Two-Level Ultimatum Bargaining with Incomplete Information: an Experimental Study
Author(s): Werner Güth, Steffen Huck, Peter Ockenfels
Source: The Economic Journal, Vol. 106, No. 436 (May, 1996), pp. 593-604
Published by: Blackwell Publishing for the Royal Economic Society
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The EconomicJournal, io6 (May) 593-604. ( Royal Economic Society I996. Published by Blackwell
Publishers, io8 Cowley Road Oxford OX4 iJF, UK and 238 Main Street, Cambridge, MA 02I42, USA.
BARGAINING
TWO-LEVEL ULTIMATUM
INFORMATION:
INCOMPLETE
STUDY*
AN EXPERIMENTAL
WITH
HuckandPeterOckenfels
Guith,
Werner
Steffen
In a two-level ultimatum game one player offers an amount to two other players who then, in the
case of acceptance, divide this amount by playing an ultimatum game. The first offer has to be
accepted by the second proposer. Only the first proposer knew the true cake size whose a prioriprobabilities were commonly known. The fact that most proposerswith the large cake offered two
thirds of the small cake has importast implications for the theory of distributive justice: better
informed parties do not question that others want a fair share and, thus, pretend fairnessby 'hiding
behind some small cake'
Previousexperimentsof ultimatumbargaining(see Roth 1995) for a recent
survey) have shown that fairnessconsiderationsare important.But why do
peoplecareaboutfairoutcomes?Experimentalstudiesof institutionallyricher
situations (e.g. Mitzkewitz and Nagel, I993; Bolton and Zwick (I995),
Hoffmanet al. I994; Rapoportet at. (in press),and Guth and van Damme,
I994) may provideclues as to whethera proposeris intrinsicallyinterestedin
the responder'swell-beingor onlyfearinga rejectionin the faceof unfairoffers.
The main effect seems to be that proposersoften become greedierif unfair
proposalsare not recognised.Our experimentsreveal a much more specific
conclusion:thosewho can hide theirgreedwant to appearto be fair,whereas
thosewho cannot hide theirgreedseem to balancetheir desireto exploit and
their fear of rejection.
The more specific hypothesesare easier to discussafter introducingour
experimentalsituation.In a two-levelultimatumgametherearethreedifferent
players,labelledX, Y, and Z, who have to distributethe monetaryamountc.
FirstX choosesthe offerx for Yand Z whichthen Ycan acceptor reject.In the
case of rejectionthe game ends with zero-payoffs.Acceptancemeans that Y
and Z proceed as in a usual ultimatum game with Y proposinghow to
distributex. WhereasX is just a proposerand Z just a responder,the novel
featureis the role of Ywho is both, a proposerand a responder.We expected
that Y, who has to encounterboth aspectsof ultimatumbargaining,will be
moreconcernedaboutfairnessthanusualproposers.Clearly,sucha bargaining
situation is empiricallyrelevant: in many real life-situationsa bargaining
outcomeis just what can be distributedin subsequentnegotiations,e.g. in the
case of multipleproductionstages.In somesensewe were able to confirmthe
conjecturethat received generosityinduces own generositywhich can be
interpretedas some kind of non-bilateralreciprocity(Mauss,I954).
Since the 'cake' c could be either large (c = e) or small (c = c) with G > c >
o and sinceonly X knewthe truecakesize,playersX with c = c couldhide their
* The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft
(DFG) and helpful comments by John D. Hey as well as by three anonymous referees.
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greed by offering a fair share of c instead of C-.'For players X with c = c this is
impossible. Since Z was not informed about the offer x, no responding player
knew the available amount, unless it was revealed by the offer. One might
expect to observe a wide range of behaviour by players Z, but we observed
clearcut behaviour by players Z: there is an acceptance threshold for Z below
which any offer is rejected and above which any offer is accepted.
Although our classroom experiments were repeated once, this does not
provide enough opportunity for learning. As part of a broader researchproject2
we wanted to explore whether certain forms of decision preparation can induce
a more mature behaviour and thereby substitute learning by repetitive play.
