Expectations and fairness in a modified Ultimatum game

Transcription

1 JOURNAL OF ELSEVIER Journal of Economic Psychology 17 (1996) Expectations and fairness in a modified Ultimatum game Ramzi Suleiman * Department of Psvchology Unicersity of Ha~fil. Haifil Israel Received 7 February 1995; revised 10 January 1996 Abstract Non-cooperative game theory predicts that Allocators in Ultimatum games will take almost all the 'cake'. Experimental evidence is in sharp contrast with this prediction, as several experiments show that the modal offer is the even split. Interpretations of such results usually make reference to two competing explanations. One invokes the notion of fairness; the second argues that in the absence of common knowledge of the rationality and beliefs of Recipients, Allocators raise their offers because they expect that non-satisfactory offers might be rejected, Although several experiments have been successful in inducing more selfish offers by manipulating the Allocators' expectations, they have done so predominantly by means of extrinsic manipulations that are not accounted for by game theory. The present study introduces a minor variation of the Ultimatum game by implanting a discounting factor, 6 (0 < 6 < 1), in the standard game. Whereas game theory is indifferent to this modification, experimental results from the modified game show that by continuously changing 6, it is possible to induce systematic changes in Allocators' and Recipients' behaviors and beliefs. These results are used to competitively test the fairness and expectations hypotheses. PsyclNFO classification." 3020 Kevwords: Bargaining: Expectations; Fairness; Ultimatum game * rsps854@uvm.haifa.ac.il, Fax: , Tel.: /96/$15.00 Published by Elsevier Science B.V. PII S (96)

2 532 R. Suleiman / Journal of E~'onomic Psychology 17 (1996) Introduction The standard Ultimatum game can be described as follows: an amount of money, M, is to be divided between two subjects: Player 1 and Player 2. Player 1 is designated the role of the Allocator and Player 2 the role of the Recipient. The Allocator can propose any split of the amount M, say M-x for himself, and x for the Recipient. The Recipient can either accept the offer, in which case the proposed split is implemented, or reject it, in which case both players get nothing. The Ultimatum game is perhaps the simplest strictly competitive game. Like the prisoner's Dilemma, it captures a minimal situation suitable for studying competitive behavior. The prediction of non-cooperative game theory for the Ultimatum game is straightforward. If the two players are rational (in the sense that their utility functions are monotonically increasing), then the Allocator should propose M- e for himself, and e for the Recipient, where e is an infinitesimally small positive number. The Recipient should agree to any amount e, no matter how small, since any e > 0 is better than nothing. In an attempt to test this prediction, Giith et al. (1982) created a situation in which all subjects were seated in one room where they could see one another, but no subject could recognize his/her partner in the dyad. The Allocators' offers and the Recipients' replies (accept/reject) were made by written messages, and all earnings were paid in cash. The results showed that very few behaved in accordance with the game theoretic prediction, and that the modal offer was the equal split. Many subsequent studies tried to gain support for the game-theoretic prediction. Typically, these studies implemented exogenous manipulations designed to motivate Allocators to exploit their strategic advantage. Giith and Tietz (1986) used the second highest price auction (Vickrey, 1961) to determine who wins the Allocator role in a subsequent Ultimatum game. Hoffman et al. (1994) tried to enforce the Allocators' 'property rights' by having subjects win the Allocator's position by scoring high on a general knowledge quiz, and by stressing their property rights through the instructions. Other manipulations for eliciting selfish demands are 'framing' the bargaining situation in economic terms of a 'buyer' and a 'seller', using a double anonymity procedure (Hoffman et al., 1994), and selecting subjects who are motivated to win money (Forsythe et al., 1988). By and large, such manipulations were successful in decreasing both the number of equal split offers and the mean proportion Allocators offered to their Recipients. Nevertheless, they all fell short of validating the game-theoretic prediction. The mean offer in the Giith and Tietz (1986) experiment ranged from

3 R. Suleiman / Journal of Economic Psychology 17 (1996) % for a DM 15 pie, to 35% for a DM 100 pie; and many studies replicated the finding of Giith et al. (1982) that the modal offer was the equal split. (cf. Gtith and Tietz, 1986; Forsythe et al., 1988; Bolle, 1990; Prasnikar and Roth, 1992). One explanation for such results invokes the notion of fairness. Based on experimental results, Kahneman et al. (1986) posit that the major reason for more equitable offers in the first stages of bargaining games can be related to a concern for the Recipients' well-being. To test the relative importance of fairness in explaining the Allocators' behavior in the Ultimatum game, Forsythe et al. (1988) and Hoffman et al. (1994) compared Allocators' offers in the Ultimatum and Dictator games. In the latter game, the Allocator makes his decision on how to split the pie and the game ends. If fairness is the major factor behind equitable allocations, then allocations should be similar in the two games. Results from the two studies cited above showed that this was not the case. Instead, they showed that subjects in the Dictator game play more egoistically, demanding significantly more for themselves. For example, in the Hoffman et al. study, 70% of the Dictator games played under the double anonymity condition ended with the first mover demanding the whole 'cake'. Hoffman et al. (1994) and Harrison and McCabe (1992) strongly reject the fairness explanation. Instead, they posit that non-strategic offers in Ultimatum games are driven primarily by the expectations held by Allocators regarding the reservation values of the Recipients. According to this explanation, a rational Allocator, who cannot assume that the Recipient is rational, might reason that positive but insufficient offers are likely to be rejected. Such reasoning will lead him to offer at least what he believes is the Recipient's reservation value. In the Hoffman et al. (1994) study, expectations were manipulated by stressing the property rights of the Allocators (primarily by having subjects 'win' the Allocator's position). Although successful, this manipulation (and similar ones like the Gfith and Tietz (1986) auctioning procedure) does not allow for a satisfactory degree of control and manipulation of the players' expectations. Such control was more convincingly achieved by Harrison and McCabe (1992), whose subjects played Ultimatum games against automata programmed to play strategically. They were thus able to manipulate the subjects' expectations, and subsequently their behaviors, closer to the game-theoretic prediction. The authors conclude by refuting the fair-share hypothesis and asserting that Allocators do not exploit their strategic advantage because they do not have common knowledge of the rationality, beliefs and motives of other players. Weg and Smith (1993) make an appealing distinction between framing and structural incentives for encouraging greedy behavior in bargaining games. According to this distinction, the auctioning procedure (Giith and Tietz, 1986),

