A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION

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1 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION BRADLEY LARSEN AND ANTHONY LEE ZHANG Abstract. Ths paper provdes a new, nonparametrc dentfcaton and estmaton approach for a varety of ncomplete nformaton games, both statc and dynamc. The approach reles on the Revelaton Prncple, explotng the ncentve compatblty of the drect revelaton mechansm correspondng to the underlyng and unspecfed game, rather than attemptng to solve for or specfy the extensve form or equlbrum strateges of the game drectly. We llustrate the approach usng smulated and actual data from barganng settngs. Keywords: Incomplete nformaton game estmaton, mechansm desgn, revelaton prncple, nonparametrc dentfcaton and estmaton, ncentve compatblty Date: May 12, Frst verson: July We thank Susan Athey, Laner Benkard, Gabrel Carroll, Evgen Drynkn, Han Hong, Mke Ostrovsky, Karl Schurter, Paulo Soman, Cao Wasman, and Al Yurukoglu, as well as partcpants at the Federal Trade Commsson, Htotsubash Unversty, Nagoya Unversty, Unversty of Tokyo, Stanford Unversty, the 2016 Calforna Econometrcs Conference, and the Emprcs and Methods n Economcs Conference for helpful comments. Larsen: Stanford Unversty, Department of Economcs and NBER; bjlarsen@stanford.edu. Zhang: Stanford Unversty, Graduate School of Busness; anthonyz@stanford.edu. 1

2 2 LARSEN AND ZHANG 1. Introducton Incomplete nformaton games have posed dffcult challenges for emprcal work n economcs. The emprcal lterature has largely proceeded by desgnng dentfcaton strateges for specfc extensve forms: for any gven extensve form, the analyst solves for a Bayes-Nash equlbrum, and uses ths equlbrum to determne the mappng between observed equlbrum strateges and players unobserved types. In some settngs, such as commonly studed aucton games, clean models of equlbrum behavor enable emprcal researchers to dentfy and estmate underlyng prmtves from observed aucton outcomes, yeldng a rch methodologcal and appled lterature. However, for a large set of extensve-form games that are mportant n practce, equlbrum characterzaton s dffcult; multple equlbra often exst, wth dfferent equlbra yeldng qualtatvely dfferent outcomes, and often no complete characterzaton of these equlbra exsts. Ths class encompasses, for example, certan types of barganng games, non-standard auctons, sgnalng games, games wth persstent prvate nformaton, nonstatonary games, and olgopoly prcng games wth ncomplete nformaton. Relatve to the rch emprcal lterature on standard auctons, proposals for dentfcaton and estmaton under many of these extensve forms have been scarce. The theoretcal lterature on mechansm desgn, poneered by Myerson (1981) and others, has proposed a dfferent approach. The revelaton prncple allows the analyst to study ncomplete nformaton games, ndependently of specfc extensve forms, by studyng revelaton mechansms the mappngs between agent types and physcal outcomes nduced by the Bayes-Nash equlbra of extensve form games. Any such mappng from types to outcomes must be ncentve compatble for all types; conversely, any ncentve compatble mappng can be supported as an equlbrum of some ncomplete-nformaton game. Hence, studyng ncentve compatble revelaton mechansms s equvalent to studyng the full class of outcomes that can be supported as equlbra of ncomplete nformaton games. Ths abstract noton of revelaton mechansms appears unamenable to emprcal settngs. A contrbuton of our

3 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 3 paper s to make a precse connecton between the mechansm desgn revelaton prncple approach and emprcal work, and then to propose an extenson of ths concept whch allows for dentfcaton and estmaton n settngs, such as sequental games, where the econometrcan often does not observe an agent s full vector of contngent actons. In ths paper, we propose a class of technques for emprcal analyss of ncomplete nformaton games, ndependently of specfc extensve forms. We suppose that the econometrcan observes multple repettons of a tradng game, and that n each repetton she observes, for each agent, the fnal allocaton (.e. whether the good traded hands and, f so, to whom), any transfers pad by each agent, and one addtonal pece of nformaton, analogous to the report made by the agent to the mechansm desgner n a theoretcal mechansm desgn framework. Ths addtonal pece of nformaton may ether be an acton taken by the agent, or some proxy a varable such as an agent s ntal acton n a sequental game or some characterstc of the agent known to the econometrcan but not to the opposng partes. The dentfcaton arguments and estmaton procedure are analogous n the observed acton and proxy cases, but the proxy case s much more general. Rather than analyzng strateges n the context of specfc extensve form equlbra, we thnk of agents actons/proxes as choces from a menu of feasble expected physcal outcomes nduced by the Bayes-Nash equlbrum of the game. We show that ths expected physcal outcome menu s suffcent to summarze agents choces n equlbrum; moreover, ths menu can be estmated by the econometrcan. If an agent s observed choosng a gven pont on the menu of feasble physcal outcomes, the margnal costs of other feasble optons on the equlbrum menu allow the econometrcan to derve bounds on the agent s unobserved type. In the case that all types play dstnct actons n equlbrum, we show that these bounds collapse to sngle ponts, so that the mappng between observed actons and agent types s pont-dentfed ndependently of the extensve form of the game. If types do not play dstnct actons, our approach gves bounds on the values of agents playng any gven acton; these bounds are the best possble, n the sense that no nformaton about the extensve form of the game can allow more precse dentfcaton of values.

4 4 LARSEN AND ZHANG For estmaton we propose procedures for both dscrete and contnuous actons cases. In the dscrete case, we propose an emprcal ronng procedure remnscent of Myerson (1981) to enforce convexty of the outcome menu n the process of estmaton. For the contnuous case, we propose a nonparametrc local polynomal regresson estmator for values, whch estmates pontwse the mappng from actons to values as the rato of two nonparametrc dervatve estmates. Addtonally, we propose a nonparametrc contnuous analog of the dscrete ronng procedure, usng constraned cubc splnes to estmate contnuous menus wth convexty constrants. We llustrate ths approach usng Monte Carlo smulatons of the blateral barganng game studed n Satterthwate and Wllams (1989). Ths game, referred to as a k double aucton, has a contnuum of qualtatvely dfferent equlbra. Wthout pror knowledge of the precse equlbrum played and the barganng power weghts, the tradtonal structural approach of nvertng player s best-response functon (as s done n frst prce auctons n Guerre, Perrgne, and Vuong (2000), for example), fals. The dentfcaton and estmaton approach we propose, on the other hand, does not requre ths pror knowledge, and we demonstrate that t performs well n practce n estmatng player s valuatons. We show that our method can be extended n a number of drectons, usng tools developed n the emprcal auctons lterature for partcular extensve forms. We show that the assumpton of ndependent values can be relaxed; n a settng wth nonndependent prvate values, one can proceed by estmatng type-contngent menus and usng ther subgradents for value estmates. We show that we can combne our approach wth tools used to analyze settngs wth unobserved heterogenety, poneered by Krasnokutskaya (2011), n a large class of generalzed bddng games, usng deconvoluton arguments to recover unobserved-heterogenety-corrected menus from observed probablty/transfer outcomes. We apply our estmaton approach to data from a secret reserve prce aucton followed by dynamc, two-sded barganng. Ths mechansm s used n busness-tobusness transactons between used-car dealers as well as other settngs (Elyakme,

5 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 5 Laffont, Losel, and Vuong (1997)). Ths game has multple equlbra and no complete theoretcal characterzaton. In the data, we observe fnal transacton prce, an ndcator for whether or not trade occurred (the allocaton), and the secret reserve prce of the seller, whch we assume s strctly ncreasng n the seller s underlyng valuaton. We combne our proxy varable approach wth our methods to correct for unobserved heterogenety to estmate the mappng from observed reserve prce offers from sellers to seller s values. We fnd that the estmated dstrbuton of values s close to bounds on seller valuatons estmated n Larsen (2014), whch used the same data but exploted outcomes of the barganng game. An advantage of the approach we propose s that t s applcable even n settngs where only the fnal outcome s observed, and not the ntermedate actons, such as ntermedate offers n a barganng game. We use our estmates to compute counterfactual revenue n a settng where, rather than partcpatng n an aucton whch ncreases competton among buyers and thus decreases market power of buyers the hgh bdder and seller face each other n a sngle, take-t-or-leave-t offer barganng game, wth the offer made by the buyer. We fnd that sellers gans from trade would decrease by approxmately $369 per car, suggestve that, n the current mechansm, seller s beneft from a substantal degree of market power. Several prevous papers n the structural estmaton lterature propose methods that rely on smlar deas to the revelaton-prncple dentfcaton we present here. In partcular, the past two decades have seen a number of nnovatons that yeld dentfcaton of prmtves of nterest by pluggng n drectly observable agent actons, choce probabltes, or outcome probabltes rather than fully solvng for equlbra of games. For example, Guerre, Perrgne, and Vuong (2000) demonstrated that valuatons n a frst prce aucton can be dentfed drectly from dstrbutons and denstes of observed bds. The approach of Guerre, Perrgne, and Vuong (2000) can be thought of as a specal case of ours, where ours generalzes the dea to arbtrary ncomplete nformaton tradng games. Our approach s also related to Tamer (2003), whch derved dentfcaton results n statc dscrete games relyng on pluggng n emprcal

