Bounded Rationality in Laboratory Bargaining with Asymmetric Information *

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1 Bounded Rationality in Laoratory Bargaining with Asymmetric Information * Timothy N. Cason and Stanley S. Reynolds Decemer 003 Summary. This aer reorts an exeriment on two-layer seuential argaining with asymmetric information that features some forces resent in multi-round monooly ricing environments. Buyer-seller airs lay a series of argaining games that last for either one or two rounds of offers. The treatment variale is the roaility of continuing into a second round. Euilirium redictions do a oor jo of exlaining levels of rices and treatment effects. As an alternative to the conventional euilirium model, we consider models that allow for ounded rationality of sujects. The uantal resonse euilirium model catures some of the imortant features of the results. Keywords and Phrases: Laoratory, durale goods monooly, logit euilirium JEL Classification Numers: C78, D8, D4, C9 Deartment of Economics, Krannert School of Management, Purdue University, 00 S. Grant Street, West Lafayette, IN , USA cason@mgmt.urdue.edu) Deartment of Economics, Eller College of Business and Pulic Administration, University of Arizona, Tucson, Arizona 857, USA * This research was funded in art y the National Science Foundation SBR-98090). The exeriments were run at the Economic Science Laoratory of the University of Arizona and the Krannert Laoratory for Exerimental Economic Research at Purdue University, using the z-tree software develoed at the Institute for Emirical Research at the University of Zurich Fischacher [8]). David Cooer, Rachel Croson, Charles Noussair, an anonymous referee, and conference articiants at the Economic Science Association and the Society for the Advancement of Economic Theory meetings rovided helful comments. Timothy O Neill Dang, Thomas Wilkening and Marikah Mancini rovided exert research assistance.

2 . Introduction The literatures on seuential argaining theory and durale goods monooly theory focus on how reeated interaction etween a single seller and one or more uyers influence market outcomes. Three tyes of forces shae outcomes in this environment. First, a uyer who has the otion of deferring a urchase until later may have an incentive to wait for a etter rice offer even if their current ayoff from making a urchase consumer surlus) is ositive. Second, a seller should anticiate strategic ehavior y uyers when choosing rices. If a uyer is waiting for a etter offer then the seller may have to set a rice elow the rice they would set in a takeit-or-leave-it situation, in order to induce the uyer to urchase earlier. Third, the ossiility of making multile rice offers may ermit a seller to ractice a form of inter-temoral rice discrimination and extract more surlus from uyers than is ossile in a static setting. Researchers have utilized a variety of game theoretic formulations to examine how these forces lay out. These formulations include models of seuential argaining with asymmetric information and dynamic market models with comlete information. These models yield euilirium redictions aout initial rice offers, changes in rices over time, and uyer urchasing ehavior. Under some assumtions, the seller s initial rice offer in euilirium falls as the discount factor rises; as the discount factor aroaches unity the euilirium initial rice offer converges to marginal cost see Stokey [], and Gul, Sonnenschein, and Wilson []. This last result seems to cature the essence of Coase s conjecture Coase [5)] on durale goods monooly ricing. However, euilirium results are uite sensitive to assumtions. If one assumes comlete information in a setting with a finite numer of uyers and discrete demands, rather than a continuum of uyers or argaining with a single rivately-informed uyer, then the Reynolds [9] descries how the durale goods monooly model has een extended in a variety of directions in recent aers.

3 euilirium may involve erfect rice discrimination over time rather than rices close to marginal cost see Bagnoli, Salant and Swierzinski []). Recent laoratory exerimental studies y Güth, Ockenfels, and Ritzerger [3], Raoort, Erev, and Zwick [8], Cason and Sharma [4], and Reynolds [9] have examined euilirium redictions from these models. By and large, euilirium redictions from game theoretic models of multi-round ricing have fared oorly in these exeriments. Prices are often far from redicted levels, and uyers urchasing ehavior fails to conform to some redictions. Perhas most significant is the tyical failure of comarative statics redictions aout the effects of changes in information, changes in the discount factor, or changes in the time horizon. Moreover, adjustments to the models to take into account risk references or references for fairness are not adeuate to exlain the results in these exeriments e.g., see the discussion in Reynolds [9]). And although rules of thum and social norms aout fairness can rovide a artial exlanation Raoort et al. [8]), no formal and systematic model has een roosed to exlain the oserved deviations from euilirium. The resent aer reorts on a new exerimental design for seuential argaining with asymmetric information. The design secifies a relatively simle decision-making environment, while maintaining enough richness to cature the forces at work in multi-round ricing environments. The simlicity of the exerimental design serves two uroses. First, it gives euilirium theory its est shot at successfully redicting sujects ehavior. This is imortant in light of the redictive failures noted aove. Second, the simlicity of the design allows us to comute redictions for some general models of ounded rationality. We have designed exeriments in which uyer-seller airs lay a series of argaining games that last either one or two rounds. The seller is uninformed aout the resale value for the

