Measuring Beliefs in an Experimental Lost Wallet Game

Ž.
Games and Economic Behavior30,163᎐1822000
doi:10.1006r game.1999.0715,available online at http:r r on
Measuring Beliefs in an Experimental Lost
Wallet Game*
Martin Dufwenberg
Department of Economics,Stockholm Uni¨ersity,SE-10691Stockholm,Sweden
E-mail:md@ne.su.se
and
Uri Gneezy
Department of Economics,Uni¨ersity of Haifa,Haifa31905,Israel
E-mail:gneezy@econ.haifa.ac.il
Received August28,1996
We measure beliefs in an experimental game.Player1may take x-20Dutch
guilders,or leave it and let player2split20guilders between the players.Wefind
Ž.
that the higher is x our treatment variable,the more likely is player1to take the
x.Out of those who leave the x,many expect to get back less than x.There is no
positive correlation between x and the amount y that2allocates to1.However,
there is positive correlation between y and2’s expectation of1’s expectation of y.
Journal of Economic Literature Classification Numbers:C72,C92.ᮊ2000Academic
Press
I.INTRODUCTION
Suppose youfind a wallet in the street.No one sees you.The wallet contains money,and some other stuff which is of apparent value to the owner but of no use to you.You can either keep the wallet,or bring it to a
*We wrote most of this paper,which was initially titled‘‘Efficiency,Reciprocity,and Expectations in an Experimental Game,’’while we were both at the CentER for Economic Research at Tilburg University.We thank Eric van Damme for his advice,Gary Bolton,Doug ¨
DeJong,Werner Guth,Jos Jansen,Jan Potters,Ariel Rubinstein,and Oded Stark for valuable comments,the referee for detailed and constructive suggestions,Tone Ognedal for suggesting the story with the lost wallet,and Jos Jansen and Wim Koevoets for assisting us during the experimental sessions.We gratefully acknowledgefinancial support from CentER and the European Union’s HCM-Network project‘‘Games and Markets,’’organized by Helmut Bester.
163
0899-8256r00$35.00
Copyrightᮊ2000by Academic Press
All rights of reproduction in any form reserved.
164
DUFWENBERG AND GNEEZY
Ž.
FIG.1.The game⌫x.
nearby police station for the owner to pick up.The police routinely register your name,and subsequently ask the wallet owner to reimburse you in the amount she considers appropriate.What would you do?
It is commonly assumed in economics that people are motivated only by material,self-centered concerns.In the above situation such an assump-tion leads to an inefficient outcome.The owner does not reimburse the finder if she picks her wallet up at the police station.Thefinderfigures this out and simply keeps the wallet.Both these persons would prefer that the owner gets back the wallet and reimburses thefinder sufficiently.
Ž.
A special instance of this situation is modeled in the game⌫x in Fig.
Ž.
1,where payoffs are in Dutch guilders f,x is an exogenously given parameter such that0-x-20,and y is chosen by player2such that 0F y F20.
If the players are motivated solely by personal monetary gain,the
Ž.Ž.
unique subgame perfect equilibrium in⌫x is Take,y s0,correspond-ing to the dismal outcome with the lost wallet.This outcome is inefficient since if1chooses Lea¨e and2chooses y such that x-y-20,then a payoff vector is realized which is better for both players.
However,much experimental evidence suggests that when humans inter-act they may be motivated by various nonmaterial considerations and not only personal monetary gain.In some cases this may eliminate inefficiency. We address related issues by investigating the nature of strategic behavior
Ž.
in an experimental game which resembles⌫x.However,for methodolog-ical reasons,we ask player2to report a strategyᎏa choice of yᎏwithout informing her about1’s choice.1We still refer to the experimental game as Ž.
⌫x.
1Ž.
The strategy approach,which goes back to Selten1967,has the advantage that we can
Žrecord2’s behavior irrespective of whether her information set is reached.See Roth1995, .
pp.322᎐323for a discussion of the strategy method.We discuss this aspect of our design further in Section IV B.
MEASURING BELIEFS IN AN EXPERIMENT165 We use x as a treatment variable taking values of f4,7,10,13,and16.2Ž.
As in⌫x,the money pie to be split by player2is heldfixed at f20in all treatments.We let participants engage in anonymous,one-shot plays of this experimental game.Our objective is to record regularities in the participants decision making and to draw conclusions about the motiva-tions behind their behavior.To shed some additional light on this,we explicitly measure some of the players’beliefs about one another’s actions and beliefs by asking the participants to make certain guesses about other participants’choices or guesses,rewarding them for accuracy.