6 6 LARSEN AND ZHANG measures of probabltes that cannot be pnned down unquely by a model (due to multplcty of equlbra). In dynamc games, Bajar, Benkard, and Levn (2007) and others proposed two-step methods n whch the frst step nvolves estmatng polcy functons drectly from observed choce probabltes rather than from solvng the model. Smlar procedures are also adopted n Athey and Nekpelov (2010) appled to search poston auctons, n Nekpelov, Syrgkans, and Tardos (2015) appled to ad auctons, and n Hortaçsu and McAdams (2010) appled to treasury auctons. Agarwal and Soman (2014) present a method for estmatng preferences from reported rankngs n a matchng game; as n our settng, the authors treat agents as choosng an expected outcome from a menu that can be estmated n the data. In contemporaneous work, Klne (2016) provdes an dentfcaton argument that s closely related to ours for tradng games wth observed actons, dervng stronger results than our paper n the non-ndependent prvate values case under an addtonal assumpton about equlbrum monotoncty. Our work dffers n focusng on both estmaton and dentfcaton, dervng results relatng to menu convexty, and applyng our approach to cases where actons are not fully observed. A number of other papers have bult on earler nsghts n Guerre, Perrgne, and Vuong (2000), Tamer (2003), and others to acheve feasble estmaton approaches n partcular settngs. For the most part these settngs have been cases n whch the equlbrum of the game can be characterzed and the extensve form s known, and the advantage of pluggng n emprcal objects n these cases s that t avods the need to solve for the equlbra. A contrbuton of our approach s that t yelds dentfcaton, and a correspondng estmaton approach, n arbtrary ncomplete nformaton tradng settngs n whch the full characterzaton of equlbra and the extensve form may be unknown. 2. Model Throughout, agents wll be ndexed by. Uppercase X wll denote random varables or vectors, lowercase x wll denote realzatons, and bold x ( ) wll denote functons. For all such objects, we wll use a subscrpt to denote the vector of

7 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 7 objects for all agents other than. For example, X (X 1... X 1, X X m ), where m s the number of agents. We consder an ncomplete nformaton tradng game. In ths secton, we consder an nformaton envronment wth asymmetrc ndependent prvate values; n Subsecton 5.1 we extend these results to allow for correlated values. Agents {1, 2,... m} have values V for a sngle ndvsble good, where each V s drawn ndependently from a contnuous bounded dstrbuton F ( ), supported on [v, v ]. Agent s value s observed only by. All agents are rsk-neutral. Let x be an ndcator representng attanng the good, and t R any net payment made by. If agent has value V = v, her utlty for the par (x, t ) s lnear n her value, as s standard n the theoretcal mechansm desgn lterature: v x t. Agents play tradng game G. The soluton concept s Bayes-Nash equlbrum. 1 We wll analyze G n normal form (thus, we do not requre refnements such as perfecton). Frst, values V are drawn from F ( ) and observed by each agent. Havng observed ther types V, agents choose (potentally mxed) strateges: s : R A, mappng values v R nto probablty dstrbutons over actons a A, where A s the space of actons avalable to. The outcome allocaton and transfers for all agents (x 1, t 1 ), (x 2, t 2 )... (x m, t m ) are calculated as a functon of all agents actons a 1... a m. We wll refer to the ndvdual allocaton and transfer functons as x (a 1... a m ), t (a 1... a m ). We assume nothng about the structure of G, except that each agent has some outsde opton ā whch leads to some outcome x, and transfer normalzed to t = 0, ndependently of the actons of other agents a. 1 A varety of processes can lead agents to play Bayes-Nash equlbra; we do not take a stance on any partcular set of assumptons underlyng the Bayes-Nash equlbrum concept.

8 8 LARSEN AND ZHANG For a gven strategy s, we defne Σ (v ) as the set of all actons a A played by type v wth postve probablty under strategy s ( ). Let s 1 (a ) = {v : a Σ (v )}, that s, s 1 (a ) s the set of types v whch play a wth postve probablty under strategy s. Two examples of ncomplete nformaton tradng games are the followng: Example 1. Aucton: Agents {1... m} partcpate n an aucton. Actons a belong to a space that depends on the rules of the aucton. For example, n a sealed-bd aucton, the actons are sealed bds n R. In an ascendng or mult-round aucton, actons are hstory-contngent bddng strateges. Agents outsde optons are to leave wthout partcpatng n the aucton, leadng to x = 0. Example 2. Barganng game: Seller (player 1) and buyer (player 2) bargan over an ndvsble good. The seller s outsde opton s x 1 = 1, and the buyer s outsde opton s x 2 = 0. Once agan, the form of the actons a depends on the specfc rules of the barganng game; the game could be a take-t-or-leave-t offer by one party or an alternatng-offer barganng game, or could follow any other barganng protocol. Assumng all other agents play accordng to ther equlbrum strateges, f type v of plays acton a, she attans some expected physcal outcome (P (a ), T (a )), defned as: P (a ) E [x (a, A )], T (a ) E [t (a, A )] that s, the expectaton of the allocaton x (a, A ) and transfer t (a, A ) over the actons A of players (whch, from s perspectve, s a random vector). The expected utlty that type v of agent attans from playng acton a, relatve to her outsde opton, s: v P (a ) T (a ) v x In Bayes-Nash equlbrum, all types v of each agent must be optmally choosng actons wth respect to the dstrbutons of opponents actons A. Ths mples that,

9 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 9 for all, v, the followng ncentve compatblty condton must hold: Or, equvalently, a Σ (v ) = a arg max v P (a a ) T (a ) v x (1) v s 1 (a ) = v P (a ) T (a ) v P (a ) T (a ) a (2) a Note that, n addton to ncentve compatblty, we requre ndvdual ratonalty: must do better than the outsde opton, so the total utlty max a v P (a ) T (a ) v x must be nonnegatve. However, ths condton wll not play a major role n our prmary dentfcaton and estmaton arguments, wth the excepton of the unobserved heterogenety correcton n Subsecton 5.2. (2) s a necessary and suffcent condton for strateges s (v ) to consttute a Bayes- Nash equlbrum. Importantly, (2) does not drectly reference ether the extensve form of the game that s, the functons x (a 1... a m ), t (a 1... a m ) or the dstrbuton of opponents actons A. Ths s because nether of the objects x (a 1... a m ) and t (a 1... a m ) enter drectly nto the utlty functon of type v of agent. From the perspectve of agent, the equlbrum of G defnes a menu of feasble expected physcal outcomes (P (a ), T (a )), ndexed by acton choces a. Ths menu s a suffcent statstc for s choce n equlbrum each type v of agent chooses the tem (P (a ), T (a )) from the equlbrum menu whch affords her the hghest utlty. We wll explot ths menu n a varety of ways below to obtan dentfcaton and estmaton results. 3. Identfcaton In ths secton we derve dentfcaton results for the model descrbed above. In Subsecton 3.1, we dscuss dentfcaton n the case n whch the econometrcan observes agents actons drectly, as n many smultaneous-move tradng games such as a sealed-bd auctons or a double aucton. Here we prove that the slope of the (P, T) menu, evaluated at the acton chosen by the agent, provdes dentfcaton

10 10 LARSEN AND ZHANG of the agent s valuaton. The arguments we present n ths case are related to a varety of arguments already known n the lterature to some extent, although, to our knowledge, ths paper, along wth Klne (2016), s the frst general exposton of how these arguments can be used for dentfcaton. We demonstrate that ths dentfcaton argument holds regardless of whether the structure of the equlbra of the game s known. We focus on these results frst because they provde the necessary backdrop for our man result. Our man result, whch s new to the lterature, s found n Subsecton 3.2. Here we generalze our approach to tradng games such as alternatng-offer barganng or other sequental barganng or multstage aucton games n whch the econometrcan cannot observe all contngent actons of agents. Here we requre that the econometrcan observe a proxy for agents types. Ths proxy may be an ntal acton of a dynamc game or a frst-stage bd, or some other feature of the data, such as demographc characterstcs about an agent that the econometrcan observes but other agents do not. In all cases, we assume the econometrcan observes multple ndependent nstances of a sngle equlbrum of the tradng game G, where nstances of G are ndexed by j. Thus, n each nstance of the game, values V j are ndependently drawn from F, and agents take actons to a sngle set of equlbrum strateges (s 1 ( ),..., s m ( )). We assume that n each nstance j of the game, the econometrcan observes outcomes x j (the allocaton) and t j (the transfer). If players are asymmetrc (.e. not exchangeable n ther ndces, ), we assume the econometrcan also observes the dentty of any player whose value s to be dentfed Fully observed actons case. In ths secton, we assume that n each nstance j of the game, n addton to observng x j and t j, the econometrcan observes agents actons a j. Examples of cases n whch the econometrcan may observe agents actons are any sealed-bd tradng game or any smultaneous-move tradng game. Ths ncludes not only frst prce or second prce auctons, where the structure of equlbra

11 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 11 s well-known n the theoretcal and emprcal lterature, but also any arbtrary sealedbd tradng game where such propertes may be less well-known or less well-behaved, such as the Medcare medan-prce aucton dscussed n Cramton, Ellermeyer, and Katzman (2015). Ths also ncludes sealed-bd tradng games wth multple equlbra, such as a k double aucton (Satterthwate and Wllams 1989). We assume nothng else about the structure of the game, or the partcular equlbrum beng played (except that the same equlbrum s played n each nstance j). In partcular, the equlbrum need not be ncreasng or n pure strateges; the medan-prce aucton s such an example. However, we wll obtan stronger results when such an ncreasng, pure-strategy equlbrum does exst. It s also worth notng that for our dentfcaton argument to hold, t need not be the case that all agents are behavng accordng to an equlbrum. In partcular, for dentfcaton of the valuaton of a partcular agent, t need only be the case that s best-respondng to other agents actons, regardless of whether those other agents actons themselves represent best-responses. In Secton 2, we argued that the expected outcome functons (P (a ), T (a )) f suffcent to summarze agents choces n equlbrum. The bass of our dentfcaton approach s that these expected outcome functons can also be estmated by the econometrcan. Whle estmaton wll be dscussed n more detal n Secton 4, we smply note here that from observng n nstances of the game, we can consstently estmate (P (a ), T (a )) by takng the emprcal averages of x j, t j condtonal on agent choosng acton a : ˆP (a ) = n j=1 x j1 aj =a n j=1 1 s j=a, ˆT (a ) = n j=1 t j1 aj=a n j=1 1 a j=a For any gven acton value a, the econometrcan can then use (2) to bound the values of any type v s 1 (a ), that s, any type v that plays a wth postve probablty n equlbrum. We state ths dentfcaton result as the followng theorem:

12 12 LARSEN AND ZHANG Theorem 1. For any a, all v s 1 (a ) satsfy: v T (a ) T (a ) P (a ) P (a ) a : P (a ) < P (a ) v T (a ) T (a ) P (a ) P (a ) a : P (a ) > P (a ) Proof. Follows mmedately from (2). The ntuton behnd ths dentfcaton result s as follows. The econometrcan observes multple nstances of equlbrum play; hence, the econometrcan can take sample averages condtonal on any observed acton value a to estmate the expected physcal outcome (P (a ), T (a )) ndexed n equlbrum by acton a, that s, the menuof expected physcal outcomes {(P (a ), T (a ))} avalable to agent n equlbrum. Rankng these actons accordng to ther expected allocaton P( ), agent s chosen acton reflects how the agent traded off P( ) and the expected transfer T( ), yeldng bounds on the agent s value. In Fgure 1, we llustrate a hypothetcal equlbrum menu n a settng where agents possble actons are a {a1,..., a5 }. If we observe an agent choosng pont a3, t must be the case that the agent s value v s 1 (a 3 ) s lower than the margnal cost T (a ) T (a 3 ) P (a ) P (a 3 ) of tems a {a4, a5 } wth P (a ) > P (a 3 ). Lkewse, the agent s value must be hgher than the margnal cost T (a 3 ) T (a ) P (a 3 ) P (a ) from tems a {a1, a2 } wth P (a ) < P (a 3 ). Thus, the value of any agent type choosng pont a 3 les between the upper and lower margnal costs from pont (P (a 3 ), T (a3 )), represented by the slopes of the green lnes labeled v (a3 ), v (a3 ) respectvely. Snce any acton played n equlbrum must be optmal for some type, the nequaltes n Theorem 1 must have nonempty ntersecton; n partcular, ths mples that

13 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 13 T (a ) Fgure 1. Hypothetcal menu a 5 v (a 3 ) a 4 a 6 a 3 v (a 3 ) a 1 a 2 P (a ) Notes: Hypothetcal menu. The slopes of the green lnes are upper and lower bounds for the value of an agent choosng acton a 3. the menu {(P (a ), T (a ))} of actons played wth postve probablty n equlbrum must be convex, rulng out the exstence of ponts such as a 6 n Fgure Pure Strateges. We can derve further results f we assume that the equlbrum of G s n pure strateges; that s, each v plays a sngle acton wth postve probablty n equlbrum, so that Σ (v ) contans only a sngle value a for any v. Then we can thnk of the strategy s (v ) as a functon mappng values to actons a A. The nformatveness of the bounds n Theorem 1 depends on the degree to whch dfferent types play dfferent actons n game G. 2 Specfcally, suppose agents wth types δ apart play strctly dfferent actons; that s, s (v + δ) s (v ) v. Then, we have 2 We wll thnk of actons a, a whch nduce the same expected physcal allocaton and transfer (P (a), T (a)) as dentcal. Hence, wthout loss of generalty, dstnct actons a, a lead to dstnct physcal outcomes.

14 14 LARSEN AND ZHANG for any v: v T (s (v + δ)) T (s (v )) P (s (v + δ)) P (s (v )) v + δ (3) v δ T (s (v )) T (s (v δ)) P (s (v )) P (s (v δ)) v (4) Hence, for any a, s 1 (a ) s an nterval wth length at most 2δ. In partcular, f s ( ) fully separates types, the ntervals s 1 (a ) all collapse onto sngle ponts, leadng to the followng result: Corollary 1. If, n game G, each type v has a dstnct best response acton s (v ), the nverse mappng s 1 (a ) from actons to types s pontwse dentfed. Proof. Follows mmedately from (3) and (4). As we demonstrate below, the menu {(P (a ), T (a ))} s convex, whch mples that the nequaltes n Theorem 1 are tghtest for those values of P (a ) whch are closest to P (a ). Supposng we have ordered actons a s.t. P (a ) s strctly ncreasng, then n the case where s ( ) s strctly ncreasng as well, we have the followng: v = lm δ 0 T (s (v )) T (s (v δ)) P (s (v )) P (s (v δ)) Dfferentable, Increasng Actons. In many examples the functons T and P are smoothly ncreasng, n addton to s ( ), and all three objects are dfferentable, n whch case ths expresson smplfes further. Corollary 2. If a R and the functons T, P, s are monotoncally ncreasng and dfferentable, we have: v = s 1 (a ) = dt da dp = T (a ) P da (a ) In Subsecton 4.2, we wll descrbe an estmaton strategy based on Corollary 2. We note n the followng example that exstng dentfcaton arguments for some easly solveable tradng games, such as frst prce auctons, are specal cases of our argument.

15 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 15 Example 3. Consder a frst prce aucton n a symmetrc IPV envronment. Bdder chooses a bd b, and the maxmum opposng bd s gven by M G( ). In ths settng, P (b ) = G(b ) and T (b ) = b G(b ). Player s value s then gven by v = dt db dp = b g (b ) + G (b ) g (b db ) = b + G (b ) g (b ) Ths expresson s equvalent to that derved n the dentfcaton argument presented n Guerre, Perrgne, and Vuong (2000). Note, however, that ths explct soluton requres knowng the rules/extensve form of the game, whereas our approach does not Theoretcal Propertes of the (P, T) Menu. We now provde several observatons about the structure of the equlbrum menu {(P (a ), T (a ))}. We frst defne a number of terms from convex analyss; see, for example, Rockafellar (1997) for more detals regardng these and related objects. 3 Let {(P (a ), T (a ))} denote the set of all (P (a ), T (a )) pars. We wll defne a subgradent of a set {(P (a ), T (a ))} at pont a as any value ν such that T (a ) T (a ) + ν (P (a ) P (a )) a, that s, a lne n p, t space of slope ν passng through (P (a ), T (a )) whch les weakly below all ponts n {(P (a ), T (a ))}. We defne the graph of {(P (a ), T (a ))} as the functon obtaned by jonng the ponts n order of ncreasng P (a ) values. Proposton 1. (1) The graph of {(P (a ), T (a ))} s convex. (2) For any a, s 1 (a ) for any a s the collecton of subgradents of {(P (a ), T (a ))} at P (a ). Each s 1 (a ) s a closed nterval, and the unon of all s 1 (a ) contans the nterval of values [v, v ]. (3) If we order actons a by the values of P (a ), s 1 (a ) s setwse ncreasng n a. For any a, a, the ntervals s 1 pont. (a ), s 1 (a ) ntersect at at most one 3 Note that our notaton s adapted to our settng, and does not correspond exactly to Rockafellar (1997)

16 16 LARSEN AND ZHANG Proof. See Appendx A.1. Proposton 1 can be nterpreted as follows. Part 1 formalzes the sense n whch we refer to the menu {(P (a ), T (a ))} as convex. Part 3 states that s 1 (a ) s hgher for values of a wth hgher probabltes P (a ). Ths s related to the classc fact n sngle-crossng mechansm desgn that mplementable allocaton rules must be monotone, assgnng hgher bundles to hgher types. Intutvely, under our convex menu nterpretaton of equlbra, convex menus have monotoncally ncreasng margnal costs, hence agents that choose bundles wth hgher P (a ) pay hgher margnal costs, and thus must have hgher values. Together, parts 2 and 3 also state that each s 1 (a ) s an nterval, and dstnct ntervals s 1 (a ), s 1 (a ) ntersect at no more than a sngle pont. Ths mples that the bounds of Theorem 1 effectvely parttons the nterval of values [v, v ]. Whle n general ths does not allow us to dentfy the exact types of each agent, ths dentfcaton result s the best possble, n the sense that dfferent types n the same nterval v s 1 (a ) are observatonally equvalent from the perspectve of the econometrcan observng x j, t j, a j, regardless of the extensve form of the game played. Thus, the bounds n Theorem 1 asymptotcally capture the full emprcal content of the ncomplete nformaton games model. 4 Ths allows us to draw a parallel to mechansm desgn: f the econometrcan observes x j, t j, a j, the extensve-form structure of the game s largely rrelevant for the queston of dentfcaton of s 1 (a ); the extensve form matters only nsofar as t affects the equlbrum mappng of types v to expected physcal outcomes {(P (a ), T (a ))} Man result: actons not fully observed. In many contexts, t s mpossble for the econometrcan to observe the entre acton vector a j n any nstance j of the game. For example, n a multple-offer barganng game, observng a would ental observng all actons contngent on all possble sequences of back-and-forth offers from 4 However, n fnte samples, estmators that explot the structure of the partcular game beng played may be more effcent than our proposed estmaton procedures.

17 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 17 other agents, or n an ascendng aucton, all bd strateges over all sequences of bds, wthn a sngle nstance j of the game. Thus, observng a would not smply mean observng what actons took, but what actons would have taken n every hstory of the game, ncludng those not reached; ths would be equvalent to observng a set of nstructons player would have gven to another agent to play on her behalf. However, n many cases the econometrcan wll be able to observe a proxy Z whch s correlated wth A, but uncorrelated wth A. In ths secton, we show that such proxes Z allow the econometrcan to derve lower bounds on the utlty of any gven type n equlbrum. Moreover, f the proxy fully separates types of the agent, we can once agan recover the entre mappng from types to actons. 5 Defnton 1. Z s a proxy for A f Z and A are not ndependent, and Z s ndependent of A. Two examples n whch ths condton s satsfed are: Example 4. Suppose that A specfy strateges n a multple-round barganng game. If always makes the frst offer n the game, the frst offer cannot depend on other actons A. So the frst offer satsfes the condtons of Defnton 1. Example 5. Suppose we observe characterstcs Z of agent, such as demographc nformaton or nformaton about the agent s behavor n other settngs, whch are correlated wth her value V (and hence her acton A ), but are unobserved by other players. Then Z must be ndependent of A, snce other agents can t condton ther actons on s prvate nformaton. So these characterstcs satsfy the condtons of Defnton 1. For example, n the settng of Ambrus, Chaney, and Saltsky (2014) that of Spansh rescue partes negotatng wth North Afrcan prates the amount of earmarked money rased by the captve s famly back home s known to the econometrcan and to the buyer (the rescue party) but s unobserved to the seller 5 The proxy varable termnology s used smlarly n the lterature on producton functon estmaton, for example, Levnsohn and Petrn (2003), n whch flexble monotonc functons of proxy varables such as nvestments or materal nputs are used to control for unobserved productvty.

