Inequality Aversion, Efficiency, and Maximin Preferences in
By DIRK ENGELMANN AND MARTIN STROBEL*
We present simple one-shot distribution experiments comparing the relative impor-tance of efficiency concerns, maximin preferences, and inequality aversion, as wellas the relative performance of the fairness theories by Gary E Bolton and AxelOckenfels and by Ernst Fehr and Klaus M. Schmidt. While the Fehr-Schmidt theoryperforms better in a direct comparison, this appears to be due to being in line withmaximin preferences. More importantly, we find that a combination of efficiencyconcerns, maximin preferences, and selfishness can rationalize most of the datawhile the Bolton-Ockenfels and Fehr-Schmidt theories are unable to explain im-portant patterns. (JEL D63, D64, C99)
Among the recent attempts to explain behav-
compare the relative importance of inequality
ior observed in economic experiments, models
aversion, concerns for efficiency, and maximin
based on inequality aversion have received spe-
preferences1 in simple distribution experiments.
cial attention. The attractiveness of these mod-
On the other hand, we compare the performance
els is based on their ability to rationalize a
of the two theories based on inequality aversion
number of well-known anomalies with just two
by Fehr and Schmidt (1999, henceforth F&S)
motives, selfishness and inequality aversion.
and Bolton and Ockenfels (2000, henceforth
The latter is understood as disutility arising
from differences between one’s own payoff and
Our first treatments that were designed to
compare ERC and F&S recognized the potential
The aim of this paper is on the one hand to
importance of efficiency and thus controlled forit. It turned out that efficiency had a majorimpact (see treatments F and E in Section III).
* Engelmann: CERGE-EI, Charles University and Acad-
This finding inspired further experiments to test
emy of Sciences of the Czech Republic, P.O. Box 882,
its robustness and to investigate to what extent
Politickych veznu 7, CZ-11121 Prague 1, Czech Republic
inequality aversion is dominated by efficiency
(e-mail: [email protected]); Strobel: Depart-ment of Economics and International Institute of Infonom-
concerns or maximin preferences. In particular,
ics, University of Maastricht, P.O. Box 616, 6200 MD
these treatments allow us to compare the ex-
Maastricht, The Netherlands (e-mail: Martin.Strobel@
planatory power of ERC and F&S to the model
infonomics.nl). We thank Gary Bolton, Colin Camerer,
by Gary Charness and Matthew Rabin (2002,
Gary Charness, James Cox, Martin Dufwenberg, Armin
henceforth C&R), that is based on efficiency
Falk, Ernst Fehr, Urs Fischbacher, Alfonso Flores-Lagunes,Simon Ga¨chter, John Kagel, Ulrich Kamecke, Georg Kirch-
concerns and maximin preferences and was in-
steiger, Wieland Mu¨ller, Hans Normann, Axel Ockenfels,
Andreas Ortmann, Frank Riedel, Arno Riedl, Klaus
Our results suggest that efficiency concerns
Schmidt, Robert Sherman, Nat Wilcox, and participants of
and maximin preferences are important in sim-
the ESA 2000 annual meeting in New York, the First WorldCongress of the Game Theory Society in Bilbao, the 8th
ple distribution experiments. While this does
World Congress of the Econometric Society in Seattle, and
not necessarily imply that they are equally im-
seminars at Humboldt-Universita¨t, CERGE-EI, University
portant in other classes of games, common in-
of Zu¨rich, University of Maastricht, UCSB, Caltech, New
terpretations of several games may well be
York University, and the University of Arizona for helpfulcomments. Parts of this research were financed by theDeutsche Forschungsgemeinschaft (through Sonderfors-chungsbereich 373 and Grant No. EN 459/1), the Interna-
1 Efficiency is here simply understood as the sum of
tional Institute of Infonomics, the DGZ-DekaBank, and an
payoffs, not in the sense of Pareto efficiency. Maximin
ESC-postdoctoral fellowship from the CERGE-EI founda-
preferences are a desire to maximize the minimal payoff in
tion. The financial support is gratefully acknowledged.
confounded with these motives. This may have
In Section I we outline the difference be-
been given too little attention in the past (see
tween ERC and F&S that we focus on. Section
Engelmann and Strobel, 2002, for a discussion).
II presents our experimental procedures, and
To illustrate, consider the following example.