One form of decision preparation was a pre-experimental questionnaire
designed to induce consideration of backward induction, the other was to
auction player positions instead of allocating them by chance. Of course,
auctioning the positions of two-level ultimatum games is likely not only to
trigger a more thorough analysis of the decision problem, but also to engender
feelings of entitlement and alter perceptions of fairness (Gtith and Tietz, I 986).
Naturally we were not only interested in the actual plays after auctioning
positions, but also in the bids themselves which can be interpreted as a prioripayoff expectations for the various player positions.
Section I describes the experimental set-up in more detail. Section II states
behavioural regularities, and Section III our essential conclusions.
I. EXPERIMENTAL
DESIGN
To avoid any misunderstandingwe summarisethe rules of two-level ultimatum
games with players X, Y, and Z and the four subsequent decision stages:
(i) Nature chooses c with probability 2 for c = = 24 6o and ' for c = c =
I 2-6o (all amounts are in Deutsch mark, DM). While the two potential values
of c and their probabilities are commonly known, only X learns which value of
c has been chosen.
(2) Knowing ce{_, E}player X chooses the offer x with c > x > o, i.e. the
amount he wants to leave for Y and Z together.
(3) Knowing x player Y can accept or reject X's offer. If Y rejects, the game
is over and all players receive nothing. Otherwise X received c- x regardlessof
what happens later on, and Y decides about his offer y to Z with x > y > o.
(4) Knowing y but neither c nor x player Z can accept or reject YPsoffer. If
Z rejects, both Y and Z receive nothing. Otherwise Z receives y and Y the
residual amount x-y which ends the game.
Before describing how this game was experimentally implemented let us
explain the 2 X 2-factorial design of decision preparation illustrated by Table I.
The pre-experimental questionnaire was composed of three parts: one part
asking for which choices one expects, another eliciting hypothetical decisions
(what would you do if you were player... ?), and the final part asking for a
1 Real life examples of such opportunities are windfall profitswhich cannot be observed by employees who
nevertheless might feel entitled to participate in them.
2 'Decisionpreparation
and boundedrationality'-initiated by W. Guith and R. Tietz and financed by the
Deutsche Forschungsgemeinschaft (DFG).
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Table I
The2 x 2-factorialDesign of DecisionPreparation
2
X
Auction?
2-design
Questionnaire?
No
Yes
Yes
II
IV
No
I
III
Table 2
Numbersof Participants,Bidders,and Plays
Session
Pilot (III)
I
Number of subjects
Number of bidders per
48
27
position
Number of plays
i6(+
i6)
9(+9)
II
i8
32
X
Y
Z
II
II
IO
4(+4)
IV
III
25
X
9
6(+6)
Y
Z
9
7
4(+4)
ranking of possible outcomes. To trigger considerations of backward induction
all questions were asked in the reversed order of play, e.g. by first asking what
Z will do at stage 4, then what Y will do at stage 3, etc.
Positions were auctioned by independent auctions for roles X, Y, and Z. All
subjects, who could bid for only one position is {X, Y,ZI, had to submit a sealed
bid. Four positions i were given to the four highest bidders at the price of the
fifth highest bid for role i. Subjects were told that under such rules it is best to
bid truthfully.
We first conducted a pilot experiment with 48 graduate students of
economics who were unfamiliar with game theory where we relied on the preexperimental questionnaire for decision preparation (cell III in Table i). For
the main experiments students from all over the campus (University of
Frankfurt am Main) were invited by posters (see Table 2 for the number of
subjects, bids and plays where the number of plays in brackets indicates the
number of plays in the repetition which do not qualify as independent
observations).
One can describe the experimental procedure for all four cells of Table i by
focusing on cell IV since all other cells just require that certain steps are
omitted:
(a) All subjects meet in a large lecture room. They are randomly partitioned
into three subgroups of equal size. These groups are led into three different
lecture rooms where they receive their code numbers.
(b) (only cells II and IV) Subjects receive written instructions about the
auction procedure including the argument that bidding truthfully is best. (It
was checked how far this advice was accepted by a pretest in the form of a
private value auction with predetermined values.)