18 18 LARSEN AND ZHANG (the prates). Ths earmarked money can serve as an proxy for the rescue party s acton Menu bounds. Supposng that Z satsfes our Defnton 1, we know that: E [(x (A, A ), t (A, A )) Z = z ] = E [E [(x (A, A ), t (A, A )) A, Z = z ] Z = z ] Snce we have assumed Z s ndependent of A, we can gnore the nner condtonng on Z : = E [E [(x (A, A ), t (A, A )) A ] Z = z ] = E [(P (A ), T (A )) Z = z ] Hence, condtonal expectatons of x, t wth respect to z recover convex combnatons of {(P (a ), T (a ))}. Snce the graph of {(P (a ), T (a ))} s convex, any such convex combnatons le above the graph of {(P (a ), T (a ))}. In other words, for any z that satsfes Defnton 1, the graph {(P (z ), T (z ))} les strctly above the graph {(P (a ), T (a ))}. Ths allows us to lower-bound the utlty of any gven type v n equlbrum: Corollary 3. For any proxy satsfyng Defnton 1, a lower bound on the equlbrum utlty of v s gven by max z v P (z ) + T (z ) Intutvely, bundles {(P (z ), T (z ))} are averages from probablty dstrbutons over outcomes {(P (a ), T (a ))} from dfferent actons a. Hence, agents can acheve the physcal outcome assocated wth any value of z by usng a mxed strategy correspondng to the dstrbuton over a nduced by z. Another way to say ths s as follows: Corollary 4. The graph of {(P (z ), T (z ))} s an upper bound for the graph of the true menu {(P (a ), T (a ))}.

19 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 19 Fgure 2. Example Proxy Menu Notes: Actual menu together wth menu generated from a nosy proxy. Whle we can thus derve an upper bound for the menu, n general, nether the true menu {(P (a ), T (a ))} nor the jont dstrbuton of proxes and values G (z, v) are dentfed. In Fgure 2, we show the proxy menu {(P (z ), T (z ))} generated from a true menu {(P (a ), T (a ))}, where we generate proxes such that V = Z + ɛ. The proxy menu s convex; thus, f we only observed {(P (z ), T (z ))}, we would be unable to dstngush the true data generatng process from an alternatve process n whch Z = V, and the menu {(P (a ), T (a ))} were exactly the observed menu {(P (z ), T (z ))} One-to-one proxes. In some cases, the econometrcan may be wllng to assume that some proxy Z s a one-to-one functon of type V, that s, V = z 1 (Z ). 6 For example, suppose the game G s a multple-round barganng game, wth a frst sealed-bd stage n whch the optmal bd s a strctly ncreasng functon of type v. In ths case, the mappng z ( ) s fully dentfed from the data. If z s a one-to-one functon of type, then (P (z ), T (z )) s exactly the physcal outcome attaned by the unque type z 1 (z). Moreover, for any other z, the physcal outcome (P (z ), T (z )) s attanable by type z 1 (z ). Also, there are types v + 6 Once agan, we wll treat dfferent values of z as dentcal f they nduce the same physcal outcome (P (z), T (z)).

20 20 LARSEN AND ZHANG δ, v δ playng dfferent actons z (v + δ), z (v δ). As n Subsecton 3.1, ths mples the followng bounds for any δ: v T (z (v + δ)) T (z (v )) P (z (v + δ)) P (z (v )) v + δ (5) v δ T (z (v )) T (z (v δ)) P (z (v )) P (z (v δ)) v (6) Thus, as n Subsecton 3.1, the bounds collapse to a sngle pont, and the entre mappng z ( ) s dentfed, whch we state as the followng extenson of Corollary 1: Corollary 5. If, n game G, each type v s one to one wth z, the nverse mappng v = z 1 (z ) from proxes to types s pontwse dentfed. Proof. Follows mmedately from Equatons 5 and 6. We also obtan the mmedate extenson of ths result, that f the (P (z), T (z)) menu, as a functon of the proxy, s dfferentable, t s slope drectly corresponds to the valuaton of agent : Corollary 6. If z R and the functons T, P, z are monotoncally ncreasng and dfferentable, we have: v = z 1 (z ) = dt dz dp = T (z ) P dz (z ) In Secton 5.3, we consder the case n whch the proxy z s not a one-to-one functon of values v. Under some restrctons on the condtonal dstrbutons G (v z), we show how we can dentfy, rather than the full jont dstrbuton G (z, v), the condtonal expectatons E [V z], and we show how ths allows us to calculate a lower bound on expected gans from trade. 4. Estmaton We now present an approach for estmatng valuatons n ncomplete nformaton tradng games. The approach follows the dentfcaton argument above closely. We focus ths secton on the case where the agent s actons are ether fully observable

21 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 21 and are ncreasng n the agent s type, or where a proxy s observed that s ncreasng n the agent s type; n ether of these two cases, the estmaton strategy wll be the same. We frst dscuss estmaton n the case of dscrete actons/proxes, and then n the case of contnuous actons/proxes. Our goal s to estmate a player s valuaton usng observatons of the game, where n each nstance of the game we observe the outcomes (allocaton and transfer) and ether an acton or proxy. Gven that the estmaton strategy s the same n the acton and proxy cases, we wll, wthout loss of generalty, refer to actons n ths secton, rather than actons/proxes. Throughout the estmaton secton, we wll focus on estmaton for a sngle agent, thus we wll omt subscrpts, wrtng for example a, v, P ( ), T ( ) to mean a, v, P ( ), T ( ) Dscrete actons. Suppose that there are a fnte number of actons, so that s {a 1... a K }, wth generc element a k. terms of ncreasng probablty P (a k ) of attanng the asset. the set of types s 1 (a k ) choosng each acton value a k. As above, we order the values of a k n We wsh to dentfy Agan, we suppose that the econometrcan observes multple nstances of the tradng game, and that n each nstance she observes the acton a j, the trade outcome x j and the transfer t j. We can construct a famly of two-step estmators as follows. Frst, we construct estmates ˆP (a k ), ˆT (a k ) as the averages of x j, t j respectvely condtonal on actons a k. Utlzng the convex structure of the set of pars {(P (a k ), T (a k ))}, we can then choose, as n Theorem 1: max k <k [ ˆT (ak ) ˆT ] (a k ) ˆP (a k ) ˆP (a k ) ŝ 1 (a k ) mn k >k [ ˆT (ak ) ˆT ] (a k ) ˆP (a k ) ˆP (a k ) Asymptotcally, all ratos ˆT (a k ) ˆT (a k ) converge to ther populaton equvalents, ˆP(a k ) ˆP(a k ) hence ŝ 1 (a k ) consstently estmates the bounds of the set s 1 (a k ). A dsadvantage of ths estmator s that, n fnte samples, the set of {(P (a k ), T (a k ))} pars may not be convex, n whch case the lower and upper bounds may cross for

22 22 LARSEN AND ZHANG some values of a. An alternatve strategy s to adopt an emprcal ronng procedure: rather than usng the {(P (a k ), T (a k ))} graph drectly, we take ts convex hull, and use the subgradents of the convex hull to estmate values. ] For a gven collecton of [ˆP ( ), ˆT ( ) pars, we defne the supportng hyperplane H (p; ν) of slope ν, as the hghest lne of slope ν whch les below all {(P (a k ), T (a k ))} pars: b (ν) max {b : T (a k ) b + νp (a k ) a k } H (p; ν) b (ν) + νp We construct the convex hull of [P ( ), T ( )] at any pont p by takng the supremum over all supportng hyperplanes: F (p) = sup H (p; ν) ν We wll estmate ŝ 1 (a k ) usng the set of subgradents of F (p) at pont P (a k ); that s, the set of slopes ν such that H (p; ν) attans the supremum at pont P (a k ): ŝ 1 (a k ) = {ν : H (P (a k ) ; ν) = F (P (a k ))} F (p) s an upper envelope of lnear functons H (p; v), so t s convex. Thus, t admts subgradents at any pont p, and the collecton of subgradents s setwse ncreasng n p. Asymptotcally, snce the true graph {(P (a k ), T (a k ))} s convex, the nferred ŝ 1 (a k ) has the same lmt as the frst estmator. However, usng the convex hull of {(P (a k ), T (a k ))} ensures that the estmator produces attanable bounds n fnte samples. In the dscrete case, estmatng the sets of values ŝ 1 (a k ) s equvalent to estmatng the subgradents of the convex graph {(P (a k ), T (a k ))}. We have descrbed a smple two-step procedure whch accomplsh ths estmaton by estmatng the ˆP (a k ), ˆT (a k ) values, then calculatng subgradents based on these. However, t s concevable that one could construct a more effcent estmator by estmatng the subgradents drectly