Section III the experimental results. Section IV
First, let person 2 choose only between alloca-
tions A and B among persons 1, 2, and 3. I. Inequality Measures in ERC and F&S
The difference between the inequality mea-
sures in ERC and F&S is represented in the
motivation or utility function. The motivation
function of ERC is given by v ( y , ), with y
denoting subject i’s own payoff and subject
i’s share of the total payoff, and v for given y
If person 2 is inequality averse she prefers B
being maximal if ϭ (1/n), n being the number
over A, but B is also her preferred choice if she
of players. F&S assumes a utility function
is driven by efficiency concerns or maximin
U ( x) ϭ x Ϫ ␣ [1/(n Ϫ 1)] ¥
preferences. Thus deriving any conclusions
from a choice of B concerning the importance
␣ Ն  Ն 0,  Ͻ 1 and x the payoff of sub-
of inequality aversion is confounded by effi-
ciency concerns and maximin preferences. One
cannot tell whether person 2 wants to redistrib-
average payoff to be as close as possible to their
ute money because she dislikes inequality, cares
own payoff while F&S assumes that subjects
for efficiency, or cares particularly for the poor-
dislike a payoff difference to any other individ-
est. Now consider the case that person 2 can
ual. According to ERC, a subject would thus be
choose from A, B, and C. A choice of B now
equally happy if all subjects received the same
clearly indicates inequality aversion, since self-
payoff or if some were rich and some were poor
ishness, efficiency concerns, and maximin pref-
as long as she received the average payoff, but
according to F&S she would clearly prefer that
In our experiments, we disentangle efficiency
all subjects get the same. In a real-life situation
concerns, maximin preferences, and inequality
F&S predicts that the middle class would tax the
aversion to compare their relative importance.
upper class to subsidize the poor, whereas ERC
In order to exclude, as far as possible, motives
like reciprocity, we chose degenerate gameslike the one above that were completely reduced
II. Experimental Procedures
to the question of distribution. Since both ERCand F&S are formulated on the basis of distri-
We conducted 13 experimental treatments in
butions only, these games seem to us the most
three sessions. These sessions were all con-
neutral playground to compare their predictive
ducted as classroom experiments at the end of a
lecture during the first weeks of introductory
In contrast to previous experiments, in sev-
economics courses at Humboldt-Universita¨t zu
eral of our treatments ERC and F&S predict
Berlin. One hundred thirty-six participants took
choices of allocations that are at the opposite
part in the first session in 1998, 240 in the
ends of the choice set. Here, F&S does better ingeneral. This, however, appears to result fromF&S being in line with maximin preferences in
theories that explicitly take intentions into account (e.g.,
this situation. For a complete explanation of our
Rabin, 1993; Armin Falk and Urs Fischbacher, 2000; Mar-
results, efficiency and maximin preferences are
tin Dufwenberg and Georg Kirchsteiger, 2004) since this
would require assumptions about beliefs concerning thechoices of subjects with whom one might be matched. Thesame holds for the full C&R model but we can shed somelight on the basic model, relying on selfishness plus quasi-
2 Other fairness theories could be applied to our setting
maximin preferences (maximizing a weighted sum of total
as well. Our experiments, however, are not suited to test
ENGELMANN AND STROBEL: SIMPLE DISTRIBUTION EXPERIMENTS
second session in 2000, and 210 in the third
different degrees of redistribution. All choices
session in 2001. We had determined a number
provide the “middle income” individual with
of seats corresponding to the desired number of
the same payoff. This is to remove any effects
participants in advance. We asked students to
of selfishness, so we can focus on motives re-
either take one of these seats or leave the class-
lated to fairness. For all treatments in this sec-
room. Each participant then received a decision
tion, the F&S prediction coincides with the
sheet with the instructions and a questionnaire.
maximin allocation. This reflects the structure
We used the questionnaires to gather some bio-
of this class of taxation problems and is not an
graphical data and to check whether the partic-
artifact of our design. The crucial property of
ipants understood the task completely. The total
these treatments is that the allocation that min-
procedure took about 20 minutes. Participants
imizes the difference between the payoffs of
were paid after the lecture in the following
person 2 and each of the other persons, maxi-
week. They were identified by a code that was
mizes the difference between the payoff of per-
noted both on the decision sheet and on a de-
son 2 and the average payoff and vice versa.
tachable identification sheet. They received the
Thus ERC and F&S predict choices of opposite
payment in a sealed envelope in exchange for
this sheet. These procedures implied anonymity
In one treatment (F), we choose payoffs so
with respect to the other participants.
that efficiency coincides with the F&S predic-
The decision sheet contained three different
tion and the maximin allocation; in another (E),
allocations of money between three persons, of
we choose payoffs so that efficiency coincides
which the subjects had to choose one. They
with the ERC prediction. This allows us to
were informed that we would randomly form
investigate the extent to which efficiency coun-
groups of three later on and would also assign
terbalances the various types of fairness con-
the three roles randomly, hence subjects faced
cerns (F&S, ERC, and maximin) and it ensures
role uncertainty. Only the choice of the partic-
that efficiency concerns do not bias the results
ipant selected as person 2 mattered.3 Two con-
in favor of either ERC or F&S.5 We also con-
trol treatments assigned fixed roles in advance,
sider a variation of treatment F, called Fx, and a
but kept the random ex post formation of
variation of treatment E, called Ex, in which the
groups. To avoid influence by computation er-
outcome predicted by ERC involves exactly the
rors we also noted the average payoffs of per-
“fair share,” i.e., 1⁄3 , for the “middle income”
sons 1 and 3 and the total payoff for each
individual. The purpose of these treatments is to
allocation in the decision sheet. The precise
allocations and the resulting predictions of the
The allocations for treatments F, E, Fx, and
different theories will be presented along with
Ex are presented in Table 1 (all payoffs are
the results for the individual treatments.4
given in DM, 1 DM corresponded to $0.45 to$0.55 by the time of the various sessions) along
III. Experimental Results
with the average payoff of persons 1 and 3, therelative payoff of person 2, and the total pay-
off. We also marked which allocations are pre-dicted by ERC, F&S, efficiency concerns, and
Details and Predictions: All of the treatments
maximin preferences, as well as the actual
in this section involve a “middle income” indi-
vidual (person 2) choosing payoffs for a “high
Each of the treatments E and F was divided
income” individual (person 1) and a “low in-
into two subtreatments that only differed by the
come” individual (person 3). One can think of
order in which the allocations were presented on
the choices as tax systems corresponding to
3 In other words, we used (a reduced form of) the strat-
5 The preferable way to prevent results from being con-
egy method. Apart from generating three times the data, it
founded with efficiency would have been that all allocations
secured that all participants were considered equally entitled
yielded the same total payoff. If the decision maker’s own
to the money since all performed the same task.