We investigate the following issues which relate to the treatment vari-able x:
ⅷAn efficient outcome obtains if and only if player1chooses Lea¨e. However,if1chooses Lea¨e his potential loss is increasing in x.Is the propensity for1to choose Lea¨e negatively correlated with x?
ⅷBy choosing Lea¨e,player1places a trust in2.To what extent does 2reciprocate and keep this trust by choosing y G x?3
ⅷOne might argue that the higher is x,the kinder is1by choosing Lea¨e since the potential loss he may incur by doing so is higher.Player2 may want to be kind in return by correspondingly choosing a higher y.Is y positively correlated with x?
Furthermore,we investigate the following issues which relate to the players’beliefs:
ⅷIs there a connection between1’s expectation of y and1’s propen-sity to choose Lea¨e?In particular,does1choose Lea¨e only if he expects to get back at least x?
ⅷIs player2’s choice of y positively correlated with her expectation of
Ž.
1’s expectation of y conditional on1choosing Lea¨e?This could happen if2were averse to‘‘letting1down,’’in the sense of not wanting to choose y below1’s expectation of y.Of course,player2cannot know1’s expecta-tion of y.Yet,the higher is her expectation of1’s expectation of y the higher she may be inclined to choose y.
Ž.
⌫x is related to the Dictator game in which one player decides how to
Ž.
divide f20between herself and another dummy player.The subgame of
2At the time of the experiment f20were worth approximately12U.S.dollars.
3The usage of the terminology‘‘place a trust,’’‘‘keep the trust,’’and‘‘reciprocate’’here is
Ž.Ž. in line with that of Berg et al.1995,p.126who study a game further discussed below which is related to ours.
166
DUFWENBERG AND GNEEZY
Ž.4⌫x where2moves,considered in isolation,has precisely such a structure. When the Dictator game is tested experimentally,with monetary payoffs controlled,‘‘the dictator’’quite often gives away more than zero,which is typically explained with reference to altruism or fairness considerations Ž.
see Davis and Holt1993,pp.263᎐269for a discussion.We suspect that
Ž.
2’s behavior in⌫x will be affected by similar concerns,but that in addition it may matter that whether2’s choice affects payoffs is at1’s discretion.To check this,we also run an experimental Dictator game in which the procedures,including the belief elicitation,were analogous to
w x
those discussed above.Letting y g0,20denote the dictator’s choice of how much to allocate to the dummy player,we investigate the following issues:
Is the choice of y lower in the Dictator game than in⌫x?
ⅷŽ.
ⅷIs the dictator’s choice of y in the Dictator game positively corre-lated with her expectation of the dummy player’s expectation of y?
We conclude this introduction by discussing some other related litera-Ž.
ture:Berg et al.1995analyze a‘‘trust game’’which shares many features Ž.
with⌫x:Player1is given a sum of money.He chooses how much to keep and‘‘sends’’the rest to player2.The amount sent is tripled and given
5Ž.
to player2who chooses how much to‘‘send back.’’Bolle1995reports
Ž. experimental results involving a game which resembles⌫x,except that an element of chance was added.A lottery was conducted to select four out of the64experimental games that were played,and only the partici-pants acting in these games were rewarded according to their decisions.6 Bolle set x equal to half the value of the pie split by player2and he did not consider the effect of changing x.In both of these studies,most
4We investigate the potential importance of some nonpecuniary concerns that arise due to a choice that precedes a dictator subgame.One can compare this to the Ultimatum game,in which an action is added that succeeds a proposed dictator division:a responder gets to accept or reject the proposed split,and in the latter case,each player gets a zero payoff.See Ž.Ž.
¨
Camerer and Thaler1995and Guth1995for detailed discussions of results in experimen-
Ž.
¨
tal Ultimatum games.See Guth and van Damme1998for a report on an experiment on a game which incorporates essential elements of both the Dictator and the Ultimatum game. 5Ž.Ž.
⌫x can be related to the game of Berg et al.as follows:In⌫x player1is given a
Ž.
certain amount of money x)0.He chooses to send all or nothing to player2.The amount sent is multiplied by a factor inversely related to x,and given to player2who makes a choice
Ž.
on how much money to‘‘send back’’to player1.⌫x may be viewed as more general than the game of Berg et al.in allowing other exogenously given multiplication factors than3,and more restrictive in not allowing player1to send intermediate amounts.
