Bounded Rationality in Laboratory Bargaining with Asymmetric Information

Transcription

1 Bounded Rationality in Laoratory Bargaining with Asymmetric Information Timothy N. Cason and Stanley S. Reynolds * Deartment of Economics, Krannert School of Management, Purdue University, West Lafayette, IN ; and Deartment of Economics, Eller College of Business and Pulic Administration, University of Arizona, Tucson, Arizona 857 June 00 This aer reorts an exeriment on two-layer seuential argaining with asymmetric information. The exerimental design is simle enough to ermit comutation of redictions of models of ounded rationality and/or learning, yet rich enough to cature the forces at work 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. Data from four different continuation roailities are used to test euilirium redictions. We also consider uantal resonse euilirium QRE) to cature ounded rationality of sujects, and this aroach catures many of the imortant features of the results.. 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 the efficiency of 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 take-it-or-leave-it one-shot) situation, in order to induce the uyer to urchase * 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. Timothy O Neill Dang, Thomas Wilkening and Marikah Mancini rovided exert research assistance.

2 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, 98, and Gul, Sonnenschein, and Wilson, 986). This last result seems to cature the essence of Coase s conjecture Coase, 97) 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 euilirium may involve erfect rice discrimination over time rather than rices close to marginal cost see Bagnoli, Salant and Swierzinski, 989). Recent laoratory exerimental studies y Güth, Ockenfels, and Ritzerger 995), Raoort, Erev, and Zwick 995), Cason and Sharma 00), and Reynolds 000) 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 Reynolds 000) descries how the durale goods monooly model has een extended in a variety of directions in recent aers.

3 aout the effects of changes in information, changes in the discount factor, or changes in the time horizon. Moreover, fairly simle 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, 000). The resent aer reorts on a new exerimental design for seuential argaining with asymmetric information. The idea is to have a design that is simle enough to ermit comutation of redictions of models of ounded rationality and/or learning, yet rich enough to cature the forces at work in multi-round ricing environments. To this end, 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 uyer they are matched with. The seller makes a rice offer 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. In the second round, the seller makes a rice offer 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 assuming 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 3

4 for straightforward comutation of erfect Bayesian euilirium redictions. More imortantly, this relatively simle setu facilitates comutations using models of learning or of ounded rationality in games. We will focus on one such model in this aer: the agent uantal resonse euilirium AQRE) model of ounded rationality or, noisy decision-making) roosed y McKelvey and Palfrey 998) for extensive form games.. 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 Ritzerger 995) 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 analyzing durale goods monooly ricing games. The results for untrained sujects were grossly inconsistent with the theory. Prices failed to conform to comarative statics redictions and rices tended to e much higher than redicted. Prices with trained sujects were closer to theoretical redictions, ut rices still failed to satisfy comarative statics redictions. Raoort, Erev and Zwick 995) reort on argaining exeriments with time discounting, one-sided incomlete information, and an infinite unlimited) time horizon. They reort 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 Güth, Ockenfels and Ritzerger 995) use redictions from a model with a continuum of uyers. Their exerimental design involves 0 uyers, each demanding a single unit, with all sujects fully informed aout uyers values. Bagnoli, Salant and Swierzinski 989) show that euilirium rices and seller rofits for this kind of discrete demand model tend to e much higher than euilirium rices and rofits for a continuum model. 4

5 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 00) 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 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. Results for the two treatments were fairly similar, in site of large differences redicted y euilirium models. 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 000) 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. However, there is a significant failure of 5

6 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. 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 game has either one or two rounds of argaining, deending on the treatment condition. Sujects interacted only through a comuter network running an alication written using the University of Zurich s z- Tree rogram. All sujects were inexerienced, in the sense that they had never articiated in a revious argaining session that emloyed this same 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. 990). 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.] 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. 6

7 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. The instructions shown in Aendix A 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 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 7

8 using an exchange rate of laoratory dollars = U.S. dollar. Salient earnings ranged from $0 to $45, with a mean of aout $3. 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. 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 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 = 8

