Structural inferences from first-price auction data Paul Pezanis-Christou
Andrés Romeu Santana
Bureau d’Economie Théorique et Appliquée Université Louis Pasteur 61, Avenue de la Forêt Noire 67085 Strasbourg, France
Departamento de Fundamentos del Análisis Económico, Universidad de Murcia, Campus Espinardo 30100 Murcia, España
December 2007 Abstract We use structural econometric methods to assess the Symmetric Bayes-Nash Equilibrium (SBNE) model of bidding with data from first-price auction experiments. We analyze the data from both empirical and experimental perspectives. Our study focuses on the sensitivity of structural inferences to SBNE’s basic assumptions and to the quality of the information available to the field researcher. It shows that once behavior is diagnosed “out-ofequilibrium”, structural estimates become highly sensitive to the information available and, most important for field studies they do not improve with the quality of information. From an experimental perspective (i.e., when taking account of bidders’ value realizations), we identify conditions to structurally test the SBNE model for homogenous or heterogeneous constant relative risk averse bidders in experimental contexts and report an overwhelming rejection of this model to explain behavior at first-price auctions with independent private values.
Key Words: first-price sealed bid auctions, structural econometric analysis of auctions, constant relative risk aversion, optimal reserve price, experiments. JEL Classification Numbers: C9, D44
We thank Olivier Armantier, Jacob Goeree, François Laisney, Harry Paarsch and seminar participants at GREQAM, the Institute for Economic Analysis (CSIC, Barcelona), the Max Planck Institute (Jena), the Henri Poincaré Institute, the Economic Science Association conferences in Strasbourg and Tucson, and the North American Summer Meetings of the Econometric Society in Durham for helpful comments and discussions; and James Cox, John Kagel and James Walker for access to their data. Financial support from the European Commission through a EU-TMR ENDEAR Network grant (FMRX-CT98-0238) and the Ministerio de Ciencia y Tecnologia (Grant BEC-2001-0980) is gratefully acknowledged. Remaining errors are ours. Pezanis-Christou e-mail Telephone
Romeu Santana
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1. Introduction The growing use of auctions to allocate assets has been paralleled in recent years by the development of an impressive literature on the structural estimation of auction models (see Paarsch and Hong, 2006, for a review). In its essence, the structural approach to data allows one to test a model under the assumption that the variables of interest are in equilibrium. In the context of games with incomplete information such as first-price sealed bid auctions, however, equilibrium models are thought to be difficult to assess because the relevant data are usually not observable (e.g., the number of active bidders, the functional form of bidders’ distribution(s) of private values, their risk preferences or the realizations of their private values). Therefore, rather than testing equilibrium models, the structural approach to auction data typically consists in recovering the unobserved information by assuming bidders to act in a given equilibrium. Inference based on this approach --- henceforth, structural inference --- is fundamental for policy recommendations such as setting a revenue enhancing optimal reserve price, or implementing an alternative selling mechanism. But the success of such recommendations, which may imply nonnegligible money transfers from one side of the market to another, remains highly conditional on the assumption that the recovered information is correct or equivalently, that the observed bids or prices are in the presumed equilibrium. While the literature focuses on identifying conditions to recover information from bid data (Athey and Haile, 2000, Perrigne and Vuong, 2007), empirical applications typically do not check that the recovered information is indeed consistent with the model’s assumptions and/or predictions. The first goal of this paper is to assess the sensitivity of structural inferences to some basic behavioral assumptions and to the information that is typically available to the field researcher. To this end, we use laboratory data and a methodology inspired from Laffont, Ossard and Vuong (1995) for the analysis of symmetric first-price auctions with private independent values. We motivate the use of experimental rather than field data by the fact that in experiments, the auction fundamentals are perfectly known to the researcher and are made explicit to the participating bidders. Our choice to study symmetric auctions further keeps the analysis simple as it requires that the bidders’ private values are all drawn (with replacement) from the same probability distribution; a crucial assumption for the determination of equilibrium predictions. Hence, by controlling the exhaustiveness of the analysis (i.e., prior check of a symmetric / homogenous bidding behavior) and the quality of the information available (e.g., information on all bids or only on winning bids), we can test the SBNE model for risk averse bidders and check how
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inferences and policy recommendations are affected when the model’s basic assumptions are not fulfilled. Our study reveals in particular that once behavior is diagnosed “out-of-equilibrium”, both structural inferences and recommendations become highly sensitive to the information available and do not necessarily improve with its quality. So far we assumed an empirical setup in that we discarded information about the bidders’ value realizations from analysis. The paper’s second goal is to build a structural model that takes explicit account of this information for the analysis of experimental data. In first-price auction experiments, behavior is typically characterized by a significant overbidding (i.e., bidding above the risk neutral Nash equilibrium prediction) and heterogeneity; both patterns being usually well or best explained by a SBNE model that assumes heterogeneous constant relative risk attitudes (Cox et al., 1982, 1983, 1988, 1996 and Chen and Plott, 1998). This assumption, however, has raised some controversy because it does not consistently organize the behavior observed at firstprice auctions (Kagel, 1995, Ockenfels and Selten, 2005, Neugebauer and Selten, 2006, and Engelbrecht-Wiggans and Katok, 2007), nor does it across various auction institutions (Kagel, Harstad and Levin, 1987, Kagel and Levin, 1993, and Engelbrecht-Wiggans and Katok, 2006). It appears to us that this controversy mostly results from the lack of a structural assessment of the SBNE model for risk averse bidders with experimental data; the literature drawing its conclusions from the estimation of reduced-form expressions of equilibrium behavior and/or from a goodness-of-fit comparison of behavioral models (Cox et al., 1988, 1996, Chen and Plott, 1998, Goeree, Holt and Palfrey, 2000, Bajari and Hortaçsu, 2005, and Armantier and Treich, 2006). In this paper, we identify conditions that allow a straightforward assessment of this model in experimental conditions and report an overwhelming rejection of it. To some extent, our analysis complements the one of Bajari and Hortaçsu (2005) who estimate structural econometric models with experimental first-price auction data. 1 While their study concentrates on fitting various behavioral models in the aggregate and on comparing the non-parametrically estimated values to the actual ones, our analysis deals instead with testing some variants of the SBNE model under information conditions that pertain to empirical or experimental setups. In the next section we sketch the theoretical predictions for first-price auctions. Sections 3 and 4 outline the econometric procedures used to test the SBNE model in empirical and experimental conditions, respectively. Section 5 reports and discusses the outcomes and Section 6 concludes.
1
See also Armantier (2002) who analyzes experimental auction data with non-parametric structural methods to decide which of the common value or the private independent value framework applies best.
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2. Theory and Model Specification 2.1. Symmetric Bayes-Nash Equilibrium Behavior In a first-price sealed-bid auction, n bidders compete for the purchase of a single commodity which is awarded to the highest bidder for a price equal to her/his bid. Each bidder is assumed to receive a private reservation value vi , which is an independent draw from a distribution F with support [v ; v ] and density f . Bidders have the same utility function u(¸) on monetary payments, with u ' 0 , u " b 0 and u(0) 0 so that a bidder with value vi who submits a bid bi has a utility of winning the auction equal to u(vi bi ) . The number of bidders, F and u(¸) are common knowledge but the value realizations vi are private information. In this context, a bidding strategy b(¸) is a SBNE strategy if for all valuations, it is a best response for bidder i to use b(¸) if all bidders j v i also use b(¸) . Maskin and Riley (2000) further show that if b(¸) is a bidder’s best response then it is monotone increasing in valuations. Therefore, if b 1(bi ) stands for the inverse of b(¸) then bidder i’s expected payoff is defined as
U (vi , bi ) u(vi bi )[F (b 1(bi ))]n 1 .
(1)
Using the Bayes-Nash best-response first-order condition and imposing a symmetric behavior (i.e., bi b(vi ), i 1,..., n ) yields the following nonlinear first order differential equation:
b '(vi )
d ln[F (vi )]n 1 d ln u(vi b(vi ))
with b(0) 0 .
(2)
This equation indicates that in equilibrium the slope of the bid function is equal to the elasticity of substitution between the probability of winning and bidder i’s payoff. Following Holt (1980) and Cox, Smith and Walker (1982), we assume bidders to display homogenous constant relative risk averse preferences so that u(w ) w r for i 1,..., n and where 1 r v 0 represents the
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buyers' common Arrow-Pratt index of constant relative risk aversion. 2 With such preferences, equation (2) yields the following SBNE bid function
bi K(vi ; F , v , r ) w vi ¨
vi
v
F (s ) ¯ ¡ ° ¡¢ F (vi ) ±°
n 1 r
ds ,
(3)
which is linear in values if the latter are uniformly distributed in [v ; v ] .