payoff is fixed, however, ERC implies indifference between
4 Sample instructions can be found in Engelmann and
all allocations if the average and thus the total payoff of the
TABLE 1—ALLOCATIONS (IN DM), PREDICTIONS BY ERC AND F&S, MAXIMIN AND EFFICIENT ALLOCATIONS, AND DECISIONS
the decision sheet.6 All other treatments were
maximin preferences. The three allocations
divided into six subtreatments, one for each
were not chosen with equal probability (p
0.001), in particular the F&S allocation waschosen significantly more often than the ERC
RESULTS: The results for treatments E and F
(including the subtreatments in the last two
results are more dispersed. Slightly more sub-
rows) as well as for Fx and Ex are presented in
jects chose the allocation predicted by ERC and
Table 1. In both treatments F and E there is
efficiency than that predicted by F&S and maxi-
virtually no difference between the two sub-
min preferences, while 23.5 percent chose the
treatments (2 ϭ 0.08, p Ͼ 0.96 for treatment E,
intermediate allocation.9 The hypotheses that all
and 2 ϭ 0.16, p Ͼ 0.92 for treatment F).7 This
three or the two extreme allocations were cho-
consistency suggests that our data are not com-
sen with equal probability cannot be rejected
The results for treatment F are very clear:
ments balance the influence of efficiency con-
83.8 percent of subjects chose the allocation
cerns, we also study the pooled data. There,
predicted by F&S, efficiency concerns, and
60.2 percent of subjects chose the allocation
6 This was done to avoid the conceivable influence of a
preference for the center or right allocation. The allocation
significance for a multinomial test of the hypothesis that all
with intermediate payoffs was always presented on the left,
allocations are chosen with the same probability, whereas
since it was the one we were least interested in.
will denote the level of significance for a (two-sided)
7 Hence we can conclude that the results are not driven
binomial test of the hypothesis that allocations X and Y are
by a preference for either the middle or the right column and
chosen with the same probability taking the number of
we pool the data from the respective subtreatments. For the
choices for the third allocation as given.
other treatments we do not report results for the subtreat-
9 The explanation that some of these subjects provided in
ments, since the number of subjects in each of the subtreat-
the questionnaires indicates that they were looking for a
compromise between efficiency and fairness. ENGELMANN AND STROBEL: SIMPLE DISTRIBUTION EXPERIMENTS
predicted by F&S, whereas 22.8 percent de-
stantial part of the data are consistent with
cided in line with ERC (p Ͻ 0.001, binomial
maximin preferences. Furthermore, since most
of the choices which are not in line with maxi-
Of the 136 choices in both treatments, 61.8
min preferences are efficient (the ERC alloca-
percent are in line with the maximization of
tion in treatments E and Ex), quasi-maximin
total payoffs while 21.3 percent minimize it
preferences (as in C&R) are consistent with
(p Ͻ 0.001, binomial test). Furthermore, the
about 85 percent of the data, if one allows for
distribution of decisions clearly differs between
treatments E and F (2 ϭ 33.07, p Ͻ 0.001).
Since the crucial difference between E and F is
the role of efficiency, we see this as substantialevidence that efficiency matters. Details and Predictions: Treatments F and E
The results for treatments Fx and Ex almost
demonstrated a major influence of efficiency.
exactly match those for F and E. (Fx: p
This inspired us to subject both theories of
inequality aversion to a more severe test, in
1; both treatments pooled, ERC vs. F&S allo-
which they predict decisions that are Pareto-
cation: p Ͻ 0.001, efficiency maximization vs.