4 534 R. Suleiman / Journal ~[' Eco,omic Psychology 17 (1996) the use of market terminology, the double-anonymity procedure, and the winning of property rights to Allocate (Hoffman et al., 1994) are all examples of framing incentives because they are external to the description of the game. On the other hand, an incentive is structural if it is embedded in the game form. Examples of simple games with structural incentives are the two-period Ultimatum game with discounting proposed by Binmore et al. (1985), and the Weg and Smith (1993) costless two-period games. In the Binmore et al. game, a rejection in the first period extends the game to a second period, during which the players alternate roles and Player 2 offers an allocation of a discounted 'cake'. The discount factor in the Binmore et al. study was 0.25, for which case the game-theoretic solution predicts that Player 1 should propose Player 2 about 25% of the 'cake'. As in many Ultimatum games, the results showed that the modal opening offer in this study was the even split. On the other hand, the results of a follow-up study in which the Recipients of the first study assumed the roles of Allocators showed that the modal opening offer was 25%, as predicted by game theory. Binmore et al. concluded that when faced with a new problem, subjects choose the 'equal division' because it is an 'obvious' and 'acceptable' compromise, and that "such considerations are easily displaced by calculations of strategic advantage, once players fully appreciate the structure of the game" (Binmore et al., 1985, p. 1180). Weg and Smith (1993) devised and tested two modified Ultimatum games, which they call the OneOne and the OneTwo Ultimatum games. The two games are two-period costless games. In the OneOne game, Player 1 is given two opportunities to propose, while in the OneTwo game, the two players alternate playing the Allocator role. A strategic analysis of the OneOne game shows that Player 1 retains his power, and can demand the whole 'cake' for himself, whereas a similar analysis of the OneTwo game shows that Player 2 should get the entire 'cake'. The experimental results of Weg and Smith contrasted sharply with the game-theoretic predictions. The authors concluded by arguing that "as long as one player can veto any offer by the other, shares tend to be more even than is predicted by the standard game-theoretic model" (Weg and Smith, 1993, p. 17). It is worthwhile to stress that although studies that used framing incentives were usually successful in encouraging greedy allocations, this success should not be taken as supportive of the strategic model, because these extrinsic manipulations are not accounted for by game theory. Auctioning the positions (Giith and Tietz, 1986), or winning the property rights to Allocate (Hoffman et al., 1994) involve, by their nature, intervening variables that are extrinsic to the game which are not alluded to by game theory. In contrast, modified games that

5 R. Suleiman / Journal of Economic Psychology 17 (1996) manipulate structural incentives, like the Binmore et al. (1985), and the Weg and Smith (1993) modified Ultimatum games, manipulate structural variables that are accounted for by standard game theory. Therefore the effects of such manipulations seem more appropriate for testing the theory. The present research constitutes an additional effort to understand the behaviors of Allocators and Recipients in the Ultimatum game. Following Weg and Smith (1993), the main objective of this research is to test the hypothesis that the expectations held by Allocators regarding the reservation values of the Recipients, as well as the Recipients' positive reservation values, constitute a 'correct' response to the structure of the game that is not captured by the subgame-perfect equilibrium. A modified ultimatum game, called the 6-Ultimatum game, is proposed and tested in the present study. Section 2 provides a brief description of the new game. Section 3 details the research hypotheses. Section 4 describes the experiment set to test the hypotheses and details its results. Section 5 is a concluding discussion. 2. The 6-Ultimatum game The modified Ultimatum game, which I call the 8-Ultimatum game, could be described as follows: Player 1 is assigned the role of an Allocator of a 'cake' of size M between himself and player 2 (the Recipient). To any offer [M- x, x] stating the Allocator's and Recipient's shares respectively, the Recipient can reply by either accepting or rejecting. If he accepts, the offer is realized. But if he rejects, he imposes the outcomes (M- x)* 8 and x * 8 (for the Allocator and Recipient respectively), where 8 is a commonly known discount factor (0_< 8_< 1). Whereas a rejection results in shrinking the shares of both players by a factor of 6, the relative share allocation x/(m-x) is unaffected by the Recipient's response. This means that although the Recipient has some control over the absolute sizes of the offers, his reply cannot affect the relative allocation. Note that unlike previous structural modifications of the Ultimatum game, the 8-Ultimatum game is a single-period game. Also note that for 6 = 0, the 8-Ultimatum game reduces to the standard Ultimatum game, while for the other extreme, 8 = 1, it reduces to the Dictator game. Thus, the two games can be viewed as special cases constituting the two end points of the continuum of all possible 8-Ultimatum games. The game-theoretic prediction for the &Ultimatum game is straightforward: common knowledge regarding the rationality of both players will lead the

6 536 R. Suleiman / Journal of Economic Psychology 17 (1996) Allocator to demand the whole 'cake' minus an infinitesimally small e (e > 0). The Recipient, being rational, must accept this offer, since a rejection from her side might cause her share to shrink by 6 ( 6 < 1). For the case where 6 = 1, the Recipient will be indifferent between accepting or rejecting any offer. The Allocator in this case is practically a Dictator, and a rejection from the Recipient does not affect the allocation. Although the subgame-perfect equilibrium is indifferent to 6, it makes sense to conjecture that the expectation of the Allocator regarding a Recipient's possible rejection will depend on the latter's efficacy in determining the final allocation, which, in turn, is a function of 6. Specifically, it is reasonable to assume that when playing under low 6 conditions, Allocators will be more afraid to make low offers than when playing under high 6 conditions, and that this fear will cause them to raise their offers. Moreover, it is reasonable to assume that the efficacy of the Recipient (as determined by 6) will influence her level of aspiration regarding what she might consider as an acceptable share, and consequently the probability that she will accept or reject any certain offer. If these conjectures are valid, then by continuously changing ~ one can manipulate the players' expectations, aspiration levels, and behaviors. The conjectures regarding the Allocators' offers are supported by results from previous experiments which compared offers made under the extreme cases of 3 = 0 (standard Ultimatum) and 6 = 1 (Dictator) conditions (Forsythe et al., 1988; Hoffman et al., 1994). It is important to stress that since 6 is a structural factor, any obtained effect will be due to structural incentives that are embedded in the game form. This implies that the confirmation of the above conjectures will call into question the appropriateness of the subgame perfectness for the 8- (and standard) Ultimatum games. 3. Hypotheses Based on the previous analysis, the following hypotheses can be formulated: HI: H2: Allocators' offers in /~-Ultimatum games are inversely related to 8, such that 'strong' Recipients playing under small 6 conditions will receive higher offers than 'weak' Recipients playing under large 6 conditions. Recipients' rates of rejection are inversely related to 6, such that for a certain offer, higher rates will occur under lower 6 conditions and vice versa.

