ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART

Transcription

1 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART J. NEIL BEARDEN Abstract. In the basic ultimatum bargaining game two players, P 1 and P 2, must divide a pie (π). P 1 proposes a division in which he gets π x and P 2 gets x. P 2 can then accept the division, in which the π is split according to P 1 s proposal, or reject the proposal, in which case neither player gets anything. The current paper reviews empirical research on ultimatum bargaining games. It covers early work starting with Güth et al. (1982), but largely focuses on more recent work (post Roth (1995)). Taken together, the research suggests that P 1 s behavior in largely in accord with game theoretic income-maximization, but P 2 s behavior cannot be easily reconciled with standard game-theoretic assumptions. Rather, P 2 seems to be driven by a sense of fairness, specifically, a desire to be treated fairly by P 1. Both P 1 and P 2 behavior are in agreement with equity theory. The most important conclusion that falls out of this review is that players motivations, which often are not the ones posited by traditional game theory (and neo-classical economics, in general), and their perceptions of others motivations are of fundamental importance in understanding strategic interaction. Other ultimatum bargaining findings are reported as well. Future research directions are suggested throughout. Date: December 18,

2 2 J. NEIL BEARDEN Contents 1. Introduction 3 2. The Ultimatum Game 3 3. Important Early Findings (And Related Matters) 4 4. Equity and Fairness 7 5. Effects of Payoff Information Player Position Determination Ultimatum and Related Games Anonymity Psychological Mediators of Bargaining Decisions Effects of π Size Real vs. Hypothetical π Individual-Group Differences Learning in Ultimatum Games Cultural Differences in Ultimatum Bargaining Formal Refinements Evolution and Norms Conclusion 49 References 51

3 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 3 1. Introduction Agents are assumed to be self-interested income-maximizers in standard game theory. Though ubiquitous in economic theory, there is considerable evidence that this assumption is false. Economists may be shocked by this, but ordinary people probably are not. Social interaction is much richer than the beautiful abstractions of game theory, and motivational factors other than income-maximization, such as fairness, anger, and spite, seem to be equally strong determinants of behavior. The current paper reviews empirical research on ultimatum bargaining games, where violations of standard theory are the norm. As will be shown, successful descriptive theories of bargaining behavior must incorporate non-economic, psychological factors. The paper is organized as follows. The first section reviews results from early ultimatum bargaining experiments. These results are included primarily to set the stage for a review of more recent findings. Camerer, and Thaler (1995), Thaler (1988), and Roth (1995) can be consulted for more detailed summaries of early ultimatum bargaining experiments (see Güth (1995) for more of an insider s view of ultimatum bargaining research). Subsequent sections review findings from more recent ultimatum game experiments, most of which were designed to determine why people violate the dictates of game theory in bargaining situations. These intermediate sections cover the effects of payoff size, player anonymity, cultural differences, learning, and many other factors that have been studied in the context of ultimatum bargaining. Finally, the ultimate section considers the themes that emerge in this review, and suggests new avenues for ultimatum bargaining research The Ultimatum Game In the ultimatum game two players must divide a pie (π), which, unless otherwise specified, we will take to be a sum of money. The first player, the allocator, hereafter denoted P 1, proposes a division in which he receives d 1 = π x, where x [0, π], and the second player, the receiver, hereafter P 2, receives d 2 = x. If P 2 accepts the offer, then π is split according to the proposal; if P 2 rejects the offer, neither player receives anything. Denoting the proposed allocation d = (d 1, d 2 ), the subgame perfect Nash equilibrium proposal for the ultimatum game 1 Before we proceed it is important to point out the obvious: Research in different labs at different universities conducted by people with different backgrounds will certainly vary in many subtle and perhaps not so subtle ways. The current review, like all reviews, must brush aside many of these differences and grossly categorize experiments.

4 4 J. NEIL BEARDEN Notation Measure d 1 Demand for P 1 d 2 Offer to P 2 d = (d 1, d 2 ) P 1 s Proposal MAO Receiver s Minimum Acceptable Offer Table 1. Notation for ultimatum game measures. is d = (π ɛ, ɛ). 2 This solution follows from the following three assumptions: Assumption 1 (A1): Each player prefers a payoff of α to β whenever α > β. Assumption 2 (A2): Both players are aware of A1. Assumption 3 (A2): P 1 can calculate the optimal offer. Since by A1 P 2 prefers any positive payoff to a payoff of 0 and P 1 knows this by A2, P 1 can use backward induction (A3) to arrive at the subgame perfect Nash equilibrium (π ɛ, ɛ). 3 Some studies ask the P 2s for their minimum acceptable offer, which we abbreviate MAO. This notation is summarized in Table Important Early Findings (And Related Matters) Early work on ultimatum bargaining focused primarily on whether real bargainers (in contrast to idealized game-theoretic bargainers) adhere to game-theoretic prescriptions. We begin by briefly reviewing the first experiments on ultimatum bargaining by Güth et al. (1982) and the important exchange between Binmore, Shaked, and Sutton (1985) and Güth and Tietz (1987) that this work caused. 4 Related subsequent work is discussed as well. This section sets the stage for what follows. 2 Actually, (π, 0) can also be a Nash equilibrium if we assume that the P 2 will accept the an offer of 0 rather then choose at random between the offer and rejection of the offer, which both result in payoffs to the receiver of 0. However, throughout this paper I will work with the (π ɛ, ɛ) equilibrium. 3 Implicit in A1 is the assumption that a player s utility function takes a single argument: the player s own payoff. As will be shown below, this is clearly a false assumption. 4 This debate is a paradigmatic example of the role of one s philosophy in science. Though both Güth and Binmore are economists, only the latter seems, at least during the period of this early debate, to be committed to defending the classical position. Greater still, as we will see, is the difference between economists and noneconomists (e.g., psychologists) in underlying philosophy and how these philosophies color the way in which research is conducted and results are interpreted.