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(c) Instructions concerning the actual game are distributed (Appendix A),
and subjects are informed about their (potential) player position X, Y, or Z.
Questions are answered in private.
(d) (only cells III and IV) Subjects have to fill out the pre-experimental
questionnaire (Appendix E).
(e) (only cells II and IV) In each room four player positions are auctioned.
The role prices are collected immediately and are publicly announced in all
three rooms.3
(f) To determine the cake size we throw a die in the room of the X-players
who have on their code cards four 'lucky digits'. When the die shows one of
these, the cake is large and otherwise it is small for that particular player. (We
never spoke about probabilities. Furthermore, the procedure guarantees that 3
of all X-players have a large cake.)
(g) X-players have to fill out their decision form (Appendix C). The decision
forms are transferredto the Y-playerswho have to fill out their decision form
(Appendix D). These are finally transferredto the Zs who either learn that Y
has already rejected X's offer or who have to decide on the offer made to them
by Y.
(h) All subjects learn about their own play, i.e. about the true cake size, all
offers and acceptances. Afterwards they have to fill out a post-experimental
questionnaire (Appendix B). Then we announce the repetition and explain
that nobody will meet the same partners.
(i) Repetition of (d) to (h) where (d) was only repeated in cell III.
According to these rules the game is in the terminology of Mitzkewitz and
Nagel (I993) an 'offer game'. Mitzkewitz and Nagal studied ultimatum games
with six possible cake sizes (I, 2 ... , 6) and restricted offers to multiples of I.
Offer games were also studied by Rapoport et al. (in press) who used a
'continuum' of cake sizes4 to study the effects of the amount of uncertainty.
The main reason for us to use only two possible cake sizes is that we want to
distinguish between the smallest possible offer and the fair share of the smallest
possible cake.5 Furthermore, our design allows players to signal the true cake
size without offering a fair share of it.
II. BEHAVIOURAL
REGULARITIES
Before analysing the actual decision data it seems worthwhile to illustrate the
game theoretic solution. We rely on two assumptions, namely that players are
guided purely by monetary incentives and that any offer implying a zeropayoff will be rejected with probability I. Let c be the smallest money unit.
Since Z accepts every positive y (and rejects y = o), the optimal offer of Y is
y* = c for all x > 26. Note that x = c implies a zero-payoff for player Y, since
Thus, decisions within cells II and IV are not independent, since subjects interacted before the play in
the auction. That explains why we will focus on bids. Announcing prices publicly may, however, enable, new
fairness norms, e.g. the 'dividend standard' according to which people aim at equal dividends instead of
equal absolute payoffs.
Of course, the usual indivisibilities of money do not allow a real continuum.
In the experiment of Mitzkewitz and Nagal both amounts were equal, namely 2.
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either he offers y = e what would imply o for himself or y = o which would be
rejected. Accordingly, Y will reject any offer x < c and accept all greater offers.
Anticipating this decision it is optimal for X to offer exactly X = 26. This
solution (which is a perfect equilibrium of the game6) therefore yields payoffs
of C- 26 for player X and e for each of players Y and Z.
II. i. ProposersX
Imagine you are a player Y and are confronted with an offer x = 3C= 8&40.
Does that not look like the offer of an X who - due to bad luck - got the small
cake and wants to be extremely fair? At least, this possibility cannot be
excluded. However, this also provides the chance for proposersX with c = e to
imitate the fairness of the poor. As a matter of fact in the first round of all
> x > 8&oo.If
sessions7
840e 9 out of I 5 X-players with c = c offered an x with
one neglects the auction experiments (cells II and IV), where due to role prices
other fairness standards might be relevant, 7 out of io X-players with c = C
chose an x which can be interpreted as pretending fairness, and two of the
remaining three players chose an x with x >,I 2'6o signalling the large cake. In
the repetition the frequency of pretending fairness actually increased to 8o0 0
(7 I 4 ?/ over all sessions of Table I). This justifies
BEHAVIOURAL REGULARITY I .