4 uyer she is matched with. The seller simly chooses a rice offer from a set of ten ossile rices in the first round and the uyer either accets or rejects the offer. If the uyer rejects the offer then the game moves into a second round with some known, exogenous roaility. If there is a second round, the seller chooses a rice offer from this same set and the uyer either accets or rejects. The treatment variale in our exeriments is the roaility of continuing into a second round. We ran exeriments with the following continuation roailities: zero ercent, 30 ercent, 60 ercent, and 90 ercent. The game theoretic euilirium rediction is that the oening rice will fall as the continuation roaility rises from zero to 30 ercent and from 30 to 60 ercent, and that the oening rice will rise as the continuation roaility rises from 60 to 90 ercent. Strategic withholding of demand y uyers is redicted on the euilirium ath) only for the case of a 90 ercent continuation roaility. The environment is ket simle y emloying a finite set of ten ossile rices for the seller, two ossile uyer values high or low), and at most two rounds in a game. This allows for straightforward comutation of erfect Bayesian euilirium redictions. More imortantly, this relatively simle setu facilitates comutations for models of ounded rationality in games. We resent results for two such models: the Noisy Nash Model Nash euilirium lay, lus random decision errors) and the agent uantal resonse euilirium AQRE) model roosed y McKelvey and Palfrey [7].. Exerimental Background Several recent studies reort on exeriments that were designed to test game theoretic redictions of multi-round monooly ricing models. The exeriments of Güth, Ockenfels and 3

5 Ritzerger [3] match a single seller with 0 uyers. Each exeriment consisted of 5 or 6 exerimental games. In each game there was a different comination of maximum numer of trading rounds two or three) and discount factors. In one exeriment sujects received rior training in durale goods monooly ricing games. The results for untrained sujects were grossly inconsistent with theoretical redictions from a model with a continuum of uyers. Prices failed to conform to comarative statics redictions and rices tended to e much higher than redicted. The levels of rices with trained sujects were closer to theoretical redictions, ut rices still failed to satisfy comarative statics redictions. Raoort, Erev and Zwick [8] reort on argaining exeriments with time discounting, one-sided incomlete information, and an infinite unlimited) time horizon. They find that: ) rice offers tended to decline over time, as redicted y the seuential euilirium SE), ) average initial rices were higher the higher the discount factor, contrary to the SE, and 3) for some discount factors the average initial rice was aove the static monooly rice, contrary to the SE. Raoort et al. suggest that the SE theory may e failing ecause some uyers are using suotimal rules of thum for making urchases and ecause of a social norm aout fair divisions of surlus. Cason and Sharma [4] focus on the role of information aout uyers values. Game theoretic analyses of durale goods monooly ricing redict large differences in outcomes deending on whether agents are informed aout uyers values. If all agents are informed aout uyers values then erfect rice discrimination can emerge. This is in shar contrast to asymmetric information environments, which tyically yield euilirium rices elow the static monooly rice. Cason and Sharma reort on infinite horizon exeriments with two uyers and one seller. In the Certain Demand treatment all agents know that one uyer has a high value and 4

6 the other uyer has a low value. In the Uncertain Demand treatment, uyers are rivately informed aout values; all agents know that the most likely outcome is that one uyer has a high value and the other has a low value, ut outcomes with two low values or two high values are also ossile. Prices in oth treatments were much closer to euilirium redictions of the incomlete information model than to the euilirium redictions of the comlete information model. Reynolds [9] reorts on exeriments with one-sided incomlete information, a finite horizon, and either one uyer argaining) or five uyers market). Bargaining exeriments were run with one, two and six trading rounds, while all market exeriments were run with six rounds. The demand withholding results were roughly in line with game theoretic redictions: withholding is much lower for one round games and for the final round of multi-round games than for earlier rounds of multi-round games. Reynolds nevertheless finds a significant failure of the euilirium ricing redictions. The erfect Bayesian euilirium redicts that oening rices will fall as the trading horizon increases from one to two to six rounds. Instead, oening rices in the argaining exeriments rise as the trading horizon increases, holding suject exerience constant. Güth, Kröger and Normann [4], however, find etter suort for rice comarative statics as the rivate) discount factors change in a two-round argaining exeriment with incomlete information regarding uyer values. 3. Exerimental Design and Predictions We conducted a total of 9 sessions, emloying 3 sujects. Fifteen or 3 sujects articiated in each session, as summarized in Tale. In each session sujects articiated in a seuence of argaining games against different anonymous oonents. Each argaining game 5

7 had either one round or two rounds of argaining etween a uyer and a seller. Sujects interacted only through a comuter network running an alication written using the University of Zurich s z-tree rogram Fischacher [8]). All sujects were inexerienced, in the sense that they had never articiated in a revious argaining session that emloyed this design. Our goal was to minimize reeated game incentives ut allow sujects to otain feedack and learn from their earlier exerience laying the game, so we used the matching scheme emloyed y Cooer et al. [6]. In this strangers design all sujects argained exactly twice with each other suject in their session once as the seller and once as the uyer and sujects never knew the identity or history of their argaining oonent. They never argained against the same oonent in consecutive eriods. Sujects alternated etween eing seller and uyer, and one suject sat out each eriod. Thus each session consisted of 30 eriods, and each of the 5 sujects argained 4 times as a seller and 4 times as a uyer. [The sessions with 3 sujects lasted for 6 eriods.] Switching roles might have heled the sujects learn how to ackward induct in this game, ut it could have also led to some initial confusion. The ordering of the sujects and the air assignments were determined randomly at the start of the session, when the sujects were randomly assigned to a comuter. The instructions see htt:// were read aloud while sujects followed along on their own aer coy. The instructions exlain the matching rotocol descried aove. The instructions also exlain that the sellers cost is zero and the uyers value is randomly determined, indeendently for each uyer. With a 40 ercent chance the uyers value is 54 and with a 60 ercent chance the uyers value is 8. An individual uyer s value is her rivate information. In each argaining round the seller chooses a rice offer from the set {5, 0, 5, 0, 5, 30, 35, 40, 45, 50}, and then the uyer either accets 6