7 R. Suleiman /Journal of Economic Psychology 17 (1996) H3: For the same offers, Allocators will give higher estimates that their offers will be rejected when playing under low 6 conditions (i.e., with a 'strong' Recipient) than under high 6 conditions (i.e., with a 'weak' Recipient). H4: 'Weak' Recipients (playing under higher /~ conditions) will be more satisfied by a certain offer than will 'strong' Recipients (playing under lower 6 conditions). 4. The experiment 4.1. Method Subjects Three hundred subjects, all students at the University of Haifa, participated in the experiment. They all responded by writing their names and telephone numbers on a notice inviting students to participate in an 'economic experiment'. They where told that they could earn as much as NIS 40 (about $13) depending on their performance Equipment A DEC PDP 11/73 computer was used to conduct the experiment. The computer controlled four CRT terminals, each located in a private and soundproof booth. A fifth terminal was used by the experimenter to monitor the experiment Procedure Subjects were invited to the laboratory in groups of four. On arrival, each subject was admitted separately, seated in a booth, and given detailed written instructions (see Appendix A). Special efforts were made to prevent subjects from meeting or even seeing each other before the experiment. Following the arrival of the fourth participant, the computerized experiment started. First, the experimenter made sure that all participants completely understood the game. Then the four participants were randomly paired into two dyads, and in each dyad one participant was randomly ascribed the role of Allocator, and the other the role of Recipient. Each dyad then played a one shot 6-Ultimatum game, for a 'cake' of NIS 40 and one of five 6 values: 6= 0, 0.2, 0.5, 0.8, and 1. Following the Allocator's offer and the Recipient's reply (accept/reject), each participant responded to ten computerized questions.

8 538 R. Suleiman / Journal of Economic Psychology 17 (1996) The questions for the Allocator were: la. What are the chances (in %), in your opinion, that the Recipient will accept the offer? 2a-9a. What is the likelihood that the Recipient would have accepted an offer of NIS x? (with x equalling 2, 4, 6, 8, 10, 12, 15, and 20, for questions 2a to 9a respectively). 10a. If you were in the Recipient's place, what amount you would have wanted to be offered? The questions for the Recipient were: lr. How satisfied are you (on a scale of 1-10) by the offer that was made to you? 2r-9r. How satisfied you would have been if the Allocator was to offer you NIS y? (with y equalling 20, 15, 12, 10, 8, 6, 4, and 2, for questions 2r to 9r respectively). 10r. If you were in the Allocator's place, what is the amount in NIS that you would have offered the other participant? Thirty dyads were randomly assigned to each one of the five 6 conditions, for a total of 150 dyads. When the experiment ended, each participant was informed via the CRT about his or her earnings. All participants were requested to wait patiently for the experimenter, who then paid each one in his or her booth, and released him or her from the laboratory. Care was taken that participants be sent out one at a time, thus preventing them from meeting after the experiment Results Analysis of the Allocators' offers HI predicted that Allocators' offers will decrease with 6. Fig. 1 depicts the frequency distributions of the offers under the five 6 conditions. The dark segments of the bars in each panel of Fig. 1 represent accepted offers, while the shaded segments represent rejected ones. As the panels show, the modal offer under all 6 conditions is NIS 20, the split. The percentages of Allocators who offered their counterparts at least 50% of the 'cake' were 54%, 60%, 54%, 34%, and 30% under 6 conditions 0, 0.2, 0.5, 0.8 and 1 respectively. Note that while for 6 = 0 and ~ = 0.2 no offers were made that were below NIS 5, there were several zero and below NIS 5 offers for = 0.5, 0.8 and As shown, almost all zero and very small offers in the

9 R. Suleiman / Journal of Economic Psychology 17 (1996) a=o0 6=02 ~ Rejected =Accepted] fg ~.,0I 6 'T i+~... =.I. Offer (NIS) I t2 ~o~! ml,,i,~.~+ ~,m, 0,, ~[~,,,I......, t6 1; 1~ Offer (NIS) I 6=05 5=08 is,- 14 lz ~-, r. '" e t O 22 Offer (NIS) ~=10 l,,,,,.,,... I,, 1 2 G G 1/; 18 1~ 2o 22 Offer (NIS) IS e 4 2 O Offer (NIS) Fig. 1. The frequency distributions of offers under each 6 condition. = 0.5 and 6 = 0.8 conditions were rejected. (No rejections could be made at the 6 = 1 condition, since this is a Dictator game.) A quantitative comparison among all five pairs of distributions in Fig. 1 was