Most X-players with large cakes pretendfairness,
via auctioning
positions.Reallyfair offerswithx
especiallyin theabsenceof entitlement
2Gare extremelrare.
This result allows a re-interpretation of the observations of Hoffman et al.
Further anonymity by introducing incomplete information does not
offset fairness considerations: proposers with large cakes still want to appear as
fair.8 They might be greedy, but they want to avoid greed being verifiable. This
motive is much more important than a real taste for equity in the sense of
x = 2j which we could observe only once. A possible reason could be the fear that
offers below 3Gwill be mostly rejected. Another reason might be that complying
- at least superficially - with a (fairness) norm has some intrinsic value: you
feel better when others do not know how greedy you are.9
Those players Xwith c = c who cannot hide their greed offer significantly less
than the others. The hypothesis that of all plays with x < 3G the offer x for
c = c is smaller than for c = c is highly significant (one-tailed MWU-test, first
rounds only, n = I 7, a = o0ooo3). Actually only 2 out of 8 players with small
cakes offered an x with 8-40 > x > 8-oo in the first rounds with the share even
decreasing in the repetition (i of 9). We sum up these observations by
(I994).
f
If one relaxes the assumption that offers yielding zero payoffs are rejected there are more perfect
equilibria.
7 The data analysis in Section II is always based on the main experiments. However, the data of the pilot
study confirm - as far as possible - the behavioural regularities, especially regularities I and 2.
S Note that Hoffman et al. introduced a zero-cake to avoid greed being verifiable. Subjects in their study
received dollar bills in an envelope and then had to allocate the money. However, it was commonly known
that one subject would receive strips of plain paper instead of money. This amounts in our notation to setting
c = o, and, therefore, a o-offer in their study could be re-interpreted as an equal split with respect to the
smallest possible cake.
9 To decide whether norm compliance in this weak sense is important a subsequent study relies - among
others - on dictator experiments with incomplete information (Guth and Huck, in press).
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X-players with c = c rarely offeran equal share to Y
and Z.
To see why this regularity is quite natural assume that both X-types, i.e. an
X with c = c and an X with c = c, offer 3_ to Y and Z: clearly, for X with c =
e the share on offer is o034 %0whereas the corresponding share for X with c =
c is o-67 %0which is nearly twice as high.
Given these observations Y-players who face offers which would be fair if the
cake were small should be very suspicious. One might even expect that Y would
reject offers x with 8-40 > x >? 8-oo and accepts lower ones. This is, however,
not supported by our data.
Let us now turn to the question of how decision preparation influenced the
X-players. The main effect of the pre-experimental questionnaire is expressed
by
REGULARITY 3. The offers of players X with large cakes are
higher
in
session III than in session I.
significantly
We tested the hypothesis for the first rounds with the one-tailed MWU-test
(n = IO, a = 0o02).1O It may seem surprising that a questionnaire structured to
induce backward induction did not trigger gamesmanship but more generous
offers. Becoming more familiar with a decision problem may also discourage
strategic thinking and induce a search for socially acceptable solutions since
considerations of backward induction make one think about others and how
they might feel. One can conjecture that thinking about the other party's
position has two countervailing effects: on the one hand, it may encourage
subjects to engage in backward induction, i.e., making them less generous. On
the other hand, it could lead to an emphatic perspective - putting oneself in the
other party's shoes - and thus promote generosity. Given that it, in fact,
promoted generous offers, perhaps the latter effect of the questionnaire was
stronger than the former.
The change from cell I to cell III of Table I suggests
BEHAVIOURAL
BEHAVIOURAL REGULARITY
4. The auction bids of potential players X are smaller
in session IV than in session II.
Since subjects had to fill out the pre-experimental questionnaire first and
then bid, the questionnaire should trigger concerns for Y and Z, and the
expected value of becoming player X - and therefore the bid for role X - should
decrease. We tested this hypothesis for the bids of the first rounds, and it was
significantly validated (one-tailed MWU-test, n = 20, ac= o0os).
