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1 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 Published by: Blackwell Publishing for the Royal Economic Society Stable URL: Accessed: 06/10/ :39 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. Royal Economic Society and Blackwell Publishing are collaborating with JSTOR to digitize, preserve and extend access to The Economic Journal.

2 The Economic Journal, io6 (May) ( Royal Economic Society I996. Published by Blackwell Publishers, io8 Cowley Road Oxford OX4 ijf, UK and 238 Main Street, Cambridge, MA 02I42, USA. TWO-LEVEL ULTIMATUM BARGAINING WITH INCOMPLETE INFORMATION: AN EXPERIMENTAL STUDY* Werner Guith, Steffen Huck and Peter Ockenfels 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 proposers with 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 fairness by 'hiding behind some small cake' Previous experiments of ultimatum bargaining (see Roth 1995) for a recent survey) have shown that fairness considerations are important. But why do people care about fair outcomes? Experimental studies of institutionally richer situations (e.g. Mitzkewitz and Nagel, I993; Bolton and Zwick (I995), Hoffman et al. I994; Rapoport et at. (in press), and Guth and van Damme, I994) may provide clues as to whether a proposer is intrinsically interested in the responder's well-being or only fearing a rejection in the face of unfair offers. The main effect seems to be that proposers often become greedier if unfair proposals are not recognised. Our experiments reveal a much more specific conclusion: those who can hide their greed want to appear to be fair, whereas those who cannot hide their greed seem to balance their desire to exploit and their fear of rejection. The more specific hypotheses are easier to discuss after introducing our experimental situation. In a two-level ultimatum game there are three different players, labelled X, Y, and Z, who have to distribute the monetary amount c. First X chooses the offer x for Y and Z which then Y can accept or reject. In the case of rejection the game ends with zero-payoffs. Acceptance means that Y and Z proceed as in a usual ultimatum game with Y proposing how to distribute x. Whereas X is just a proposer and Z just a responder, the novel feature is the role of Y who is both, a proposer and a responder. We expected that Y, who has to encounter both aspects of ultimatum bargaining, will be more concerned about fairness than usual proposers. Clearly, such a bargaining situation is empirically relevant: in many real life-situations a bargaining outcome is just what can be distributed in subsequent negotiations, e.g. in the case of multiple production stages. In some sense we were able to confirm the conjecture that received generosity induces own generosity which can be interpreted as some kind of non-bilateral reciprocity (Mauss, I954). Since the 'cake' c could be either large (c = e) or small (c = c) with G > c > o and since only X knew the true cake size, players X with c = c could hide 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. [ 593 1

3 594 THE ECONOMIC JOURNAL [MAY 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 research project2 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 misunderstanding we summarise the 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 regardless of 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 YPs offer. 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 profits which cannot be observed by employees who nevertheless might feel entitled to participate in them. 2 'Decision preparation and bounded rationality' -initiated by W. Guith and R. Tietz and financed by the Deutsche Forschungsgemeinschaft (DFG).? Royal Economic Society I996

4 I996] TWO-LEVEL ULTIMATUM BARGAINING 595 Table I The 2 x 2-factorial Design of Decision Preparation 2 X 2-design Auction? Questionnaire? No Yes No I II Yes III IV Table 2 Numbers of Participants, Bidders, and Plays Session Pilot (III) I II III IV Number of subjects i8 25 Number of bidders per X Y Z X Y Z position II I I IO Number of plays i6(+ i6) 9(+9) 4(+4) 6(+6) 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.) K Royal Economic Society i996

5 596 THE ECONOMIC JOURNAL [MAY (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 transferred to the Y-players who have to fill out their decision form (Appendix D). These are finally transferred to 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. ( Royal Economic Society I996

6 I996] TWO-LEVEL ULTIMATUM BARGAINING 597 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. Proposers X 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 proposers X with c = e to imitate the fairness of the poor. As a matter of fact in the first round of all sessions7 840e 9 out of I 5 X-players with c = c offered an x with > x > 8&oo. If 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 8o00 (7 I 4?/ over all sessions of Table I). This justifies BEHAVIOURAL REGULARITY I. Most X-players with large cakes pretend fairness, especially in the absence of entitlement via auctioning positions. Really fair offers with x 2G are extremel rare. This result allows a re-interpretation of the observations of Hoffman et al. (I994). 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 3G will 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 If one relaxes the assumption that offers yielding zero payoffs are rejected there are more perfect f 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). ( Royal Economic Society I996

7 598 THE ECONOMIC JOURNAL [MAY BEHAVIOURAL REGULARITY 2. X-players with c = c rarely offer an 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 %0 whereas the corresponding share for X with c = c is o-67 %0 which 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 BEHAVIOURAL REGULARITY 3. The offers of players X with large cakes are significantly higher in session III than in session I. 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 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) 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 increases with 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.? Royal Economic Society I996