5 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 5 Güth, Schmittberger, and Schwarze (1982) conducted the first experimental study of ultimatum bargaining, and found that P 1s demands did not adhere to subgame perfect predictions, and that P 2s were willing to reject non-trivial offers. In the easy condition, subjects played two rounds of ultimatum bargaining separated by one week against different opponents with π from DM 4 to DM Of the 21 proposals in the first round, only 2 were consistent with the game theoretic prediction, and of these only 1 was accepted. More than a quarter (6 of 21) of P 1s offered a 50:50 division. None of these offers were rejected. The second round outcomes were not dramatically different from the first, though the rejection rate was slightly higher. They also compared ultimatum bargaining behavior within individuals. Using π = DM 7, they had 37 subjects give both their demands d 1 and MAO. A large number of the subjects, 17, offered more than their MAO (d 1 > MAO), 5 offered less (d 1 < MAO), and 15 gave offers consistent with their own minimum (d 1 = MAO). Again, the results were inconsistent with game theoretic predictions: Only 2 subjects demanded nearly all of π for themselves and only 2 were willing to accept very small offers (MAO DM.10.). The authors conclude: [S]ubjects often rely on what they consider to be a fair or justified result. Furthermore, the ultimatum aspect cannot be completely exploited since subjects do not hesitate to punish if their opponent asks for too much (p. 384). This conclusion from the early ultimatum game results has sparked a whole industry of research on bargaining behavior. Skeptical of the failure of game-theoretic predictions in ultimatum bargaining, Binmore, Shaked, and Sutton (1985) set up a two-period ultimatum game in which first period rejections lead to a game in which the size of π was reduced to δπ in period 2, where δ = The authors found that in the second game, in which P 2 from the first game played the role of P 1, the first period offers were close to equilibrium, which in this game is.75π, whereas the modal first period offers in the first game were around.50π. 7 The authors claimed that 5 The reader may consult the original paper for a description of the complicated condition. It does not concern us here. 6 One can use standard backward induction reasoning to find the subgame perfect equilibrium in multi-stage games. Consider a two-period game in which the players swap roles in period 2. Using our single stage reasoning, we know that in the second period the equilibrium outcome is (δπ ɛ, ɛ); thus, in period 1 P 2 (who is P 1 in period 2) will prefer a payoff of δπ to δπ ɛ, which is the best she could obtain by going on to period 2. Hence, in this game the equilibrium payoffs are ((1 δ)π, δπ). 7 In criticizing this study, many have pointed out the instructions used by the authors. The authors encouraged the participants to play according to the dictates

6 6 J. NEIL BEARDEN the one-period ultimatum game is a special and dangerous case from which to draw strong conclusions, and that the two-period game has the proper virtues needed to truly test the predictive accuracy of game theory. It looked as if game theory had been partially saved by the multi-period ultimatum game. Güth and Teitz (1987, 1988) used a two-stage ultimatum game as well but with conditions in which δ =.10 and δ =.90. In contrast to Binmore et al. s (1985) experiment, the game-theoretic outcomes were almost never observed. The authors argued that Binmore et al. s experiment itself was a rather special case, and the reason that responses were not dramatically far from equilibrium was because the equilibrium payoffs were not dramatically unfair (with respect to the 50:50 criterion), as they were in the cases where δ =.10,.90 i.e., where equilibrium predictions fared poorly. 8 To date, the most systematic study of the effects of δ on ultimatum bargaining behavior was conducted by Ochs and Roth (1989). The details of the results were reported in Roth s (1995) review and will not be covered here. The bottom line was that P 1s try to exploit their position and almost always demand at least.50π, but their demands are tempered by a fear that P 2 will reject insultingly low offers. Spiegel et al. (1994) analyzed results from a large number of alternating bargaining experiments conducted by other researchers, as well as from a new experiment of their own. They found that P 1s with a large advantage i.e., those whose first round equilibrium payoff was substantially greater than P 2s tended to demand more than.50π, but those with a disadvantage tended to demand around.50π. Subjects tended to take advantage their position when they had a legitimate right to do so (according to subgame perfection), and tried to hedge their position when they were disadvantaged. A summary of these results is shown in Table 2. Ochs and Roth (1989) looked at previous experiments on multiperiod ultimatum bargaining and discovered one of the most informative regularities in the ultimatum bargaining literature: P 2 rejections are often followed by demands that give P 2 less than he rejected. Rates of disadvantageous counteroffers from several experiments are shown in Table 3. Tables 2 and 3 tell an important part of the ultimatum game story: People do not like unfairness; in particular, they do not like unfairness of game theory by instructing them: YOU WILL DO US A GREAT FAVOR IF YOU SIMPLY MAXIMIZE YOU WINNINGS. 8 See Binmore, Shaked, and Sutton (1988) for a retrospective look at this exchange.