11.2. Players Y
Analysing the Y-decisions is more complicated than the X-decisions since the Ys
typically face different offers x. There are, however, some interesting patterns
as
BEHAVIOURAL REGULARITY
5. y increaseswith x.
lo Although n = Io is relatively small, the value ca= 0'02 is rather impressive. Furthermore, Konigstein
and Tietz (1994) replicated our result in another experiment.
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There is a clear (linear) trend between y and x. Excluding one player Y (who
in both rounds faced x = 8-4o and chose first y = o and then y = 8-40) the
linear regression y = 0o47 + o035x has a highly significant coefficient of x (t =
7-56) which can be interpreted as a measure of equity considerations. Players
Y do not share x equally (in the sense of y = x) but claim about 2/3 of x.11
Surprisingly, this corresponds closely to the mean demanded quota in normal
ultimatum games (see Roth, I995). The linear regression can also be seen to
measure how received generosity induces own generosity in a kind of nonbilateral reciprocity (Mauss, I954).
With respect to I's rejection behaviour three observations seem worth
mentioning: first, offers x with x > 8-4o are never rejected. Secondly, offers x
with 8-40 > x > 8&ooare only rejected in 3 out of 22 cases, i.e. pretended
fairnessis quite successful.Thirdly, for offers x with x < 8&oothe probability of
rejection increases dramatically to 35 00 We can sum up these observations by
BEHAVIOURAL REGULARITY
6. When it becomesobviousto Y that X has not offered
a fair share,i.e. whenx < 8&oo,thefrequencyof rejectionincreasesdramatically.
Regularities 2 and 6 together imply that for small cakes bargaining more
often ends in conflict than it does for large cakes.
Concerning the effects of the 2 X 2-design (Table I) on the behaviour of the
Y-playersnothing much can be said. Since the pre-experimental questionnaire
enhances X's generosity, this reduces according to regularity 6, the rejection
rate:
reducesthe
questionnaire
7. The pre-experimental
REGULARITY
BEHAVIOURAL
positionsincreasesthenumberof conflicts.
numberof conflicts,whereasauctioning
The second part of this behavioural regularity is obviously due to the
dramatically low offers x caused by the extremely high prices of position X
which were I 200 and I 260 (repetition).
Z
II.3. Responders
At first sight players Z seem to encounter the most difficult situation. If Z has
to decide, he knows hardly anything except y: so, how should he decide?
Surprisingly,we observed a very constant behaviour acrosssubjects, namely an
acceptance threshold determined by prominence considerations:
andeveryy <
BEHAVIOURAL REGULARITY 8. Everyy withy> 2 00 was accepted,
rate
was
the
threshold
200
33 ?/0
rejection
2 00 was rejectedat theacceptance
III.
CONCLUSIONS
What appears as fair might not be fair! This may disappoint those who believe
in the ubiquity of fairness. And who has not been intrigued by the vision of a
society relying on fairness?On the other hand fairnessclearly matters: subjects
try to behave in a way such that others cannot be sure that they are greedy.
Will a similar tendency hold outside the laboratory? We believe so. Beside
windfall profits one can think, for example, of employers complaining about
" If one excludes those Y-players with x > c, the regression becomes y =
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bad cash flows before wage bargaining. These situations have one feature in
common: the better informed parties do not question that others want a fair
share and, thus, pretend fairness by 'hiding behind some small cake'.
Another interesting result is that generosity is enhanced when proposers
become more familiar with the conflict situation after answering a preexperimental questionnaire forcing them to think about the concerns of
others.12In a subsequent study (Guithand Huck, forthcoming) we try to find
out whether this is due to an intrinsic concern for others or to purely strategic
reasoning based on the anticipation of higher rejection rates.
HumboldtUniversity
HumboldtUniversity
GoetheUniversity
Date of receiptoffinal typescript:AugustI995
REFERENCES
Bolton, G. E. and Zwick, R. (I995).
Anonymity versus punishment in ultimatum bargaining. Gamesand
EconomicBehaviour,vol. I0, pp. 95- I2I.
Guth, W. and Huck, S. (in press). 'From ultimatum bargaining to dictatorship - an experimental study of
four games varying in veto power.' Metroeconomica
(forthcoming).