23 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 23 from the observed data, rather than estmatng the ˆP (a k ), ˆT (a k ) functons as an ntermedate step Contnuous actons. In many cases of nterest the equlbrum strateges (or the transformatons between proxes and values) are smooth functons of values, and the equlbrum P (a), T (a) are also smooth. In ths case, we can estmate the mappng from actons to values usng nonparametrc regresson. In partcular, assume that the mappngs P (a), T (a) are dfferentable, and the functon v = s 1 (a) s contnuous. Corollary 2 mples that: s 1 (a) = If we can nonparametrcally estmate the dervatves ˆT (a), ˆP (a) as functons of actons a, ther rato s a consstent estmator for s 1 (a). Nonparametrc dervatve estmaton of smooth functons can be done usng local polynomal regresson (Fan and Gjbels, 1996). The local polynomal regresson estmator for T (a) at a gven dt ds dp ds pont a wth degree p, bandwdth h, kernel K h s: [ˆβ0 (a), ˆβ 1 (a)... ˆβ p (a)] = arg mn β [ t j j ] 2 p β k (a j a) k K h (a j a) In ths expresson, p represents the degree of the local polynomal ft; Fan and Gjbels suggest usng even polynomal orders p = k + 2m + 1 for estmatng frst dervatves, hence local quadratc regresson wth p = 2 s approprate for our case. K h ( ) s a kernel functon of bandwdth h; common kernel functons nclude Gaussan or Epanechnkov kernels. The coeffcent β k estmates the kth dervatve of T. Therefore, an estmate of the frst dervatve ˆT (a) s gven by performng a local polynomal regresson of the observed transfer, t j, on the observed acton, a j, and takng the coeffcent on the lnear term n (7), ˆβ 1. Smlarly, an estmate of the frst dervatve ˆP (a) s gven by performng a local polynomal regresson of the observed k=0 (7)

24 24 LARSEN AND ZHANG allocaton, x j (.e., an ndcator for whether the player won), on the observed acton, a j, and takng the coeffcent on the lnear term n the regresson. We note here that Fan and Gjbels, chap. 4.2, descrbe the followng rule-of-thumb bandwdth selecton procedure for local quadratc regresson. Frst, one fts a global quntc polynomal by standard OLS: ˆT (a) = α 0 + α 1 a... + α 5 s 5 Let the resdual varance from the regresson be s 2. The rule of thumb bandwdth s then equal to the followng varance components -lke formula: ĥ = C ν,p (K) n =1 s 2 2 (ˆT (a )) 1 7 Where C ν,p (K) s a kernel-specfc constant, whch s approxmately 1 for the Gaussan kernel and 2 for the Epanechnkov kernel. Intutvely, ths procedure chooses smaller bandwdths for functons that can be ftted better by polynomals. A smlar approach apples to estmaton of ˆP( ). As n Subsecton 4.1, ths estmaton procedure may result n a nonconvex {(P (a), T (a))} menu, and t may be desrable to ron the emprcal menu functon, constranng t to be convex durng estmaton. In addton, t s often desrable to enforce monotoncty of the P (a) functon. In a manner smlar to Judd (1998) and Schumaker (1983), we propose a splne-based procedure to nonparametrcally estmate the P ( ), T ( ) functons whle mposng convexty of the [P ( ), T ( )] menu. In Appendx A.5, we descrbe the constructon of the quadratc and cubc splne bases shown n Fgure 3. Constranng the quadratc (cubc) splne coeffcents to be nonnegatve ensures that the target functon s nondecreasng (convex). By constructon, the quadratc splnes have two contnuous dervatves, and the cubc splnes three contnuous dervatves. Estmaton then proceeds n two stages: frst, P (a) s nonparametrcally estmated as a smooth functon of a, possbly constraned to be monotonc usng quadratc splnes. Then, T ( ) s estmated as the composte functon ˆT (P (a)), where ˆT ( ) s

25 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 25 Fgure 3. Quadratc/cubc splne bass Notes: Quadratc and cubc splne bass functons, wth knots at x = (1, 2, 3, 4, 5, 6). In addton, the quadratc splne bass ncludes an ntercept term, and the cubc splne bass ncludes slope and ntercept terms. constraned to be a convex cubc splne. Snce ˆT (p) s a cubc splne, the estmated mappng s 1 (a) = dt dp s guaranteed to be contnuous and dfferentable Smulatons. To llustrate our method, we choose a settng that prevously exstng methods are ncapable of handlng: a k double aucton. A k double aucton s a blateral barganng game of ncomplete nformaton n whch both partes smultaneously submt sealed offers. If the buyer s offer (p B ) exceeds that of the seller (p S ), trade occurs at prce p = kp S + (1 k)p B, where k [0, 1]. The parameter k can be consdered a barganng power weght. A k double aucton wth k = 1 corresponds to the seller-optmal mechansm (a take-t-or-leave-t offer by the seller) and a k double aucton wth k = 0 corresponds to the buyer-optmal mechansm (a take-t-or-leave-t offer by the buyer). As demonstrated n Satterthwate and Wllams (1989), ths game has nfntely many equlbra that can be qualtatvely qute dfferent. Therefore, t s mpossble to back out buyer and seller valuatons from observed offer data usng equlbrum frst order condtons, as s done n frst prce auctons n Guerre, Perrgne, and Vuong (2000) and the follow-on lterature, for example, where the equlbrum s unque. 7 7 It s mportant to note that, as n much of the structural lterature, we requre that the same equlbrum be selected at all observatons n a gven sample. Ths assumpton does not mply that

26 26 LARSEN AND ZHANG Also, even f the model were to have a unque equlbrum, solvng for equlbra n k double auctons s somewhat more nvolved, as descrbed below. The mechansm desgn approach we propose heren solves both of these ssues by dentfyng/estmatng valuatons through explotng the ncentve compatblty constrants that must be satsfed by players observed choces, rather than actually specfyng or solvng for the equlbrum. Satterthwate and Wllams (1989) demonstrated that, for any k = [0, 1], a contnuum of ncreasng, dfferentable equlbra exst satsfyng the followng lnked dfferental equatons: ( p B( 1) (p S (s)) = p S (s) + kp S (s) s + F ) s(s) f s (s) ( p S( 1) (p B (b)) = p B (b) + (1 k)p B (b) b 1 F ) b(b) f b (b) where s F s s the seller s value, b F b s the buyer s value, and p B( 1) ( ) and p S( 1) ( ) are the nverses of the buyer s and seller s strateges. (8) (9) Satterthwate and Wllams (1989) provded an approach for solvng for equlbra numercally gven knowledge of the dstrbutons of player types. A pont (s, b, p) s chosen n the set P = {(s, b, p) : s s p b b, s s, b b}, and then a one-dmensonal manfold passng through ths pont s traced out usng dfferental equatons defned by (8) and (9). Ths path traces out an equlbrum. An example of a soluton path crossng through a pont n P s shown n Fgure 4. Ths approach does not allow for dentfcaton of players value dstrbutons, only for solvng for equlbra gven knowledge of the dstrbutons. We use ther approach to solve for an equlbrum and smulate data from equlbrum play, then apply our method for estmatng the underlyng valuatons to llustrate the estmator s performance. We draw 5,000 realzatons of buyer valuatons from a N(0.6, 0.3) and seller valuatons from a N(0.5, 0.2), wth each dstrbuton truncated to le between [0, 1]. We set k = 1/2. We choose an equlbrum passng through the pont (s, b, p) = the researcher knows whch equlbrum s selected, only that t be the same n each realzaton of the game observed n the data.

27 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 27 Fgure 4. An Equlbrum n the k Double Aucton Notes: A soluton to the k = 1/2 double aucton, lyng wth the tetrahedron P = {(s, b, p) : s s p b b, s s, b b}. Ths soluton passes through the pont (s, b, p) = (0.375, 0.625, 0.5). Buyer valuatons are drawn from a N(0.6, 0.3) and seller valuatons from a N(0.5, 0.2), wth each dstrbuton truncated to le between [0, 1]. (0.375, 0.625, 0.5), whch s the equlbrum path llustrated n Fgure 4. We solve for ths equlbrum usng the Satterthwate and Wllams (1989) approach, and then use the smulated draws of buyer and seller valuatons to smulate offers and outcomes (the allocaton and transfer). We treat these 5,000 realzatons of the buyer offer, allocaton, and transfer as our data and estmate the (P, T) menu and nfer valuatons. We focus on estmatng buyer valuatons for ths exercse. For estmaton, we apply the local polynomal approach descrbed n Secton 4.2. The estmated menu for the buyer s dsplayed n Fgure 5. As wth the llustratve, hypothetcal menu dsplayed n Fgure 1, the horzontal axs s the expected probablty of trade correspondng to dfferent offers and the vertcal axs s the expected transfer at these offers. The expected probablty of trade and expected transfer are estmated n separate local polynomal regressons. The estmates are dsplayed n red dots. Dashed lnes dsplay pontwse 95% confdence bands computed from a nonparametrc bootstrap constructed by resamplng from the data 200 tmes and performng estmaton on each bootstrap sample. The menu s estmated qute precsely.

28 28 LARSEN AND ZHANG Fgure 5. Estmated (P, T) Menu from k Double Aucton Notes: (P, T) menu estmated from local polynomal regressons. Red dots ndcate estmates, dashed lnes ndcate 95% confdence bands from 200 bootstrap replcatons. It s mportant to note that Fgure 5 the menu s only a dsplay of data; the menu by tself does not yet mpose the structure of our method. That structure comes nto play when we nterpret the slope of the menu as provdng nformaton about buyer valuatons. Estmates of ths slope are gven by the lnear terms n the local polynomal regressons. In Fgure 6 we plot the observed buyer offers on the horzontal axs and the estmated buyer values (n red dots) on the vertcal axs. Dashed lnes correspond to pontwse 95% confdence bands. represents the true valuatons. The sold blue lne The estmated values reflect the true values qute well, wth the 95% confdence bands contanng the truth over most of the range of offers. We also remark here that ths estmaton exercse dd not explot any nformaton about the value of k (the barganng power), the offers made by the seller, or the partcular equlbrum beng played. Recall that any pont n the tetrahedron dsplayed n Fgure 4 has an equlbrum passng through t, and these equlbra wll vary dependng on the value of k. Indeed, the generated data n our smulaton exercse could have come from any fxed value of k and any fxed equlbrum, and the mechansm desgn approach would stll have returned reasonable estmates of valuatons based solely on the observed buyer offer, the allocaton, and the transfer.