9 redicted effect of an increase in the discount factor in the infinite horizon exeriments run y Raoort, Erev and Zwick 995). 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] 0.375, 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 Tale resents the mean and median offer rices in each treatment, ooled across eriods. The rightmost column is discussed later, in Section 5.) 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. 4 3 The derivation of this euilirium is similar to the derivation in Fudenerg and Tirole 983). The main difference is that the rice sace is discrete in our setting, whereas Fudenerg and Tirole have rices chosen from a continuous interval. 4 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 9

10 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) 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) 5) 30 ercent Purdue).9) 5) 60 ercent ) 0) 90 ercent * ) 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. 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 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 0

11 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 lower than euilirium 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. 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 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.

12 eriods ecause they alternated etween uyer and seller roles. In the following discussion, eriod air refers to eriods and, eriod air refers to eriods 3 and 4, and so on. As shown in Tale, the PBE redicts that rices in round one should vary sustantially as the continuation roaility varies. Consistent with the visual imression rovided y Figure, however, in many 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. 5 Also consistent with euilirium, rices are higher in the zero ercent continuation roaility treatment than the 60 ercent continuation roaility treatment in eriod airs 4, 6, 7, and through 5. Contrary to the PBE, however, rices are also significantly lower in the Arizona 30 ercent continuation roaility treatment than in the 60 ercent continuation roaility treatment in eriod airs, 6 and 8 through 5, and they are significantly lower than the Purdue 30 ercent continuation roaility treatment in eriod airs 3, 4, 5 0,, 3, 4 and 5. 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 reject 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 5 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 3, 4, and 6 through 5; and are lower than in the 90 ercent continuation roaility treatment in eriod airs through 6, 8 through, 3 and 4 all five-ercent significance level, one-tailed tests).

13 roaility is 30 ercent; and c) median rices eual 50 when the continuation roaility is zero ercent. 6 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 4 and 4. 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 7, 8, 0, and 4). 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. A uick look at the zero ercent continuation roaility results at the to of this tale indicates how results from this game are strikingly different from the tyical rejection ehavior in the well-studied ultimatum game. In the ultimatum game uyers freuently reject offers that give them less than half of the exchange surlus. But our 6 The one excetion is eriod air 5 for the 30 ercent continuation roaility treatment at Purdue, which is not significantly different from 40. 3

14 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

15 Tale 3 continued): 60 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

16 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 see, e.g., Fehr and Schmidt, 999, for a summary of ultimatum game results). Recall that our comuter rogram 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). 5. Bounded Rationality A natural uestion that arises is whether game theoretic redictions are essentially useless for redicting ehavior in reasonaly comlex settings such as these argaining exeriments, or whether game theoretic redictions can e modified in some fruitful way. One aroach that has received attention recently involves the concet of uantal resonse euilirium QRE), develoed y McKelvey and Palfrey 995). In a sense this aroach is a relaxation of the assumtion of erfect rationality on the art of decision-makers. It osits that agents in a game setting do not always select a strategy that maximizes their exected utility, ut rather that agents 6

17 ick actions that yield higher exected ayoffs with higher roaility. This work is related to an earlier analysis y Rosenthal 989) that examined the imlications of roailistic choice y oundedly rational layers in games. McKelvey and Palfrey s 995 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 kind of choice framework may e modeled y secifying the ayoff associated with a choice as the sum of two 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 998). 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. Aendix B 7

18 resents the alication of McKelvey and Palfrey s logit-aqre model to our exerimental environment. The model secified in Aendix B rovides the asis for maximum likelihood estimation reorted elow. There are features of oth uyer ehavior and seller ehavior that suggest that the AQRE concet 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. Also, 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 that secify that the roaility of urchase rises gradually as consumer surlus rises. A smooth 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. In our alication 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. 7 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 game a round of the exeriment) and α is the index of relative risk aversion. 8