2.2. Empirical and Experimental Frameworks We will consider two estimation frameworks, both of which assume the actual number of participating bidders to be common knowledge for the bidders and to be perfectly known by the researcher. The first framework pertains to empirical studies in that it assumes the researcher to be unaware of the bidders’ private values and to only have partial information about the distribution F . The second framework pertains to experimental studies in that the researcher is assumed to know both the functional form of F and the realizations of bidders’ private values. Henceforth, we will refer to these frameworks as the empirical and the experimental framework, respectively. We already note that although each framework implies its own econometric model specification, both will assume that the distribution F belongs to the Beta family which nests a wide variety of distributions such as the uniform as well as other asymmetric ones, with only two parameters. We further motivate the use of a parametric approach rather than a non-parametric one à la Guerre, Perrigne and Vuong (2000) by the fact that the true distribution F from which values are drawn is perfectly known in experimental setups, so that it is natural to use and control this information to evaluate the sensitivity of structural inferences. 3 Since the Beta distribution is defined on the unit interval, we normalize the support of values to this domain such that v
0 and v
1 , and for notational convenience, we suppress the v
2
Such preferences suit better the study of risk aversion in experimental settings than constant absolute risk averse (CARA) preferences because they encompass risk neutrality as a special case. 3 The parametric approach also avoids the conduct of an additional test to check the stochastic equivalence of bidders’ estimated and actual values (cf. Bajari and Hortaçsu, 2005), and it allows a straightforward distinction between systematic errors (measured by the difference between the actual and the estimated distributions and characterizing an out-of-equilibrium behavior) and non-systematic ones (represented by the usual zero-mean error term and characterizing a trial-and-error behavior) whereas the non-parametric approach cannot disentangle ‘trialand-error’ from ‘out-of-equilibrium’ types of behavior.
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argument of K(vi ; F , v , r ) and refer to F by its parameter vector R , so that the bid function eventually reads K(vi ; R, r ) .
3. Inference within the Empirical Framework 3.1. Characterization of bids and prices. In this framework we assume the researcher to only have access to bid data, and that his/her objective is to estimate the SBNE model’s unknown parameters. We consider the following two situations which an empirical researcher typically has to deal with. WINBIDS: The researcher only has information about winning bids. The expected winning bid
bW t in auction t conditional on R and r is then given by the following equation:
mW R1, R2, r E (btW R, r )
1
¨0 K(R, s, r ) dFR (s)n .
(4)
ALLBIDS: The researcher has information on all submitted bids bit . The expected value of bidder i’s bid in auction t conditional on R and r then takes the following expression:
mA R1, R2, r E (bit R, r )
1
¨0 K(R, s, r )dFR (s) .
(5)
3.2. Conditions for identification. In both the WINBIDS and ALLBIDS situations, the question arises if the use of first order moments in (4) and/or (5) is enough to estimate the parameters r and R . The literature has already discussed conditions for the identification of these parameters in structural first-price auction models. 4 Hereafter, we discuss the identification conditions needed within our parametric estimation setting. 4
Within a non-parametric estimation setting, Perrigne and Vuong (2000) define conditions (to be fulfilled by the distribution of bids) to ensure that a consistent estimate of the distribution of values can be recovered from bids alone when assuming risk neutrality, i.e., r 1 . These conditions essentially require FR (¸) to behave smoothly at the tails. Campo et al. (2002) and Perrigne and Vuong (2007) further show that when assuming risk averse bidders (r v 1) , some additional restrictions are needed both on the distribution of bids and on the bidders’ utility function to ensure identification.
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Let E denote the vector of parameters R1, R2,r with parameter space and consider the following two non-linear estimation models for the WINBIDS and ALLBIDS cases: W bW i mW R1, R2 ,1 Fi
biA
mA R1, R2,1 FiA
(6)
where risk neutrality has been imposed. Even with such a restriction, the next proposition states that an additional restriction on the R parameter is needed.
Proposition 1: The non-linear models in (6) are not first-order identified. Partial identification can however be achieved under the restriction r R2 1 , in which case (3) takes the following expression n 1 K(v; R1,1,1) v n 1 1/ R1
Sketch of proof: The proof proceeds by showing that the expectation of bids or final prices is not uniquely determined by a single vector R1, R2 . See the Appendix. Note that this partial identification result refers to first-order moments and does not address the question of identifying the three parameters of the SBNE model with higher order moments; this being impossible without imposing risk neutrality (Athey and Haile, 2000).
3.3. Tests and Procedures. We use experimental first-price auction data within this empirical framework to report on the following issues:
Testing for a homogenous bidding behavior. Admittedly, this is a very demanding hypothesis for empirical data to fulfill since real-world auctions are rarely attended by perfectly symmetric bidders. 5 Yet, this assumption is fundamental to the model’s predictions and since lies under the null of a SBNE behavior, it should be tested. We check if the bidders participating in a given session display a homogenous behavior by estimating (6) with individual dummy variables for the parameter R1 . We denote by R1i the R1 parameter of bidder i (i = 2,…4 and check whether K(v; R1i ,1,1) K(v; R1 j ,1,1) for all i v j by defining
5
See for example Laffont, Ossard and Vuong (1995).
6
dummies which take value one if the bid corresponds to individual i and specify 4
R1i R1 Ei 1\i j ^ for i 1 . Homogenous bidding implies Ei 0 for all i 1 . We j 2
refer to the null hypothesis of this test as H 0E 0 . Note that rejecting the null of a homogenous behavior cannot reasonably mean that bidders are acting in some asymmetric equilibrium since it is common knowledge to them that their respective values are all drawn from the same probability distribution; a condition that leads to a unique equilibrium bidding strategy (Maskin and Riley, 2003). 6 Also, bidders cannot be said to be asymmetric in their risk attitudes in the sense of Li and Riley (1999) because the equilibrium predictions then assume that the bidders’ risk aversion indices are common knowledge, which is not the case here. 7 For these reasons, rejecting the null of a homogenous behavior in our setting is tantamount to reject the SBNE model of bidding.
Testing the informational equivalence of bids and prices. The distinction between the ALLBIDS and WINBIDS frameworks allows one to test the appropriateness of the SBNE model to organize the observed behavior. Notice first that if individuals are homogenous and bid in equilibrium, then the T1 estimates in ALLBIDS and WINBIDS (i.e., using (5) and (4), respectively) are consistent in the sense that they should converge to the same value as the sample of observations increases. However, with data samples of finite size we expect differences to be observed simply because the winning bid is only the maximum statistic of the sample of bids \b1, b2, ...,bn ^ so that it contains less information about the parameter vector (R1, R2, r ) than \b1,b2,..., bn ^ . 8 As the researcher has less information in WINBIDS than in ALLBIDS, the estimates are expected to be more efficient in the latter case: both estimation frameworks should yield similar estimates but with smaller variance in ALLBIDS than in WINBIDS. To this extent, testing the equality of ALLBIDS and WINBIDS estimates and the stochastically larger variances of WINBIDS estimates provides a necessary condition for the SBNE model to explain the observed behavior. 9 Also, since the equilibrium bid function is monotonically increasing in values, the vector of observed bids \b1,b2,..., bn ^
6
Maskin and Riley (2003) actually show that uniqueness requires that the bidders’ common distribution of values has a positive, even infinitesimally small, mass at v v to cope with a singularity at that point. 7 See Campo (2002) for an empirical study of this type of asymmetry in procurement auctions. 8 See Deltas (2004) for the empirical implications of the small sample properties of winning bids in first- and second-price auctions. 9 Clearly, we cannot build a Haussman-type of test for this necessary condition because the estimates would be biased both in ALLBIDS and in WINBIDS.
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contains as much information as the vector of value realizations \vi b 1(b1 ), i 1,...n ^ regarding the estimation of (R1, R2, r ) . 10 To this extent, we further conjecture that any difference between the estimates obtained from (5) and those obtained from an environment in which bidders’ private values are observed, must asymptotically disappear if bidders play in equilibrium. This last conjecture provides an easy way to assess the effect of improving the quality of information on structural inferences.