29 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 29 Fgure 6. Estmated Buyer Values from k Double Aucton Notes: Estmated buyer values from dervatves of (P, T) menu estmated from local polynomal regressons. Red dots ndcate estmates, dashed lnes ndcate 95% confdence bands from 200 bootstrap replcatons, and sold blue lne ndcates true values. We conjecture that, n some settngs, modelng a barganng game as k double aucton n ths fashon may serve as a reasonable alternatve to mposng a Nash barganng structure on the game. Such a framework would allow the presence of ncomplete nformaton, unlke Nash barganng, and would allow for barganng power (k n ths case) to be flexble. The method would requre repeated realzatons of the game, wth observatons of the fnal transfer, the allocaton (both from cases where trade occurred and dd not occur), and some ntal offer (such as a lst prce or ndcatve bd) or proxy for the player s valuaton. We leave further exploraton of ths possblty for future work. 5. Extensons We now provde a number of extensons to our dentfcaton arguments provded above. In Subsecton 5.1, we show that our approach generalzes to non-ndependent prvate value settngs. In Subsecton 5.2, we extend our approach to allow for unobserved game-level heterogenety. In Subsecton 5.3, we show condtons under whch lmted nformaton about the jont dstrbuton of proxes and values n partcular, the condtonal expectatons functons E [V z] are dentfed.

30 30 LARSEN AND ZHANG 5.1. Non-ndependent prvate values. In ths subsecton, we relax the assumpton that values of dfferent agents are ndependent. Suppose that agents values V 1... V m are drawn from some jont dstrbuton F (v 1... v m ), whch s common knowledge to all agents. Ths ncorporates and generalzes, for example, the afflated prvate value model of frst-prce auctons analyzed by L, Perrgne, and Vuong (2002). As above, we suppose that the agents play tradng game G. We assume that the equlbrum of the game s separatng: equlbrum strateges are descrbed by the s (v ), where each s s nvertble. We show that, as n Subsecton 3.1, we can derve bounds on the nverse functons s 1 ( ) for each a. Let s ( ) denote the equlbrum strategy of agent. Snce values are not ndependent, equlbrum actons wll be gven by some jont dstrbuton G (a 1... a n ), derved from F (v 1... v n ) and the equlbrum strategy s ( ). Fx any gven value v of player ; condtonal on v, the dstrbuton over values of agents s some F (v v ). Ths condtonal dstrbuton of values, combned wth the equlbrum strateges of other players s, nduces a condtonal dstrbuton over opponents actons G (a v ). Thus, n equlbrum, f type v of agent plays acton a, she attans the physcal outcome [P v (a ), T v (a )], defned as the expectaton of the physcal outcomes x (a, A ), t (a, A ) when A G (a v ). That s, P v (a ) = E [x (a, A ) A G (a v )] T v (a ) = E [t (a, A ) A G (a v )] In order for type v to play acton a = s (v ) n equlbrum, as n Subsecton 3.1, s (v ) must then satsfy: s (v ) = arg max v P v a (a ) T v (a ) v x (10) As n Subsecton 3.1, ths allows us to bound s 1 (a ), the unque type that plays a n equlbrum.

31 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 31 Proposton 2. for each a value, the unque v = s 1 (a ) satsfes: v T v =s 1 (a ) P v =s 1 (a ) v T v =s 1 (a ) P v =s 1 (a ) (a ) T v =s 1 (a ) (a ) (a ) Pv =s 1 (a ) (a ) (a ) T v =s 1 (a ) (a ) (a ) Pv =s 1 (a ) (a ) {a : P v =s 1 {a : P v =s 1 (a ) (a ) } (a ) < Pv =s 1 (a ) (a ) (11) } (a ) > Pv =s 1 (a ) (a ) In the case that the dstrbuton F (v 1... v n ) has full support, these bounds collapse to a sngle pont. (12) Proof. Follows from (10). If the dstrbuton F (v 1... v n ) has full support on the rectangle [mn v 1, max v 1 ] [mn v 2, max v 2 ]..., then the equlbrum probablty dstrbuton over acton tuples G (a 1... a n ) wll lkewse have full support on the product rectangle of actons played; thus, by observng multple ndependent repettons of G, the econometrcan can consstently estmate both the equlbrum acton dstrbuton G (a 1... a n ), and the outcomes condtonal on all acton tuples: P (a 1... a n ) = E [x (a 1,... a n ) a 1,... a n ] T (a 1... a n ) = E [t (a 1,... a n ) a 1,... a n ] Note that G ( a v = s 1 (a ) ), the equlbrum acton dstrbuton condtonal on v = s 1 (a ), nvolves the unknown quantty s 1 (a ). However, ths s equvalent to the condtonal dstrbuton G (a a ), whch can be derved from G (a 1... a n ). Thus, for any a, the econometrcan can estmate the functons: P v =s 1 (a ) (a ) = E [x (a, A ) A G (a a )] T v =s 1 (a ) (a ) = E [t (a, A ) A G (a a )] These functons, plugged nto the equatons n Proposton 2, allow us to dentfy the unque s 1 (a ).

32 32 LARSEN AND ZHANG Ths approach s related to that of L, Perrgne, and Vuong (2002), although t s more general, as t apples to ncomplete nformaton tradng games more broadly, rather than just auctons, showng that our dentfcaton strategy apples to prvate value settngs wth correlated values. The argument utlzes the fact that any type v must play an equlbrum acton that s a best response to the dstrbuton of opponents actons condtonal on her type. These condtonal dstrbutons can be estmated by the econometrcan, allowng us to dentfy types essentally as n Subsecton 3.1. Our approach n ths secton requres that the equlbrum strategy s (v ) s strctly separatng. Ths assumpton s necessary because t allows us to estmate the dstrbuton G (a v ) for the unque v = s 1 (a ) usng the observed G (a a ). If s s not nvertble, n general s 1 (a ) s a set of v values; thus, the observed G (a a ) s a mxture over dstrbutons G (a v ) for dfferent values v s 1 (a ). We thus cannot use G (a a ) to consstently estmate P v v when s s not nvertble. (a ), T v (a ) for any gven type 5.2. Unobserved heterogenety. We now consder an extenson of our dentfcaton arguments to a settng wth unobserved game-level heterogenety, smlar to the unobserved aucton-level heterogenety n the model of Krasnokutskaya (2011), but appled to the general ncomplete nformaton tradng games we consder here, rather than only statc frst prce auctons. We refer to the class of games we study here as generalzed bddng games, although, as before, these games need not be auctons; many barganng games would also ft nto ths class. The mportant feature of games n ths class s that actons consst of a prce offer. Throughout ths secton, we refer to the observed actons case, although these results apply to the proxy case as well. We defne the class of generalzed bddng games as follows. In the frst stage, common component W s drawn from H ( ) and commonly observed by all agents 1... m, but not the econometrcan. In the second stage, agents prvate values V are drawn ndependently from dstrbutons F ( ); agents values are then Ṽ = V + W

33 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 33 In the thrd stage, agents take actons ã R, whch are observed by the econometrcan. We requre the game to satsfy the followng property: Defnton 2. We say that game G satsfes the generalzed bddng game property for all ã 1... ã m,, and for all : x (ã 1 +, ã 2 +,... ã m + ) = x (ã 1, ã 2,... ã m ) (13) t (ã 1 +, ã 2 +,... ã m + ) = t (ã 1, ã 2,... ã m ) + (x (ã 1, ã 2,... ã m ) x ) (14) In the case of an aucton, (13) mples that f all agents ncrease bds by a constant amount, the prce pad by the wnnng bdder ncreases by. Snce the common component W s observed by all agents pror to agents acton choces, agents can condton ther strateges on the common component W; thus, we can thnk of agents strateges n generalzed bddng games as functons s (v, w) mappng common components and prvate values nto actons. Bayes-Nash equlbrum n the full game requres that agents strateges consttute Bayes-Nash equlbra condtonal on any value of w. Fxng a gven value of w, the game s dentcal to that of Subsecton 3.1. Let A w denote the random varable representng s equlbrum acton when the common component s w. As n Subsecton 3.1, we defne the expected probablty and transfer that acheves when playng a n equlbrum as: P w (a ) E ( ( x a, A )) w, T w (a ) E ( ( )) t a, A w s: Type v of s expected utlty from playng a when the common component s w (v + w) P w (a ) T w (a ) (v + w) x Analogous to Subsecton 3.1, equlbrum n a generalzed bddng game wth common component w requres that s strategy s (v, w) maxmzes her utlty, n expectaton over the dstrbutons of other agents actons A w. That s, fxng w, for all, v, we requre

34 34 LARSEN AND ZHANG s (v, w) arg max a (v + w) P w (a ) T w (a ) (v + w) x In the followng proposton, we show that the equlbra of generalzed bddng games have a translaton nvarance property wth respect to w j f actons a 1... a n are equlbrum actons condtonal on w, actons a + w w are equlbrum actons under w. Proposton 3. Fx some value of w, and suppose that bddng strateges: s 1 (v 1, w)... s m (v m, w) consttute an equlbrum. Then, for any common component w, bddng strateges: s 1 (v 1, w ) = s 1 (v 1, w) + w w. s m (v m, w ) = s m (v m, w) + w w consttute an equlbrum. Proof. See Appendx A.2. Motvated by ths theorem, we wll defne markup equlbra by requrng that agents play the same equlbrum for any common component w: Defnton 3. A markup equlbrum s a set of markup strateges s (v ), such that: s (v, w) = s (v ) + w and s (v ) consttute equlbrum strateges for w = 0. Suppose w j = 0. We wll defne the markup functon as the expected transfer for w = 0: M (a ) = E [ T 0 (a ) ] = E [ ( )] t a, A 0