6Ž.
See Bolle1990for a discussion of whether such a setup skews incentives relative to the case where all participants are paid.
MEASURING BELIEFS IN AN EXPERIMENT167 participants did not behave according to the subgame perfect equilibrium with only self-interested material considerations affecting payoffs.7
The paper is organized as follows:Section II explains the experimental procedure.Section III presents the hypotheses we test,and the experimen-tal results.Section IV contains a discussion of our mainfindings.Appen-dices1᎐3contain the experimental instructions.
II.EXPERIMENTAL PROCEDURE
The participants were recruited via an ad in the weekly students’newspaper at Tilburg University and via posters on campus.These an-nouncements invited participants to come to our offices and‘‘sign up’’for an economic experiment on decision making.We indicated that the participants’earnings would depend on these decisions,and approximately how much money was at stake.
Ž.
In total we hadfive sessions of experimental⌫x games and two sessions of experimental Dictator games.Twelve different pairs of students
Ž.
interacted in each session.In the⌫x games,x wasfixed within a session and was changed between sessions to f4,7,10,13,16.For each session we had invited13participants to Room A,13participants to Room B,and4 extra participants to a third room to cover for no-shows.Afterfilling
Ž
Rooms A and B with13participants using participants from the third
.Ž
room room if necessary these where given an‘‘Introduction’’see Ap-.
pendix1.Then,they took an envelope at random.In each room,12
Ž
envelopes contained12different numbers A1,...,A12in Room A and
.
B1,...,B12in Room B.These numbers were called‘‘registration num-bers.’’One envelope was labeled‘‘Monitor,’’and determined who was the person who checked that we did not cheat.That person was paid the average of all other participants in that session.After opening the en-
Ž
velopes the second part of the instruction was distributed see Appendix .2.At this point it was stressed by the experimenter that the game would be played only once.
ŽParticipants in Room A read the instruction for this part see Appendix .Ž.
2.In the⌫x sessions,they were then asked to go to the experimenter, one at a time.They got an envelope with x in it,and then had to go behind a curtain.Over there they had to decide whether to take the money
7Several other authors have conducted experimental studies in which aspects of trust,
Ž.Ž. reciprocity,and efficiency are key features.See,e.g.,Fehr et al.1993,Fehr et al.1997,Ž.Ž.Ž.
¨
Guth et al.1994,van der Heijden et al.1997,and McKelvey and Palfrey1992.However,
Ž
these experiments are relatively less closely related to ours various real world market
. institutions are mimicked,there is no‘‘Dictator subgame,’’or there are more stages.
168
DUFWENBERG AND GNEEZY
out of the envelope or not,then to write their registration number on a note,to put this note in the envelope,and to put the envelope in a box near the experimenter.
ŽParticipants in Room B also read the instructions for this part see .
Appendix2.They were asked to write down how much they would give to
Ž.
their anonymous counterpart in Room A i.e.,to choose y,conditional on
Ž.
him r her choosing to leave the x in the envelope in the⌫x treatments. The participants’choices,sealed in envelopes,were put in a box near the experimenter.
Then part2started.The participants in Room A received new instruc-tions.They were asked to guess the average y chosen by participants in
Ž
Room B in part1,and were rewarded according to accuracy see Appendix .3.Our intention was to provide incentives for them to state their expecta-tion of their co-player’s choice of y.8
Ž
The participants in Room B also received new instructions see Ap-.Ž.
pendix3.In the⌫x sessions they were asked to guess the average guess of the participants in Room A who chose to leave the money in the envelope in part1.In the Dictator game sessions they were asked to guess the average guess of the participants in Room A.Meanwhile,an experi-menter and the two monitors checked and recorded the envelopes of
ŽRoom A,and matched them each with an envelope from Room B as
.
described in the instructions.In the end,all the payoffs from parts1and2 were calculated and the participants were privately paid.9
III.HYPOTHESES AND RESULTS
We now present the hypotheses we test and report on our experimental
Ž.
findings.Wefirst discuss the experimental game⌫x and then the experimental Dictator game.
8We want to measure1’s expectation of the y chosen by his co-player,but nevertheless ask 1to guess the a¨erage choice of y in the whole session.We believe this creates a superior measure.Say,for example,that a participant believes that the co-player chooses y s f0with 1
probability and otherwise chooses y s f10.Such a participant has an expectation of f5.
