently in Deterministic10
24vs NonCompetitive10 (p = .06) and in Deterministic10 vs Competitive10
(p = .03), but not in Competitive10 vs NonCompetitive10. This strengthens
the results expressed above that in general Deterministic is different from
Competitive and NonCompetitive.
Likewise, there is no significant difference between Deterministic-30 and
Competitive-30, nor between NonCompetitive-30 and Competitive-30, al-
tough there is a significant (p = .04) difference between Deterministic-30 and
NonCompetitive-30. It might be the case that the results in NonCompetitive-
30 are rather peculiar, and do not fit well with the whole frame: although
not always significant, an overall comparisons of all treatments for dictators
exhibit a monotonic effect of prize and risk (see table 3). It may also be
that the average behavior is so close to all tokens kept that no significant
difference can be shown with this sample size. Yet, focusing on prosocial
dictators leads to the same observations as in the previous paragraphs: for
the (30; 10) prize, Deterministic is different from Competitive (p = .08) and
from NonCompetitive (p = .01), and Competitive and NonCompetitive are
at best weakly different, if not identical.
It is also interesting to note that, while being shifted towards more gener-
ous choices as explained before dummies’ choices exhibit essentially identical
patterns of treatment effects. As can be seen in Table 4, choices become
more selfish when own relative prize is high (except for the deterministic
condition) and when the game type involves more risk.
5.3 Controls and Demographics
Another way to look at the data is to use a multivariate regression. Ta-
ble 8 reports a tobit analysis (left-censored at 0 and clustered by subjects)
of the number of tokens given on a number of covariates, based on the ex-
perimental design and subjects demographics. All demographic variables are
self-reported. This analysis, restricted to dictators, is consistent with findings
put forward before: subjects are more selfish in the probabilistic treatments
25(especially in Competitive) than in the baseline deterministic condition. It is
worth noting that Competitive is different from Deterministic at a high level
of significance (the corresponding p−value is actually .003), but NonCom-
petitive is at best very weakly different (p = .17), reinforcing the already
formulated suggestion of mixed effects of this game type. Own prize has a
strong negative impact on the number of tokens given.
’Given tokens’ Coefficient [95% Conf. Interval]
competitive -1.26*** (.43) [−2.10; −.42]
non competitive -.48 (.35) [−1.16; .21]
prize -.11*** (.03) [−.16; −.06]
male -1.21* (.68) [−2.55; .12]
dutch .91 (.70) [−.47; 2.28]
religious -2.15*** (.81) [−3.75; −.55]
economist -2.52*** (.81) [−4.10; −.94]
siblings .063 (.20) [−.33; .46]
education mother -.04 (.12) [−.29; .20]
education father -.34** (.17) [−.68; .01]
income parents/1000 .11*** (.02) [0.08; .15]
own income/1000 .14 (.87) [−1.16; 1.85]
experiments -.50*** (.07) [−.64; −.36]
errors -.19 (.24) [−.67; 28]
example 1 -.63 (.94) [−2.47; 1.21]
example 3 -1.12 (.95) [−2.98; .74]
period in block -.18 (.17) [−.51; .14]
block -.77*** (.22) [−1.20; −.33]
Standard error in parenthesis. *= significant at the .1 level, **= significant at the .05
level, ***= significant at the .01 level.
Table 8: Tobit Analysis
New insights concern the impact of other variables: subjects appear to
be somewhat more generous at the beginning than later on (block order
matters). Yet, the general findings seem robust enough to still hold when
controlling for this (see footnote 10). It is also reassuring to see that other
design-based variables have no significant influence on dictators’ behaviors:
the order within block (”period in block”) as well as errors made in the
26control questions in the corresponding block (”errors”) do not seem to mat-
ter, and the different examples given in the instruction, which were more or
less equality-based, appear not to affect the results (”example1” and ”exam-
ple3”).
Some demographic variables are likely to play a role, and most of these
effects are in line with previous studies: economics students seem to be more
selfish (Carter and Irons 1991), male tend to give less (Eckel and Grossman
1998). Lab experience seems to lead subjects to more selfish behaviors, as
reported before. Some new findings may also be of importance: religious
subjects appear to be less pro-social, and ’social status’ also plays a role,
although apparently an ambiguous one: the number of given tokens increases
with parents’ income12 but decrease with father’s education.