Implementing Optimal Reserve Prices: The structural analysis of auction data is principally motivated by normative reasons like the design of market institutions and the setting of optimal reserve prices. The seller’s expected revenue being equal to the expected winning bid, it is determined as
ER(p * ; F , n, r )
v
¨p K(F , s, p*, r ) dF (s)n , *
(7)
where the integration is taken with respect to the distribution of the sample’s maximum valuation, F (s )n , and where p * p v stands for the seller’s reserve price provided that her reservation value for the commodity is lower than v . Clearly, p * is optimal if it maximizes (7). Actually, p * is the solution to 0 p * ¢ 1 F (p * ) ¯± / f (p * ) when r 1 , and it must be numerically determined when r v 1 . An interesting property of this optimal reserve price is that it decreases with risk aversion (Riley and Samuelson, 1981, Proposition 5). 11 Hence it is of crucial importance for the seller to know the type of preferences buyers have, for setting an optimal reserve price that assumes risk neutral bidders would decrease the seller’s expected revenue if bidders are risk averse. Given the identification conditions just discussed, however, determining optimal reserve prices is impossible to achieve without some prior assumption on the bidders’ risk preferences or an exogenous assessment of their risk attitudes. To this extent, and to evaluate the possible effects of assuming a risk averse 10
This is because if a statistic S is sufficient for a parameter vector (R1, R2 , r ) and h(·) is bijective, then h(S ) is
also sufficient (see Gourieroux and Monfort, 1995). In our case, as the inverse bid function vi b1(bi ) is bijective it meets this condition. This however is not the case for the relationship between the winning bid bW t and the bid sample \b1, b2 ,..., bn ^ . 11 The intuition for this is that highly risk averse bidders submit bids that are almost equal to their private values so that the rents the seller extracts from bidders by setting an optimal reserve price do not outweigh the risk of lowering competition.
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behavior on the seller’s revenues when optimal reserve prices are to be set, we use the result of Proposition 2 and compare the outcomes of a scenario where bidders are assumed to be risk neutral with values distributed according to a Beta distribution (henceforth, the BETA scenario) to those of a scenario where bidders are constant relative risk averse with uniformly drawn values (henceforth, the UNI scenario).
4. Inference within the Experimental Framework 4.1. Data Generating Process for bids conditional on valuations. When conducting auction experiments, the researcher has full information about the number of bidders, the distribution(s) of values and the bidders’ value realizations. In most experiments the distribution of values is chosen to be a uniform on [v ; v ] so that by normalizing values and bids to the [0,1] interval, (3) yields the following SBNE bidding function, which is linear in values: 12
bit w K(vit ,1,1, r )
n 1 v . n 1 r it
(8)
Our approach consists in encompassing this linear model into a general parametric model that allows for non-linear (monotone increasing) bid functions by using the fact that a uniform distribution defined on [0,1] is a particular Beta distribution with R1 R2 1 . That is, estimating (8) reverts to conducting a test over R1 and R2 that checks that the estimated distribution is indeed uniform, i.e., that R1 R2 1 . 13 When the bidders’ value realizations are observed, we will thus consider the following estimating equation:
bit vit ¨
vit
0
12
FR (s ) ¯ ¡ ° ¢¡ FR (vit ) ±°
n 1 r
ds Fit K(vit , R, r ) Fit ,
(9)
We normalize values vit and bids bit as vit (vit v ) (v v ) and bit (bit v ) (v v ) , respectively.
13
In the Results section of the paper, we will also report on the case where the estimated bidding functions can only be linear in values, i.e., when R2 is set equal to 1.
9
where Fit stands for a non-systematic error term with E (Fit vit ) 0 . This random term is a reduced-form error that captures deviations from equilibrium behavior that are not explained by
K(vit , R, r ) .
4.2. Identification and testing. We already showed that within the empirical framework two parameters should be restricted (namely r and T 2 ) to achieve partial identification. The next proposition shows that fewer restrictions are needed within an experimental framework simply because the researcher has more information than within an empirical framework (i.e., bidders’ value realizations and the functional form of the distribution F ).
Proposition 2: The non-linear models in (9) is not first-order identified. Partial identification requires the restriction r 1 . With such a restriction, if R2 1 then an estimate of r obtains from the condition r 1/ R1 . Proof: See the Appendix. It thus follows from Proposition 2 that by imposing r 1 in (9) we get
bit vit ¨
vit
0
FR 0 (s ) ¯ n 1 ¡ ° ds Fit , ¡¢ FR0 (vit ) °±
(10)
which can be estimated using standard non-linear least squares where the integral must be computed using quadrature numerical procedures. Notice also that if we cannot reject the null that R2 1 , then r 1/ R1 and the estimated bid function is linear in valuations with a slope equal to (n 1) /(n 1 r ) , as predicted by the model for uniformly drawn values. Non-linear bid functions are characterized by R2 v 1 and are interpreted as evidence against the SBNE model with uniform values. In what follows, we outline the tests of the SBNE model with homogeneous and with heterogeneous risk attitudes in experimental conditions.
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Homogenous risk attitudes: We have seen that imposing r 1 ensures identification of the
R vector parameter, and it implies R2 1 and R1 1/r if the SBNE model holds. Therefore, we can assess the SBNE model by testing R10 ¬ 1/ r0 ¬ H 0 : 0 R2 ® 1 ®
vs
0¬ R1 1/ r0 ¬ H 1 : 0 v R2 ® 1 ®
(11)
The null hypothesis in (9) defines a function h : \ l \ 2 such that if the model holds and r0 is the true constant relative risk aversion parameter, then there is an implicit restriction on the parameter space of R which is given by R 0 h(r0 ) . In the Appendix, we show that to test (11), one only needs to conduct a standard normal test of the significance of the second logparameter of the estimated Beta distribution.
Heterogeneous risk attitudes: So far the SBNE model assumed bidders with homogenous constant relative risk averse preferences. The model proposed by Cox et al. (1982) assumes instead that the bidders’ utilities are defined as ui (w ) w ri , where ri is i.i.d. according to
G() ¸ which has a positive continuous density g() ¸ on (0, rmax ] , and where rmax stands for the risk parameter of the least risk averse bidder. As behavior at first-price auction experiments is typically characterized by a significant overbidding, risk neutrality is often considered as a natural lower bound on bidders’ risk aversion so we assume rmax 1 . Notice that this
¸ , g() ¸ and (0, rmax ] to be common knowledge to bidders, and that model assumes G() although bidders have heterogeneous risk parameters, they are still ex-ante symmetric. A useful property of this model is that if bidders’ values are uniformly distributed, then the individual equilibrium strategies are linear up to the maximum possible bid, b * , of the least risk averse bidder, and they are not defined for bids greater than b * . Cox et al. (1988) and Van Boening, Rassenti, Smith (1998) show in particular that b * (n 1) /(n 1 rmax ) and that the equilibrium bidding functions have the following expression
bit K(vit ;1,1, ri )
n 1 v n 1 ri it
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for vit [0; K1(b * (rmax );1,1, ri )] .
A structural test of this model with heterogeneous risk attitudes therefore consists of checking if i) bidders display a homogenous behavior or not, ii) individual bidding functions are all linear for bids smaller than b * (rmax ) and iii) the estimated risk aversion parameters are all smaller or equal to rmax .
5. Results We test our models with the data of Isaac and Walker (1985) which consist of 10 sessions of first-price auctions. Each of these sessions involves 4 different bidders who play for 25 rounds and who have their valuations drawn with replacement from a uniform on < 0,10 > . We also use the data of Dyer, Kagel and Levin (1989), which Bajari and Hortacsu (2005) used for their study, to put in perspective some differences between parametric and nonparametric estimation routines. These data consist of sessions where the actual number of bidders participating in an auction was made random (it could either be 3 or 6); a necessary condition for the nonparametric identification of the Bayes-Nash model for risk averse bidders. Given that these experiments involve only three sessions, we chose to only report the detailed estimation results for the data of Isaac and Walker (1985). Otherwise, in both experiments the winning bid was revealed to the participating bidders at the end of each round of play. 14 We conduct our estimations separately for each session of a dataset and assume that any test outcome for a session s is a Bernoulli variable Rs that takes the value 1 if the null is rejected (at B .05 , two-tailed and 0 otherwise, so that Rs B(1; 0.05) . To draw our conclusions for a
given dataset of
sessions, we run a one-tailed binomial test on the number of rejections given
a probability of rejecting the null (“success”) of .05, so that } B(10; 0.05) with }
Rs . s 1
The null hypothesis of this test is that the probability of wrongly rejecting the null in a given session is B .05 and the alternative is that B .05 , or equivalently that the null tested in each individual session is globally rejected. Thus, the null of this global test is rejected if the probability of observing at least } rejections is smaller than .05 . This simple test allows a
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The provision of this information is important as it may induce a feeling of regret to losers which induces them to bid higher (see Engelbrecht-Wiggans, 1989, Engelbrecht-Wiggans and Katok, 2006, 2007, and Ockenfels and Selten, 2001). These data were also analyzed by Cox et al. (1988), Cox and Oaxaca (1996) and Ockenfels and Selten (2001).