dominated. This situation is represented by
minimization: p Ͻ 0.003; Ex vs. Fx: 2 ϭ
treatment N, where the payoff to person 2 is
again intermediate and kept constant. In this
In treatments Fx and Ex the ERC prediction
treatment F&S predicts a choice of C, which is
is much more salient than in F and E. Since the
Pareto-dominated by the ERC prediction B,
results changed only marginally (distributions
which is in turn Pareto-dominated by allocation
are far from significantly different: 2 ϭ 0.69,
A (see Table 2 which is structured in the same
p Ͼ 0.7 for Ex vs. E and 2 ϭ 0.34, p Ͼ 0.84
way as Table 1). We call these games envy
for Fx vs. F) and not in favor of ERC, we
games, because envy could lead the middle
conclude that the poor performance of ERC in
class to take money from the poor, only to be
our original treatments cannot be attributed to
nonsalient differences in relative payoffs. Argu-
We also used this treatment as a baseline to
ably, these are still not huge, but if nonsalience
test the robustness of our results with regard to
was the issue, then the performance of ERC
the monetary incentives for person 2. To test
should improve at least somewhat compared to
whether subjects were willing to give up own
payoff for their desire to increase efficiency or
Explaining their decisions in treatments E
to reduce inequality, we let the payoff of person
and F, 17 of the 18 subjects who explicitly
2 vary across allocations in the treatments Nx,
referred to fairness chose according to F&S and
Ny, and Nyi (see Table 2). Since both F&S and
one chose the intermediate allocation. Effi-
ERC also take selfishness into account, their
ciency concerns were stated by 12 subjects as
predictions depend on the weight assigned to
the reason for their decision. Only one subject
selfishness relative to inequality aversion (see
referred to relative payoffs in the explanation,
Table 2). The purpose of these treatments is to
but contrary to ERC, this subject stated that he
test whether our results in the other treatments
wanted to maximize his own share. In treat-
might be artifacts of the irrelevance of the
ments Ex and Fx all 15 subjects who explicitly
choice for the decision maker’s own payoff, not
referred to fairness chose the F&S allocation.
to measure precisely the value subjects attach to
Efficiency concerns were mentioned by 16 sub-
jects, and 6 indicated maximin preferences. Thus among the subjects who explicitly men-
RESULTS: In treatment N, 70 percent chose
tioned fairness as a motivation, F&S did much
the Pareto-efficient allocation (which is consis-
better than ERC and a substantial part of sub-
tent with quasi-maximin preferences) and ERC
jects explicitly stated efficiency concerns.
Hence, we conclude for the taxation games
10 We do not claim that the motivation that leads subjects
that F&S outperforms ERC and that efficiency
to behave in that way is in fact envy, which corresponds to
clearly influences choices. Since the F&S pre-
the ␣-component of F&S. It only seems a likely influence in
diction is always the maximin allocation, a sub-
this class of games. Hence our choice of name.
TABLE 2—ALLOCATIONS (IN DM), PREDICTIONS BY ERC AND F&S, MAXIMIN AND EFFICIENT ALLOCATIONS, AND DECISIONS
clearly outperforms F&S, but with the aid of
payoff, but it is minor.12 Hence the relative
importance of the different motives does not
seem to change fundamentally if selfishness be-
In treatment Nx we added 1 DM for person 2
in allocation A and subtracted 1 DM in C. As
Note that Ny and Nyi are the only treatments
expected, this increased the number of choices
where F&S makes a unique prediction (C) for
all subjects, including those which are not in-
equality averse, since the decision maker’s own
tracted 1 DM (0.5 DM) in allocation A and
payoff is maximal and inequality minimal. But
added 1 DM (0.5 DM) in C. As expected, this
this prediction only covers one-sixth of deci-
increased the number of choices of C some-
what. However, again the majority chose A,
We conclude for the envy games that F&S
whereas the choices of B are reduced (Ny:
Ͻ 0.001, p Ͻ 0.001, p Ͻ 0.001; Nyi:
dominance and that ERC does somewhat better
Ͻ 0.011, p Ͻ 0.011, p Ͻ 0.044). Thus
but not well, whereas the basic C&R model
the results in these treatments are qualitatively
well in line with the constant-own-payoff treat-ment N with deviations as expected by standard
economic theory.11 This result suggests that our
Note that in Ny 76.7 percent of subjects give up 22
percent of their own payoff, apparently to satisfy quasi-
results in the other treatments are not plain
maximin preferences. While this share corresponds to only a
artifacts of the constancy of the decision mak-
relatively small absolute payoff, it is often considered strong
er’s payoff. There is an (expected) effect of
evidence against selfishness if subjects are willing to give up
small variations in the decision maker’s own
20 or 25 percent of their payoff to achieve, e.g., equality.
13 The envy games also provide an example that the
predictive power of F&S can in some cases substantially beimproved by abstracting from the linear form. If the disutil-
11 The effect should be larger in treatment Ny than in Nyi
ity is assumed to be, e.g., quadratic in inequality, F&S could
and the number of choices for A should not increase in Ny.
also explain choices of B. In addition, if the restriction  Յ
These deviations, however, can be attributed to randomness
␣ is relaxed, then F&S can be consistent with choices of A.
in the data, that naturally follows from the random alloca-
Hence the results can be seen as evidence against some
tion of the subjects to the treatments. No pair of distribu-
forms of inequality aversion but not as evidence against all
tions is significantly different at 5 percent, 2-test.