10 540 R. Suleiman / Journal of Economic Psychology 17 (1996) ~ m+ + 15i... :: C m 14~ ;\ ; 13~.~ ~.~] I Condition Fig. 2. Mean offers as a function of 6. performed using the Kalmagorov-Smirnov test. The comparisons revealed significant differences (p = 0.05) between each of the two distributions corresponding to offers under smallest 6 conditions (6 = 0 and 6 = 0.2) and the distribution under the 6 = 0.8 condition. When tested under a significance level of p = 0.1 (instead of 0.05), the two smallest 6 distributions are also significantly different from the distribution corresponding to 6 = 1. Fig. 2 depicts the mean offer as a function of 3. Although Fig. 2 looks somewhat like an inverse s-shaped function, its main tendency is supportive of H1. Notwithstanding the violations at both ends of the graph, the mean offer exhibits a noticeable decrease as 6 increases from 0.2 to 0.5, and then to 0.8. The mean offers for the five 6 conditions 0, 0.2, 0.5, 0.8, 1 are NIS 16.70, 17.03, 13.47, 10.77, and respectively. A one-way ANOVA on the offers with 6 as the independent variable revealed a significant effect (F(4,145) = 5.24, p < 0.01). A Student-Newman- Keuls test revealed significant differences between each of the mean offers for the lowest 3 conditions (6 = 0 and 6 = 0.2) on one hand, and the mean offers for the two highest 6 conditions (6 = 0.8 and 6 = 1). It is interesting to note that the mean proportion offered under the standard Ultimatum condition (3 = 0) is 16.7/40 = , an almost identical result to the one reported by Giith et al. (1982), who reported a mean offer to Recipients of It is also very close to the results obtained by Kahneman et al. (1986) of 0.476, 0.447, and 0.421, for the Psych/Psych, Psych/Commerce, and Commerce/Psych conditions respectively. Hence, the robustness of the mean offer result is shown to hold under a variety of designs, 'cake' sizes, and cultural backgrounds.

11 R. Suleiman / Journal of Economic Psychology 17 (1996) Analysis of the Recipients' replies Table 1 summarizes the rejection rates (number of rejections divided by number of offers) by condition, 6, and size of offer, x, in NIS. For the sake of convenience, the offers are grouped in categories of NIS 3 each, except for the last category, which includes all rejections for offers of NIS 12 or more. H2 predicted that for a given offer, higher rates of rejection will occur under lower 6 conditions. Unfortunately, the rejections data do not allow this hypothesis to be tested, because no small offers were made under low 6 conditions. The results presented in Table 1 show that the overall rejection rate for a -- 0 is only about 7%, a result quite similar to the 8.3% reported by Hoffman et al. (1994) under the random condition. As 6 increases, the rejection rate increases monotonically, reaching a relatively high rate of about 37% under the 6 = 0.8 condition. A chi-square test on the frequencies of rejections resulted in the rejection of the null hypothesis stating that the rejections are evenly distributed across the 6 conditions (X2(3)= 10.4, p < 0.05). A simple examination of Table 1 (or Fig. 1, middle panels) shows that the high rejection rates at the two largest 6 values (0.5 and 0.8) occurred for relatively small offers, in the range of 0-2. Two complementary explanations may account for this result. First, one can argue that Allocators erroneously expected that Recipients would recognize their 'weak' position and accept very small offers. Second, one can argue that Allocators were not seriously concerned by the possibility that Recipients would reject small or even zero offers, since this would still leave them with a considerable amount of the 'cake'. This is especially true for the 'almost Dictator' condition at 6 = 0.8, under which an almost certain rejection of a zero offer resulted in a reduction of the Allocator's maximal gain by only NIS 8. A highly self-interested and risk-averse individual may be willing to pay that price for the 'riskless option' of offering zero to the Recipient (and expecting certain Table 1 Rejection rates by condition (6) and size of offer (x) Offer (x) (NIS) Condition (6) Overall /0 0/0 6/7 8/8 14/15 (93%) / 1 0/0 0/0 1/3 2/4 (50%) 6-8 0/1 1 / 1 0/0 0/0 1/2 (50%) /3 2/4 2/4 2/6 6/17 (35%) /25 1/25 1/19 0/13 3/82 (4%) Overall 2/30 4/30 9/30 11/30 26/50 (17%) (7%) (13%) (30%) (37%)

12 542 R. Suleiman / Journal ~" Economic Psychology 17 (1996) rejection), rather than allocating more fairly while incurring the risk that such offers may still be rejected. The first explanation, alluding to faulty expectations, is partially supported by the estimates data analyzed in the following section, but it seems less likely for the case of zero offers, which constitute the bulk of rejected offers as can be seen from Fig Analysis of Players' estimates Question 10a elicited the Allocator's aspired demand (AD) from the 'cake', had he been in the Recipient's place, and Question 10r elicited the Recipient's hypothetical offer (RO) to his counterpart had he played the role of the Allocator. Table 2 depicts the means and standard deviations of AD, RO, and the actual offer x, for each 6 condition and across conditions. The table shows that although the means of the Allocator's demand (AD) and the Recipient's hypothetical offer (RO) exhibit some tendency to decrease with 6, this tendency is considerably less than the decrease in the actual offers. The results of two separate one-way analyses of variance for AD and RO showed that the effect of 3 on the means of AD and RO was not significant (p > 0.1). The Pearson correlation coefficients between x and RO, and between AD and RO, were zero. The Pearson correlation between x and AD was It seems that although there was some consistency between Allocators' offers and their expectations regarding how much they would expect to get as Recipients, they aspired that they would be treated more fairly than they actually treated their counterparts. This is especially evident in conditions ~ = 0.8 and 1, under which the Recipient can barely influence the final allocation. In the same line, the Recipients' 'generous' hypothetical offers (RO's) under all 6 conditions do not seem highly credible when compared with actual offers. Question 1 to the Allocator asked her to estimate the likelihood (in %) that the Recipient would accept her offer, and Question 1 to the Recipient asked her Table 2 Means (and standard deviations) of Allocators' aspired demands and Recipients' hypothetical offers Condition ( 6 ) Overall Offer (x) (4.47) (4.07) (8.02) (8.05) (4.46) (7.07) Allocator's aspired demand (AD) (3.52) (3.95) (7.06) (8.63) (7.92) (6.59) Recipient's hypothetical offer (RO) (5.85) (5.28) (6.11) (8.95) (7.31) (6.88)