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straightforward assessment of a given hypothesis by summarizing the evidence in a series of fully independent observations.
5.1. Empirical Framework In this section we assess homogeneity in behavior and the SBNE model with the estimating equations (4) and (5), and we explore the effects of different information conditions on inferences and the seller’s expected revenues. x
Testing for Symmetric Bayes-Nash Equilibrium behavior
We first check if the bidders of a given session display a homogenous behavior by estimating equation (6) with individual dummy variables for the parameter R1 and by checking whether the
G1i parameters in the specification R1i R1 E1i for i 2, 3, 4 are jointly non significantly different from zero ( H 0E 0 ). The statistics in Table 1 indicate that we reject this hypothesis in four sessions under WINBIDS and in one session under ALLBIDS so that according to the binomial test, we reject the null of a homogenous behavior in WINBIDS ( p .0010 ) but not in ALLBIDS ( p .4013 ). We proceed with checking the necessary condition for the observed behavior to be labeled a SBNE one, i.e., that the WINBIDS and ALLBIDS frameworks yield stochastically equivalent R1 estimates, with larger standard deviations in WINBIDS than in ALLBIDS. The conduct of a nonparametric randomization test on the Rˆ1 statistics of Table 1 indicates that both the estimates and their standard deviations are significantly larger in WINBIDS than in ALLBIDS ( p .0003 and p .0000 , one-tailed, respectively) so that the symmetric benchmark model for either constant relative risk averse or risk neutral bidders would not be appropriate to recover the unknown distribution of values from this data. Note that if we overlooked the homogeneity hypothesis and if had we not checked this necessary condition, we would have concluded that bidders are acting in equilibrium and are either i) risk neutral with values drawn from a Beta distribution with parameters (R1,1) (with an average R1 estimate of 1.44 in WINBIDS and of 1.19 in ALLBIDS) or ii) risk averse with a common average risk parameter r 1/ R1 ( r .84 in WINBIDS and r .69 in ALLBIDS) and uniformly drawn values on < 0,1 > .
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In what follows, we explore the consequences of wrongly not rejecting the SBNE model (i.e., committing a Type II error) on the seller’s expected revenues following the implementation of “optimal” reserve prices. x
Implementing Optimal Reserve Prices
Table 1 reports the expected revenue increase (in %) for each session when optimal reserve prices are implemented without first checking for a homogenous behavior or for the equivalence of WINBIDS and ALLBIDS estimates. Although expected revenues increase in both cases, they remain well below the 2.08% predicted by the SBNE benchmark for risk neutral bidders, and on average, they are greater in ALLBIDS than in WINBIDS (i.e., 1.40% versus .90%, respectively). Such lower-than-expected predictions are due to the bidders’ tendency to overbid, which shifts the estimated distributions towards higher valuations on the unit interval. Indeed, as the R1 estimates are all greater than 1, the mean and standard deviation of the estimated Beta distribution are respectively higher and lower than those of a uniform defined on [0,1]. With such right-shifted distributions, the implementation of optimal reserve prices extracts less of buyers’ information rents than if values were uniformly distributed, so that the seller’s expected revenues increase by less. Given our previous finding that the WINBIDS estimates are significantly greater than the ALLBIDS ones, this also explains the lower predictions in WINBIDS than in ALLBIDS. Next, to evaluate the possible effects of assuming risk averse preferences on the seller’s revenues when optimal reserve prices are to be set, we compare the expected revenues obtained under the BETA and the UNI scenarios defined in section 3.3. Figure 1 shows how the seller’s expected revenues relate to reserve prices for three different degrees of risk aversion; the first two of which correspond to the average estimates for the WINBIDS and ALLBIDS cases in Table 1. In panel A, expected revenues with zero reserve prices are equal to .6456 if the BETA scenario applies with R (1.19,1) and to .6249 if the UNI scenario applies with r 1/1.19 .84 . Once “optimal” reserve prices are determined and implemented, expected revenues rise to .6545 and .6289 , respectively, leading to increases of 1.38% and 0.63%. Hence, if the exact distribution of the bidders’ values is unknown the seller could be advised, on the grounds of expected revenue maximization, to assume the BETA scenario for setting optimal reserve prices or equivalently, to assume risk neutral rather risk averse bidders.
14
The plots of panels B and C refer to higher degrees of risk aversion ( r .69 and .50 , respectively) and indicate that the difference in expected revenues between the two scenarios increases with risk aversion and that the benefits from implementing optimal reserve prices in the UNI scenario vanish with risk aversion. For example, when r .50 , a mid-range figure reported by several studies, the optimal reserve price would be equal to .023 and would yield virtually no expected revenue increase; the change being of the order 106 . Eventually, the plots of panel C reveal that setting a reserve price that would be optimal in scenario BETA, would also disappoint the seller’s revenue expectations by -14.47% if the actual scenario is UNI with r .50 (cf., a in panel C). Implementing such a reserve price would also lead to an expected revenue decrease of -4.63% when compared to setting a reserve price that assumes risk averse bidders with uniformly drawn values (cf., b in Panel C). That is, choosing the risk aversion hypothesis would imply lower optimal reserve prices than assuming risk neutral bidders and would minimize the risks of reducing competition by implementing an inadequately high reserve price. This choice, however, would cost the seller in terms of foregone expected profits if bidders are actually neutral to risk. One possible way of deciding which scenario applies best would be to compare the observed average revenues to those predicted in each scenario and to cast the one that fits the data best. 15 Figure 2 reports the observed and predicted revenues for each session when no reserve price is implemented. The plots indicate that the BETA scenario fits the data best and that within each scenario, inferences from WINBIDS (empty markers) fit the observed values almost always better than those from ALLBIDS (solid markers). This, however, is not surprising given that the seller’s revenues are determined by winning bids. For this reason, with finite samples of data the inferences from ALLBIDS can only be noisier than those from WINBIDS or, as it is the case here, biased by subjects’ out-ofequilibrium play. In sum, we have used the SBNE model to determine optimal reserve prices although we have shown that it is not supported by the data. To decide whether bidders should be assumed ‘risk averse’ or ‘risk neutral’, we compared actual to expected revenues in either case and concluded that optimal reserve prices should assume risk neutral bidders, i.e., the BETA scenario should be used. Also, as we could not reject homogeneity in the ALLBIDS case (at p .4013 ), inferences and recommendations should account of all bids to fulfill the model’s assumptions, although the 15
Such a comparison is possible only if the seller’s expected revenue is always greater in one of the scenarios, which is the case here.
15
WINBIDS case outperforms the ALLBIDS case in terms of goodness-of-fit (cf. Figure 3). Unfortunately, the presence of an out-of-equilibrium play implies that there is no way to decide which of the alternatives ‘choosing the model which basic assumptions are fulfilled by data’ or ‘choosing the one that fits data best’ would be most appropriate for the determination of revenue enhancing reserve prices. Further experimental investigations would be needed to determine which of these alternatives tends to outperform the other in terms of generated revenues. In what follows, we check to what extent the SBNE model for homogenous risk attitudes, or its augmented version for heterogeneous risk attitudes, explains bidding behavior at first-price auction experiments.
5.2. Experimental Framework x
Linear Bidding functions: Constrained Estimation.