possible forms of inequality aversion. ENGELMANN AND STROBEL: SIMPLE DISTRIBUTION EXPERIMENTS
TABLE 3—ALLOCATIONS (IN DM), PREDICTIONS BY ERC AND F&S, MAXIMIN AND EFFICIENT ALLOCATIONS, AND DECISIONS
FOR THE RICH AND POOR GAMES, AS WELL AS FOR TREATMENT EY
The envy games emphasize the importance of
mediate payoff. Our treatments R and P study
efficiency and maximin preferences if they
situations where the decision maker receives
combine to Pareto-dominance. Even then, how-
either the highest payoff (i.e., is “rich,” treat-
ever, they do not capture all choices and thus
ment R) or the lowest payoff (i.e., is “poor,”
there is a potential role for other motives like
treatment P), which is again constant (see Table
3). Since F&S aggregates over all persons richer
In the questionnaires, references to (Pareto)
or poorer than oneself, it predicts the same as
efficiency are more prominent in treatment Nx
ERC in these situations. So these treatments do
(21 subjects) than in N (11) or Ny and Nyi (15
not allow us to distinguish between F&S and
in total). In all envy games together fewer sub-
ERC. They allow us, however, to contrast effi-
jects mention fairness (7) than maximin prefer-
ciency, maximin preferences, and inequality
ences (11) and selfishness (13). One subject
aversion. In treatment R person 2 can choose for
states preferences in line with ERC.
the other subjects payoffs that are relativelyequal (C) or that are maximal in sum (A). Both
F&S and ERC predict a choice of the efficientallocation A, whereas maximin preferences pre-
Details and Predictions: In the preceding eight
dict C. In contrast, in treatment P inequality
treatments person 2 always obtained an inter-
aversion predicts a choice of the least efficientallocation C. The minimal payoff is constant, somaximin preferences cannot influence the re-
14 Our results in treatment N do not necessarily imply
sults. Hence this treatment allows us to contrast
that 30 percent of subjects are inequality averse rather than
efficiency and inequality aversion in a frame
motivated by efficiency or maximin. The pattern of ob-
served proportions declining with the efficiency and maxi-min rank of the allocations well fits a random utility version
At this point we also study our last treatment
of quasi-maximin preferences. Error rates nearly this
Ey. It is identical to Ex except that the alloca-
high have been estimated from retest reliabilities in two-
tor’s payoff is 9 instead of 12. Ey has the basic
alternative lottery choice tasks (see, e.g., T. Parker Ballinger
structure of the taxation games, but it does not
and Nathaniel T. Wilcox, 1997) and in our treatments the
share the crucial property of the taxation games
error rates might be higher since they involve the choiceamong three alternatives.
that allowed a comparison of F&S and ERC.
The ERC prediction is shifted from A to C. Not
only ERC and F&S, but also maximin and
against a primary importance of inequality aver-
hence all fairness motives under consideration
sion in general form, not just the specific for-
predict the choice of the least efficient alloca-
mulations of F&S and ERC. According to the
tion. Therefore, this treatment serves the same
axiomatic characterization of F&S provided by
purpose as the poor game, namely the com-
William S. Neilson (2002), a choice of C in
parison of efficiency concerns and fairness
treatment R only contradicts a combination of
inequality aversion and linearity.17 A choice ofA in treatments N, Ny, and Nyi contradicts a
combination of inequality aversion and posi-
and F&S predict the efficient allocation A, only
tional asymmetry (which is reflected by the
26.7 percent of the choices were in accordance,
condition ␣ Ն ). In contrast, in treatments P
whereas 53.3 percent of the subjects chose C
and Ey, a choice of A is inconsistent with the
inequality aversion property alone18 as well as
treatment P, where both ERC and F&S predict
with non-self-centered inequality aversion and
allocation C, 60 percent chose the efficient al-
ERC. In both treatments fewer subjects chose
the allocation predicted by all versions of in-
more subjects chose the efficient allocation
equality aversion than the efficient allocation,
when it is not minimizing inequality compared
although the former is also consistent with com-
to the case when it does (p Ͻ 0.08). The distri-
petitiveness and in Ey even with maximin pref-
bution of choices differs significantly between
erences, motives that appear to be of substantial
R and P (2 ϭ 7.23, p Ͻ 0.03).
The comparison indicates that maximin pref-
Treatment P also shows the limits of quasi-
erences are important. In R the minimal payoff
maximin preferences, since for any positive
is maximized in allocation C,15 which was cho-
weight on efficiency quasi-maximin preferences
sen by the majority of subjects, whereas in P the
imply a choice of A, chosen by only 60 percent
minimal payoff is constant, so maximin prefer-
of the subjects. A third of the subjects instead
seems to be guided by either inequality aversion
The results of treatment Ey show roughly a
tie between the efficient allocation A (40 per-
It is conceivable that the role uncertainty that
cent) and the least efficient, but supposedly fair
subjects faced in the preceding treatments might
allocation C (36.7 percent). These results are
have enhanced their concerns for efficiency.
well in line with treatment P, since the lower
They were clearly confronted with the possibil-
number of efficient choices and the marginally
ity to end up in any of the three roles and this
higher number of choices for C are consistent
might have increased their concern for the well-
with a positive influence of maximin prefer-
being of the subjects in the other roles. It also
ences.16 The fundamental difference between
might have increased in particular the concern
the treatments Ey and Ex is the ERC prediction.
for the subject with the lowest payoff and hence
The results are essentially identical (even mar-ginally against ERC), which indicates that ERCis irrelevant in this context.