7 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 7 Study Equilibrium(d 1 /π) Mean(d 1 /π) Rounds Forsythe et al. (1988) Güth & Tietz (1986) Güth et al. (1982) Güth & Tietz (1987) Binmore et al. (1985) Harrison & McCabe (1990) Neelin et al. (1988) Bolton (1991) Ochs & Roth (1989) Harrison & McCabe (1990) Ochs & Roth (1989) Spiegel et al. (1994) Bolton (1991) Bolton (1991) Spiegel et al. (1994) Spiegel et al. (1994) Güth & Tietz (1987) Table 2. Average (normalized) first round proposed allocation d 1 /π. Parts adapted from Spiegel et al. (1994). perpetrated against them. Subjects in the experiments from which these data are taken offered less when they felt they could do so and get away with it, and made disadvantageous counterproposals to avoid being treated unfairly i.e., to avoid getting a substantially lower payoff than their opponent. Over and over we will see that P 1s try to exploit their strategic position when they can, and P 2s try to avoid being exploited, even if doing is economically disadvantageous. 4. Equity and Fairness Until now, we have used the term fair to mean an equal (50:50) division of π. This seems to be a rather strong definition. For example, is it fair for P 1 to get twice as much π if P 2 sacrificed a thumb in order to for the two players to be able to bargain over π? Is it fair for P 1s to get the same amount of π as P 2s if they somehow earn the right to be P 1s? Clearly, there are no objective answers to these questions. If we wish to use these terms fair and fairness, however, we need to try to sharpen what we mean, even though any definition we give them will be inadequate in some regard.

8 8 J. NEIL BEARDEN Study Rejections DC Güth et al. (1982) Binmore et al. (1985) Neelin et al. (1988) Ochs & Roth (1989) Bolton (1991, cell 1) Bolton (1991, cell 4) Bolton (1991, cell 6) Table 3. Proportion of first-offer rejections and disadvantageous counteroffers (DC) in multi-stage bargaining games. Parts adapted from Roth (1995). Our solution, which has been used in the context of ultimatum bargaining before by Güth (1988), is to use ideas from equity theory (Adams, 1963, 1965; Homans, 1961, 1974; Walster, Berscheid, & Walster, 1973); that is, we will essentially equate fairness and equity. Equity theory can be summarized by the following propositions (Walster, Walster, & Berscheid, 1978): Proposition 1: Individuals attempt to maximize their own outcomes. Proposition 2: Inequitable relationships between individuals cause distress. Proposition 3: Individuals in distress seek to restore equity. Proposition 4: Groups will evolve conventions for maintaining equity. This involves rewarding equitable individuals and punishing inequitable ones. 9 Note that Proposition 1 is consistent with standard game theoretic assumptions of maximization. Under equity theory, according to Walster, Walster, and Berscheid (1978): So long as individuals perceive 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 (p. 16). Thus, the theory is not one of benevolent beings, just as game theory is not; rather, it is one of self-interested individuals who exist within a social structure with mechanisms for the maintenance of equity. The forces embodied in the propositions interrelate and interact. For example, an individual may act to maximize his own outcome (Proposition 1) even if the action 9 I will save discussion of this proposition until the more speculative part of the current paper.

9 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 9 is inequitable, but his actions may be dampened due to Proposition 2 and Proposition To determine if equity obtains one must compare individuals relative gains. Denoting outcomes φ i and investment ψ i the relative gain for an individual i is expressed as (1) Φ i = φ i ψ i ψ i. The equity principle states that the relative gains should be equal for all individuals, and there will be distress if Φ i Φ j for some i, j. Furthermore, distress is assumed to increase as Φ i Φ j increases. 11 In the context of bargaining games, it is sensible to set φ i equal to P i s payoff. The hard problem is measuring P i s investment ψ i (Selten, 1978). Bartos (1978) argued that in abstract bargaining games all players have the same investment. Thus, an ultimatum bargaining division would be equitable (fair) only if d 1 = d 2, i.e., if the division were 50: However, rather than assume what is fair and equitable, we will treat Eq. 1 as a measurement device and try to determine what ultimatum bargainers find equitable. For example, assuming that a proposal would be rejected only if Φ 1 > Φ 2, we can begin to estimate P 2 s belief about P 1 s relative investment in ultimatum bargaining. To avoid confusion and to be consistent with (much of) the literature, I will use fair throughout this paper to mean equitable. 10 On first reading, this may sound contradictory. However, similar relationships exist between basic physical principles. Consider a dampened pendulum, for example, and the principles that govern its behavior. 11 It is important to point out that the parameters of Eq. 1 must be estimated by an observer, and that these estimates are based on the observer s beliefs about the situation. And two observers may have different beliefs regarding the same situation. Below we will examine the covert perspectives of ultimatum bargainers by looking at their overt bargaining decisions through the lens of Eq. 1. It is precisely the way in which the parameters of Eq. 1 vary across observers that interests us. The measurement properties of the parameters in Eq. 1 are unknown; so we must be very careful in how we interpret it. Our solution is to make only weak qualitative claims based on ordinal properties of relative gains that we can infer from observed behavior such as P 2 rejections. 12 Binmore et al. (1991) had subjects give fairness judgments for different games, and found that subjects responses were correlated with the ostensible power of each player. More specifically, in a bargaining game with outside alternatives, subjects responded that the subject with the more attractive (i.e., high paying) outside alterative should receive more of π, despite the fact that the outside option paid less than the subgame perfect payoffs.