8 or rejects the offer. When an offer is acceted, seller rofits simly eual the rice, and uyer rofits eual their value minus the rice. If no offer is acceted for a articular eriod oth the uyer and seller earn zero. The rogram did not allow uyers to accet rice offers that result in negative rofits. Sujects were recruited from the undergraduate oulation at the University of Arizona and Purdue University. Uon arrival to the la sujects were aid a $5 show-u fee. Including instructions, sessions reuired aroximately 00 minutes to comlete. Payoffs were converted using an exchange rate of laoratory dollars = U.S. dollar. Total earnings er suject ranged from $5 to $50, with a mean of aout $36. Perfect Bayesian euilirium PBE) redictions for risk neutral agents are summarized as follows. The PBE for the zero ercent continuation roaility is straightforward. The seller sets the highest ossile rice 50) and the high value uyer accets this rice offer. The exected ayoff for the seller from this rice = 0) exceeds the ayoff from setting a rice that oth uyer tyes would accet 5). There is no demand withholding in this euilirium. The form of the PBE is similar for the 30 and 60 ercent continuation roaility treatments. The high-value uyer accets the round one offer, and if the game goes to a second round, the low-value uyer accets the round two offer of 5. The seller s rices must satisfy the following incentive constraint for the high- value uyer with value v = 54 ): IC) v δ v ) In this ineuality, δ is the continuation roaility. The rice is the highest rice that a lowvalue uyer will accet; i.e., = 5. The seller chooses the highest rice in the first round that satisfies the IC. For δ = 0. 3 this rice is 40 and for δ = 0. 6 this rice is 30 = =. There is no demand withholding along the euilirium ath. Demand withholding would occur only for 7

9 rices off the euilirium ath e.g., if δ = 0. 3 and the seller sets 45 then the PBE strategy for a high-value uyer is to reject this offer even though it rovides ositive consumer surlus). Note that as the continuation roaility increases from zero to 30 to 60 ercent, the PBE initial rice offer falls from 50 to 40 to 30. As high-value uyers ecome more likely to receive an attractive second round offer after a rejected first offer, the seller resonds y lowering the initial rice offer. The logic for this result is essentially the same as the logic ehind the redicted effect of an increase in the discount factor in the infinite horizon exeriments run y Raoort, Erev and Zwick [8]. When the continuation roaility gets closer to one in a two round game, the character of euilirium changes. The seller will not set a rice of 5 in the second round with roaility one, and therefore a high-value uyer cannot count on receiving a low rice offer if she rejects the initial rice offer. When δ = 0. 9 the PBE is as follows: 3 a) = 50, ) high-value uyer uses a mixed strategy for round one decision with Pr[ accet = 50] 0.357, c) seller uses a mixed strategy in round two, with {5,50} and Pr[ = 50] The PBE rice redictions are summarized in Tale. = 4. Results: Comarison with PBE Predictions When δ = 0.9 there is no initial offer that is consistent with a screening euilirium an euilirium in which the high value uyer accets the initial offer and the low value uyer accets the second offer). When δ = 0.9 a high value uyer will reject any initial offer aove 5, if they exect a second offer of 5. An euilirium must involve a second offer aove 5 with ositive roaility. 3 The derivation of this euilirium is similar to the derivation in Fudenerg and Tirole [9]. The main difference is that the rice sace is discrete in our setting, whereas Fudenerg and Tirole have rices chosen from a continuous interval. 8

10 Tale resents the mean and median offer rices in each treatment, ooled across eriods. The rightmost column is discussed later, in Section 5.) Figure elow indicates that the time trends for these rices are not sustantial, and there is no evidence for significant time trends in the second half of the sessions eriods 6-30). 4 In this tale and in the suseuent analysis we ool across sites for the zero, 60 and 90 ercent continuation roaility treatments, since results did not differ significantly at Arizona and Purdue for these treatments. For reasons we are unale to exlain, however, rices tended to e lower at Arizona than Purdue for the 30 ercent continuation roaility treatment, so to err on the side of caution we do not ool those datasets. 5 In the first round the euilirium rices are highly sensitive to the continuation roaility, falling from 50 to 40 to 30, and then rising ack to 50 as the continuation roaility rises from zero to 90 ercent. Oserved oening offer rices, y contrast, do not vary much across the continuation roaility treatments. Although as redicted) mean rices are highest in the zero ercent continuation treatment, in this treatment 47 ercent of the oening offer rices are less than or eual to 30, while only 3 ercent eual the rediction of 50 and 0 ercent are 45). Even more striking are the low oening offer rices for the 90 ercent treatment. Fully 74 ercent of these rices are less than or eual to 30, and only 4 ercent are either 45 or 50. It is temting to attriute the mean rice level to fairness, since rices in the range of 5 to 30 4 To evaluate whether significant time trends exist in the data, we regressed rices on eriod numer and in an alternative secification, on /eriod). The coefficient estimates on these time trend variales are not significantly different from zero in any treatment when droing the first 5 eriods. 5 In articular, according to nonarametric Wilcoxon Signed-Rank tests conducted for individual airs of eriods so that each suject contriutes one rice offer to each test), oening offer rices are lower in UA30- than in PU30- at the 5 ercent significance level in over half of the eriod airs. According to this same test, oening offer rices in the two Arizona sessions UA30- and UA30- are not significantly different in any eriod air. For the other session-site comarisons in the other three continuation roaility treatments i.e., UA00- vs. PU00-, UA60- vs. PU60-, and UA90- vs. PU90-), these offer rices are significantly different at the 5 ercent level in only 5 of the 43 eriod airs. Similar tests for second round rices are only ossile in the 60 and 90 ercent continuation roaility treatments, since few oservations exist for second round rices in individual eriod airs when the continuation roaility is 30 ercent. For these tests in the 60 and 90 ercent continuation roaility treatments, the rices are generally not significantly different across exeriment sites. 9