13 R. Suleiman / Journal of Economic Psychology 17 (1996) to state her level of satisfaction (on a 1-10 scale) with the offer she received. Table 3 summarizes the means and standard deviations of the Allocators' Estimates of Acceptance (EA), and the Recipients' Levels of Satisfaction (LS) for each 6 condition, and across conditions. In addition to total means, the table details separate results for accepted and for rejected offers. The total means for the Estimates of Acceptance (EA) show that on average, the Allocator's estimate that the Recipient will accept her offer decreases as 6 increases. This is quite consistent with the decrease in the mean offer. The Pearson correlation coefficient between the Allocator's offers and her estimates of acceptance is.59 (0.53 and 0.59 for accepted and rejected offers, respectively). A comparison of the overall mean Estimates of Acceptance for accepted and rejected offers (82.6% and 55.6%, respectively) indicates that, to some degree, Allocators were able to predict their counterparts' replies. To test the above observations, a 4 X 2 MANOVA was performed on EA, with ~ and the Recipient's reply (accept/reject) as independent factors. The analysis revealed two significant main effects (F(3,112) = 3.36, p < 0.05, and F(1,112) = 15.75, p < 0.01 for 3 and the reply respectively). The interaction between the two variables was not significant (F(3,112) = 1.11, p > 0.1). A Student-Newman-Keuls test for comparison between the means revealed significant differences between each of the Estimate of Acceptance means corresponding to the two lowest 6 (6 = 0 and 8 = 0.2) and the two Estimates of Table 3 Means (and standard deviations) of Allocators' estimates of acceptance and Recipients' levels of satisfaction Condition ( 6 ) Overall Allocator estimate of acceptance (EA) in % Accepted offers ] a 85.5[26] ] ] Rejected offers 85.0 [2] 71.3 [4] 47.1 [9] 51.6 [11] Total (16.7) b (18.8) (33.5) (35.4) (28.1) Recipient let,el of satisfaction (LS) (Scale 0 10) Accepted offers 7.1 [ [26] 7.9 [21] 6.6 [191 - Rejected offers 1.5 [2] 1.0 [4] 1.4 [9] 0.6 [11] - Total (3.3) (3.2) (4.2) (3.9) (3.9) The figures in brackets indicate frequencies of accepted or rejected offers. b The figures in parentheses are standard deviations. 7.4 (2.9) 3.0 (3.6) 5.8 (3.8)

14 544 R. Suleiman / Journal of Economic Psychology 17 (1996) Acceptance corresponding to the higher 6 values ( 6 = 0.5 and 6 = 0.8). Recall that a similar significance pattern was found between the mean offers. Turning to the Recipients' mean Level of Satisfaction (LS), it is immediately noticeable that they rank in exactly the same order as the mean offers. The Pearson correlation coefficient between the Level of Satisfaction and the offer is 0.88 (0.86 for accepted offers and 0.82 for rejected offers). A $ by Recipient's reply MANOVA revealed a significant effect for the reply (accept/reject) (F(1,141) = 81.31, p < 0.01), as well as for 6 (F(4,141) = 4.71 p < 0.01). The interaction between the two variables was not significant (F < 1). The comparison between mean Levels of Satisfaction (LS) for the various 6 conditions using the Student-Newman-Keuls test revealed significant differences between the Level of Satisfaction means corresponding to 6 = 0 and 6 = 0.2, and the means corresponding to 6 = 0.8, and 6 = 1. The picture emerging from the analysis of Allocators' Estimates of Acceptance and the Recipients' Levels of Satisfaction is fairly simple. Under the larger 8 conditions, Allocators offer, on average, smaller portions of the 'cake' and consistently report lower likelihoods that their offers will be accepted. These estimates are quite consistent with the Recipients' Levels of Satisfaction. I turn now to summarizing data from questions 2a to 9a, concerning the Allocators' estimates that the Recipients will accept each one of eight hypothetical offers in the range 2-20, and from questions 2r to 9r, concerning the Recipients' self-reported Levels of Satisfaction with the same offers. Fig. 3 depicts, for each ~ condition, the means of the Allocators' Estimates of Acceptance for Hypothetical Offers (EAHO) as a function of the Hypothetical 100~... ~ ~ 60-0 m E 3o!/ ii ~ J~- 1 t~ lo! o!! Size of hypothtical offer (NIS) Fig. 3. Means of Allocators' estimates of acceptance for each ~ condition, as a function of hypothetical offers.

15 R. Suleiman / Journal of Economic Psychology 17 (1996) _=_ C 0.2 o 0.5 ~ 0.8 ~ 1.0 "i i Size of hypothetical offer (NIS) Fig. 4. Means of Recipients' level of satisfaction for each ~ condition, as a function of hypothetical offers. Offer Size (HO), and Fig. 4 depicts the means of the Recipients' Levels of Satisfaction with the same offers (LSHO). Fig. 3 clearly shows that the mean Estimate of Acceptance for Hypothetical Offers (EAHO) for ~ = 0.8 is noticeably larger than the corresponding mean EAHO obtained under the remaining 6 conditions. A 6 X Hypothetical Offer 4 X 8 analysis of variance on EAHO, with 6 as a between subject condition, and Hypothetical Offer as a within subject condition, revealed a significant effect for 6, in addition to the obvious main effects for HO (F(3,116)= 3.76, p < 0.01, and F(7,812) = , p < 0.01 respectively). The interaction between these two factors was also significant (F(21,812)= 2.35, p < 0.05). A comparison between EAHO means for the different ~ conditions using the Neuman-Keuls method showed that the mean EAHO for 6 = 0.8 is significantly higher (p = 0.05) than the mean EAHOs for the other three 6 conditions (6 = 0.0, 0.2, and 0.5). Separate comparisons for each one of the eight hypothetical offer values replicated the above finding for Hypothetical Offer values NIS 2, 4, 6, and 8. For Hypothetical Offer values NIS 10 and NIS 12, the comparisons revealed a significant difference between the mean EAHO for 6 = 0.8 and the mean EAHOs for 6 = 0.2 and 0.5, but not for the differences between the means for 6 = 0.8 and 6 = 0.0. All other differences were not significant. A similar analysis conducted on the Recipients' Levels of Satisfaction with Hypothetical Offers (LSHO) revealed a significant effect for the hypothetical offer, as expected (F(7,1015) = , p < 0.01). On the other hand, the effect of 6 was barely significant (F(4,145)= 2.15, p < 0.07), and the interaction