Guth, W. and Tietz, R. (1986). 'Auctioning ultimatum bargaining positions - how to act if rational decisions
are unacceptable.' In CurrentIssuesin West GermanDecisionResearch(ed. R. W. Scholz), pp. I73-85.
Frankfurt.
Guth, W. and van Damme, E. (I 994) -'Information, strategic behavior and fairnessin ultimatum bargaining
- an experimental study. Working Paper, CentER/Tilburg.
Hoffman, E., McCabe, K., Shachat, K. and Smith, V. A. (I994). 'Preferences, property rights, and
anonymity in bargaining games.' Gamesand EconomicBehavior,vol. 7, pp. 346-80.
Konigstein, M. and Tietz, R. (I994). 'Profit sharing in an asymmetric bargaining game.' Working Paper,
Berlin/Frankfurt.
Mauss, M. (I954). The Gift: Formsand Functionof Exchangein ArchaicSocieties.Glencoe/Ill.
Mitzkewitz, M. and Nagel, R. (I993).
'Experimental results on ultimatum games with incomplete
information.' International
Journalof GameTheory,vol. 22, pp. I7I-98.
Rapoport, A., Sundali, J. A. and Potter, R. E. (in press). 'Ultimatums in two-person bargaining with onesided uncertainty: offer games.' International
Journalof GameTheory,forthcoming.
Roth, A. E. (I995).
'Bargaining experiments.' In Handbookof ExperimentalEconomics(ed. J. Kagel and
A. E. Roth), pp. 253-348. Princeton, NJ.: Princeton University Press.
APPENDIX
A
Game Instructions
Please read these instructionscarefully.
If you have questions,please ask the experimentersprivately.
You are participating in a very simple 3-person-game, in which a positive amount of
money has to be allocated among 3 players. The rules are most easily explained by
telling what the three parties have to do. Let us call them X, Y, and Z. X moves first,
then Y, followed by Z. Please, note: All players have receivedthe same instructions!
Player X
On the back of his code card containing his 5-digit code number, every player X finds
four additional digits in the range from I to 6. These digits determine the amount that
has to be allocated. Two amounts are possible: either DM 24-60 or DM I2-6o. The
12
Keep in mind that experience gained by playing the game did not enhance generosity.
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chance move works as follows: The experimenters throw a dice. If the dice shows one
of the four 'lucky digits', the large amount is chosen, otherwise only the small amount
can be allocated. Since every player X has four digits on his code card, the large 'cake'
will be distributed in 4 of 6 cases. It is important that only player X knows the actual
'cake size'. Players Y and Z only know the mechanism of choosing the cake.
Knowing the true cake size player X has to decide which amount he wants to leave
for Y and Z (and thus the residual amount which he demands for himself) by writing
down his offer on his decision form. Note: He will only write down his offer but not the
amount he demands for himself!
Player Y
Player Y will receive player X's decision form on which X has written down his offer.
He only knows this amount which-has to be distributed among himself and player Z.
He neither knows how much player X demanded for himself nor the true cake size. He
only knows the possible amounts and their chances as well as the offer of X.
Given this information player Y has to decide if he wants to accept or reject the offer.
He marks his decision on his decision form. If he rejects, the game will end, and no
player (including X) will receive a payoff. If he accepts, player X receives the difference
between the true cake and his offer. Y then has to decide which amount of the accepted
offer by X he wants to give to Z. He writes down this offer to Z on his decision form
which will be passed on to player Z. Hence, player Z will not know what Y demanded
for himself.
PlayerZ
Z receives - unless Y has rejected - Y's decision form with the monetary amount that
Y has offered to him. Note: Player Z neither knows the true cake size nor what Y
demanded for himself. Z has to decide whether to accept or reject the offer by Y. He
marks his decision on the same decision form. If he accepts he will receive the amount
offered by Y, and Y will receive what he demanded for himself. If he rejects, both Y
and Z will receive no payoff, while X, of course, gets what he demanded for himself.