11 rovide a roughly eual slit of the exchange surlus etween the seller and the value=54 uyer. But as we discuss elow, the sujects in this exeriment aear sustantially less sensitive to fairness concerns than are sujects in other related exeriments. The lower anel of Tale resents the summary statistics for the second round of rice offers. For the 30 and 60 ercent continuation treatments the second round mean rices are within 5 laoratory dollars of the rediction of 5. For the 90 ercent continuation treatment the second round mean rice is greater than the mean rices for the other treatments, ut it is far elow the PBE redicted rice of 50. Figure dislays the time series of mean oening offer rices for the four continuation roaility treatments. This figure ools adjacent eriods ecause sujects alternated etween eing uyers and sellers, so each suject contriutes at most one rice offer) oservation to each adjacent air of eriods. In the oening eriods the mean rices are highest in the 90 ercent continuation roaility treatment. Mean rices tend to fall in the 90 ercent continuation treatment, however, while mean rices rise for the first third of the sessions in the zero ercent continuation treatment. Prices also tend to fall in the 30 ercent continuation roaility sessions conducted at Arizona. For statistical tests we cannot ool non-indeendent) choices made y the same suject, so we conduct conservative nonarametric Wilcoxon tests searately for each air of eriods. As noted aove, each suject contriutes at most one rice oservation to each adjacent air of eriods ecause they alternated etween uyer and seller roles. In the following discussion, eriod air refers to eriods and, eriod air 4 refers to eriods 3 and 4, and so on. Tale shows that the PBE rices in round one vary sustantially as the continuation roaility varies. Consistent with the visual imression rovided y Figure, however, in many 0

12 eriod airs the rices are not significantly different across treatments. Prices in the Arizona 30 ercent continuation roaility treatment are often significantly lower than rices in the zero and 90 ercent continuation roaility treatments, consistent with the PBE. 6 Also consistent with euilirium, rices are higher in the zero ercent continuation roaility treatment than the 60 ercent continuation roaility treatment in eriod airs 8,, 4, and through 30. But contrary to the PBE, rices are also significantly lower in the Arizona 30 ercent continuation roaility treatment than in the 60 ercent continuation roaility treatment in eriod airs, and 6 through 30, and they are significantly lower than the Purdue 30 ercent continuation roaility treatment in eriod airs 6, 8, 0 0,, 6, 8 and 30. Prices are not significantly different at the five-ercent level for 53 of the 58 other airwise comarisons with a rice difference redicted y the PBE i.e., etween zero and Purdue 30, zero and 60, Purdue 30 and 60, Purdue 30 and 90, and 60 and 90 continuation roaility treatments). A series of Wilcoxon signed-rank tests also clearly rejects the PBE null hyothesis for oening round rices in three of the four treatments. Figure indicates that mean rices tyically range etween 0 and 35, while PBE rices vary etween 30 and 50. In every eriod air, the data reject at the five-ercent level the null PBE hyotheses that a) median rices eual 50 when the continuation roaility is 90 ercent; ) median rices eual 40 when the continuation roaility is 30 ercent; and c) median rices eual 50 when the continuation roaility is zero ercent. 7 The data reject the PBE hyothesis that median rices eual 30 when the continuation roaility is 60 ercent at the five-ercent significance level) only in eriod airs 8 and 8. 6 In articular, nonarametric Wilcoxon tests indicate that rices are significantly lower in the Arizona 30 ercent continuation roaility treatment than in the zero ercent continuation roaility treatment in eriod airs 6, 8, and through 30; and are lower than in the 90 ercent continuation roaility treatment in eriod airs through, 6 through, 6 and 8 all five-ercent significance level, one-tailed tests). 7 The one excetion is eriod air 5 for the 30 ercent continuation roaility treatment at Purdue, which is not significantly different from 40.

13 The second round offer rices rovide etter suort for the euilirium redictions, although only the 60 and 90 ercent continuation roaility treatments have sufficient oservations for individual eriod air statistical tests. Consistent with the PBE, mean second round offer rices are higher for every eriod air in the 90 ercent treatment than in the 60 ercent treatment, and the differences are statistically significant in 5 of the last 8 eriod airs airs 4, 6, 0, 4 and 8). Moreover, second round offer rices are not significantly different from the PBE rediction of 5 in any eriod air of the 60 ercent continuation roaility treatment. Prices are, however, significantly less than the PBE rediction of 50 in every eriod air of the 90 ercent continuation roaility treatment. Finally, consider the demand withholding rice rejection) choices y uyers. As we show in the next section, rices do not vary significantly over time in the second half of the sessions. Tale 3 therefore resents the freuency distriution of oening rice offers and accetances for these later eriods. Recall that we did not allow uyers to accet rice offers that result in negative rofits, which is why the accetance rate for the value=8 uyer is uniformly zero for all rices greater than 5. The value=8 uyers usually accet offers less than or eual to 5 in the zero ercent and Arizona 30 ercent continuation roaility treatments, ut in the other treatments they accet offers of 5 less freuently. The value=54 uyers accet most rice offers in the zero ercent and Arizona 30 ercent continuation roaility treatments, ut in the other treatments they often reject high rice offers. These rejections for oth tyes of uyers are contrary to the PBE. Such demand withholding is less common in the second round not shown); in the second round uyers reject only out of 65 accetale offers 3 ercent). Although the oening round offer rejections are not consistent with the PBE, they are also not consistent with the social fairness norms that have een modeled and calirated to