17 A choice of C would, however, only be consistent with
unrealistically extreme forms of inequality aversion thathave absurd implications. Even if the disutility was cubic inthe payoff difference, B would still be preferred over C.
15 Nine of ten subjects who mentioned fairness chose C,
18 The results in P would be consistent with inequality
only two subjects explicitly indicated maximin preferences.
aversion if the utility function was highly convex in the
16 From this comparison, though, this influence seems
inequality, but this property is just the opposite of what is
rather weak. Furthermore, maximin does worse in compar-
necessary to reconcile results in R and the basic dictator
ison to efficiency than in treatment R (distributions are,
game with inequality aversion. Choices for A in Ey are even
however, far from significantly different, 2 ϭ 1.8, p Ͼ
inconsistent with this form of inequality aversion.
0.4). A possible explanation is that the trade-off between
19 Charness and Brit Grosskopf (2001) also study pure
efficiency and the minimal payoff is more favorable to
distribution experiments and they find that about 10 percent
maximin in R than in Ey. Thus the difference is consis-
of choices can clearly be attributed to competitive prefer-
tent with reasonable parameter distributions in the C&R
ences. Falk et al. (2000a) find even 19 percent competitive
ENGELMANN AND STROBEL: SIMPLE DISTRIBUTION EXPERIMENTS
increased the role of maximin preferences.20
tory variables, with x the payoff to person k in
We have conducted control treatments for Ex
and P (with 90 subjects each), where subjectsknew their role in advance. Only subjects in the
role of person 2 were asked to choose an allo-
cation and they knew that their choice would be
implemented.21 Treatment P allows us to studythe isolated effect on efficiency, treatment Ex
possible effects on both efficiency and maximinpreferences.
The control treatments do not provide any
indication that our results are primarily driven
by the role uncertainty method. In both treat-
ments without role uncertainty the number of
choices for the efficient allocation decreases byone-sixth (see Engelmann and Strobel, 2002,
for the detailed results). This is in line with the
hypothesis that role uncertainty favors effi-ciency, but the differences are small and far
from significant (Ex: 2 ϭ 1.22, p Ͼ 0.54, P: 2
0.65, p Ͼ 0.72). There is also no indication
that the role uncertainty increased the focus on
maximin preferences (if anything, the data point
in the opposite direction). Charness and Rabin
(2001) conducted control treatments for 11
games to test whether the role reversal theyemployed in Charness and Rabin (2002) affects
Then according to the conditional logit model
behavior. They do not find significant or sub-
the probability that person i chooses allocation j
D. The Relative Importance of the Differentg ʦ ͕A,B,C͖
In order to better understand the relative in-
fluences of the different motives we pool the
Since we only have one decision per subject, we
data and estimate a conditional logit model (our
cannot take into account any individual differ-
situation is captured by McFadden’s choice
ences. Hence with this approach we estimate the
model, see, e.g., G. S. Maddala, 1983).
preferences of an “average subject” and all het-
For each allocation j ʦ {A, B, C} that person
erogeneity is incorporated in the error. i can choose we define the following explana-
Considering the ␣ and  components of F&S
separately allows us to investigate for both com-ponents individually whether they explain anyof the variance. This, however, causes a col-
20 On the other hand, the role uncertainty could also
linearity problem because in all of our treat-
enhance the role of inequality aversion since this method
ments FS␣ ϭ FS Ϫ 1⁄2 Eff ϩ 3⁄2 Self. To
underlines that all players are a priori in the same situation,
overcome this problem, in a first approach we
so that no one deserves more or less than the others.
exclude Self, because we are not primarily in-
The subjects who were assigned the roles of person 1
or 3 were asked how they would have chosen if they had
terested in the role of self-interest. In a second
been in the role of person 2 and what choice they expected
approach, we include a strict version of F&S,
person 2 to make. Neither the distribution of the hypothet-
FSstrict ϭ FS␣ ϩ FS, replacing the separate
ical choices nor of the expectations differs significantly
components by an aggregate measure of in-
from the distribution of actual choices for any group of
equality that assumes equal weights assigned to
subjects or treatment ( Ͻ 3.1, p Ͼ 0.21 for all pairwise
disadvantageous and advantageous inequality.
TABLE 4—ESTIMATED ODDS RATIOS FOR THE CONDITIONAL
separate F&S components yields qualitatively
LOGIT MODEL AND RESULTS OF LIKELIHOOD RATIO TESTS
important insight. We now find a highly signif-
icant positive effect of FSstrict and a highly
significant negative impact of the ERC motive.