19 Tale 4 reorts maximum likelihood estimates of two arameters: the risk aversion index α and the noise arameter λ. 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., 000, or Cara et al., 00). 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. There do not aear to e any trends in rices after eriod 5 in any of the exeriments. 8 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 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. 000) estimate α ˆ = 0. 5 in QRE estimates for indeendent rivate value first-rice auction exeriments. 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 = 90.4 and 5.6, resectively), ut the estimates are consistent with risk neutrality in the other datasets. The large d.f. 7 Risk aversion was also advanced as an exlanation of rices for the one-shot ricing exeriments reorted y Reynolds 000). 8 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. 9

20 difference in the estimated risk aversion index for the 30 ercent Arizona and Purdue data is surrising. This seems to e due mainly to lower oening rices in the Arizona data. We do not have a good exlanation for this difference. Tale 4: Maximum Likelihood Parameter Estimates* data after eriod 5) δ : Continuation Proaility zero 30 ercent Arizona 30 ercent Purdue 60 ercent 90 ercent Pooled Data α : Risk Aversion Index ) ) ) ) ) 0. λ : Noise Parameter ).0 0.0).0 0.5).9 0.8).0 0.).7 Log-likelihood Restricted α=0) risk neutral model λ : Noise Parameter ) ).0 0.5).9 0.). 0.).9 Log-likelihood # 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. 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 998) estimate λ coefficients that range etween one and two. 9 The 9 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 0

21 estimated λ for the zero ercent continuation treatment is 4.4, more than twice as high as 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. Although the estimated λ λˆ =.0) is lower in the Arizona 30 ercent exeriments than in the other two round exeriments, note that sujects act as if they are more risk averse in this treatment. The utility function with arameter α ˆ = 0. 6 tends to raise utility ayoffs over the relevant range of er eriod ayoffs $-$3) y aout 30 to 00 ercent. E.g., the utility of a $ ayoff is 0.4 / ) Conseuently, the lower λ estimate in the Arizona 30 ercent continuation treatment arises largely from this scaling effect, so the level of noise estimated in all treatments with two-round argaining aears similar. Parameter estimates for the ooled data shown on the right of Tale 4 indicate a fairly 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. 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. µ=/λ, Cara et al. 999 and 00) 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 utility, and the er-eriod earnings differ across these exeriments anyway, even in real terms.

22 Figures through 6 demonstrate that the AQRE model 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. 0 The AQRE also imlies more offers of 45 and 50 than oserved, excet in the one-round game. Overall, however, these AQRE distriutions tend to e consistent with the ualitative roerties of the results articularly when comared to the redictive ower of 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. This is not too surrising in light of the failures of this kind of euilirium theory in related exeriments. We find that the agent uantal resonse euilirium AQRE) model of noisy decisionmaking in games, catures imortant features of the results. In articular, this model redicts two 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 0 The fit would e imroved sustantially using the treatment-secific rather than the ooled arameter estimates. For examle, the greater risk aversion estimated in the zero and Arizona 30 ercent continuation treatments leads to a more ronounced mode on 5 for the AQRE in those treatments.

23 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. There are clearly a numer of things left to investigate with these data. Buyer ehavior has not een examined in detail. The AQRE aears to cature imortant features of uyer ehavior, ut this has not een checked systematically. We also lan to comare the erformance of the AQRE to alternative models of suject ehavior, such as the noisy Nash model McKelvey and Palfrey, 998) and models of suject learning. 3

24 Figure : Mean Oening Prices y Period Pair for Four Continuation Proaility Treatments Offer Price Mean Oening Price, 0% Continuation Mean Oening Price, 30% Continuation, Arizona Mean Oening Price, 30% Continuation, Purdue Mean Oening Price, 60% Continuation Mean Oening Price, 90% Continuation Period Pair 4

25 80 Figure : Actual Oening Prices and AQRE Price Distriution using Pooled ML Estimates: 0% Continuation, Data after Period 5 70 Freuency AQRE Acceted) AQRE Rejected) Actual Acceted) Actual Rejected) Price 80 Figure 3: Actual Oening Prices and AQRE Price Distriution using Pooled ML Estimates: 30% Continuation Arizona), Data after Period 5 Freuency AQRE Acceted) AQRE Rejected) Actual Acceted) Actual Rejected) Price 5