As a transition from an empirical to an experimental framework, we consider a hybrid framework where the researcher still has limited information about the actual distribution of bidders’ values (as in the previous section), but has full information about the bidders’ value realizations (as is the case in experiments). The purpose of this framework is to assess the effect of knowing the bidders’ value realizations on inferences when the actual distribution FR (¸) is still imperfectly known. Such a setting implies the estimation of a variant of (5) which includes the bidders’ value realizations and which can be obtained from (10) by imposing the constraints
R2i 1, i 1,..., 4 . That is, we restrict i’s bid function to be linear with a slope equal to (n 1)/(n 1 1 / R1i ) . For sessions that do not reject homogeneity ( H 0E 0 ), we estimate R1 in (10) without individual dummies whereas for sessions that reject this hypothesis, we estimate the four individual R1i parameters of a session after discarding all bids greater than or equal to b * (rmax ) . The estimation and test results are reported in Table 2 and refer to all submitted bids. The statistics indicate a rejection of homogeneity in eight sessions so that the basic hypothesis of the SBNE model is rejected ( p .0000 , according to the binomial test). The estimated risk parameters for sessions with either homogenous or heterogeneous bidders average at .42 , which is in line with the estimates reported by Cox and Oaxaca (1996) ( r .33 ), by Kagel, Harstad and Levin (1986) for six-bidder auctions ( r .49 ) and by Goeree et al. (2000) and Armantier and Treich (2005)
16
for two-bidder auctions ( r .48 and .39 , respectively). 16 Although the relative invariance of these estimates to the number of bidders may be seen as supportive of the SBNE model for constant relative risk averse bidders, we recall that the linear restriction that we imposed only allows an evaluation of the model’s goodness-of-fit, not a test of it; the latter requiring a joint test that the bidding functions of a group of subjects are all linear over a given range of bids. The lack of consistency between the estimates of Table 1 and those of Table 2 confirms the presence of an out-of-equilibrium behavior and reveals that structural inferences then become highly sensitive and do not necessarily improve with the quality of the information available. To illustrate the latter point, Figure 2 reports the expected revenues associated to the Rˆ1 estimates of Table 2 when assuming a BETA or a UNI scenario (cf. HYBRID markers in Figure 3). When compared to the inferences from ALLBIDS, the UNI scenario now appears to outperform BETA in explaining the data. That is, the seller should now assume risk averse rather risk neutral bidders: the opposite of what was reported when the value realizations were not taken into account. x
Non-Linear Bidding functions: Unconstrained Estimation.
We now estimate (11) with individual dummies for both parameters so that we allow for nonlinear (monotone increasing) bidding functions. The test procedure is essentially the same as the one just described: for each session, we first check for a homogenous behavior by testing H 0E 0 . 17 If we cannot reject this hypothesis, we estimate R1 and R2 in (10) without individual dummies and test the SBNE model by checking that the common R2 parameter estimate is equal to 1 ( H 0R2 1 ). If we reject homogeneity, we check for linearity in the estimated bidding functions by
testing that the four R2i estimates of a session are jointly equal to 1 ( H 0R2i 1 ). Finally, we explicitly test the SBNE model for heterogeneous risk attitudes by testing that the four R2i estimates are still jointly equal to 1 after having discarded all bid observations greater than or *
equal to b * (rmax ) ( H 0R2i 1 ), and by checking that the estimated risk parameters are all smaller or 16
Our risk estimates suggest a significantly lower degree of risk aversion than those of Cox and Oaxaca (1996) who used the same dataset ( p .0234 according to a two-tailed Randomization test with 10 paired observations). Such different estimates could be due to the fact that Cox and Oaxaca (1996) estimate affine bid functions which happen to have significantly negative intercept terms whereas we estimate bid functions without intercept terms to comply with the SBNE model. The presence of negative intercepts, which tend to increase the estimated slopes of non-affine linear bid functions, could therefore explain the observed difference. 17 The null hypothesis of this test is that R1i R1 and R2i R2 , i 2, 3, 4.
17
equal to rmax . If these hypotheses hold for a particular session, then we do not reject the SBNE model for that session. Our conclusion for the ten sessions is again based on the results of a binomial test, as previously outlined. The estimation and test results are reported in Table 3. A comparison of the predictive accuracy of the models estimated in Tables 2 and 3 indicates that allowing for non-linear bidding functions considerably improves the model’s adjusted goodness-of-fit measure (AIC). The test results in Table 3 also indicate a rejection of homogeneity in nine sessions ( p .0000 , according to the binomial test) so that we focus analysis on the SBNE model for heterogeneous *
risk averse bidders. The H 0R2i
1
test statistics reveal that we reject this model in six sessions so
that we can reject it as a possible explanation of the observed behavior ( p .0000 , according to the binomial test). 18 We reach the same conclusion if we overlook the homogenous bidding hypothesis and only check for bidding functions that are linear up to b * : the model would still be rejected in four sessions ( p .0010 , according to the binomial test). 19 Finally, in the last two columns of Table 3, we report the estimation outcomes when we allow for risk loving bidders (i.e., rmax 2 instead of rmax 1 ). While the effect of such an assumption translates into a decrease in the number of rejections to three, our conclusions remain unchanged as we still reject the model ( p .0115 , according to the binomial test). 20 In sum, although there is an improvement in terms of goodness-of-fit when one allows for nonlinear bidding functions, the SBNE model is nevertheless overwhelmingly rejected by the data when we check for the consistency of subjects’ bidding functions with the model’s predictions. We reach the same conclusions for the data of Dyer et al. (1989) for which Bajari and Hortaçsu (2005) report an aggregate estimate for three- and six-bidder auctions of r .27 when the data are analyzed with Logit Equilibrium models, and of r .158 when non-parametrically estimating the SBNE model. When we separately estimate risk parameters for auctions with 18
To test the model we had to discard 110 observations with b b * (rmax 1) from analysis, which is less than
half the expected number of observations that we would have to discard, had we truncated the data at vi* .75 , as in Kagel and Levin (1985). See Cox, Smith and Walker (1992, pp 1399-1400) for a discussion on this truncation issue. As all bidders who display linear bidding functions on [0, b * ] also have ri b 1 at B .05 , we do not report the outcomes of this test. 19
*
The risk parameter estimates of sessions 6 and 8, which did reject H 0R2i 1 but not H 0R2 1 , are r .33 and .63 .
Note that the expected number of observations to be discarded (i.e., with b b * (rmax ) ) increases with rmax so that the power of inference reduces as rmax increases. With rmax 2 , we had to discard 335 observations with bids greater than .60, which represent 225 more observations than with rmax 1 . 20
18
three and six bidders, we find an aggregate estimate of r .43 for n 3 and of r .72 for
n 6 . As in these experiments each subject submitted two bids per period of play (one for a market with n 3 and one for a market with n 6 ), these estimates are clearly not invariant to the number of bidders and suggest an out-of-equilibrium play. Also, when we estimate an aggregate risk parameter for these data with our approach we find r .493 ; which further suggests that parametric and non-parametric procedures may lead to very different outcomes when bidders do not act in the SBNE equilibrium. x
Alternative explanations.
While the SBNE model with constant relative risk aversion does not appear to consistently organize human behavior at auction experiments, alternative explanations remain scarce; the most successful of them cast features of bounded rationality and require specific experimental designs to deal with the identification problems raised. Engelbrecht-Wiggans (1989) explains deviations from risk neutral bids in terms of bidders’ regret from losing or from winning an auction round. A winning bidder may feel regret if the second highest bid is revealed (i.e., s/he could have won with a lower bid) whereas a losing bidder may feel regret if the winning bid is revealed and is smaller than his/her private value (i.e., s/he could have won with a higher bid). Engelbrecht-Wiggans and Katok (2007) experimentally test a reduced form of this model by manipulating bidders’ feedback information to induce these forms of regret and find that they fit individual behavior well. 21 An interesting feature of their model is that when one considers the relative weight of these forms of regret, the model yields the same symmetric best-response strategy as for a constant relative risk averse SBNE bidder with risk parameter r. To this extent, our results apply and suggest a rejection when their model is structurally tested in a standard auction setting; with bidders competing against each other rather than against robots programmed to submit risk neutral SBNE bids. Figure 3 reports individual bid data and the estimated bidding functions for each of the ten sessions. Besides an obvious heterogeneity in behavior, these plots usually indicate a less-thanproportional bidding at high values which is not consistent with SBNE bidding. We find a similar pattern in the data used by Bajari and Hortaçsu (2005) and by Cox et al. (1988) for
21
See also Engelbrecht-wiggans and Katok (2006) and Filiz and Ozbay (2007). Ockenfels and Selten (2004) and Neugebaeur and Selten (2005) also study a form of regret grounded in bounded rationality (Impulse Balance Equilibrium) and reach similar conclusions.