2
With the incentive scheme we use he should indeed guess f5.Had we asked him to guess his co-players choice he should guess either f0or f10,however.
9Note that while the participants were anonymous to each other,the experimenter could
Ž.
learn each one’s decision.The study of Hoffman et al.1996shows that it is then likely that dictators give away more than with subject᎐experimenter anonymity.Probably a similar Ž.
remark applies to⌫x,but the importance of‘‘social distance’’concerns in general games is
Ž.Ž.
a matter of some controversy.See Hoffman et al.1994and Bolton and Zwick1995for
Ž.
some partly conflicting evidence,and Roth1995,pp.298᎐302for further discussion.
MEASURING BELIEFS IN AN EXPERIMENT169Ž.
A.⌫x
Ž.
The experimental raw data concerning⌫x is given in Tables I and II. The following two hypotheses shouldfind support if participants behave
Ž
according to the‘‘classical solution’’subgame perfect equilibrium when
.
each player’s payoff depends only on his monetary reward:
H:All players1choose Take.
H:All players2choose y s0.
1
Recall that12different pairs of participants interacted in each treat-ment.Table III summarizes for each treatment how many participants
Ž
behaved according to the classical solution e.g.,in the f4treatment,none out of the12players1chose Take.In the f16treatment,3players2chose .
y s0.
It is clear by inspection of the table that the hypotheses H and H do
01 notfind much support.
We now move toward investigating what other patterns of behavioral regularities show up in the data.Wefirst focus on player1,and then on player2.
An efficient outcome results if and only if1chooses Lea¨e.Is the propensity for player1to choose Lea¨e related to the size of x?By
TABLE I
a
Ž.
Raw Data on Player1in⌫x
x s4x s7x s10x s13x s16 Participant Choice Guess Choice Guess Choice Guess Choice Guess Choice Guess 1L4T 2.5T 1.25T0T0 2L5T3T5T0T1 3L8T 4.5T5T0T 1.65 4L8T6T10T 3.5T2 5L8T7.5L6T4T 2.5 6L8T8L7T 5.5T3 7L8L0.75L8T 6.25T4 8L8.45L4L8T9T4 9L8.5L 4.75L10L4T 5.5
10L8.5L6L10L6T7
11L10L8L10L9T10
12L10L9L10L9L16.05 Average12L r127.876L r12 5.338L r127.524L r12 4.691L r12 4.73 aŽ.
For each treatment,thefirst column indicates strategy choice T s Take,L s Lea¨e, and the second column indicates the guess of the average y.
170
DUFWENBERG AND GNEEZY
TABLE II
a
Ž.
Raw Data on Player2in⌫x
x s4x s7x s10x s13x s16 Participant y Guess y Guess y Guess y Guess y Guess
10 6.50 4.504000 4.5
245080 4.50705
34608110013011
44809.5561622
567271075535
61052910878410
7108.575108.58388
8101087101088107.5
9101097.5101088.45109
1010101081010107.51010
111010108125108.51010
12101010912.5916.57.51212
Average7.338.00 4.837.547.547.67 6.12 6.83 5.757.83
a For each treatment,thefirst column indicates the strategy choice,and the
second column indicates the guess of the average guess of y made by the
players1who chose Lea¨e.
TABLE III
Number of Choices Made According to the Classical
Ž.
Solution in Each Treatment in⌫x
x s4x s7x s10x s13x s16࠻of Take064811
࠻of y s014223
inspection of Table III one immediately sees that the number of cases
Ž.
where1chooses Lea¨e is apart from the f7treatment decreasing in x.A logistic regression supports this observation.As can be seen from Table IV,this effect is highly significant.
Next we investigate whether monetary efficiency is achieved only when player1expects to earn at least x.In that case the following hypothesis shouldfind support:
H:1chooses Lea¨e only if1’s expectation of y is at least x.
2
The procedure for measuring expectations is described in Section II and Appendix3.The relevant data are summarized in Table V:
MEASURING BELIEFS IN AN EXPERIMENT171
TABLE IV
a
Lea¨e Choices and the Size of x
Parameter Wald Pr) Variable DF Estimate Error Chi-Square Chi-Square Intercept1y3.14050.861313.29330.0003 Value of x1 1.01400.262314.94400.0001
a Results of a logistic regression.
TABLE V
Ž.
Efficiency and H for Each Treatment in⌫x
2
x s4x s7x s10x s13x s16
Ž.