26 40 Figure 4: Actual Oening Prices and AQRE Price Distriution using Pooled ML Estimates: 30% Continuation Purdue), Data after Period 5 35 Freuency AQRE Acceted) AQRE Rejected) Actual Acceted) Actual Rejected) Price 80 Figure 5: Actual Oening Prices and AQRE Price Distriution using Pooled ML Estimates: 60% Continuation, Data after Period 5 70 Freuency AQRE Acceted) AQRE Rejected) Actual Acceted) Actual Rejected) Price 6

27 70 Figure 6: Actual Oening Prices and AQRE Price Distriution using Pooled ML Estimates: 90% Continuation, Data after Period 5 60 Freuency AQRE Acceted) AQRE Rejected) Actual Acceted) Actual Rejected) Price 7

28 REFERENCES Bagnoli, M., S. Salant, and J. Swierzinski, Durale-Goods Monooly with Discrete Demand. Journal of Political Economy, Vol ), Cara, C. M., J. Goeree, R. Gomez and C. Holt, Anomalous Behavior in a Traveler s Dilemma? American Economic Review ), Cara, C. M., J. Goeree, R. Gomez and C. Holt, Learning and Noisy Euilirium Behavior in an Exerimental Study of Imerfect Price Cometition, International Economic Review 00), forthcoming. Cason, T. and T. Sharma, Durale Goods, Coasian Dynamics and Uncertainty: Theory and Exeriments, Journal of Political Economy 00), forthcoming. Coase, R. H., Duraility and Monooly Journal of Law and Economics 5 97), Cooer, R., D. DeJong, R. Forsythe, and T. Ross, Selection Criteria in Coordination Games: Some Exerimental Results, American Economic Review, 80: March 990), Fehr, E., Schmidt, K. M., A Theory of Fairness, Cometition, and Cooeration, Quarterly Journal of Economics 4 999), Fudenerg, D., and J. Tirole, Seuential Bargaining with Incomlete Information. Review of Economic Studies, Vol ), Goeree, J., C. Holt, and T. Palfrey, Quantal Resonse Euilirium and Overidding in Private- Value Auctions, Caltech working aer 000). Gul, F., H. Sonnenschein, and R. Wilson, Foundations of Dynamic Monooly and the Coase Conjecture Journal of Economic Theory ), Güth, W., P. Ockenfels, and K. Ritzerger, On durale goods monoolies: An exerimental study of intraersonal rice cometition and rice discrimination over time Journal of Economic Psychology 6 995), McKelvey, R., and T. Palfrey, Quantal Resonse Euiliria in Normal Form Games Games and Economic Behavior 7 995), McKelvey, R., and T. Palfrey, Quantal Resonse Euiliria for Extensive Form Games Exerimental Economics 998),

29 Raoort, A., I. Erev, and R. Zwick, An Exerimental Study of Buyer-seller Negotiation with One-sided Incomlete Information and Time Discounting Management Science 4 995), Reynolds, S., Durale-Goods Monooly: Laoratory Market and Bargaining Exeriments, RAND Journal of Economics 3 000), Rosenthal, R., A Bounded-Rationality Aroach to the Study of Noncooerative Games International Journal of Game Theory 8 989), Stokey, N., Rational Exectations and Durale Goods Pricing Bell Journal of Economics 98),