10 10 J. NEIL BEARDEN 5. Effects of Payoff Information Several studies have looked at the effects of players information regarding their own and their opponents payoffs on bargaining behavior. The data suggest that ultimatum bargainers are not motivated by a sense of altruism. Kagel, Kim, and Moser (1996) had subjects bargain over 100 chips that had different exchange rates for P 1 and P 2 using the standard ultimatum bargaining procedure. The authors manipulated the value of the chips to each player and the information the players had about their own and their opponent s exchange rates. Whenever a chip was worth $.10 ($.30) to P 1, it was worth $.30 ($.10) to P 2 (P 1). The information conditions were: P 1 knew the exchange rate for both players, but P 2 knew neither; P 2 knew both exchange rates, but P 1 knew neither; and both players knew both exchange rates. The subjects played 10 periods with different opponents. Note that the subgame perfect equilibrium for this game is (99 chips, 1 chip) in all conditions. The authors were primarily interested in whether non-equilibrium ultimatum offers are based on P 1 s desire to be fair to P 2 i.e., to give P 2 his fair share of π or more precisely.50π. When P 1 had the higher conversion rate and knew his opponent s conversion rate, the altruistic 50:50 payoff offer was (25 chips,75 chips). However, when P 1 knew the conversion rates and P 2 did not, the average offer to P 2 was 46.9 chips (which gives P 1 roughly 3 times the payoff of P 2). The P 2 rejection rate as.08. Under the same information condition when P 1 had the lower conversion rate, the average offer was 31.4 chips, and rejection rates nearly tripled, going to.21. P 1 offers were not very different when only P 2 knew the conversion rates. 13 When only P 2s knew the conversion rates, rejection rates were nearly twice as high in the low P 2 conversion rate condition. 14 The pieces fall into place when one considers the conditions in which both players were fully informed. When P 1 had the higher conversion rate, the (early period) offers were near 50:50 division of the chips. It 13 This is based on comparing the session 7 and session 8 data. Comparing all responses obtained under these conditions, it is actually the case that offers were substantially lower when P 2 had the higher conversion rate, even though P 1 did not know either player s conversion rate. Unless P 1s have ESP, this makes little sense. The authors report that the differences may have been due to practice period offers. Sessions 7 and 8 were run to replicate the conditions in which only P 2 knew both payoffs but without the practice periods. I report the results of these sessions in the body of the paper because they make substantially greater sense. 14 These authors found no significant difference in the bargaining behavior between students from undergraduate economics and psychology classes.

11 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 11 Conversion Information (P 1, P 2) P 1 P 2 Both ($.30,$.10) ($.10,$.30) Table 4. Average first period offers by P 1 to P 2 in Kagel et al. (1996). Note: Offers reported in the P 2 information condition are from sessions 7 and 8. is as if P 1s defined fair as equal division of chips when this was in their best interest. 15 However, P 2 rejection rates drove the offers up toward equal payoff divisions over the course of the 10 periods. Early offers were much closer to payoff equality when P 2 had the upper hand, with P 1s only averaging slightly more than 24 in their first period offers. These offers were constant over the 10 periods, as P 2s seemed to have been relatively content with equal monetary splits. 16 Mean first period offers are shown in Table 4. Most importantly, these results show that players tend to define fairness rather egocentrically, and that P 1s exploit their strategic position when they can (see Knez & Camerer, 1995, for additional interesting evidence of egocentric bias in ultimatum bargaining). Consider the P 1 information condition. P 1s were willing to offer half of the chips when they had the higher conversion rates, which made them appear fair to their naïve opponents, but demanded more chips for themselves when they had the lower conversion rates. 17 P 1s acted is if equal chip divisions were equitable only when it was in their financial best interest. In sum, proposers with high value chips went for equal splits, while those with low value chips allocated according to equal monetary payoffs. Using a standard ultimatum game, Straub and Murninghan (1995) had subjects give offers, make P 2 acceptance decisions, and make minimal acceptable offers (MAOs). One set of offers was made to P 2s who knew neither the size of π nor the distribution from which it was taken. 15 Different division framings occur frequently in wage discussions, where lowincome individuals advocate equality, while high-income individuals argue for wages based on productivity (Elster, 1989). 16 The full information condition results are consistent with those of Nydegger and Owen (1974) who had two subjects divide an even number of chips. The conversion rate for P 1 was twice that of P 2. Subjects divided the chips such that each got the same monetary payoff. It is important to note that each player knew his own and the other player s conversion rate. 17 P 1s playing such that their offers appear fair occurs quite often in ultimatum bargaining games with incomplete information (cf. Dufwenberg, et al., 2000; Güth, Huck, & Ockenfels, 1996).