16 546 R. Suleiman / Journal of Economic Psychology 17 (1996) between these two factors was not significant (F(28,1015) , p > 0.1). Separate comparisons for each one of the eight Hypothetical Offer values (despite the non-significant interaction), revealed a significant difference at HO = NIS 8 between the Recipient's mean Level of Satisfaction at 3 = 1 and the corresponding means obtained at t5 = 0.2 and ~ = 0.5. Another significant difference emerged for HO = NIS 10 between the mean LSHO at 6 = 1 and the corresponding means obtained at 6 = 0.0, 0.2, and 0.5. The results of the above analysis are only partially supportive of H3 and H4. The prediction that compared to Allocators playing under low 6 conditions, Allocators playing under high 6 conditions would estimate that Recipients would be more accepting of a certain offer received partial support. Specifically, this prediction held true only for the comparison of the estimates given by Allocators playing under the 6 = 0.8 condition with the estimates of Allocators playing under all the lower ~ conditions. Similarly, only Recipients playing under the t5 = 1 (Dictator) condition, and to some extent Recipients playing under the ~ = 0.8 condition, reported Levels of Satisfaction that were significantly higher than the Levels of Satisfaction reported by Recipients playing under smaller ~ conditions. 5. Discussion Studies of Ultimatum games make reference to three major hypotheses: the game-theoretic hypothesis predicting that the Allocator will behave strategically and demand the whole 'cake' for himself; the fair-share hypothesis predicting that the Allocator will split the 'cake' equally between himself and the Recipient; and the 'expectations' hypothesis predicting that the Allocator will base his behavior on his expectations and common knowledge regarding likely responses by the Recipient. Giith and Tietz (1986), and more recently Hoffman et al. (1994) have shown that equitable or 'fair' offers can be significantly reduced by asserting the property rights of the Allocators. Based on their results, Hoffman et al. concluded that the Allocators' offers are "due to strategic and expectations considerations" (Hoffman et al., 1994, p. 346). It is argued here that in spite of the success of property rights manipulations in inducing more strategic offers, their appropriateness for testing noncooperative game theory predictions regarding Allocators' offers is questionable. As noted by Hoffman et al., "noncooperative, nonrepeated game theory is about strangers with no shared history... they meet, interact strategically in their individual self-interest according to well-specified rules and payoffs, and never meet again. These stark conditions

17 R. Suleiman / Journal of Economic Psychology 17 (1996) are necessary to assure that the noncooperative, nonrepeated game theoretic prediction for the interaction is not part of a sequence with a past and a future" (Hoffman et al., 1994, p. 347). With this in mind, it seems reasonable to assume that property rights manipulations violate these conditions, since they inevitably introduce some form of 'history' to the players' relationships even when the game is played once and the players remain completely anonymous to each other. Property rights manipulations are also inappropriate for testing the expectations hypothesis regarding the Allocators' offers in standard Ultimatum games. This is due to the fact that the establishment of the first mover property right requires that both players update their aspiration levels, and consequently their expectations, in favor of the right holder. As such, property rights manipulations induce different expectations than those derived intrinsically from the structure of the game. The modified ultimatum game implemented in the present study overcomes the two shortcomings of property rights manipulations. The 6 parameter introduced to the game constitutes a mere structural modification that does not violate the basic assumptions of noncooperative game theory. Not only does this modification not impose any 'history' or future on the game form, but it also enables to manipulate the Allocators' expectations intrinsically, thus allowing a more appropriate test of the expectations hypothesis. The results of the present study strongly refute the prediction of the noncooperative game theoretical model, and add evidence to a considerable body of research that clearly shows that the standard strategic model simply does not work. Allocators offered substantial portions of the 'cake' under all 3 conditions and their offers were strongly dependent on the size of 3, all in contrast to the game-theoretic prediction of practically zero offers under all 6 conditions. As in many previous studies (Giith and Tietz, 1986; Forsythe et al., 1988; Bolle, 1990; Carter and Irons, 1991; Prasnikar and Roth, 1992), the modal allocation in the present study was the split under all 6 conditions. The mean proportional offer for the standard Ultimatum condition was about 42% of the 'cake', a result that seems quite stable across various experimental conditions (Gtith et al., 1982; Kahneman et al., 1986; Bolle, 1990). Although these results can be taken as supportive of a fairness explanation, a careful examination of the results raises at least two reservations regarding such a conclusion. First, note that although the equal split was the modal offer for all 6 conditions, the percentage of Allocators who offered an even split decreased significantly with 6 (from 54% for 6 = 0.0 and /~ = 0.5, and 60% for 6 = 0.2 to 34% for 6 = 0.8 and 30% for 6 = 1). If fairness alone were the motive behind such offers, then the percentage of Allocators offering even splits should be

18 548 R. Suleiman / Journal c~]" Economic Psychology 17 (1996) invariant with 6. Secondly, although Allocators playing under conditions 6 = 0.0 and 0.2 made no offers below NIS 5 (12.5% of the 'cake'), many offers made under higher 6 conditions were zero or very small (7 (23%) for 6 = 0.5 and = 1, and 8 (27%) for ~ = 0.8). This indicates that when playing with 'weak' Recipients, with whom Allocators do not have to fear the consequences of rejection, a substantial number of Allocators showed no benevolence at all. Notwithstanding the fact that norms of fairness cannot be ruled out as an explanation, most of the results of the present study seem to favor an 'expectations' explanation for the Allocators' offers. As indicated earlier, the results clearly show that the Allocators' mean offers, as well as the percentage of Allocators who offered equal splits, were strongly related to 6. Although less conclusive, the analysis of the Levels of Satisfaction with Hypothetical Offers (LSHO) adds support to the expectations hypothesis by showing that Allocators predicted that Recipients playing under the ~ = 0.8 conditions would be more satisfied with a certain offer than would Recipients playing under smaller 6 conditions (see Fig. 3). The results of the Recipients' Levels of Satisfaction from Hypothetical Offers were partially supportive of H4, that predicted that 'weak' Recipients playing under higher ~ conditions would be more satisfied by a certain offer than would 'strong' Recipients playing under lower 6 conditions. Unlike the data from the Allocators' offers, the Recipients' data were insufficient for a satisfactory analysis of their behavior, especially for testing the hypothesis regarding the inverse relationship between their rejection rates and 6. The results in Table 2 show that the highest number of rejections (14 out of 26) occurred for very small offers in the range 0-2 NIS, and under high 6 conditions (0.5 and 0.8). It is reasonable to assume that even 'weak' Recipients playing under these conditions were willing to pay the relatively low cost of one or two NIS in order to punish 'unfair' Allocators who offered them almost nothing. The second highest number of rejections occurred for offers in the range 9-11 NIS (see Table 2) but their small number and the corresponding pattern of rejection rates across the 6 conditions make it hard to use this data to explain the Recipients' behaviors. Taken together, it seems fair to conclude that the picture emerging from the Recipients' results remains largely unclear, and certainly insufficient for the understanding of Recipients' behaviors in ultimatum situations. An important question related to Allocators' behaviors in ultimatum and other resource allocation situations concerns the issue of fairness. A frequently voiced stand in this respect posits that what might look like a fairness-driven offer in ultimatum situations is not necessarily so. In support of this position, Harrison