14 related games. For examle, the zero ercent continuation roaility results at the to of Tale 3 indicate how results from this game are strikingly different from the tyical rejection ehavior in the well-known ultimatum game. In the ultimatum game uyers freuently reject offers that give them less than half of the exchange surlus. By contrast, our uyers accet offers of less than 7 ercent of the surlus over 80 ercent of the time e.g., rice offers of 5 to a value=8 uyer, and rice offers of 45 to a value=54 uyer). High value uyers accet an offer of 7.4 ercent of the surlus i.e., a rice of 50) two-thirds of the time, whereas in the ultimatum game such unfair offers are almost always rejected. Fehr and Schmidt [7] rovide a summary of ultimatum game results and model the fairness concerns that lead to euitale outcomes in that and other games. It is straightforward to show that their aroach does a oor jo descriing the uyer choices in this exeriment. In Fehr and Schmidt s [7] model references include ineuity aversion, so that sujects refer higher earnings for themselves ut also more eual earnings across layers. Utility is eual to monetary ayoffs less ineuity costs that rise as the difference etween a suject s own and the other s monetary ayoff increases. 8 Fehr and Schmidt also derive arameter distriutions for the relative tradeoff of monetary gains and ineuity aversion that descries ehavior across a variety of games, which we can use to assess the effectiveness of this aroach in descriing the rejection rate data reorted here. Alying their distriution of references to our sujects, it is straightforward to show that in the 0 ercent continuation treatment, 70 ercent of the value=8 uyers should reject a rice of 5. 9 Tale 3, however, indicates that these uyers only reject 5 ercent of these offers. Likewise, in this treatment the Fehr-Schmidt model and arameters 8 In articular, for a two-erson game layer i s utility is Ui x) = xi αi max{ xj xi, 0} βi max{ xi xj, 0 }, i j, where x k denotes monetary earnings k = i, j ), αi β j, and > β i 0. 9 For this calculation one only needs the distriution of layers disutility from disadvantageous ineuality i.e., the α arameter in the revious footnote), ecause the uyer s earnings are lower than the seller s earnings. We use Fehr and Schmidt s distriution of α ={0, 0.5,, 4} in roortions of {0.3, 0.3, 0.3, 0.}. 3

15 indicate a 70 ercent rejection rate for the value=54 uyers when rices are 45 or 50; ut the oserved rejection rate is only 9 ercent. Why don t we see the ronounced fairness effects that others often oserve? We conjecture that there are two main reasons. First, we have uncertainty aout uyer tye, which could have weakened the erceived fairness norm. For examle, a articular rice might e fair to one tye of uyer ut not to the other tye Raoort et al. [8]; Güth et al. [4]). Second, in this exeriment sujects alternated etween uyer and seller roles, which could also reduce the imortance of fairness concerns articularly the unfairness of highly asymmetric distriutions of exchange surlus in different rounds. Sujects essentially were ale to take turns taking advantage of the rivileged seller) osition Thaler []). Future exeriments can systematically vary the role alternation and the incomlete information aout uyer tyes in order to test these conjectures. We close this section with a few remarks aout the imlications of risk aversion. The PBE redictions shown in Tales and 3 are ased on an assumtion of risk neutrality. In the next section we estimate a model of oundedly rational, risk averse agents, so it is useful to first check whether risk aversion of sujects alone can account for deviations of oserved lay from PBE redictions. We show elow that the estimated value of the Constant Relative Risk Aversion CRRA) index is 0. for these data. If we recomute PBE redictions using this estimated risk aversion index, then PBE redictions change very little. Euilirium redictions for 0 and 60 ercent continuation roailities do not change, and the structure of the euilirium for the 90 ercent continuation roaility treatment remains the same, with only small changes in euilirium mixing roailities. 0 The initial rice rediction for the 30 0 In articular, the roaility that a high-value uyer accets an initial rice offer aove 5 dros from to 0.9. The roaility that the seller sets a low rice in eriod = 5) rises slightly, from.3% to.%. 4