12 12 J. NEIL BEARDEN They found that the offers in the uninformed condition $U, in which P 2 did not know the size of π, were lower than those in the informed (complete information) condition $I. However, it was still the case that offers in $U increased in π. (Similar results were obtained in Pillutla and Murninghan (1995).) Again, we find that P 1s take strategic advantage of their position. In $U 64% P 2s said they would accept offers as low as $.01, whereas fewer than 25% of the subjects in the informed condition would accept $.01 when π = $10. Their subjects were willing to accept small offers when the offers were potentially fair but not when they were clearly unfair. This can be seen in Table 5 where we observe that the mean MAO increased in π. Note that all of the offers where π $100 are hypothetical; so it remains to be seen if, in fact, a poor college student would really reject an ultimatum offer of, for example, $1000 when π = $1, 000, 000. The current author would not. However, it is important to note that 10 of 49 respondents said that they would accept an offer of $.01 when π = $1, 000, 000. I doubt the validity of this finding as well. In a second experiment, Straub and Murninghan (1995) simply offered subjects different amounts of money, no strings attached. They wanted to alleviate the interdependence embodied in the ultimatum game in order to estimate the effect of this interdependence on responses. They found that subjects were willing to reject low offers even when their responses affected only themselves: 36 of 90 subjects rejected $.01 no stings attached, 55 of 90 rejected it in the partial information condition ultimatum game, and 70 of 90 did in the complete information game. The authors then conclude that wounded pride rather than fairness motivates rejections in the ultimatum game. In other words, their subjects would rather have nothing than accept a humiliating handout. Consistent with many ultimatum bargaining studies, in Straub and Murninghan (1005) the expected payoff from offering.50π was always near the expected payoff for the optimal offer. In their sample offequilibrium offers were more profitable than equilibrium offers, suggesting, perhaps, that non-equilibrium offers were motivated by profit maximizing concerns rather than by fairness. Croson (1996) manipulated the salience of the fairness of offers by having P 1s make offers in terms of, what she called, absolute (dollars) and relative (percentage) payoffs. She factorially combined these framings with the information available to P 2s about the size of π. Consistent with others (e.g., Pilluta & Murninghan, 1995; Straub & Murninghan, 1995), she found that P 1 offers were smaller when P 2s did not know the size of π, and were smallest when reported in terms

13 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 13 Information Partial Complete π d 1 MAO d 1 MAO , , 000, , , Table 5. Average offers and minimal acceptable offers (MAO) by amount and information condition in Straub and Murninghan (1995). Note: All entries are in US$. Values superscripted with an asterisk (*) are responses to hypothetical ultimatum offers. of dollars. P 1s offered more in the informed condition when the offers were in terms of dollars, but offered more in the uninformed condition in terms of percentage. Statistically, the differences between the percentage offers were no different in the informed and uninformed condition, though the mean offer in the latter was smaller. Probably the most interesting finding in Croson is that P 2 rejections were substantially greater in the informed percentage condition than in any other. That is, the rejection rates seem disproportionately high in this

14 14 J. NEIL BEARDEN P 2 Information Informed Uninformed $ Offer $4.50 $3.57 (.07) (.04) % Offer $4.20 $3.92 (.21) (.03) Table 6. Average offers under two different P 2 information conditions for dollar ($) and percentage (%) framings from Croson (1995). Rejection rates are shown in parentheses. condition given the more modest difference between the offers across conditions. 18 A summary of these data is shown in Table It is important to note that the uninformed conditions discussed above involved ignorance on the part of P 2 (Croson, 1995; Pilluta & Murninghan, 1995; Straub & Murninghan, 1995) because the subjects did not know anything about the potential values of π. Alternatively, one can inform the subjects about the distribution from which π is sampled to create a situation akin to what is referred to as a risky situation in the decision theory literature. The next set of experiments I report use this procedure. Mitzkewitz and Nagel (1993) used two different variants of the ultimatum game to look at the effects of P 2 information on bargaining behavior. In the offer game, P 1 offers an amount to P 2 without P 2 knowing the residual left for P 1. The demand game is very similar 18 I think it is worth reiterating that this is a bizarre finding. Technically, the dollar and percentage offer distributions were not statistically indistinguishable at the α =.05 level, e.g., but were for α =.10. Assume they are no different. (Like anyone who has thought about this enough I do not take these cutoffs seriously, but I will manipulate the αs nonetheless to direct the reader to this interesting result. Too bad I cannot footnote my footnotes.) Then, it is the case that, when the size of π ($10) was known, these bright young Harvard students were more likely to reject offers stated in terms of percentages than those stated in dollars. 30% of $10 is $3 and that seems pretty clear. It seems unlikely that the difference in responses was due to the bounded rationality of the subjects, in the sense that the they could not convert x% of $10 to dollars. My point: The percentage framing had a shocking effect on the subjects; they seem to have been much more concerned with fairness under the percentage framing, even though the frame is fully transparent. 19 Croson (1996) offers Tversky et al. s (1988) contingent weighting model as an explanation for these results. According to this model, which has been used to explain preference reversals, e.g., people make marginal trade-offs between gambles that depend on the methods used to elicit preference for the gambles.