30 APPENDIX A Instructions for Sujects The instructions given to sujects for the 30 ercent treatment are elow. The screen dislays referred to are not included. INSTRUCTIONS This is an exeriment in the economics of decision making. The instructions are simle and if you follow them carefully and make good decisions you will earn money that will e aid to you in cash. In this exeriment you will articiate in a seuence of market trading eriods. In each trading eriod sujects meet as uyers and sellers to trade a fictitious commodity. Buyers and sellers will conduct transactions using a comuterized trading system. Matching Buyers and Sellers At the eginning of each trading eriod sujects are matched together to form airs, with one uyer and one seller in each air. Each uyer will have an oortunity to make one urchase from the seller they are matched with during the eriod, according to trading rules that are exlained elow. You will e notified via your comuter whether you are articiating as a uyer or as a seller or, whether you are sitting out, in the current trading eriod. During the course of the exeriment you will e matched with every other suject two times; once as a uyer and once as a seller. This means that you will not e matched with any other suject more than once as a uyer, and similarly, you will not e matched with any other suject more than once as a seller. Buyers' Resale Values A uyer s earnings for a trading eriod deend on their resale value for the eriod. A uyer s resale value for a eriod is selected randomly y the comuter. There is a 40 % chance that the resale value is 54 and a 60 % chance that it is 8. A uyer is informed of their resale value at the start of a trading eriod; no other suject oserves the uyer's value. A new resale value for each uyer will e randomly selected at the eginning of every trading eriod during the exeriment. A uyer's resale value determines the rofit the uyer earns y making a urchase. Suose for examle that your resale value as a uyer is 54 and that you have een offered a rice of 5 for the good. The screen dislay you will oserve is on the next age. By making a urchase you ay a rice of 5 and receive revenue of 54 y reselling the good to the exerimenter. Your earnings for making a urchase in this situation are: 30

31 Earnings = 54-5 = 49 Suose instead that your resale value is 8 and that you have een offered a rice of 5 for the good. Then your earnings for making a urchase are: Earnings = 8-5 = 3 If the rice offer exceeds the uyer's resale value then the comuter will not ermit the uyer to make a urchase that is, a uyer will not e ermitted to have negative earnings). Sellers and Trading Rules A seller has the oortunity to make at least one, and ossily two, rice offers during each trading eriod. A seller chooses a rice from the following list of ten ossile rices: 5, 0, 5, 0, 5, 30, 35, 40, 45, 50. The screen dislay you will oserve as a seller is on the next age. A seller chooses a rice offer at the eginning of each trading eriod. The uyer that the seller is matched with then decides whether or not to accet this rice offer. If this offer is acceted then the eriod is over. If this offer is not acceted then the seller may e ale to make a second offer during the trading eriod. There is a 30 ercent chance that a second rice offer will e ermitted. You will receive a message on your comuter screen informing you of whether a second rice offer is ermitted, if the first offer is not acceted. If the uyer accets a rice offer then: ) the uyer's earnings for the eriod are eual to their resale value minus the rice, as exlained aove, and ) the seller's earnings for the eriod are simly eual to the rice. Suose, for examle, that you offer a rice of 5 as a seller and your offer is acceted. Then your earnings for the eriod are: Earnings = 5 Suose instead that you offer a rice of 50 as a seller and your offer is acceted. Then your earnings for the eriod are: Earnings = 50 A seller can make at most one sale during a trading eriod. If a seller's first rice offer is acceted during a trading eriod then the seller will not have the oortunity to make a second rice offer. If no rice offers are acceted during a trading eriod then oth the uyer and the seller earn zero for the eriod. Exchange Rate 3

32 At the end of the exeriment you will e called individually and aid in cash in rivate y the exerimenter. The amount of your cash ayment will deend on your earnings in the exeriment. For every units of exerimental earnings you will e aid one dollar in cash. Note that the higher your exerimental earnings are, the higher your cash ayment will e. Recording Earnings The last age of the instructions is a sheet where you should record your values if you are a uyer), rice offers, acceted offers, and your earnings for each eriod. Your total earnings for the exeriment are the sum of your earnings for all trading eriods. Summary Buyer earnings = resale value - rice Seller earnings = rice There is a 40 % chance that a uyer's resale value is high 54); a 60 % chance that it is low 8). Buyers and sellers are matched into airs at the eginning of each trading eriod. A seller can make at least one rice offer er eriod; a second offer can e made with a 30 % chance if the first offer is not acceted. A uyer can accet at most one rice offer er eriod. The exchange rate is units of exerimental earnings er dollar of cash. 3