19 R. Suleiman / Journal of Economic Psychology 17 (1996) and McCabe (1992) provide an interesting analysis of data from Carter and Irons (1991) study. This analysis suggests that rational and self interested Allocators may choose to make fair allocations not because of cooperative or altruistic motives, but out of pure egoistic ones. In the Carter and Irons study, subjects were requested to enter strategies for both an Allocator and a Recipient, and the designation of roles actually played was made after all strategies were submitted. The allocation was made by randomly pairing the subjects, and then letting the winner of a word-game be the Allocator. Based on these data, Harrison and McCabe computed the Allocators' expected payoffs from demanding 100%, 90%, 80%, 60%, or 50% of the 'cake'. Their results showed that an expected payoff maximizer should choose the 50% (equal) split. Another convincing evidence in this respect was given by Kravitz and Gunto (1992) who conducted two experiments designed to study the replies and expectations of Recipients in a one-shot standard Ultimatum game. Their study contained a pre-test question that elicited the minimum acceptable offer by each subject. The authors used the cumulative distribution of the minimum acceptable offers to calculate the expected value for the Allocator from each possible offer between $0.5 and $2.5. The resulting expected value function shows that the optimal offer guaranteeing a maximal EV to the Allocator is $2 (40% of the entire cake), a size that is not far from the equal split. Yet when the subjects were asked what offers they would make if they knew the other would accept anything greater than zero, only 11% said they would offer at least an even share. The modal response (42%) was 1 cent. The authors conclude by quoting from Walster et al. (1973) that "so long as individuals perceive that they can maximize their outcomes by behaving equitably, they will do so, should they perceive that they can maximize their outcomes by behaving inequitably, they will do so" (Walster et al., 1973, p. 153). Did Allocators in the present study try to maximize their expected payoff? And if so, can a maximization of expected payoff model account for even splits as well? To answer both questions, I first computed for each Allocator the prediction of a Subjective Expected Values (SEV) model. Each Allocator's estimates for the likelihood that the Recipient would accept each one of several hypothetical offers (see questions 2a-9a 1) was used to calculate his respective expected payoffs. The SEV model prediction was chosen as the offer that yields i In addition to the expected values for the offers NIS 2, 4, 6, 8, 10, 12, 15, and 20, the expected value for a zero offer was also calculated, under the reasonable assumption that such an offer would be rejected with certainty.

20 550 R. Suleiman / Journal g/" Economic Psychology 17 (1996) Table 4 Frequencies of correct predictions by the SEV and FS models for each 6 condition Model Condition ( 6 ) SEV 19 (15) 15 (13) 12 (9) 8 (5) - 63% 50c/~ 40% 27% FS 17 (15) 18 (18) 16 (16) 10 (10) 10 (9) 57% 60c~ 53% 33% 33% SEV and FS 12 (11) 12 (12) 4 (4) 0 (0) - 40% 40cA 13 % 0% Neither SEV nor FS 6 ( 11 ) 9 ( I 1 ) 6 (9) 12 (15) - 20% % 40% Total % 100'/, 100% 100% 100% the highest expected value. Table 4 summarizes the results of the SEV model predictions, and contrasts them with the predictions of a simple Fair-Share (FS) model. Each entry with no brackets indicates the number of Allocators for whom the prediction of the respective model(s) was accurate up to a NIS 2 deviation (5% of the entire 'cake') from the actual offer. The corresponding numbers within the brackets, when appearing, give the results of accurate point predictions. The first two rows give the results of correct predictions for the SEV model and the FS model respectively. The third row gives the numbers of Allocators for whom both models gave correct predictions, and the fourth row presents numbers of Allocators for whom neither model yielded correct predictions. An examination of Table 4 shows that both models perform fairly well for most 6 conditions. The percentages of correct prediction of the SEV model are 63%, 50%, 40%, and 27%, for conditions 6 = 0.0, 0.2, 0.5, and 0.8 respectively, while the predicted percentages for the Fair-Share model are 57%, 60%, 53%, 33%, and 33% for conditions 6 = 0.0, 0.2, 0.5, 0.8, and 1 respectively. More interesting are the results for joint correct predictions. These results show that for the standard Ultimatum game (6 = 0), 40% of the offers (12 out of 30) are correctly predicted by each of the two models. One way to interpret this result is that 71% (12 out of 17) of the split offers are also optimal in that they maximize the Allocator's SEV. A similar conclusion applies to the 'near Ultimatum' (8 = 0.2) condition (see Table 4). For the mid-range case of 6 = 0.5, only 13% (4 out of 30) of the offers are jointly predicted by the two models. These constitute only 25% of the splits. The cell corresponding