16 ercent continuation roaility treatment rises from 40 to 45. Thus, overall a change from risk neutrality to CRRA with a modest level of risk aversion results in fairly small changes in PBE redictions, and what change there is tends to e in a direction away from what we oserve. There are ualitative changes in PBE redictions for higher levels of risk aversion. For examle, in the 0 ercent continuation treatment, if the CRRA risk aversion index is aove 0.4 then the initial euilirium rice offer dros from 50 to 5. There are similarly large changes in PBE redictions for high indices of risk aversion for the other continuation treatments. However, these changes in redictions are generally not in the direction toward what we oserve in the exeriments. 5. Bounded Rationality For this argaining game the erfect Bayesian euilirium redictions erform oorly in oth uantitative and ualitative dimensions. Are game theoretic redictions essentially useless for redicting ehavior in settings such as these argaining exeriments, or can game theoretic redictions e modified in some fruitful way? In this section we examine the redictions of two models of ounded rationality, as alternatives to standard euilirium models. The first model is the Noisy Nash Model NNM), which is descried in McKelvey and Palfrey [7]. The NNM osits that each agent lays their euilirium PBE, for our analysis) strategy with roaility γ and randomizes uniformly) over all strategies with roaility γ. The NNM allows for mistakes and suotimal lay on the art of sujects; the NNM generalizes the euilirium model in a way that allows for variation in the data around the euilirium oint redictions. Of course, the PBE is a secial case of the NNM, 5

17 with γ =. Since this model uts eual weight on all noneuilirium strategies, it cannot exlain why certain deviations from euilirium are more common and more lausile than others. The second model is the uantal resonse euilirium QRE) model, and it is much more intuitively aealing ecause it can otentially exlain the attern of deviations from euilirium McKelvey and Palfrey [6]). Agents in the QRE model do not always select a strategy that maximizes their exected utility, ut they choose actions that yield higher exected ayoffs with higher roaility. In contrast to the NNM, the roailities of non-nash) choices in a QRE are sensitive to the exected ayoffs for these choices. The QRE is related to an earlier analysis y Rosenthal [0] that examined the imlications of roailistic choice y oundedly rational layers in games. It can cature the notion that strategies which are nearly otimal might e chosen almost as freuently as the otimal strategy. For examle, as shown at the to of Tale 3, uyers acceted the euilirium rice offer of 50 in out of the 69 times a seller offered it in the later eriods, resulting in an average seller ayoff of 5.9. In these same later eriods uyers acceted the rice offer of 5 in 60 out of the 66 times sellers offered it, resulting in an average seller ayoff of 3.6. This similarity in average ayoffs might exlain the similar freuency of these two, very different, rice choices. McKelvey and Palfrey s [6] aer develos the idea of QRE for normal form games with finite strategy sets. A uantal resonse is a smoothed-out est resonse, in the sense that a layer chooses actions that yield higher exected ayoffs with higher roaility ut does not choose a est resonse with roaility one. The QRE is calculated as a fixed oint in roaility sace. Each agent s exected ayoffs determine the agent s choice roailities. These exected ayoffs are ased on the choice roailities of the other agents. This choice framework may e modeled y secifying the ayoff associated with a choice as the sum of two 6

18 terms. One term is the exected utility of a choice, given the choice roailities of other layers. The second term is a random variale that reflects idiosyncratic asects of ayoffs that are not modeled formally. The logit-qre is derived y assuming that these random variales are indeendent and follow an extreme value distriution. In a logit-qre each agent s choice roailities follow a multinomial logit distriution with arameter λ. As λ increases each agent uts less weight on choices that yield su-otimal exected ayoffs. As λ aroaches zero, each agent s strategy converges to a mixed strategy with eual choice roailities for each ossile action. This concet is extended to extensive form games which allow for asymmetric information) in McKelvey and Palfrey [7]. The euilirium concet in this setting is termed an agent uantal resonse euilirium AQRE). At each information set a layer decides on the roailities of different actions that he/she can take. A logit-aqre may e derived in which choice roailities at each information set follow a multinomial logit distriution. The aendix resents the alication of McKelvey and Palfrey s logit-aqre model to our exerimental environment, which rovides the asis for maximum likelihood estimation reorted elow. There are features of oth uyer ehavior and seller ehavior that suggest that ounded rationality might e a useful way to interret the exerimental results. Buyers are oserved to withhold sometimes in the last round even though the ayoff would have een higher if a urchase was made. In addition, uyers do not use the kind fixed cut-off rule for making urchases that is rescried y the PBE e.g., when the continuation roaility is 60 ercent, the PBE strategy for a high value uyer is to urchase with roaility one if 30 and urchase with roaility zero if > 30 ). Instead, as Tale 3 indicates uyers aear to use decision rules in which the roaility of urchase rises gradually as consumer surlus rises. A smooth 7

19 resonse such as a roit or logit distriution descries uyers decision rules etter than a cutoff rule. Seller ehavior also aears to e noisy. Sustantial variation in rice choices exists oth within and across sujects, even in a relatively simle environment such as the zero continuation roaility treatment. Tale 4 reorts maximum likelihood estimates for the NNM and AQRE models of ounded rationality. A single arameter is estimated for each model: γ for the NNM and λ for the AQRE. Below each arameter estimate we reort the standard error and the log likelihood Λ*) for the model. Following the standard ractice for laoratory analyses using the QRE model, for these estimates we only include data from the later art of the sessions e.g., see Goeree et al. [] or Cara et al. [3]). In articular, we exclude data from eriods -5. During the early eriods of an exeriment sujects are likely to e learning aout how their rivals lay the game and adjusting their own choices in resonse to what they learn. For some treatments there are trends in oening rices during the early eriods. But as noted at the start of Section 4, there do not aear to e any trends in rices after eriod 5 in any of the exeriments. Parameter estimates are reorted for each treatment individually and also for ooled data. The hyothesis that the site for the exeriment Arizona or Purdue) yields different results was rejected for all values of δ excet δ = 30 ercent. We therefore reort arameter estimates for 30 ercent Arizona and Purdue datasets searately. The estimated value of γ for the NNM is well elow / for all treatments, and is essentially zero for the 30 ercent and 90 ercent continuation roaility treatments. For the ooled data the estimated γ is 0.. This means that under the hyothesis that the NNM is the correct model, only ercent of choices corresond to Nash erfect Bayesian) euilirium choices with the remaining 88 ercent of choices est characterized as random. 8