8 I996] TWO-LEVEL ULTIMATUM BARGAINING 599 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&oo are only rejected in 3 out of 22 cases, i.e. pretended fairness is quite successful. Thirdly, for offers x with x < 8&oo the probability of rejection increases dramatically to We can sum up these observations by BEHAVIOURAL REGULARITY 6. When it becomes obvious to Y that X has not offered a fair share, i.e. when x < 8&oo, the frequency of rejection increases dramatically. 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-players nothing much can be said. Since the pre-experimental questionnaire enhances X's generosity, this reduces according to regularity 6, the rejection rate: BEHAVIOURAL REGULARITY 7. The pre-experimental questionnaire reduces the number of conflicts, whereas auctioning positions increases the number of conflicts. 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). II.3. Responders Z 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 across subjects, namely an acceptance threshold determined by prominence considerations: BEHAVIOURAL REGULARITY 8. Every y with y> 2 00 was accepted, and every y < 2 00 was rejected at the acceptance threshold 200 the rejection rate was 33?/0 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 fairness clearly 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 = 0-2 I + 0o39x (t = 4 85). K Royal Economic Society I996

9 6oo THE ECONOMIC JOURNAL [MAY 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.12 In a subsequent study (Guith and 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. Humboldt University Humboldt University Goethe University Date of receipt offinal typescript: August I995 REFERENCES Bolton, G. E. and Zwick, R. (I995). Anonymity versus punishment in ultimatum bargaining. Games and Economic Behaviour, 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 Current Issues in West German Decision Research (ed. R. W. Scholz), pp. I Frankfurt. Guth, W. and van Damme, E. (I 994) -'Information, strategic behavior and fairness in 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.' Games and Economic Behavior, vol. 7, pp Konigstein, M. and Tietz, R. (I994). 'Profit sharing in an asymmetric bargaining game.' Working Paper, Berlin/Frankfurt. Mauss, M. (I954). The Gift: Forms and Function of Exchange in Archaic Societies. Glencoe/Ill. Mitzkewitz, M. and Nagel, R. (I993). 'Experimental results on ultimatum games with incomplete information.' International Journal of Game Theory, 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 Journal of Game Theory, forthcoming. Roth, A. E. (I995). 'Bargaining experiments.' In Handbook of Experimental Economics (ed. J. Kagel and A. E. Roth), pp Princeton, NJ.: Princeton University Press. Game Instructions APPENDIX Please read these instructions carefully. If you have questions, please ask the experimenters privately. 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 received the 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 or DM I2-6o. The 12 Keep in mind that experience gained by playing the game did not enhance generosity. C Royal Economic Society I996 A

10 I996] TWO-LEVEL ULTIMATUM BARGAINING 60I 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. Player Z 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 Post-Experimental Questionnaire Please answer the following questions. You have Io minutes to fill 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? B C Royal Economic Society I996

11 602 THE ECONOMIC JOURNAL [MAY Decision form APPENDIX C Code number of player X. Player X, please write down, which amount you offer to Y and Z. Is I Decision form APPENDIX D Code number of 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 number of player Z. Player Z, please mark, whether you accept or reject Y's offer. O I accept. O I reject. Pre-Experimental Questionnaire APPENDIX EI/E2 Please answer the following questions carefully. You have 20 minutes to fill out the questionnaire. [Ei asks also for the following Personal data: Age, Sex, Major Subject at the University, and Number of Semesters] [All answers had to be inserted into little boxes.]? Royal Economic Society I996

12 I996] TWO-LEVEL ULTIMATUM BARGAINING 603 Questions concerning the experiment What do you think will 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 would you 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? Player (XI Y/Z), how would you rate the possible outcomes of the game? 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? (I2) At which possible payoff would you prefer the conflict with o-payments (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

13 604 THE ECONOMIC JOURNAL [MAY I996] APPENDIX F: Play Data (Participants are numbered as in Appendix G.) Subjects Cake size Subjects ist round 1 = large Accept Accept 2nd round Cake Accept Accept (X/Y/Z) s = small x +/- y +/- (X/Y/Z) size x +/- y +/- Session I I/i/i I/2/ /2/2 1 8.oo - 2/3/4 1 8.oo /3/3 s /4/5 s /4/ /5/ /5/ /6/ /6/6 s 5.8o - 6/7/8 s /7/ /8/ /8/ /9/ I I.55-9/9/9 s 6.oo /I/2 s Session II 2/2/2 S I/I/ /3/ /5/4 s /5/6 1 8.oo /7/6 1 8.oo - 7/7/9 1 8.oo /9/9 s 6.oo Session III I/i/i I i I/2/3 1 i oo + 2/2/2 S /3/4 s /3/3 1 I /4/5 1 8.oo /4/ /5/ /5/5 s /6/i s /6/ /I/ Session IV I/2/ I/3/ i0 + 4/3/3 I /6/3 s /5/4 s /8/ /8/7 s /9/2 S AP E NDIX G: Auction Bids (Participants are nurnbered as in Appendix F.) Subjects Subjects Subjects bidding ist 2nd bidding ist 2nd bidding 1st 2nd for X round round for Y round round for Z round round Session II I I2.00 i6.oo I I I I3.I3 II.I I II I 2.60 I I 0.5I o I 3.5I 6 IO.02 I ,20 5.0I 6 8.oo I2.26 I I IO.50 I oo 6.oo I I.00 i i.8o o50 9 I oo I I I0 2. I I I 6.oo 6.oo I I oo Session IV I I5.I5 I5.I5 I 3.00 I.00 I I I I I I5.00 I I IO oo 5 Io I0.00 I I0 3.I I I I0.00 I I0.00 I2.00? Royal Economic Society I996