15 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 15 Figure 1. Mean demand (d 1 ) by P 1 from all offer and demand games in Mitzkewitz and Nagel (1993). Figure 2. Acceptance rates for offers in demand and offer games in Mitzkewitz and Nagel (1993). except that P 2 only knows P 1 s payoff and does not know her own. These games have the same equilibrium outcome as the standard ultimatum game, viz. (π ɛ, ɛ). In the experiment π {1, 2,..., 6} in units of a fictitious currency. For a given game, π was determined with the role of a six-sided die, and only P 1 was informed of this value. P 2 knew only the probability distribution over π values. The authors observed that both P 1 and P 2 behavior is very different in the two games, and also different from behavior in standard ultimatum games with no uncertainty. They found that the proportion of π demanded by P 1 increased with π in the offer game, but decreased in the demand game. In offer games the acceptance rate increased in π, but decreased in π in the demand game. Figures 1 and 2 show the P 1 requests and P 2 responses, respectively. In the both games P 1s took advantage of P 2 s uncertainty. Since P 2 has no information about P 1 s payoffs in the offer game, any feasible offer is potentially fair, and this potentiality increases with P 1 s offer to P 2. But in the demand game, the likelihood that P 1 s keep is unfair increases in the size of the keep, becoming unambiguously unfair when it exceeds 3. Thus there are strategic factors and limitations in these games that P 1s attended to well. As in most ultimatum game experiments, P 1s in offer and demand games do considerably better than P 2s. Rapoport et al. (Rapoport & Sundali, 1996; Rapoport, Sundali, & Seale, 1996) looked at the effects of P 2 uncertainty on offer and demand game play by manipulating the variance of the distribution from which π was sampled. As the variance increased and therefore P 2 s uncertainty increased, the average offers (proportion of π) to P 2 decreased in both games. Rejection rates were largely unaffected by the distribution of π in the games. Using a two-stage ultimatum game with incomplete information, Güth, et al. (1996) found that first-stage offers were almost always consistent with the proposer presenting an offer that was fair if the lowest offer from the offer distribution had been selected. Again, P 1s were more concerned with portraying fairness than with being fair.

16 16 J. NEIL BEARDEN Roth and Malouf (1979) had subjects P a and P b bargain over lottery tickets that determined the probability that each respective player would earn his potential payoff (π a or π b, respectively, for P a and P b). In contrast to predictions of standard game theory (see, e.g., Nash, 1950), they found that when the players knew their own and their opponent s payoffs, the divisions were in the direction of equal expected payoffs, whereas when players knew their own but not their opponent s payoffs, the divisions were in the direction of equal lottery tickets (i.e., equal probability of winning). Roth (1985) discusses other experiments along these same lines that suggest the existence of focal points in bargaining games. Consistent with the results reported above from Kagel, Kim, and Moser (1996), he found that bargaining disagreement increased as the difference between the payoffs to P a and P b increased. These findings can be summarized very compactly: P 1s exploit their information advantage when they can in order to increase their own payoffs, and P 2s are resistant to unambiguously unfair offers from P 1 (see Huck, 1999, for a detailed treatment of P 2 behavior in bargaining games with incomplete information). Consistent with equity theory, P 1s try to exploit (i.e., try to maximize their own outcomes), and P 2s try to avoid being exploited (i.e., try to avoid inequity). 6. Player Position Determination Smith (1991) argued that some of the results typically obtained in bargaining experiments may be due to the standard use of random determination of roles. Often subjects are assigned to the roles of P 1 and P 2 with the flip of a coin (or by an equivalent procedure). Thus, the very plausible argument goes, P 1s may feel like their power is not legitimate and are therefore more likely to behave in a way consistent with the notion of fairness. Also, we know that people spend unearned money differently than earned money (Arkes et al., 1995; Keasey & Moon, 1996; Thaler & Johnson, 1990); in particular, they are substantially less frugal with unearned money. Ultimatum selflessness, then, may simply be the result of the spontaneous generosity that results from the receipt of unearned money. This section reviews work aimed at addressing these two potential explanations. To induce feelings of entitlement to positions, Güth and Tietz (1986) auctioned P 1 positions in an ultimatum bargaining game using a second highest bid auction. 20 Auction winners received whatever they 20 In a second-price sealed bid auction (a.k.a., Vickrey auction), which is the variety used by Güth and Tietz to auction ultimatum player positions, the optimal bid is one s own true value for the auctioned good. This was explained to their