35 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 35 In the equlbrum condtonal on w = 0, T (a ) = M (a ), hence we have as before n Subsecton 3.1: s 1 (a ) M (a ) M (a ) P (a ) P (a ) a : P (a ) > P (a ) s 1 (a ) M (a ) M (a ) P (a ) P (a ) a : P (a ) < P (a ) Thus, f we can recover the functon P (a ), M (a ), we can bound values as n Subsecton 3.1. Agan, we suppose that the econometrcan observes multple ndependent observatons of a generalzed bddng game. The econometrcan can estmate probablty and transfer functons condtonal on observed actons whch we wll refer to as: ( )] ( )] P (ã ) E [x ã, Ã, T (a ) E [t ã, Ã Proposton 4. P ( ), M ( ) are unquely determned by P (ã ), T (ã ), f W, f a. All of these objects are dentfed from observng multple ndependent repettons of a generalzed bddng game. Thus, P ( ), M ( ) are dentfed. Proof. See Appendx A.3. The ntuton behnd our dentfcaton result s as follows. Snce the unobserved prvate value components v are ndependent by assumpton, any correlaton n observed actons ã must be caused by to the unobserved heterogenety W. Usng a method smlar to Krasnokutskaya (2011), we can thus separately recover the dstrbuton f W of the unobserved heterogenety term W, and the dstrbuton of actons f a generated by the markup strateges, from the observed dstrbuton of actons fã, through a deconvoluton argument. We then show that the dstrbutons f W, f a allow us to recover P (a ), M (a ) from the functons P (ã ), T (ã ) through a seres of deconvolutons and convolutons aganst H ( ). The functons P (ã ), T (ã ) are expected values of observables, hence they are dentfed; thus, the functons P (a ), M (a ) are dentfed.

36 36 LARSEN AND ZHANG Once we have recovered P (a ), M (a ), we can bound the value s 1 (a ) of any agent playng the unobserved acton a. We can then calculate the dstrbuton of values condtonal on any observed acton ã by ntegratng aganst the dstrbuton H ( ) Identfyng E [V z]. In Secton 5.3, we showed that, for general nosy proxes, the jont dstrbuton of proxes and values G (z, v) s not dentfed. However, n ths secton, we wll show that under certan condtons, t s possble to approxmately estmate the functon E [V Z = z], whch we wll wrte n short as E [V z]. Ths wll allow us to construct an approxmate lower bound for gans from trade from the mechansm. Throughout ths secton, we suppress subscrpts, wrtng for example z to mean z. Moreover, n the remander of ths Subsecton, we assume that: Z s real-valued and supported on a contnuous bounded nterval P (v), T (v) are everywhere strctly ncreasng and dfferentable G (v z) s dfferentable n v for any z G (v z) s contnuous n z for almost all v G (v z) s ordered by stochastc domnance: z 2 > z 1 = G (v z 2 ) > FOSD G (v z 1 ). These assumptons are stronger than we need besdes the stochastc domnance assumpton, all other assumptons can be relaxed wthout sgnfcantly affectng the results. However, these assumptons make our results easer to state and prove. We am to fnd condtons under whch: T (z) P (z) E [V z] To begn wth, Proposton 5 characterzes the rato T (z) n terms of the condtonal P (z) dstrbutons G (v z) and the equlbrum menu P (v), T (v).

37 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 37 Proposton 5. We have: Proof. See Appendx A.4 T (z) P (z) = v ( 0 v ( v 0 dg(v z) dz dg(v z) dz ) P (v) dv ) P (v) dv (15) ( Snce both ( uct dg(v z) dz dg(v z) dz ) dg(v z) and P (v) are postve and fnte by assumpton, ther prod- dz ) P (v) can be thought of as a weghtng functon; hence the expresson on the rght hand sde of (15) can be thought of as a weghted average of values v wth respect to dg(v z) P (v). Thus, ntutvely, f the weghtng functon ( ) dz ( ) P (v) behaves smlarly to the probablty densty functon g (v z), the rato T (z) wll be close to the condtonal expectaton E [V z] = v P (z) 0 v g (v z) dv. Ths s only true under farly restrctve assumptons on the functons G (v z) and P (z); however, these assumptons are satsfed by certan classes of condtonal dstrbutons commonly used n practce. We descrbe two such sets of condtons below. Corollary 7. Suppose that: G (v, z) s translatonally nvarant, that s, G (v, z) = Ḡ (v z) P (v) s constant and equal to C Then, dg(v z) dz = g (v z), hence, ( v 0 v dz ( T (z) P (z) = v 0 dg(v z) dg(v z) dz ) P (v) dv ) P (v) dv = v 0 v 0 v g (v z) Cdv = E [V z] g (v z) Cdv Note that, whle the result n Corollary 7 s an equalty, Equaton 15 s smooth n all arguments, suggestng that a smlar statement about approxmate equalty holds: f G (v, z) s close to translatonally nvarant locally at some z, and P (v) s close to constant n the relevant neghborhood, T (z) P (z) wll not attempt to formally prove such a statement here. wll be close to E [V z]. However, we

38 38 LARSEN AND ZHANG Corollary 8. Suppose G (v z) has compact support: G (v z) [ν (z) δ, ν (z) + δ] for some δ, where ν (z) s contnuous. Then, ( ) v T (z) P (z) = 0 v dg(v z) P (v) dv dz ( = v 0 dg(v z) dz ) P (v) dv Elementary arguments show that T (z) P (z) [ν (z) δ, ν (z) + δ]. Hence, ν(z)+δ ν(z) δ v ( ν(z)+δ ν(z) δ ( dg(v z) dz dg(v z) dz ) P (v) dv ) P (v) dv [ν (z) δ, ν (z) + δ]. Lkewse, E (V z) T (z) E (V z) P (z) 2δ Once agan, f the support of G (v z) s not compact, but the majorty of the probablty mass s concentrated near ν (z), Corollary 8 wll stll hold approxmately. Intutvely, ths corollary can be nterpreted as sayng that, f z s a farly good measure of v n the sense that the resdual varaton n v condtonal on observng z s small the dervatves T (z) P (z) wll be farly close to E (V z) Approxmate bounds on gans from trade. If we are wllng to make the assumptons requred for ether corollary 7 or 8, so that we have: T (z) P (z) E (V z), we can lower-bound the expected gans from trade for agents. Total expected welfare from trade s: ] E v F [max [V P (v ) T (v ) V x], v where, the maxmzaton s performed pontwse for each realzaton of V. Usng the law of terated expectatons, ths s: [ [ = E E max v ]] [VP (v ) T (v ) V x] Z. As we prove n Corollary 4, the convex hull of the (P (z), T (z)) pars s an upper bound for the true menu (P (v), T (v)). Hence, for all z, [ E max v ] [VP (v ) T (v ) V x] Z = z

39 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 39 Fnally, we have [ E max z ] [VP (z ) T (z ) V x] Z = z [ ] E max [VP (z ) T (z ) V x] Z = z z (16) max E [V Z = z] P (z ) T (z ) E [V Z = z] x (17) z Mathematcally, ths nequalty follows because any value attanable on the RHS s also attanable on the LHS. An ntuton for ths nequalty s that, fxng a feasble outcome P (z), T (z), welfare gans from trade are lnear n values V. Hence, the average maxmal utlty across agent types s at least the maxmal utlty of the average agent type. In fact, as long as the menu s nontrval and the sgnal Z s not a one-to-one functon of V, ths nequalty wll be strct. We can estmate the proxy menus P (z ), T (z ), and f we are wllng to use the assumptons of ether Corollary 7 or 8, we can approxmate E [V Z = z], whch we prevously wrote as E [V z], usng the dervatve T (z). Hence, we can evaluate the P (z) term max E [V Z = z] P (z ) T (z ) E [V Z = z] x z pontwse for each value of z. Takng expectatons wth respect to the emprcal dstrbuton of proxy values z, we then get the desred lower bound for total gans from trade: [ E max v ] [ [V P (v ) T (v ) V x] E max z ] E [V Z] P (z ) T (z ) E [V Z] x Equatons 16 and 17 are nequaltes that are strct for nontrval menus and nosy proxy varables. Thus, even f we are unsure whether the approxmaton E [V z] T (z) P (z) s vald, as long as we beleve that the assumptons of Corollares 7 or 8 are even approxmately satsfed, we mght vew ths as a generally conservatve approach to estmatng expected welfare gans from trade.

40 40 LARSEN AND ZHANG 6. Applcaton to secret reserve aucton wth barganng In ths secton, we apply our approach to estmate the valuatons of used-car sellers n wholesale used-car markets. In wholesale used car markets, used-car dealers sell cars to other used-car dealers at aucton houses. The mechansm employed by the aucton houses conssts of a secret reserve prce, set by the seller, followed by an ascendng prce aucton between multple potental buyers. If the secret reserve prce s not met, the hghest bdder and the seller enter nto an alternatng-offer barganng game. Whle the full equlbrum of ths game s dffcult to characterze, Larsen (2014) proves that the seller s optmal secret reserve prce s a strctly ncreasng functon of her value. Hence, the secret reserve prce satsfes our condtons n Defnton 1 for a proxy whch s a one-to-one functon of type. In addton, whle the game does not exactly satsfy our defnton of generalzed bddng games n Subsecton 5.2, Larsen (2014) shows that equlbra of ths barganng game satsfy the equlbrum translaton property of Subsecton 3. Thus, we can combne the one-to-one proxy approach descrbed n Subsubsecton wth the unobserved heterogenety correctons descrbed n Subsecton 5.2 to estmate the equlbrum mappng from sellers reserve prce offers to sellers values. The data conssts of 135,000 realzatons of the aucton/barganng game. For each game, the prmary varables we observe are the seller s reported secret reserve prce, the fnal transacton prce, the fnal allocaton (.e. an ndcator for whether the car sold), as well as the hgh bd from the aucton Descrptve evdence. In Fgure 7, we show the behavor of the normalzed prces and bds, as well as the probablty of trade, as a functon of the normalzed reserve prce. Wnnng aucton bds are correlated wth reserve prces, volatng the ndependence assumpton of the baselne model of Secton 2. Ths suggests that t wll be mportant to account for unobserved heterogenety. We also see that sellers who post hgher reserve prces sell wth lower probabltes, but are able to attan hgher prces condtonal on sale. In partcular, the dfference between the average prce condtonal on sale and the average bd s ncreasng n the reserve prce. The average

41 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 41 Fgure 7. Average sale probablty, bds, and prces by reserve prce Notes: All seres from local lnear regresson estmates. All data wth reserve prce percentles between 1 to 99 are shown. Prces are averages condtonal on successful sale, whereas bds nclude all observatons regardless of whether the car was ultmately sold. Prces, bds and reserve prces are n unts of thousands of US dollars, relatve to the bluebook prce of the car. bd roughly measures the value of unobserved car-level heterogenety condtonal on the reserve prce. 8 Thus, f the prce-bd dfference s ncreasng n the reserve prce, ths suggests that sellers who post hgh reserve prces are forgong some probablty of sellng the good n order to obtan hgher markups on the sale prce of the car above the common value of the cars. 8 Formally, snce b ] = b+w, r = r+w, and b, r, w are ndependent, we have that E [ b r = E [w r].