20 The freuency of these aarently random choices does aear to e related to their exected ayoffs, however. The lower half of Tale 4 indicates that the AQRE model erforms etter than the NNM in the sense that the log likelihood is higher for the AQRE for every treatment. The estimated values of the noise arameter λ aear reasonale comared to other estimates for exerimental games in the literature. For examle, in most of the games they study McKelvey and Palfrey [7] estimate λ coefficients that range etween one and two. The estimated λ for the zero ercent continuation treatment is somewhat higher than in the other treatments. This imlies less noise in sujects est resonse ehavior for the one-round argaining game, which is not surrising since this game is less comlex than the two-round games. In order to imrove the redictive ower of the AQRE we allow for the ossiility of suject risk aversion. Prices in the zero roaility continuation treatment were much lower than the risk neutral euilirium rediction of 50; indeed, 47 ercent of the rices in this treatment were 5 or less. Risk aversion on the art of seller sujects could account for such low rices. In our comutation of the AQRE we osit a constant relative risk averse utility function for each α = suject of the form, u c) c / α), where c is the dollar ayoff for the one round or two round) game and α is the index of relative risk aversion. Comarisons of λ across exeriments are of limited value, however, ecause λ is sensitive to the scale of ayoffs, and the scale often varies across studies. For examle, in a similar secification ut with a ower functional form of µ=/ λ, Cara et al. [] and [3] estimate µ=8.3 and µ=6.7, resectively, in the Traveler s Dilemma and a Bertrand rice cometition game. The imlied λ of /8.3=0. and /6.7=0.5 are an order of magnitude lower than our estimates, ut this is due artly to the differences in ayoff scales. Cara et al. exress ayoffs in cents while we exress ayoffs in dollars, and the er-eriod earnings diffe r across these exeriments anyway, even in real terms. Risk aversion was also advanced as an exlanation of rices for the one-shot ricing exeriments reorted y Reynolds [9]. An alternative strategy is to introduce other-regarding reference arameters, as in Goeree and Holt [0]. As noted in the revious section, however, fairness considerations aear less imortant in our data comared to other studies. Moreover, Goeree and Holt need to introduce three new arameters to extend the QRE model for Fehr-Schmidt [7] ineuality aversion utility, so this strategy is less arsimonious than the one ursued here. 9

21 Tale 5 reorts estimates for the AQRE model with risk aversion. The estimated values of the risk aversion index α vary from a risk neutral level of zero for the 60 ercent continuation treatment to moderately risk averse α ˆ = 0. 5 ) for the zero ercent continuation treatment, to highly risk averse α ˆ = 0. 6) for the 30 ercent continuation treatment conducted at Arizona. Using the same functional form for the utility function, Goeree et al. [] estimate α ˆ = 0. 5 in QRE estimates for indeendent rivate value first-rice auction exeriments. The variation in the risk aversion index is an indication that this model and utility function are too simle and are misecified, ut the variation arises largely from one or two anomalous treatments. A likelihood ratio test for our estimates strongly rejects the null hyothesis of risk neutrality in the zero and the Arizona 30 ercent continuation treatments χ d.f. estimates are consistent with risk neutrality in the other datasets. = 90.4 and 5.6, resectively), ut the Parameter estimates for the ooled data shown on the right of Tale 5 indicate a relatively small amount of risk aversion and a moderate value for the noise arameter. Although this ooling averages out the significant differences across treatments discussed aove, we use these ooled estimates to summarize the imlications of this AQRE model with risk aversion. The rightmost column of Tale lists mean and median oening offer rices for the AQRE ased on ML arameter estimates for the ooled data in Tale 5. These redicted mean and median rices are hardly sensitive to changes in the continuation roaility. In fact, median oening offer rices for the AQRE are eual to 30 for each continuation roaility. This is consistent with our result Section 4) that these rices are tyically not significantly different across continuation roaility treatments. 3 3 Median oening offer rices are also 30 for all treatments according to the fitted Noisy Nash Model with the ooled estimate of γ ˆ =0.. But the NNM does not fit the imodal distriution of offers as well as the AQRE model does, and the NNM rovides an inferior fit for the second round offer rices. In articular, the fitted second round 0