This means that if we ignore the maximin mo-
tive, F&S appears to be a much better model of
distributional preferences than ERC. This pro-
vides a deeper understanding of why F&S
clearly outperforms ERC in the taxation games,
but does poorly in the other games. The superior
performance of F&S in the taxation games
seems to result from being in line with maximin
there, but not from being a more accurate model
IV. Conclusion
Bolton (1998) suggests three building blocks
to explain behavior in games: motivation, learn-ing, and strategic reasoning. In the presentexperiments we have completely isolated distri-
We also conducted another run excluding MM.
butional preferences from issues such as learn-
The results are reported in Table 4 along with
ing, intentions, and strategic reasoning, because
the results of likelihood ratio tests of hypothe-
distributions are given the central role in F&S
ses that certain subsets of the motives are
and ERC. We are thus able to provide a pure
test both for the comparison of ERC and F&S
If we include both components of F&S sep-
and for the relative importance of inequality
arately, we find that efficiency and especially
aversion, efficiency, and maximin preferences
maximin preferences have a clear significant
as components of the motivation block. It turns
influence. In contrast, neither component of
out that inequality aversion does not seem to be
F&S has significant impact, with the ␣ compo-
a major part in a complete explanation in this
nent having a positive impact and the  com-
setting. F&S and ERC are unable to explain
ponent a negative. Hence the motivation to
important patterns in our data. In contrast, a
increase poorer subjects’ payoffs is entirely cap-
combination of efficiency concerns, maximin
tured by the maximin motive. The ERC motive
preferences, and selfishness (which amounts to
has a negative, marginally significant impact.
the basic C&R model) can rationalize most of
Likelihood ratio tests reveal that both F&S com-
the data. The conditional logit analysis of the
ponents together do not explain additional vari-
pooled data shows that the basic C&R model is
ance (p Ͼ 0.3) and that F&S and ERC jointly
virtually sufficient to explain the data. While
add only marginally to the explanation (p Ͼ
F&S and ERC do not account for additional
0.1). Including FSstrict and Self instead of the
variance, both efficiency and maximin do. Thisis consistent with results for similar simple dis-tribution games in Charness and Grosskopf
22 The odds ratio denotes the factor by which the odds
[P /(1 Ϫ P )] are multiplied if the corresponding indepen-
dent variable increases by one unit. Choosing the negativeof the inequality as measured by F&S and ERC as explan-
23 All results reported in this section are robust to the
atory variables implies that estimating an odds ratio Ͼ1
exclusion of treatments E, F, and Ey (see Engelmann and
amounts to an influence in line with F&S or ERC. Note that
Strobel, 2002, for details and the motivation for excluding
the odds ratios for different explanatory variables are in
these treatments). We excluded the control treatments for
general not directly comparable because the variables are
Ex and P from the analysis because they were run with a
ENGELMANN AND STROBEL: SIMPLE DISTRIBUTION EXPERIMENTS
(2001), Alexander Kritikos and Friedel Bolle
three respects. First, in most treatments the
The superior performance of F&S over ERC
allocator’s payoff is not affected. Second, there
in the taxation games, which we consider the
is role uncertainty. Third, there is no strategic
most neutral playground for the comparison of
F&S and ERC, appears to be driven by the fact
Concerning the first two issues, our treat-
that F&S is in line with maximin preferences.
ments Nx, Ny, and Nyi as well as the control
Hence the results cannot be interpreted in a way
treatments for Ex and P provide no indication
that more subjects have F&S preferences than
that the absence of monetary incentives or the
ERC preferences but that F&S takes into ac-
role uncertainty substantially change the rela-
count that subjects (other things being equal)
tive importance of inequality aversion, effi-
care about the minimal payoff in the group. It
ciency, and maximin preferences. Therefore, we
appears a limitation of ERC that it does not
can at least clearly refute the claim that our
results are entirely driven by these factors.
A further deficiency of both F&S and ERC is
that they do not explicitly consider intentions (a
change subjects’ decisions, we see no obvious
matter that we deliberately designed out of our
reason why they should change the relative im-
experiments), as is demonstrated by the exper-
iments of, e.g., Sally Blount (1995), Falk et al.
The remaining issue is the absence of strate-
(2000a, b), and John H. Kagel and Katherine
gic interaction in our experiments. It is conceiv-
able that apart from the influence of reciprocity,
The degenerate games we study are certainly
strategic interaction alone might change the im-
of a special kind. Hence at the current stage, our
portance of different distributional motives. It is
results do not discard inequality aversion as a
difficult to disentangle this potential effect from
motive in general. Both F&S and ERC are,
effects of perceived intentions and to the best of
however, exclusively formulated on the basis of
our knowledge there is yet no persuasive evi-
distributions and interaction and intentions
dence on this matter.26 It is an issue of substan-
should rather appear as confounding factors.
tial importance. If the relative importance of
We conclude that theories that are based on
different distributional preferences depends on
distributions should, in general, carefully clarify
the presence and the nature of the strategic
under which conditions they are appropriate.