17 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 17 earned in the ultimatum game less their auction fee. 21 The procedure lead to a dramatic reduction in the number of 50:50 offers from P 1; in fact, none of the P 1 s who earned their position in the auction offered equal splits. On average, P 1 demanded 2 π in the auction conditions. 3 A ( 2π, 1 π) division, though, is not dramatically different from the average division in ultimatum games with coin-flip assignment. Perhaps 3 3 the P 1 auction winners did not feel completely entitled to π because P 2s had to purchase their positions as well. The average price paid for the P 1 position was twice that paid for the P 2 position; so presumably subjects recognize that P 1 has greater entitlement than P 2 in the game. In the framework of equity theory, then, one might argue that a division in which P 1 gets twice the payoff of P 2 is equitable, though from the naïve 50:50 fairness standpoint the division is unfair. Hoffman and Spitzer (1982, 1985) had two players bargain over the split of $14. In one group, one of the bargainers in each pair was chosen at random to be the controller; in another group, players earned the right to be the controller in a contest. If the pair could not agree on a division of the $14, the controller received $12 and the other received nothing. All of the divisions (12 of 12) in the group in which the controller was determined at random were 50:50, whereas only 4 of 12 divisions in the earned group were equal. The average payoff to the controller was $12.52 in the latter group. Following up on the findings of Hoffman and Spitzer (1982, 1985), Hoffman et al. (1994) conducted what is the most comprehensive and has become the most oft-cited set of experiments on the role of player position legitimacy in ultimatum bargaining games. They compared ultimatum and dictator game outcomes when the players were randomly assigned to player positions and when they earned their positions. In addition they compared play when the game was framed in standard ultimatum bargaining terms and when it was framed as a buyer-seller exchange. 22 subjects and pre-experimental data suggest that the subjects understood. Thus, the actual bids for the ultimatum positions can be taken quite seriously. 21 One of the gems of experimental research is that Güth and Tietz (1986) actually had their subjects pay out of their own pockets when their auction fee exceeded their ultimatum earnings. Fortunately, they report, Subjects with negative payoffs never complained about having to pay their bill (p. 181). 22 In the dictator game, P 1 proposes a division of π and P 2 must accept it. Thus, technically, it is misleading to call it a game, since it is really a single-person decision problem. Results from dictator games will be discussed in more detail below.

18 18 J. NEIL BEARDEN In the entitlement conditions, subjects could earn the role of P 1 in ultimatum and dictator games by outperforming their fellow subjects on a current events quiz. 23 The ultimatum games were given either the standard framing of simply dividing $10 or framed as a buyer-seller exchange. In the latter, the seller (P 1) could select a price x between $0 and $10. The buyer (P 2) could then accept the offer in which case she would get $10 x and the seller would get x. Rejected offers gave both $0. This game is simply the standard ultimatum game in different clothing i.e., it has the same subgame perfect equilibrium. However, being in the seller role, most would agree, confers (ostensible) legitimacy on P 1 and therefore, it was hypothesized, gives P 1 the right to demand more than he would have if his role in the game were arbitrarily labelled Player 1, for example. The seller simply set a selling price in the dictator version. P 1 offers to P 2 were lower in the buyer-seller game than in the standard ultimatum game, and were also lower when the players earned their respective roles. Offers of $4 or more occurred in fewer than 45% of the earned buyer-seller games; but in the random standard ultimatum game over 85% of the offers equaled or exceeded $4. In the earned buyer-seller dictator game, only 4% of P 1s offered $4 (none offered more). Still, more than 50% of offers in the earned buyer-seller dictator game were non-zero. The rejection rates were negligible across conditions, and did not significantly vary between conditions (see p. 369 for a discussion of their unusually low rejection rates). Hoffman et al. also found that subject anonymity affected dictator giving; specifically, they found that P 1 offers were less altruistic when the offers were fully anonymous (i.e., using a double-blind procedure). 24 The results on P 1 legitimacy and ultimatum offers are mixed. Sonnegård (1996) used four different methods to assign subjects to P 1 positions: random assignment; score on a dice game; score on a general knowledge test; and score on a computer game. All participants knew how assignments were made. He found no difference in the P 1 first round allocations as a function of assignment method. He also compared property right assignment to a neutral assignment that did not emphasize property rights. In the former, subjects were told that 23 Note that this procedure does not affect the size of the end-game payoffs for the players; it only affects player position legitimacy. In Güth and Tietz (1986) both the players legitimacy and net payoffs were affected by the auctioning procedures. Furthermore, since P 1 did not now P 2 s auction fees (and vice versa) both had an additional degree of uncertainty that is not present in the performance assignment procedure of Hoffman et al. 24 Anonymity effects findings will be reported below in more detail.

19 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 19 they had earned the right to be P 1, and that this offered them a significant advantage. (Note that the last part suggests a social norm for the game to the subjects.) The manipulations had no effect. Why did these two relatively similar studies obtain different results? There are a couple methodological differences between Sonnegård (1996) and Hoffman et al. (1994) that must be pointed out. 25 Most importantly, I think, Sonnegård used a two-period ultimatum bargaining game with δ =.20. Perhaps the risk of going to the second period gave P 1 the impression that P 2 was empowered as well, thereby washing out the effects of earned entitlement. Also, subjects in Sonnegård s study were Swedish, whereas Hoffman et al. s were (presumably) primarily American. Roth et al. (1991) found that bargaining behavior varied across cultures. 26 In each of these studies, the authors tried to manipulate the degree to which P 1s earned their position in the ultimatum game. The tobe-divided money was still provisionally provided to both players. So, despite P 1s earning their position in the game, π was still a windfall income to both players. Cherry (2001) eliminated the windfall perception by having subjects earn money by choosing between different gamble portfolios, and then playing their chosen gambles. Once the subjects knew their portfolio payoffs, they played dictator games with others who, they were informed, had not had an opportunity to earn money by choosing between gambling portfolios. In the control condition P 1s were endowed with amounts similar to those earned by the subjects in the gambling condition, and made dictator offers to anonymous P 2s. The dictators who earned their money offered positive amounts of money to P 2 24% of the time, whereas positive offers were made 74% of the time in the control condition; and none of the subjects who had earned their money offered.50π, but 14% of the 25 Of course saying that these two studies found different results is a bit misleading. Logically, we are justified in saying that they authors reached different conclusions: They did. But we cannot say that their results are inconsistent since Sonnegård found null results, that is, he simply failed to find sufficient evidence that his null hypotheses, that all groups were sampled from populations with the same parameters, were false. Before reaching strong conclusions we need to consider the power of his tests, which is not reported in his paper. This is an elementary point but nonetheless it is worth pointing out. See Forsythe et al. (1994) and Slonim and Roth (1998) discussions of statistical power in the context of ultimatum bargaining data. 26 Interestingly, Sonnegård does not offer an explanation for the differences between results from his own study and from Hoffman et al. s, though he does explicitly state that they are different.