5.4 Risk attitude and satisfaction
One interesting issue is to know whether the differences between the num-
ber of tokens given in different treatments can be explained by risk attitude
in general. The second part of the experiment allows us to do so. First, we
note that our average results are in agreement with previous literature: the
certainty equivalent of a 10 percent chance is higher than expected value,
while it is lower for probabilities 50 percent and 90 percent, consistent with
the inverted-S probability weighting function. This can be seen in the box
plots (Figure 3).
One of the most immediate ways to test the effect of risk attitude on be-
haviors in our probabilistic DG, is to check whether certainty equivalent for
a .50 probability of earning 20 euros is related to changes in the number of
given tokens between the deterministic treatment and the probabilistic ones,
typically the Competitive one. Indeed, the lower the certainty equivalent,
12
It is worth noting that in the case of parents’ income some outliers, namely the reported
values that were equal to 50000 euro a month, which was the upper boundary, were
transformed into missing values. Keeping them as reported by subject reinforces the
significance of the effect.
2720
15
10
5
0
equiv10 equiv50
equiv90
Figure 3: Certainty Equivalents
the stronger risk aversion and therefore the more costly it is to give away to-
kens to other subjects. The median certainty equivalent for the intermediate
gamble (‘equiv50’) is 8; it is natural to classify subjects into two categories,
depending on whether their certainty equivalent is lower or greater than
the median. A Wilcoxon rank-sum test on the difference of tokens given in
Deterministic-20 and Competitive-20 by these two categories does not give
any significant difference, even when restricting the test to dictators. When
we take into account the effect of income (the earnings from the first part
may have influenced the assessment of certainty equivalent in the second
one), some difference may exist (p = .10), and when restricting to prosocial
dictators, the effect is highly significant (p = .01): pro-social dictators cat-
egorized as more risk-averse were relatively less generous in Competitive-20
than in Deterministic-20.
28Unfortunately, this difference may not be very robust: categorizing sub-
jects into two classes with a threshold of 10 rather than 8–which would be
consistent with the idea that the utility for money being concave or con-
vex explains attitude towards risk–does not bring any significant result, even
when controlling for roles and previous earnings. Yet, we obtain that the dif-
ference between Deterministic-20 and NonCompetitive-20 is significant when
restricting to prosocial dictators (p = .04) when risk-seeking is defined with
a threshold of 10. When the prize is 10 or 30 for the dictators, these tests
do show at best some weak significance (p = .11 for Deterministic-10 vs.
Competitive-10) and in almost all cases no significance. Testing overall de-
cisions by blocks suggest an effect of risk attitude, of similar magnitude for
Competitive and NonCompetitive treatments vs. Deterministic. A Wilcoxon
test on the difference in the total number of tokens kept by risk attitude is
weakly significant (p = .07 and p = .06 resp.) when restricting attention
to prosocial dictators with some previous positive earnings. Albeit mixed
evidence, these results suggest some effect of risk attitude.
But studying this relation more detail reinforces this view. Especially, in a
generalized expected utility context, what may chiefly matter is the shape of
the probability weighing function in the range [.5, 1], corresponding to 5 to
10 tokens kept in the probabilistic treatments. This can be investigated by
assuming that for the range of monetary amounts considered here (roughly
between 8 and 16 euro on average) utility functions are linear whereas the
(cumulative) probability weighting function (PWF) is linear on the interval
(.5,.9), consistent with findings in the literature, indicating that the slope of
the PWF does not change much, except in the proximity of 0 and 1. With
these assumptions we can estimate the slope of the probability weighting
function between .5 and .9 and then using a dummy variable to classify sub-
jects in two groups on the basis of this value, we obtain a weakly significant
effect of this risk attitude in this range [.5, .9] for dictators who have given
between 5 and 8 included in Deterministic-20: a Wilcoxon rank-sum test
gives a p−value of .08. The low slope group gives on average 2 tokens in
Competitive-20, to be compared with 3.31 in Deterministic-20, whereas the
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