interaction, then the whole approach to test dis-
Inequality aversion may do better in situations
tributional preferences in one strategic situa-
involving perceived intentions, because in these
tion, to understand the results in another,
games reciprocity may coincide with inequality
appears to be problematic. There are, how-
aversion and hence the latter may serve as a
ever, also important situations, which may
black box model of the former, as Fehr and
well not be perceived as strategic interaction,
Schmidt (1999) suggest. This, however, may be
and for these our results are thus more di-
an artifact of the classes of games that have
rectly applicable. An example would be vot-
been the focus of experimental research so far
(in particular those where a player who treats
As long as there is no conclusive evidence
another player unfairly has a higher payoff, as
that the relevance of our results is entirely con-
fined to noninteractive situations, they also havesome general implications. In interpreting ex-perimental results one should keep efficiency
24 Our results are also consistent with the purely distri-
concerns and maximin preferences in mind as
butional model by James C. Cox et al. (2002). A similar
alternative explanations. They are consistent
model is studied by James Andreoni and John H. Miller(2002) and they show that it fits the data of dictator gameswell.
25 Inequality aversion might also be more important
when perfect equality is an option. Werner Gu¨th et al.
26 Evidence on this issue so far indicates primarily that
(2001) show that in mini-ultimatum games the availability
subjects become more selfish when part of the responsibility
of only nearly instead of perfectly equal allocations sub-
for the outcome can be attributed to the other subject
stantially increases the rate of unfair proposals and reduces
(Bolton and Rami Zwick, 1995, and Charness and Rabin,
2001, who call this “complicity effect”).
with many results that are readily interpreted as
Bolton, Gary E. “Bargaining and Dilemma
evidence for other motives.27 For example, in
the investment game (Joyce E. Berg et al.,
1995) sending money by the first mover appears
to reflect trust, but as shown by Cox (2004) in a
Bolton, Gary E and Ockenfels, Axel. “ERC—A
comparison with dictator control experiments,
Theory of Equity, Reciprocity, and Competi-
to a large part it can be attributed to efficiency
concerns. Similarly, a positive relation between
the amount sent and the amount returned by the
Bolton, Gary E and Zwick, Rami. “Anonymity
second mover suggests reciprocity or inequality
aversion, but might as well be driven by maxi-
Deviations from pure selfishness have been
Charness, Gary and Grosskopf, Brit. “Relative
interpreted that subjects are better people (i.e.,
more altruistic or fair), but maybe they are just
better economists. It is surprising that for econ-
omists the goal in designing economic institu-
Charness, Gary and Rabin, Matthew. “Expressed
tions is to maximize efficiency, while as
Preferences and Reciprocity in Experimen-
experimentalists, when designing economic ex-
tal Games.” Working paper, University of
periments, they tend to ignore that subjects
. “Understanding Social Preferences Andreoni, James and Miller, John H. “Giving Cox, James C. “How to Identify Trust and Rec-
of the Consistency of Preferences for Altru-
Cox, James C.; Sadiraj, Klarita and Sadiraj, Vjollca. “A Theory of Competition and Fair- Ballinger, T. Parker and Wilcox, Nathaniel T.
ness for Egocentric Altruists.” Working pa-
“Decisions, Error and Heterogeneity.”
Dufwenberg, Martin and Kirchsteiger, Georg. “A Berg, Joyce E.; Dickhaut, John W. and McCabe, Kevin. “Trust, Reciprocity and Social His- Engelmann, Dirk and Strobel, Martin. “Inequal-
ity Aversion, Efficiency, and Maximin Pref-
Blount, Sally. “When Social Outcomes Aren’t
erences in Simple Distribution Experiments.”
Fair: The Effect of Causal Attributions on
No. 2002-013, University of Maastricht, 2002,
Falk, Armin; Fehr, Ernst and Fischbacher, Urs.
“Informal Sanctions.” University of ZurichWorking Paper No. 59, 2000a. . “Testing Theories of Fairness—
Intentions Matter.” University of Zurich
Charness and Rabin (2002) present a similar
28 In Engelmann and Strobel (2002) we present an ex-
Falk, Armin and Fischbacher, Urs. “A Theory of
tensive review of classical experiments and discuss to what
Reciprocity.” University of Zurich Working
extent the results are consistent with quasi-maximin prefer-
ences. Furthermore, we discuss other experiments that com-
Fehr, Ernst and Schmidt, Klaus M. “A Theory of
pare different fairness motives and point out limitations ofquasi-maximin preferences.
Fairness, Competition, and Cooperation.”
ENGELMANN AND STROBEL: SIMPLE DISTRIBUTION EXPERIMENTSGu¨th, Werner; Huck, Steffen and Mu¨ller, Wie- Maddala, G. S. Limited-dependent and qualita- land. “The Relevance of Equal Splits in tive variables in econometrics. Cambridge:
Neilson, William S. “An Axiomatic Character- Kagel, John H. and Wolfe, Katherine Willey.
ization of the Fehr-Schmidt Model of Ineq-
“Tests of Fairness Models Based on Equity
uity Aversion.” Working paper, Texas A&M
Considerations in a Three-Person Ultimatum
Rabin, Matthew. “Incorporating Fairness into Kritikos, Alexander and Bolle, Friedel. “Distribu-

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