20 20 J. NEIL BEARDEN controls did. Regards for P 2 s welfare decreased dramatically in the dictator game when P 1s were giving away their own money. These studies have looked primarily at P 1s behavior as a function of their legitimacy. Ruffle (1998) looked at P 1s behavior toward P 2s in dictator games in which P 2s determined the size of π. In the skill condition, he had P 2s compete on general knowledge questions to determine the size of π to be allocated by P 1. Winning P 2s participated in dictator games in which π = $10, while losing ones participated in π = $4 games. In a control condition the size of π was determined by the flip of a coin. Results showed that offers to winning P 2s in the skill condition were significantly greater than offers in the $10 control condition (the lucky condition). Losing P 2s were offered less than control recipients in the $4 condition. Ruffle looked at dictator and ultimatum games under these same conditions and found that the reward effects were diminished in the ultimatum game and that the punishment effect was completely wiped out. He concluded that offers to skillful P 2s were motivated by fairness and not by strategic considerations. These findings are clearly consistent with equity theory (Homans, 1961; Walster et al., 1978). That is, the skilled recipients inputs could be considered greater than the unskilled recipients (i.e., the losers in the general knowledge games), and therefore the former were entitled to more than the latter. Schotter, Weiss, and Zapater (1996) compared ultimatum and dictator games when P 1s faced selection pressure. In one condition, P 1s were selected to go onto a second game according to their earnings in the first game. The top half of earners went on to game two. Subjects in the control condition did not have the opportunity to go onto a second game, and were therefore not under selection pressure. Results showed that P 1 ultimatum and dictator offers were lower in the selection condition, and that the difference between the control and selection condition offers was greatest in the dictator game. On average, dictator offers were lower than ultimatum offers. Also, P 2s were (in the ultimatum game) also more likely to accept small offers in the selection condition. The authors argued that P 2s were more willing to accept small offers in the selection condition because the pressures on P 1s to go to game two made small offers to P 2 less unfair. Though the results on the effects of P 1 entitlement on ultimatum offers are mixed, there is some evidence that P 1s demand more π when they feel they have earned their position. In the language of equity theory, this finding is consistent with P 1s believing that their investment is greater than P 2s, thereby legitimately endowing them with the right

21 ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART 21 to more π. Unfortunately, the data on P 2 reactions to P 1 entitlement are inconclusive. 7. Ultimatum and Related Games Distinguishing between fairness or equity and strategic concerns in ultimatum games is difficult. An offer of more than ɛ of π by P 1 to P 2 could be a result of P 1 wishing to be equitable, or it could be that P 1 fears that P 2 would reject an offer of ɛ, thereby reducing P 1 s payoffs to 0. An entirely self-interested, game-theoretic, income-maximizing P 1 would propose the division d that maximizes his subjective expected payoff. If there is doubt that P 2 is rational (according to gametheoretic axioms), then P 1 ought not necessarily offer ɛ. The problem is that two proposals d 1 and d 2 made by two different agents can be identical even when the agents are motivated by different concerns. Both might offer 50:50 divisions, for example, though one might do so out of equity concerns and the other to maximize his own payoffs. Two questions are: Are ultimatum offers motivated by fairness or by strategic concerns? And how can we distinguish between the two? Testing whether fairness can account for non-zero offers in ultimatum games Forsythe et al. (1994) compared offers in dictator and ultimatum games. 27 In the dictator game P 1 simply determines how much of π P 1 and P 2 receive; P 2 has no say in the matter. They found that 27 Aside from the importance of the empirical findings in this paper, the authors also focus on appropriate tests for bargaining data. They say: The hypotheses are all stated in terms of testing the invariance of the distribution of proposals rather than particular characteristic of the distribution such as mean and variance. This is done because conventional theory predicts that proposals will be concentrated at a single point...since theory does not predict a distribution of proposals, it provides little guidance about which functionals of the distribution should be tested. Invariance of the distribution has the appealing property of implying that all functionals are invariant. (p. 351) They then compare Cramer-von Mises, Anderson-Darling, Kolmogorov-Smirnov, and Wilcoxen rank-sum tests power, and conclude that the first two are more powerful than the latter two. The procedures used by Forsythe et al. should be noted by psychologists, in particular, many of whom tend to cram any hypothesis test into the GLM framework. Relatedly, these authors should be lauded for the way in which they report their results. They include histograms, quantile plots, and all of the raw data, in addition to their test statistics. This information is extraordinarily important for the cumulative growth of knowledge. See Roth (1995) for a nice quote regarding the journal editors and the reporting of raw data. (Footnote 16)