21 R. Suleiman / Journal of Economic Psychology 17 (1996) to joint prediction for the 'near Dictator' case (6 = 0.8) is empty. Nevertheless, under this condition a sizable minority of 33% (more than the number of SEV maximizers) offered a split. Not surprisingly, an identical proportion of Allocators offered the split under the Dictator condition. Based on the above analysis it seems reasonable to conclude, like Harrison and McCabe (1992), and Kravitz and Gunto (1992), that when playing with a 'strong' recipient (under small 6 condition), the likelihood that unfair offers will be rejected caused many allocators to offer the split, and that most of them did so not only because this was the fair thing to do, but also, and maybe primarily, because it paid to do so. On the other hand, the 'near Dictator' condition provided clear evidence that a sizable minority split the 'cake' fairly, even when as rational payoff maximizers they could do much better. Although nonequilibruim 'fair' offers can be explained as rational self-interested behaviors by Allocators who fear that Recipients might reject relatively small offers, Recipients' concerns for fairness may stem from completely different motives. Based on a series of ultimatum experiments, Murnighan and Pillutla (1995) concluded that the reasons for nonequilibrium replies are essentially not strategic, but rather psychological and moral. The results of the present study do not allow a deep analysis of Recipients' motives. They do, however, show (a) that Allocators expected to be treated more fairly (if they were to play as Recipients) than they actually treated their counterparts, and (b) that Recipients report that they would be 'generous' if they were to play as Allocators. Both of these results are consistent with Murnighan and Pillutla's conclusions. Notwithstanding the advantages of the 6-Ultimatum game alluded to before, the game suffers from at least one methodological disadvantage. As implemented in this study, both the efficacy of a Recipient (i.e. his 'power' to affect the Allocator's outcome if he chooses to reject the latter's offer) and his personal cost in doing so, are determined by the same parameter. Under the 6 conditions tested in this study, a Recipient could inflict a higher loss on the Allocator under ~ = 0 than under ~ = 0.2, but contingently, his personal loss would be higher under 6 = 0 than under 6 = 0.2. A rational Allocator might recognize this dependency and feel less threatened by his Recipient when playing under the 6 = 0 than under the condition. Although not significant, the mean offers for 6 = 0 and 6 = 0.2 (see Fig. 2) are consistent with such reasoning. An improved variant of the 6-Ultimatum game is easily obtained by ascribing different 6 values (61, 62) for the two players. Such improvement allows for separation of the Recipient's potential 'threat' to reject an offer from the personal cost that he incurs if he chooses to do so.

22 552 R. Suleiman / Journal of Economic Psychology 17 (1996) Acknowledgements This research was supported by research grant SES from NSF. I wish to thank Orly Kloner and Irk Bach for their assistance in data collection and Yuval Samid for his help in the data analysis. I also wish to thank David Boninger, David V. Budescu, Faith Gleicher, David Messick, Sam Rackover, Larry Samuelson and two anonymous reviewers for their helpful remarks. Appendix A. Instructions (A) The following are the instructions for the ~ = 0.5 condition. The instructions for the 6 = 0.0, 0.2, and 0.8 conditions are identical, except for the value of 8. You are about to participate in an economics experiment which involves only two persons. In addition to you, there are three other participants in the adjacent booths. You will be randomly paired to only one of them. The remaining two will participate in an experiment that will run parallel to the one that you will participate in. This arrangement was adopted so that no one of the four participants will be able to tell who his partner is. In other words, you will not be able to know who out of the three others is your partner for the experiment. Also your partner for the experiment will not be able to know who you are. The experiment will be conducted as follows: a sum of NIS 40 will be allocated to each pair, and one of the two participants will be requested to propose how to divide it between himself and the other participant. All that the Allocator has to do is to specify: (1) how much from the overall amount he/she wants to give to himself/herself, and (2) how much he/she wants to give to the other participant (this amount should equal NIS 40 minus the amount that the Allocator wanted for himself/herself). This is done by typing the two amounts on the computer keyboard. When the Allocator keys in his/her proposal, it will appear on the other participant's computer screen, and he/she will have to reply by stating whether he/she accepts it or rejects it. This is done by using one of two keys designated for this purpose. If the Recipient chooses to accept the proposal, then the total amount will be divided exactly as proposed by the Allocator. If the Recipient chooses to reject the proposal, then each one will get an amount that equals the amount proposed by the Allocator multiplied by a factor of 0.5.

23 R. Suleiman / Journal of Economic Psychology 17 (1996) For example, if the Allocator proposes to give himself/herself A (NIS), and the Recipient B (NIS) (A + B = 40), and the Recipient agrees, then the Allocator will be given A (NIS), and the Recipient B (NIS). But if the Recipient rejects the offer, then the Allocator will be given 0.5 * A (NIS), and the recipient 0.5 * B (NIS). In addition to the above, each participant will be requested to answer a short computerized questionnaire by keying in his/her answers on his/her terminal keyboard. The assignment of the role of Allocator to one participant and Recipient to another will be made before the experiment begins, by a random draw to be carried out by the computer. The results of the draw will be shown immediately on all the PC screens. When the experiment ends, each participant will be paid according to his/her gain. To prevent any contact between participants, each one will be dismissed alone. When you finish, please wait patiently for the experimenter to come in. (B) Instructions for the 6 = 1 (Dictator) condition. You are about to participate in an economics experiment which involves only two persons. In addition to you, there are three other participants in the adjacent booths. You will be randomly paired to only one of them. The remaining two will participate in an experiment that will run parallel to the one that you will participate in. This arrangement was adopted so that no one of the four participants will be able to tell who his partner is. In other words, you will not be able to know who out of the three others is your partner for the experiment. Also, your partner for the experiment will not be able to know who you are. The experiment will be conducted as follows: a sum of NIS 40 will be allocated to each pair, and one of the two participants will be requested to propose how to divide it between himself and the other participant. All that the Allocator has to do is to specify: (1) how much out of the overall amount he/she wants to give to himself/herself, and (2) how much he/she wants to give to the other participant (this amount should equal NIS 40 minus the amount that the Allocator wanted for himself/herself). This is done by typing the two amounts on the computer keyboard. When the Allocator keys in his proposal, it will appear on Recipient's computer screen. This proposal determines how much each participant will gain in the experiment. In addition to the above, each participant will be requested to answer a short computerized questionnaire by keying in his/her answers on his/her terminal keyboard.