19
auctions with 3 and 5 bidders (which also rejected the models tested).22 Although such misbehavior has been reported in several auction experiments, it remains largely unexplained. Battigali and Siniscalchi (2003) show that it is rationalizable in terms of bidders’ beliefs even if they have risk neutral preferences. Their argument produces a sizeable less-than-proportional bidding in auctions with two bidders and uniformly drawn values, but it becomes hardly noticeable when there are four bidders or more. Goeree et al. (2002) study two-bidder auctions within a Logit equilibrium framework and consider the case where bidders misperceive their winning probabilities: the less-than proportional bidding at high values would then witness their overconfidence in winning upon receiving high values; a well-documented phenomenon in decision-making theory and psychology (Prelec, 1998). The outcomes of their experiments indicate however a significant underestimation of winning probabilities leading to overbidding over the whole range of values rather than a less-than-proportional bidding at high values. This is confirmed by Armantier and Treich (2006) who adapt the experimental setup of Goeree et al. to disentangle the effect of risk aversion from the one of probability misperception and show that accounting for both effects improves the model’s goodness-of-fit and makes risk aversion less salient ( r .73 instead of .39 ). Within our SBNE estimation framework, any difference between the estimated and the actual distribution F could also be modeled in terms of probability misperception provided that bidders are assumed to be SBNE ones and to have risk neutral preferences. Such an interpretation, however, would still suppose a homogenous behavior and the equivalence of ALLBIDS and WINBIDS estimates; both assumptions we know are not supported by the data.
6. Conclusion We analyzed laboratory data on first-price private value auctions with structural econometric methods pertaining to either empirical or experimental settings. For both settings, our study sheds a new light on the symmetric Bayes-Nash equilibrium model to explain first-price auction data. From an empirical standpoint --- when not taking account of the bidders’ value realizations and assuming limited knowledge about the bidders’ distribution of values --- we use two basic implications of the model as tests for equilibrium behavior; an assumption that is typically 22
Kagel and Roth (1992) study auctions with 6 or 10 bidders and report an overbidding that is negatively and significantly correlated to bidders’ values which also suggests a less than proportional bidding at high vales. Pezanis-Christou and Sadrieh (2004) find that such a less-than-proportional bidding is also present in two-bidder auctions when each bidder is asked to submit a two piecewise linear bid function before receiving his/her private value.
20
imposed in the structural estimation of symmetric auction models but rarely cross-checked. The first relates to the homogeneity of bidding behavior which is fundamental to the determination of a symmetric equilibrium strategy whereas the second relates to the finite sample properties of different estimators. 23 Passing these tests provides some warrant that both predictions and policy recommendations will be reliable. Their non-fulfillment, however, suggests the presence of an out-of-equilibrium behavior. Our study has shown how structural inferences may then lead to different conclusions, depending on the prior check of the game’s fundamental assumptions and on the quality of the information available. Most important for field studies, it has shown that inferences do not necessarily improve with the quality of information when behavior is out-ofequilibrium. As for the risk aversion hypothesis which provides an additional degree of freedom in the analysis of bid data, our study has highlighted the consequences of casting this hypothesis on ones inferences and recommendations. From an experimental standpoint --- when taking account of all the information available --- our study indicates a rejection of the symmetric Bayes-Nash model for either homogenous or heterogeneous constant relative risk averse bidders. The outcomes indicate in particular that although such a model may be found to fit various auction datasets, it is rejected by the data when tested jointly for all the participating bidders in a session. Finally, our examination of the data from both empirical and experimental standpoints with similar estimation procedures also put in perspective the overall effect of knowing the bidders’ true distribution of values. It suggests in particular that the main limitation of non-experimental data, i.e., not knowing the bidders’ distribution of values, makes it less likely to reject models than when one uses experimental data, especially if their basic assumptions are not checked.
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23
Similar conditions can be derived for the structural assessment of asymmetric auction models. The Bayes-Nash equilibrium predictions for asymmetric first-price auctions predict not only how bidders should act in equilibrium but also how each type of bidder should best-respond to the other type(s) (cf. Lemma 3.1 and Proposition 3.3 of Maskin and Riley, 2000, or Propositions 9 and 10 of Li and Riley, 1999). Such properties of the equilibrium could therefore be seen as necessary conditions to be fulfilled by data for the observed behavior to be labeled “in equilibrium”.
21
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22
Engelbrecht-Wiggans, R. (1989), “The effect of regret on optimal bidding in auctions”, Management Science, 35(6), 685-692. Engelbrecht-Wiggans R. and Katok E., (2006), “Regret and feedback information in first-price sealed-bid auctions”, mimeo, Penn State University. Engelbrecht-Wiggans R. and Katok E., (2007), “Regret in auctions: theory and evidence”, Economic theory, 33(1), 81-101. Filiz E. and E.Y. Ozbay (2007), “Auctions with anticipated regret: theory and experiment”, American Economic Review, 97(4), 1407-1418. Goeree J.K., C.A. Holt and T.R. Palfrey (2002), “Quantal response equilibrium and overbidding in private-value auctions”, Journal of Economic Theory, 104(1), 247-272. Gouriéroux C. and A. Monfort (1995), Statistics and Econometrics Models, Vol.2, Cambridge University Press. Guerre E., I. Perrigne and Q. Vuong (2000), “Optimal nonparametric estimation of first price auctions”, Econometrica, 68(3): 525-574. Holt C.A. (1980), “Competitive bidding for contracts under alternative auction procedures”, Journal of Political Economy, 88(3), 433-445. Isaac R.M. and J.M. Walker (1985), “Information and conspiracy in sealed bid auctions”, Journal of Economic Behavior and Organization, 6, 139-159. Kagel J.H. (1995), “Auctions: a survey of experimental research” in Handbook of Experimental Economics, Kagel J.H. and Roth A.E. (eds), Princeton University Press. Kagel J.H. and D. Levin (1985), “Individual bidder behavior in first-price private value auctions”, Economics Letters, 19(2), 125-128. Kagel J.H. and D. Levin (1993), “Independent private value auctions: bidder behavior in first-, second-, and third-price auctions with varying number of bidders”, Economic Journal, 103, 868879. Kagel J.H., R.M. Harstad and D. Levin (1987), “Information impact and allocation rules in auctions with affiliated private values: a laboratory study”, Econometrica, 55(6), 1275-1304. Kagel J.H. and A.E. Roth (1992), “Theory and misbehavior of first-price auctions: comment”, American Economic Review, 82(5), 1379-1391. Laffont J.-J., H. Ossard and Q. Vuong (1995), “Econometrics of first-price auctions”, Econometrica, 63(4), 953-980. Li H. and J. Riley (1999), “Auction Choice”, mimeo, UCLA.
23
Maskin E. and J. Riley (2000), “Asymmetric auctions”, Review of Economic Studies, 67, 413438. Maskin E. and J. Riley (2000), “Equilibrium in sealed high bid auctions”, Review of Economic Studies, 67, 439-454. Maskin E. and J. Riley (2003), “Uniqueness and equilibrium in sealed high bid auctions”, Games and Economic Behavior, 45, 395-409. Ockenfels A. and R. Selten (2005), “Impulse balance equilibrium and feedback in first-price auction”, Games and Economic Behavior, 51(1), 155-170. Neugebaeur T. and R. Selten (2006), “Individual behaviour of first-price sealed auctions: the importance of information feedback in experimental markets”, Games and Economic Behavior, 54(1), 183-204. Paarsch H.J. and H. Hong (2006), An Introduction to the Structural Econometrics of Auction Data, MIT Press. Perrigne I. and Q. Vuong, (2000), “Structural Econometrics of First-Price Auctions: A Survey of Methods”, Canadian Journal of Agricultural Economics, 47, 203-223. Perrigne I. and Q. Vuong, (2007), “Identification and estimation of bidders’risk aversion in firstprice auctions”, American Economic Review (Papers and Proceedings), 97(2), 444-454. Pezanis-Christou P. and A. Sadrieh (2004), “Elicited bid functions in (a)symmetric first-price auctions”, mimeo, Tilburg University. Prelec D. (2006), “The probability weighting function”, Econometrica, 66, 497-527. Riley J.G. and W.F. Samuelson (1981), “Optimal auctions”, American Economic Review, 71(3), 381-392. Van Boening M.V., S.J. Rassenti and V.L. Smith (1998), “Numerical computation of equilibrium bid functions in a first-price auction with heterogeneous risk attitudes”, Experimental Economics, 1, 147-159.