These are the rules of the experiment. You will receive your payoff in real money
according to these rules. Furthermore, we assure you that none of your decisions can
be identified personally:
APPENDIX
B
Post-ExperimentalQuestionnaire
Pleaseanswerthefollowingquestions.Youhave Io minutestofill out the questionnaire.
(i) How do you rate the outcomes of your game? Are you satisfied with your payoff?
Do you think any players were unfair? If yes, who and why?
[Question 2 was only asked after the second round.]
(2) What had the first and second trial in common? What was different in the second
trial?
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C
Decisionform
Code numberof playerX.
Player X, please write down, which amount you offer to Y and Z.
I
Is
APPENDIX
D
Decisionform
Code numberof player Y.
Player Y, please mark, whether you accept or reject X's offer.
O I accept.
O I reject.
Player Y, when you have accepted, please write down, which amount you offer to Z.
I
I
Code numberof playerZ.
Player Z, please mark, whether you accept or reject Y's offer.
O I accept.
O I reject.
APPENDIX
EI/E2
Pre-ExperimentalQuestionnaire
Please answer thefollowing questionscarefully. You have 20 minutestofill out the questionnaire.
[Ei asks also for thefollowing Personal data:
Age, Sex, Major Subject at the University, and Number of Semesters]
[All answers had to be insertedinto little boxes.]
? Royal Economic Society
I996
I996]
TWO-LEVEL
ULTIMATUM
BARGAINING
603
theexperiment
Questionsconcerning
Whatdo you thinkwill happen?
(i) What will be the lowest offer of Y that an average player Z will accept?
(2) Which share of the remaining cake will an average player Y demand for himself?
(3) What will be the lowest offer of X that an average player Y will accept?
(4) Which share of the cake will an average player X demand for himself
(a) when the cake is large?
(b) when the cake is small?
How wouldyou decide?
(5) Suppose you were player Z. Which offer by Y would be just acceptable
(a) if all players knew that the cake is large?
(b) if all players knew that the cake is small?
(c) if all players only knew the mechanism determining the cake?
(6) Suppose you were player Y. Which share of the remaining cake would you demand
for yourself if the offer of X were between
(a) DM o and DM 5?
(b) DM s and DM io?
(c) DM io and DM I5?
(d) DM I5 and DM 24.60?
(7) Suppose you were player Y. Which offer by X would be just acceptable
(a) if all players knew that the cake is large?
(b) if all players knew that the cake is small?
(c) if all players only knew the mechanism determining the cake?
(8) Suppose you were player X. Which offer would you make
(a) when the cake is large?
(b) when the cake is small?
of thegame?
Player (XI Y/Z), howwouldyou ratethepossibleoutcomes
fThe question differed for X or Y and Z, respectively. Instead of the questions below
players X were only asked the (a) and (b) questions as in question (8). Furthermore
they were not asked question (I 2).]
(g) Which payoff would you rate as fully satisfying
(a) if all players knew that the cake is large?
(b) if all players knew that the cake is small?
(c) if all players only knew the mechanism determining the cake?
(i o) Which payoff would you regard as agreeable
(a) if all players knew that the cake is large?
(b) if all players knew that the cake is small?
(c) if all players only knew the mechanism determining the cake?
(i i) Which payoff would you rate as just acceptable
(a) if all players knew that the cake is large?
(b) if all players knew that the cake is small?
(c) if all players only know the mechanism determining the cake?
At which possible payoff would you prefer the conflict with o-payments
(I2)
(a) if all players knew that the cake is large?
(b) if all players knew that the cake is small?
(c) if all players only knew the mechanism determining the cake?
C Royal Economic Society I996
604
THE
[MAY
JOURNAL
I996]
F: Play Data (Participants are numbered as in Appendix G.)