࠻of Lea¨e choices efficent outcomes126841
࠻of Lea¨e choices by players who122401
expect to get back at least x
Proportion of violations of H0r124r64r84r40r1
2
In the f4treatment every player1chose Lea¨e and in all cases the
player expected to get back more than x,so H was never violated.In the
2
f16treatment we have only one observation,which is in line with H.
2 However,in each of the f7,10,and13treatments H is violated on four
2
occasions.
In the remainder of this section we focus on player2.Her choice has bearing on monetary payoffs if and only if1chooses Lea¨e.Thereby1 risks losing the x he could have for sure,and he gives2a shot at a positive payoff.To what extent does player2reciprocate in the sense of choosing y G x?By inspection of Table III,one sees that this happens quite often in the f4,7,and10treatments,but only happens once in the other treat-ments.
A related aspect is that,arguably,the higher is x,the kinder is1by choosing Lea¨e since the potential loss he may incur by doing so is higher. Player2may want be kind in return by correspondingly choosing a higher y.We expect an effect of this kind to motivate participants in making their choices,and therefore test the following hypothesis which we expect to be able to reject:
H:y and x are uncorrelated.
3
We use the Wilcoxon test to investigate whether the samples of y come from populations with the same distribution.We do a pairwise comparison by treatments.The nonparametric Wilcoxon test is appropriate because
172
DUFWENBERG AND GNEEZY
Ž
the distributions are clearly not normal in fact,using the skewness and
kurtosis test for normality we can reject the hypothesis that y is normally
.
distributed at a significance level of0.0007.In Table VI we report test results.A number in the intersection of a row and a column indicates,for the corresponding pair of treatments,the probability of getting at least as
extreme absolute values of the test statistic as we observe,given that H is
3Ž
true.The last row refers to the results in the Dictator game sessions to be
.
discussed in the next subsection.Table VI conveys a result wefind
surprising.At the5%level,H is not rejected for any of the pairs of
3
treatments.
Finally,we ask whether there is a positive correlation between y and2’s
Ž
expectation of1’s expectation of y conditional on1choosing Lea¨e;we
.
henceforth suppress this qualification.This would be in line with the idea that2might be‘‘averse to letting1down’’in the sense that she does not want to give1less than1expects to get.Of course,2does not know1’s expectation,which is why we focus on2’s expectation of this.
H:y is positively correlated with2’s expectation of1’s expectation 4
of y.
Ž.
Wefirst use the Spearman rank correlation coefficient r to test for
S
the existence of correlation between y and2’s expectation of1’s expecta-tion of y.We run the test for the entire60observations because,as shown above,the hypotheses that the choices of y in different treatments come
from the same distribution cannot be rejected.Wefind that r s0.40,and
S
that H cannot be rejected at the5%level.We interpret this as support 4
for H.After ensuring the existence of correlation,we measure the degree 4
of correlation:There is a positive correlation of0.35between y and2’s expectation of1’s expectation of y.
The connection between y and2’s expectation of1’s expectation of y is
Ž
illustrated in the diagrams of Fig.2.For each treatment including the
TABLE VI
Ž.Ž<< Wilcoxon Tests with Pairwise Comparisons of y by Treatments in⌫x Prob)z,
.
Where z Is the Test Statistic
x s4x s7x s10x s13x s16 x s4ᎏ0.11240.60330.34080.3865
x s7ᎏ0.09410.72900.4705
x s10ᎏ0.22530.3123
x s13ᎏ0.3123
x s16ᎏ
Dictator0.28210.44230.1523 1.00000.9455
FIG.2.Player2’s choice of y and2’s guess of1’s guess of y.
FIG.2ᎏContinued
.
Dictator game to be discussed in the next subsection the choices of the participants in the player2position are plotted in increasing order,with the relevant second-order expectation of y plotted alongside.
B.The Dictator Game
The experimental raw data concerning the Dictator game is given in Tables VII and VIII.
Ž.
In⌫x2’s subgame is reached only if1chooses Lea¨e.If2is motivated by reciprocity considerations,she might choose y higher than she would as
Ž
a dictator in a Dictator game where the choice of y has payoff conse-
.
quences independently of1’s behavior.Then the following hypothesis should be rejected:
Ž.
H:The same y is chosen in the Dictator game and in⌫x.
5
Refer back to Table VI.H is not rejected for any value of x.Wefind
5
Ž. this surprising although perhaps less so given that H was not rejected.