24
Tables and Figures
Table 1: Empirical framework WINBIDS Session 1 2 3 4 5 6 7 8 9 10
a H 0E 0
Rˆ1
ALLBIDS b
s.d.
.8115 1.42 .0179 1.54 .1251 1.17 .0121 1.48 .5596 1.50 .1309 1.38 .0855 1.80 .0040 1.19 .0007 1.51 .0946 1.44 a : Rejection probability; H0: R1i
p
'ER
*c
H 0E 0
d
Rˆ1
s.d.
p*
'ER
.1253 .5367 0.89 .7199 1.14 .0876 .5131 1.53 .1175 .5459 0.72 .9773 1.24 .0831 .5218 1.25 .1170 .5157 1.43 .6472 1.11 .0787 .5103 1.63 .1223 .5414 0.80 .5732 1.25 .0836 .5227 1.22 .1160 .5429 0.77 .7536 1.25 .0849 .5227 1.22 .1197 .5335 0.95 .4876 1.11 .0846 .5103 1.63 .1619 .5644 0.47 .6663 1.29 .0892 .5261 1.13 .1321 .5175 1.38 .0339 1.05 .0865 .5048 1.86 .1438 .5436 0.75 .9235 1.20 .0826 .5184 1.35 .1335 .5382 0.85 .5310 1.26 .0840 .5236 1.20 R1 j , i v j ; b: Newey-West standard deviations; c: Optimal reserve
price assuming B(Rˆ1,1) and r 1 ; d: % change in Expected Revenues when p * is implemented %ER 100 q [ER(p * ) ER(p * 0)]/ ER(p * 0) .
Table 2: Experimental framework: Constrained estimation H 0E 0
Rˆ1
s.d.
p*
a Rˆ1*
r
b
AICc
.0000 2.47 .0779 .6043 2.43 [.42] -109.87 .2347 3.08 .2942 .6335 5.74 .32 -250.38 .1944 1.78 .1366 .5630 2.14 .57 -167.76 .0384 2.93 .2701 .6268 5.03 [.37] -229.02 .0000 3.49 .1318 .6503 6.42 [.29] -27.46 .0000 1.92 .1273 .5723 2.04 [.54] -202.94 .0000 5.50 .2208 .7115 5.35 [.20] -129.31 .0000 1.51 .1825 .5436 1.64 [.75] -190.03 .0000 2.82 .1276 .6217 2.88 [.36] -31.32 .0000 2.87 .1828 .6241 5.45 [.35] -216.43 a : Parameter estimated after truncation of bids greater than b; b : Average estimated risk averse parameter r 1/ Rˆ1 if bidders are homogenous and < r >
4
i 1 1/ 4Rˆ1*i if they are
heterogeneous; c: Akaike Information Criteria.
25
Table 3: Experimental framework: Unconstrained estimation Rˆ1
Rˆ2
H 0E 0
a
b
c
*
d
e
*
AIC r H 0R2i 1 H 0R2i 1 r rmax = 1 rmax = 2 1.98 0.61 .0521 .1076 .4340 .9999 .62 -99.58 .9999 .66 1 ø ø ø 15.19 6.89 .0000 .0185 .0000 .0000 --168.04 .9998 [.13] 2 4.01 2.91 .0127 .0000 .0000 .0000 ---136.23 .0000 --3 10.23 5.09 .0096 .0007 .0000 .0000 ---207.21 .0000 --4 12.79 5.49 .0000 .0000 .0000 .0000 --67.49 .9637 [.18] 5 2.57 1.56 .0127 .1571 .0013 .0064 ---188.53 .9999 [.41] 6 4.81 0.71 .0000 .6095 .9198 ø .9828 ø [.22] 95.89 .2560 ø [.17] 7 ø ø ø 1.81 1.32 .0000 .3251 .0002 .0008 ---95.75 .9177 [.80] 8 3.02 1.13 .0000 .7513 .0005 .3727 [.28] -8.02 .0000 --9 10.43 5.22 .0130 .0000 .0099 .0947 [.16] -197.02 .1587 [.16] 10 Note: We purged outliers from the data of Sessions 2, 7 and 8 to allow the estimation procedure to converge (ø). a b c d * : Rejection probabilities; H 0E 0 : R1i R1j and R2i R2 j , i v j ; : H 0R2 1 : R2 1 ; : H 0R2i 1 : R2i 1 , i ; : H 0R2i 1 : Sess.
H 0R2 1
H 0R2i 1
e
R2i * 1 , i (assuming rmax = 1); : Average of individual risk parameters for sessions that did not reject H 0E 0 [or *
H 0R2i 1 ].
26
Figure 1: Expected revenues, reserve prices and risk aversion
ER
ER
(A)
ER
(B)
0.8
0.8
0.8
0.75
0.75
0.75
0.7
0.7
0.7
0.65
0.65
0.65
0.6
0.6
0.6
0.55
0.55
0.55
0.5
0.5
0.5
0
0.5
1
0
0.5
1
(C)
a b
0
0.5
Reserve Price
Reserve Price
Reserve Price
UNI[r = 1]
UNI[r = 1]
UNI[r = 1]
UNI[r = 0.84]
UNI[r = 0.69]
UNI[r = 0.50]
BETA[1.19,1]
BETA[1.44,1]
BETA[2.00,1]
1
Note: Panels A, B and C report the seller’s expected revenues corresponding to different reserve prices, and different assumptions about the bidders’ distributions of values (UNI: Uniform, BETA: Beta) and risk preferences (r). Filled dot markers stand for optimal reserve prices (given the distribution of values and risk preferences). In Panel C, a stands for the decrease in expected revenues when setting a reserve price that is optimal for risk neutral bidders with values drawn from a Beta distribution B(2,1) whereas bidders are risk averse (with r = 0.5) and have their values drawn from a uniform distribution on [0,1]. b measures the difference in expected revenues between the case where the wrong optimal reserve price has been implemented (i.e., for risk neutral bidders with beta distributed values) and the case where a reserve price for risk averse bidders with values drawn from a uniform has been implemented.
Figure 2: Observed and expected revenues Exp. Rev. 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 1
2
3
4
5
6
7
8
9
10
Average
Session Observed UNI [ALLBIDS]
BETA [WINBIDS] BETA [HYBRID]
UNI [WINBIDS] UNI [HYBRID]
BETA [ALLBIDS]
Note: The figure reports observed and expected revenues under various conditions. BETA[·] assumes risk neutrality and values drawn from a Beta distribution (with the parameter estimates of Table 1 or 2). UNI [·] assumes risk aversion and values drawn from a uniform distribution (with the parameter estimates of Table 1 or 2). WINBIDS: estimates use information on winning bids. ALLBIDS: estimates use information on all bids. HYBRID: estimates use information on all bids and on value realizations.
Figure 3: Estimated Individual Bidding functions 1
1 Session 1
bid
0.9 0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
Session 2
bid
0.9
0.1
value
0
value
0 0
0.2
0.4
0.6
0.8
1
0
1
0.2
0.4
0.6
0.8
1
1 Session 3
bid
0.9 0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
Session 4
bid
0.9
0.1
value
0
value
0 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
1
1 Session 5
bid
0.9 0.8
0.8
0.7
0.7
0.6
0.6
Session 6
bid
0.9
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
value
value
0
0
0 0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
1
Estimated Bid Functions
RNNE
29
b* = 0.75
b=v
0.8
1
Figure 4: Estimated Individual Bidding functions 1
1 Session 7
bid
0.9 0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
Session 8
bid
0.9
0.1
value
0
value
0 0
0.2
0.4
0.6
0.8
0
1
1
1 Session 9
bid
0.9
0.2
bid
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.6
0.8
1
Session 10
0.9
0.8
0.4
0.2
0.2
0.1
0.1
value
0 0
0.2
0.4
0.6
0.8
value
0 0
1
Estimated Bid Functions
RNNE
30
0.2
0.4
b* = 0.75
0.6
b=v
0.8
1
Appendix : Proof of Propositions Proof of Proposition 1: Take R20 1 and any finite k 0 . Take any R10,r 0 such that R10 /r 0 k .
Say
E 0 w R10 , R20 , r 0 .
Then,
K v; E 0 v
1
v
¨0 s k (n 1)ds .