22 Figures and 3 illustrate that the AQRE model with risk aversion is also consistent with the wide range of offer rices we oserve. These figures show that the freuency distriution of oening offer rices tyically varies etween 0 and 50, and that the AQRE distriution ased on the arameter estimates from the ooled data also varies widely. The AQRE has a significant fraction of offers at 5 the highest rice that oth uyer tyes can rofitaly accet), ut the mode of 5 for the oserved offer rices is significantly stronger in all ut the 90 ercent continuation treatment not shown). 4 The AQRE also imlies more offers of 45 and 50 than oserved, excet in the one-round game Figure ), and it also imlies a smoother and more uniform distriution of rices than we oserve. Overall, however, these AQRE distriutions are consistent with many of the ualitative roerties of the results articularly when comared to the erfect Bayesian euilirium. 6. Conclusion This exeriment exlores the redictive ower of a relatively standard ut rather sohisticated game theoretic euilirium concet erfect Bayesian euilirium in multiround ricing games. The results reveal significant failures of this theory in terms of levels of rices, comarative statics redictions, and uyer urchasing ehavior. These rediction failures occur in site of an exerimental design that secifies a relatively simle decision-making environment. We find that the agent uantal resonse euilirium model of noisy decision-making in games catures imortant features of the results. In articular, this model correctly redicts two NNM mean and median rices range etween 5 and 30, and are always further than the AQRE model from the oserved second round offers. 4 The fit dislayed in these figures is imroved sustantially if we use the treatment-secific rather than the ooled arameter estimates. As Haile, Hortaçsu and Kosenock [5] have recently emhasized, however, it is imortant to leave the estimated value of the QRE arameter unchanged across treatments to make comarative statics exercises informative.

23 features of the exerimental data: ) variations in the continuation roaility have relatively little imact on oening rices and ) a wide and in some cases imodal) distriution of oening rices. Our imlementation of this model also includes the ossiility of suject risk aversion, and sujects aear to ehave as if risk averse in some treatments. Other-regarding references have ecome a oular exlanation of deviations from the standard euilirium in settings like this two-erson argaining environment e.g., Fehr and Schmidt [7]). But an imlication of our results is that an aeal to fairness considerations is not the only way to exlain these deviations, and that ounded rationality and risk aversion alone are consistent with a reasonaly large roortion of the deviations from euilirium oserved in this game.

24 Tale : Summary of Exerimental Sessions Session Name Continuation Proaility Session Site Numer of Sujects Numer of Periods UA00-0 ercent Univ. of Arizona 5 30 UA30-30 ercent Univ. of Arizona 5 30 UA30-30 ercent Univ. of Arizona 3 6 UA60-60 ercent Univ. of Arizona 5 30 UA90-90 ercent Univ. of Arizona 3 6 PU00-0 ercent Purdue Univ PU30-30 ercent Purdue Univ. 5 a 30 PU60-60 ercent Purdue Univ PU90-90 ercent Purdue Univ. 5 8 a Although this session emloyed 5 sujects, we discovered after the session that one of the sujects had reviously articiated in session PU00-. Data from this exerienced suject are excluded from the analysis. The software crashed after eriod 8 in this session. 3

25 Tale : Summary Statistics for Price Offers for Each Continuation Proaility Treatment Continuation Proaility Oservations Panel A: Oening Offer Prices Perfect Bayesian Euilirium Offer 0 ercent ercent Arizona) 30 ercent Purdue) ercent ercent Panel B: Second Round Offer Prices Mean Offer Std. Error) ) ) ) ) ) Actual Offers Median Offer AQRE Model Mean Median) ) ) ) 9. 30) 8. 30) 0 ercent ercent Arizona).09) 30 ercent Purdue).9) 60 ercent ) 90 ercent * ) ).0 5) 4.3 0) 6.9 5) * The second round rice of 50 is aroximate. The seller sets a rice of 50 with roaility 98.8 ercent and a rice of 5 with roaility. ercent. 4

26 Tale 3: Oening Offer Price Accetances for Late Periods Periods 6-30) Continuation Proaility Buyer Value Prices ercent 8 Price Freuency ercent 8 Numer Acceted ercent 8 Accetance Rate ercent 54 Price Freuency ercent 54 Numer Acceted ercent 54 Accetance Rate ercent Arizona) 8 Price Freuency ercent Arizona) 8 Numer Acceted ercent Arizona) 8 Accetance Rate ercent Arizona) 54 Price Freuency ercent Arizona) 54 Numer Acceted ercent Arizona) 54 Accetance Rate ercent Purdue) 8 Price Freuency ercent Purdue) 8 Numer Acceted ercent Purdue) 8 Accetance Rate ercent Purdue) 54 Price Freuency ercent Purdue) 54 Numer Acceted ercent Purdue) 54 Accetance Rate

27 Continuation Proaility Tale 3 continued): Buyer Value Prices ercent 8 Price Freuency ercent 8 Numer Acceted ercent 8 Accetance Rate ercent 54 Price Freuency ercent 54 Numer Acceted ercent 54 Accetance Rate ercent 8 Price Freuency ercent 8 Numer Acceted ercent 8 Accetance Rate ercent 54 Price Freuency ercent 54 Numer Acceted ercent 54 Accetance Rate Notes: The erfect Bayesian euilirium rice offer is shown in the ox. This model redicts accetance rates of.00 in the old highlighted regions, and accetance rates of 0.00 elsewhere, excet for =50 in the 90 ercent continuation roaility treatment, in which the accetance rate for the high value uyer is

28 Tale 4: Maximum Likelihood Estimates for NNM and AQRE Models* data after eriod 5) δ : Continuation Proaility zero 30 ercent Arizona 30 ercent Purdue 30 ercent Pooled 60 ercent 90 ercent Pooled Data Noisy Nash Model γ S.E ) ) ) ) ) ) ) Λ* AQRE Model λ S.E ) ).0 0.5) ).9 0.). 0.) ) Λ* # oservations * Standard errors are in arentheses. Standard errors were comuted using the ootstra method. The samle size for the ootstra was eual to the samle size of the data see the last row in the tale). 50 iterations were run to create a samle of ootstra estimators. 7