42 42 LARSEN AND ZHANG 6.2. Estmaton. We handle observed and unobserved heterogenety as n Larsen (2014); we summarze these aspects only brefly here. Frst, we control for observed heterogenety followng the homogenzaton approach of Hale, Hong, and Shum (2003) by estmatng a lnear regresson of reserve prces and aucton hgh bds on a large set of observable characterstcs and treatng the resduals from ths regresson as homogenzed reserve prces/aucton bds. Let r = r + w represent the resdualzed reserve prce, where w s an addtvely separable, game-level, unobserved heterogenety scalar term as n Subsecton 5.2 and r s the reserve prce net of any observed/unobserved heterogenety. We estmate the denstes of w and r, f w and f r, usng a lkelhood approach, modelng each as normal dstrbutons. Wth these denstes n hand, our man estmaton steps are then the followng: (1) Nonparametrcally estmate the functons P ( r), T ( r) (2) Usng the estmated denstes f r, f w and the estmated P ( r), T ( r), correct for unobserved heterogenety to estmate the underlyng menu functons P (r), M (r) (3) Take dervatves of the menu P (r), M (r) to get value estmates v (r) We descrbe each step n turn Nonparametrc estmaton of P ( r), T ( r). For step 1, we use local lnear regressons of x j, t j on r j to estmate the functons P ( r), T ( r). We use normal kernels wth bandwdth 500. Ths s larger than the statstcally optmal bandwdth, as later steps of the estmaton beneft from smoothness of the P ( r), T ( r) functons n ths stage Unobserved heterogenety. In Proposton 4, we showed that M (r), P (r) are dentfed from the functons T ( r), P ( r), fr ( ), f w ( ) In partcular, P (r) solves: P ( r) = r= P (r) f r (r) f w ( r r) dr fr (r) f w ( r r) dr

43 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 43 and, M (r) solves T ( r) E (w P r) = r= M (r) f r (r) f w ( r r) dr fr (r) f w ( r r) dr Gven the estmates of f r ( ), f w ( ), P ( r), T ( r), we solve for P (r), M (r) usng mnmum weghted dstance, usng our shape-constraned splne bass functons. We model P (r) as a quadratc splne wth 9 knots, constraned to be nondecreasng. We choose splne coeffcents to mnmze the followng objectve functon: [(ˆ mn ˆP( ) ) ( ˆ ˆP (r) f r (r) f w ( r r) dr P ( r) )] 2 f r (r) f w ( r r) dr Wth the estmated P (r) functon, we can then plug ths n to (24) n Appendx A.3 to estmate the term E (w P r) as: ˆ E (w P r) = (P (r) 1) ( r r) f r (r) f w ( r r) dr Where, we have used that for the seller, the outsde opton s x = 1. We model M (r) ndrectly as the functon M (P (r)), where M ( ) s a cubc splne wth 7 knots, constraned to be convex. We wll choose M ( ) to mnmze: [(ˆ mn ˆM( ) In all cases, we use standard gradent descent methods to perform splne optmzaton. ) ) ˆM (P (r)) f r (r) f w ( r r) dr ( T (ˆ )] 2 ( r) E (w P r) f r (r) f w ( r r) dr In Fgure 8, we show the local lnear estmates of P ( r), T ( r), as well as the unobserved heterogenety corrected estmates P (r), T (r). Intutvely, the unobserved heterogenety correctons work as follows. For probabltes, the P ( r) functon s essentally a nosy verson of the P (r) functon; thus, correctng for unobserved heterogenety wll mply that P (r) s steeper than P ( r). For transfers, unobserved heterogenety necesstates two correctons to the T ( r) functon. Frst, we subtract from T ( r) the term E (w P r), whch represents the expected value of the unobserved heterogenety condtonal on r. Intutvely, for hgher values of r, we wll observe

44 44 LARSEN AND ZHANG that trades tend to happen at hgher prces, but much of ths s due to the unobserved heterogenety term w beng hgher on average, rather than the markup M (r) beng hgher. Comparng the locpoly lne to the debased lne, correctng for E (w P r) makes the slope of T ( r) sgnfcantly less negatve. Secondly, M (r) s essentally a de-nosed verson of T ( ) ( r) E w P r, and thus the slope and concavty of M (r) are both larger n absolute value than that of T ( r) E (w P r). The net effect s that M (r) s much less negatvely sloped and somewhat more concave than the orgnal nonparametrc estmate T ( r) Value estmaton. Snce our menu M (P (r)) s represented as a convex sum of splnes, we can analytcally take ts dervatves, gvng us the fnal estmated mappng v (r) from reserve prces to values Results and counterfactual. In the left panel of Fgure 9, we show the estmated [P (r), M (r)] menu. In the rght panel, we show the estmated mappng v (r) between the reserve prce r and the nferred value v (r). The estmated reservevalue mappng v (r), combned wth the dstrbuton f r of reserve prces, gves us an estmated dstrbuton F v of sellers values, and we plot ths n Fgure 10. We use the value estmate to compute a smple counterfactual measurng the how the seller s expected gans from trade would decrease, relatve to the current mechansm, f all market power were gven to buyers. The current mechansm, wth a frst-stage aucton followed by a second stage of alternatng-offer barganng, may award the lon s share of market power to the seller, as competton between buyers n the aucton reduces market power on the buyer sde. We smulate a counterfactual mechansm where, nstead, the hgh-bdder from the aucton and the seller meet n a one-tme, take-t-or-leave-t offer barganng game, wth the offer made by the buyer. Before dscussng the results, t s necessary to comment on ndvdual ratonalty (IR) constrants. Throughout our dentfcaton arguments and estmaton process, we have only used the ncentve compatblty condtons of sellers that s, the condton that outcomes under the reserve prces chosen are preferred to the outcomes from any other possble choce of reserve prce. However, n our estmated menu, we fnd that

45 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 45 Fgure 8. Deconvoluton graphs Notes: Locpoly lnes represent the local polynomal regresson estmates of P ( r), T ( r). Decon ( lnes are ) the splne estmates of P (R), T (r). The debased lne represents T ( r) E w ˆP r. Ftted lnes are the mnmum-dstance fts of P (r), M (r) to target functons. the IR constrant s volated for the roughly 14% of seller types wth the hghest values. For our counterfactual exercse, we enforce the IR constrant by takng the menu to be the convex hull of the orgnal menu and the outsde opton (P, T) = (1, 0); as a result, under the adjusted menu, all seller types acheve nonnegatve utlty n the mechansm. The red lne n Fgure 11 shows agents total utlty from the barganng mechansm as a functon of type. As expected, agents wth low values make the hghest gans

46 46 LARSEN AND ZHANG Fgure 9. Menu and value estmates Notes: The left panel shows the fnal estmated menu. The rght panel shows the estmated mappng from reserve prces to values. The dashed red lne shows the reserve prce tself for comparson. Shaded regons show pontwse 95% confdence bands from bootstrappng wth 500 repettons. Fgure 10. Estmated value CDF/PDF Notes: Value CDF, estmated from the CDF of reserve prces combned wth the mappng shown n Fgure 9. Shaded regons show pontwse 95% confdence bands from bootstrappng wth 500 repettons. from trade. Gans from trade on average across sellers are $1079. The blue lne shows a counterfactual n whch the buyer who wns the aucton can make a sngle take-t-or-leave-t offer to sellers; ths reduces sellers utlty by $369 on average.

47 A MECHANISM DESIGN APPROACH TO IDENTIFICATION AND ESTIMATION 47 Fgure 11. Utlty from mechansm vs counterfactual Notes: Net utlty from partcpatng n the mechansm by type. Actual shows utlty from the actual mechansm, from the IR-constraned menu and value estmates from Fgure 9. CF shows utlty from a hypothetcal mechansm n whch the wnner of the aucton makes a take-or-leave prce offer to sellers. 7. Concluson Ths paper provded a new, nonparametrc dentfcaton and estmaton approach for tradng games of ncomplete nformaton. The approach reled on explotng the ncentve compatblty of the drect revelaton mechansm correspondng to the actual, underlyng (and unknown) extensve form game, rather than attemptng to solve for or explot the equlbrum of ths game drectly. The man result demonstrated how ths approach can be appled n settngs where players actons may not be observable. We beleve the approach has the potental to be a useful dentfcaton and estmaton tool n a number of ncomplete-nformaton settngs where closed-form equlbrum solutons may not exst, or where players actons may be dffcult to fully observe, such as ncomplete nformaton sequental barganng games.