3
Ž
Finally we test the following hypothesis motivated along the same lines .
as H:
4
H:In the Dictator game,y is positively correlated with2’s expecta-6
tion of1’s expectation of y.
Ž. Again,wefirst use the Spearman rank correlation coefficient r to test
S
for the existence of correlation between y and2’s expectation of1’s
TABLE VII
a
Raw Data on the Dictator in the Dictator Game
Participant y Guess Participant y Guess
1001373
2031477
3041578
42 3.516712
5351786
63818103
74519105
85520105
957.5211010
10510221010
1167231010
1276241010
Average 6.08 6.38
a For each participant,thefirst column indicates the strategy choice,and the
second column indicates the guess of the dummy players’average guess of y.
TABLE VIII
a
Raw Data on the Dummy Player in the Dictator Game
Participant Guess Participant Guess
10138
20148
32158
42168.3
521710
631810
731910
8 3.152010
952110
1052210
1172310
127.52415
Average 6.54
a For each participant,the numbers indicate the guess of
the average y.
expectation of y.Wefind that r s0.44,and that H can not be rejected
S6
at the5%level.We interpret this as support for the hypothesis that y and 2’s expectation of1’s expectation of y are correlated.After ensuring the existence of correlation,we measure the degree of correlation:There is a positive correlation of0.51between y and2’s expectation of1’s expecta-tion of y.
In the last diagram of Fig.2the dictator choices of y are plotted in increasing order,with the relevant second-order expectation of y plotted alongside.
IV.DISCUSSION
In this section we discuss our results,focusing on players1and2in turn.
A.Results on Player1
Ž
The higher is x,the fewer players1choose Lea¨e apart from in the f7 .
treatment.Wefind this result quite intuitive,since the potential loss that 1may experience by choosing Lea¨e is increasing in x.
It is perhaps more surprising that several players1choose Lea¨e even Ž.
when our estimate of their expectation of y was lower than x.Experi-ments in which participants chose to give up money to other participants are not new in the literatureᎏsee the discussion in the Introduction about
the Dictator game literature,or witness many participants’behavior in the player2position of our game.However,as far as we know,there is little documented evidence indicating that players are willing to give up money in a way which increases monetary efficiency in situations where they expect a co-player to treat them unfavorably.10
B.Results on Player2
We have reported that player2quite often reciprocates in the sense of choosing y G x in the f4,7,and10treatments,while this almost never happens in the f13and16treatments.This result seems consistent with
Ž.Žthefindings by Berg et al.1995that in their experimental game cf.
.
footnote5above many players2send back no less than their counterpart sent them.11One possible explanation of our results could be that player2 is reluctant to give1more than one-half of the f20that may be split.For x F f10,reciprocity in the sense that y G x can then be achieved while maintaining the no-more-than-one-half constraint.Since reciprocity is mutual,players2are more likely to reciprocate at x less than or equal to f10,since2assumes that1understands the constraint.This explains the bifurcation of the data at x s f10.
Wefind no correlation between x and y in the experimental data. Indeed,the behavior of player2looks much like in a Dictator game.This
Ž.
result may be compared to thefinding of Berg et al.1995that there appears to be no correlation between the amount sent and the amount 12Ž.
sent back.Van der Heijden et al.1997also report similar results. Arguably,the more money is sent,the kinder is player1.Our setup is
Ž. different in that player1can only be kind in one way by choosing Lea¨e, but,in a sense,we control for how kind1is by using x as a treatment variable.This difference between the designs turns out to be unimportant. We note,however,one issue that may have bearing on this result.In our experiment player2never faces the fait accompli of1choosing Lea¨e, since we use the strategy method to record2’s behavior.This setup may
10A caveat to this result is that,with the belief elicitation method used,a risk avert player1 may have an incentive to understate his expectation of y in order to cover himself in case2 gives back less than he expects.Note,however,that this cannot account for the positive correlation between beliefs and actions wefind when testing H.4
11Ä4
Lack of reciprocity when x g13,16cannot be taken as evidence against this similarity,
Ž.
because in the game of Berg et al.the‘‘multiplication factor’’cf.footnote5is always3and hence never as low as20r13or20r16.
12Ž
In the‘‘social history’’treatment of Berg et al.in which participants were informed
.
about the choices made in earlier sessions before making choices theyfind‘‘an increase in
Ž.
the correlation between amounts sent and payback decisions’’p.135,which suggests that this result is sensitive to the social setting.。