APPENDIX
Cake size
1 = large
s = small
Subjects
ist round
(X/Y/Z)
ECONOMIC
Accept
+/-
x
y
Accept
+/-
Subjects
2nd round
(X/Y/Z)
Cake
size
x
Accept
+/-
y
Accept
+/-
Session I
1
2/2/2
1
3/3/3
4/4/4
5/5/5
6/6/6
7/7/7
8/8/8
9/9/9
Session II
s
1
1
s
6.30
8.oo
8.40
8.40
8.40
5.8o
1
8.20
1
s
8.40
6.oo
2/2/2
S
5.00
3/3/4
4/5/6
7/7/9
Session III
1
1
1
6.40
8.oo
8.oo
I/i/i
I
2/2/2
S
i6.40
4.60
1
I4.00
I/i/i
+
+
+
+
+
+
+
1
8.40
s
1
6.20
8.40
I/2/2
1
4/3/3
6/5/4
9/8/7
s
s
6.30
I4.00
8.40
7.00
+
+
+
AP
E NDIX
Subjects
bidding
for X
+
+
+
I/2/3
1
1
s
1
1
s
6.30
8.oo
8.40
8.40
8.40
3.60
+
+
+
+
+
8.40
1
1
8.20
+
3.50
8.40
I .55
+
+
+
+
+
+
-
2.00
-
2.66
+
3.00
4.00
+
+
2/3/4
3/4/5
4/5/6
5/6/7
6/7/8
7/8/9
8/9/ I
2.00
+
9/I/2
s
4.00
+
+
2.00
+
I/I/2
1
3.50
3.50
+
+
2/5/4
4/7/6
8/9/9
s
1
s
7.60
3.60
8.oo
6.oo
+
+
2.00
-
6.20
+
I/2/3
1
+
+
s
1
i6.40
4.60
8.oo
8.oo
2/3/4
3/4/5
4/5/6
5/6/i
+
+
+
+
+
2.40
0.00
3.20
+
+
+
+
+
+
+
3/3/3
4/4/4
5/5/5
6/6/6
Session IV
2.00
4.20
5.00
4.20
4.00
2.30
3.00
3.50
-
4.20
2.00
2.00
4.00
1
8.40
+
4.20
+
7.50
8.40
+
+
3.00
3.00
+
+
6.30
7.60
8.40
7.00
+
+
+
+
3.i0
+
+
+
+
+
6/I/2
s
1
+
+
I/3/4
3/6/3
4/8/3
1
s
1
9/9/2
S
+
3.40
3.70
3.00
3.50
G: AuctionBids (Participants are nurnbered as in Appendix F.)
ist
round
2nd
round
Subjects
bidding
for Y
ist
round
2nd
round
Subjects
bidding
for Z
1st
round
2nd
round
Session II
I
I2.00
i6.oo
I
2
20.00
20.00
2
I0.00
3
I3.I3
II.I3
3
4
5
6
7
8
9
I 2.60
I3.00
0.5I
I2.60
I2.50
I5.00
4
5
6
7
8
9
I0
II
0-5I
IO.02
I2.26
IO.50
I I.00
8.20
6.oo
i i.8o
8.20
6.oo
6.50
I
4.00
2.00
5.00
2
8.20
4.20
9.2I
4.00
0.36
5.00
6.o0
3
4
5
6
7
8
9
3.II
4.50
4.30
3.2 I
4.30
3.5I
5.00
3.00
2.20
5.00
7.07
4,20
I4.50
6.oo
4.20
I0
3.00
II
5.00
5.0I
9.00
6.oo
6o50
3.00
6.oo
8.oo
3.00
2.20
I0
2. I0
6.oo
2.60
2.00
3.50
3.00
3.50
5.00
5.00
2.20
2.00
4.I0
3.I0
I.50
2.00
I2.00
Session IV
I
I5.I5
2
I.00
3
4
5
6
7
8
0.00
I5.00
3.00
9
I0.00
5.00
4.20
3.00
? Royal Economic Society
I5.I5
2.20
I0.00
I I.00
8.oo
5.00
4.20
6.30
I0.00
I996
I
3.00
2
I5.00
I 7.23
3
4
5
6
7
8
9
3.50
Io070
I0.00
0.00
I0.00
I0.00
I.00
I
3.00
2
8.20
3
4
5
7
8
5.00
2.50
I0.00
2.00
5.00
I2.00
3.IO
3.60