Since v parameters R10,r 0 only enter the previous equation as the ratio, there exists infinitely many parameter vectors E such that K v; E 0 K v; E for all v . k (n 1)
Proof of Proposition 2: The non-identification result is due to the fact that we are using a single first-order moment to estimate two parameters. The proof is shown for the WINBIDS case. The ALLBIDS case mimics the steps. Take R10, R20 with R20 v 1 . The model in (6) is firstorder identified if 1
¨0 K v; R1, R2,1 dB v; R1, R2
n
mW R10, R20,1
(A1)
implies R1 R10 and R2 R20 . However, if we take R2 1 and operate in (A1) we obtain 1 mW R10, R20 ,1
2n 1 1/ R1 1 n R1(n 1)
(A2)
A solution in R1 exists since the left hand of (A2) is continuous in its domain and maps on the unit interval. Therefore, one could take R2 1 and R1 such a solution and (A1) holds but it is not true that R1 R10 and R2 R20 .
Proof of claim about testing the null hypothesis in (11): The null hypothesis in (11) defines a function h : \ l \ 2 such that if the model holds and r0 is the true constant relative risk aversion parameter, then there is an implicit restriction on the parameter space of R which is given by R 0 h(r0 ) . To test this implicit restriction, we consider the distance between a consistent estimate of R 0 under both the null and the alternative hypotheses, Rˆ , and Rˆ0 , which is consistent only under the null, and we test whether this distance is statistically significant. A consistent estimator under the null hypothesis is given by Rˆ0 h (rˆ) , where rˆ is a consistent estimate of the true risk aversion parameter r0 . This consistent estimator can be found by estimating the linear model bit J vit H it by OLS, in which case rˆ (n 1)(1 Jˆ) Jˆ . Write Kit R short for vit ¨
vit
0
FR 0 (s ) ¯ n 1 ¡ ° ds . A consistent estimator Rˆ can be found ¡¢ FR0 (vit ) °±
by NLLS, i.e.,
31
Rˆ ArgMin : SnT R w R
(bit Kit (R))2 , i,t
which is an asymptotically normal estimator so that nT (Rˆ R 0 ) l N (0, J 01I 0J 01 ) , where I0 is the asymptotic variance of the gradient of the objective function, and J0 stands for the asymptotic Hessian evaluated at the true parameter value. Following Gourieroux and Monfort (1995), it can be shown that nT (Rˆ Rˆ0 ) is asymptotically normal with a variance-covariance matrix given by V M hJ 01I 0J 01M h' , where Mh is the orthogonal projection of sh(r0 )/ sr on the space spanned by the columns of J0. Notice that since the only effective restriction in (11) is the one affecting R20 , the 2 q 2 matrix V is singular. Therefore, an asymptotic test of (11) at the required confidence level can be computed using the fact that
nT (Rˆ Rˆ0 )'V (Rˆ Rˆ0 ) l Dk2q , where V denotes any generalized inverse of V. If the variance matrix 8 of the non-systematic error term wit is assumed to be a scalar matrix, I 0 can be consistently estimated using the outer product of the gradients s RKit (R)s RKit (R) ' . In general, however, one would expect some i,t
degree of both heteroskedasticity and autocorrelation (both of unknown form) in the error term. 24 Therefore we use a Newey-West estimator of I 0 , which we denote by Iˆ . 25 Let Jˆ s2SnT Rˆ / s2R denote the estimate of the Hessian, then the test of the SBNE model is computed as follows:
ˆˆ1M h' ) (Rˆ Rˆ0 ) l Dk2q , nT (Rˆ Rˆ0 )'(M hJˆ1IJ with + denoting the Moore-Penrose generalized inverse. The next proposition states that this test reduces to a standard normal test of the significance of the second log-parameter of the estimated Beta distribution, which significantly simplifies the computation of the test in (11): Proposition: Call Rˆ2 the estimate of the log-parameter R2 and TˆRˆ the estimate of its standard 2
deviation based on the asymptotic distribution. Under the null hypothesis in (11), the t-statistic t Rˆ2 Tˆˆ is asymptotically normal with zero mean and unit standard deviation. R2
24
Heteroskedasticity may be due to the fact that bids are typically less dispersed at low values than at high values. Autocorrelation instead may be due to the bidders’ possible trial-and-error behavior which can induce timedependence in the residuals wit . 25 If the only source of heteroskedasticity was due to an increasing dispersion of bids with respect to values, then an alternative approach would be to use Generalized Least Squares (which also assumes non-autocorrelated residuals). We conducted such GLS estimations and found no significant change in our conclusions to report.
32
To simplify notation, T will denote the log-parameters which are denoted by T in the text. Taking a Taylor expansion of SnT (T 0 ) around Tˆ , we get 1 1 w 2 S nT (T 0 ) S nT (T 0 ) # nT (Tˆ T 0 ) 2 nT wT nT
(B1)
By the central limit theorem, the left hand side of (B1) converges to N(0,I0). Further, since 1 w 2 SnT (T 0 ) a.s. o J 0 we have nT w 2T d nT (Tˆ T 0 ) o N (0, J 01 I 0 J 01 ) (B1’) * Under the null hypothesis in (5), it holds that h( r0 ) T 0 . Defining SnT (r ) SnT (h(r )) , a Taylor * expansion of S nT (r ) around r0 yields
0
Defining hr0
* * * (r ) (r0 ) (r0 ) 1 wS nT 1 wS nT 1 w 2 S nT (rˆ r0 ) # wr wr nT nT nT wr wr '
w r h( r0 ) , using
(B2)
1 wSnT (T 0 ) a.s. o 0 and re-arranging terms in (B2), we get nT wT
nT (rˆ r0 )#(hr0J 0hr'0 )1hr0
1 s S (R 0 ) nT R nT
(B3)
Using (B1), we have
nT (rˆ r0 )#(hr0J 0hr'0 )1hr0J 0 nT (Rˆ R 0 ) Defining Tˆ 0
h( rˆ) , a Taylor expansion of
(B4)
nT (h(rˆ) h(r0 )) around r0 yields
nT (Tˆ T 0 ) # hr0 nT (rˆ r0 )
(B5)
Hence, from (B5) and (B3), Tˆ0 and Tˆ are linearly related so that nT (Rˆ0 R 0 )
hr'0 (hr0J 0hr'0 )1hr0J 0 nT (Rˆ R 0 ) (B6) Ph
Since
nT (Tˆ Tˆ 0 )
nT (Rˆ R 0 )
nT (Tˆ T 0 ) nT (Tˆ 0 T 0 ) , re-arranging terms in (B6) yields nT (Tˆ Tˆ 0 )
M h nT (Tˆ T 0 )
33
(B7)
Now, T 0 , i.e. the true value of the parameters, is a vector (T10 ,T 20 ) , and under the null in (5) it must hold that T 20 0 , while, T10 ln( r0 ) . Thus, hr0 ' (1/ r0 , 0) and
Mh
§ ¨0 ¨ ¨0 ©
j12 · j11 ¸ ¸ 1 ¸¹
(B8)
where jik stands for the element in row “i” and column “k” of J0. From (B7) and (B8) it follows that
§ Tˆ Tˆ 0 · nT ¨ 1 1 ¸ ¨ ˆ ˆ0 ¸ ©T2 T2 ¹
nT Tˆ2 T 20
§ ¨ § j12 · ¨ j ¸ ~ N ¨ 0, V 2 Tˆ2 ¨ ¨ 11 ¸ ¨ 1 ¸ ¨ © ¹ ¨ ©
ª§ j · 2 «¨ 12 ¸ «© j11 ¹ « j 12 « j «¬ 11
j12 º ¸· » j11 » ¸ »¸ 1 » ¸¸ ¼» ¹
(B9)
Define
ª§ j · 2 «¨ 12 ¸ j nT (Tˆ1 Tˆ10 ,Tˆ2 Tˆ20 )V Tˆ 2 «© 11 ¹ 2 « « j12 ¬« j11
j12 º » j11 » » 1 » ¼»
§ Tˆ1 Tˆ10 · ¸ ¨ ¨ Tˆ Tˆ0 ¸ © 2 2¹
From (B9), the statistic follows a chi-squared distribution with one degree of freedom (which is the rank of the variance matrix in (B9)) under the null. Using (B7) and (B8) and applying the definition of the Moore-Penrose generalized inverse, we eventually get ª Tˆ º « 2 » «¬ V Tˆ2 »¼
2
the square root of which follows a N(0,1) distribution.
34
