Optimal Allocation with Costly Verification 1

Transcription

1 Optmal Allocaton wth Costly Verfcaton 1 Elchanan Ben-Porath 2 Edde Dekel 3 Barton L. Lpman 4 Frst Prelmnary Draft October 2011 Current Draft March We thank Rcky Vohra, Benjy Wess, numerous semnar audences, and Andy Skrzypacz and three anonymous referees for helpful comments. We also thank the Natonal Scence Foundaton, grants SES and SES (Dekel) and SES (Lpman), and the US Israel Bnatonal Scence Foundaton (Ben-Porath and Lpman) for support for ths research. Lpman also thanks Mcrosoft Research New England for ther hosptalty whle the frst draft was n progress and Alex Poterack for proofreadng. 2 Department of Economcs and Center for Ratonalty, Hebrew Unversty. Emal: benporat@math.huj.ac.l. 3 Economcs Department, Northwestern Unversty, and School of Economcs, Tel Avv Unversty. Emal: dekel@northwestern.edu. 4 Department of Economcs, Boston Unversty. Emal: blpman@bu.edu.

2 Abstract A prncpal allocates an object to one of I agents. Each agent values recevng the object and has prvate nformaton regardng the value to the prncpal of gvng t to hm. There are no monetary transfers, but the prncpal can check an agent s nformaton at a cost. A favored agent mechansm specfes a value v and an agent. If all agents other than report values below v, then receves the good and no one s checked. Otherwse, whoever reports the hghest value s checked and receves the good ff her report s confrmed. All optmal mechansms are essentally randomzatons over optmal favored agent mechansms.

3 1 Introducton Consder a prncpal wth a good to allocate among a number of agents, each of whom wants the good. Each agent knows the value the prncpal receves f he gves the good to, but the prncpal does not. (Ths value need not concde wth the value to of gettng the good.) The prncpal can verfy the agents prvate nformaton at a cost, but cannot use transfers. There are a number of economc envronments of nterest that roughly correspond to ths scenaro; we dscuss a few below. How does the prncpal maxmze the expected gan from allocatng the good less the costs of verfcaton? We characterze optmal mechansms for such settngs. We construct an optmal mechansm wth a partcularly smple structure whch we call a favored agent mechansm. There s a threshold value and a favored agent, say. If each agent other than reports a value for the good below the threshold, then the good goes to the favored agent and no verfcaton s requred. If some agent other than reports a value above the threshold, then the agent who reports the hghest value s checked. Ths agent receves the good ff hs clams are verfed and the good goes to any other agent otherwse. In addton, we show that every optmal mechansm s essentally a randomzaton over optmal favored agent mechansms. In ths sense, we can characterze the full set of optmal mechansms by focusng entrely on favored agent mechansms. By essentally, we mean that any optmal mechansm has the same outcomes as such a randomzaton up to sets of measure zero. 1 An mmedate mplcaton s that f there s a unque optmal favored agent mechansm, then there s essentally a unque optmal mechansm. Fnally, we gve a varety of comparatve statcs. In partcular, we show that an agent s more lkely to be the favored agent the hgher s the cost of verfyng hm, the better s hs dstrbuton of values n the sense of frst order stochastc domnance (FOSD), and the less rsky s hs dstrbuton of values n the sense of second order stochastc domnance (SOSD). The standard mechansm desgn approach to an allocaton problem s to construct a mechansm wth monetary transfers and gnore the possblty of the prncpal verfyng the agent s nformaton. In many cases obtanng nformaton about the agent s type at a cost s qute realstc (see examples below). Hence we thnk t s mportant to add ths opton. In our exploraton of ths opton, we take the opposte extreme poston from the standard model and do not allow transfers. Ths obvously smplfes the problem, but we also fnd t reasonable to exclude transfers. Indeed, n many cases they are not used. In some stuatons, ths may be because transfers have effcency costs that are gnored n 1 Two mechansms have the same outcome f the nterm probabltes of checkng and allocatng the good are the same; see Secton 2 for detals. 1

4 the standard approach. More specfcally, the monetary resources each agent has mght matter to the prncpal, so changng the allocaton of these resources n order to allocate a good mght be costly. In other stuatons, the value to the prncpal of gvng the good to agent may dffer from the value to agent of recevng the good, whch reduces the usefulness of monetary transfers. For example, f the value to the prncpal of gvng the good to and the value to agent of recevng t are ndependent, then, from the pont of vew of the prncpal, gvng the good to the agent who values t most s the same as a random allocaton. For these reasons, we adopt the opposte assumpton to the standard one: we allow costly verfcaton but do not allow for transfers. We now dscuss some examples of the envronment descrbed above. A frm may need to choose a unt to head up a new, prestgous project. A venture captal frm may need to choose whch of a set of competng startups to fund. A government may need to choose n whch town to locate a new hosptal. A fundng agency may have a grant to allocate to one of several competng researchers. A dean may have a job slot to allocate to one of several departments n the unversty. The head of personnel for an organzaton may need to choose one of several applcants for a job wth a predetermned salary. In each case, the head of the organzaton wshes to allocate a good to that agent who would use t n the way whch best promotes the nterests of the organzaton as a whole. Each agent, on the other hand, has hs own reasons for wantng the good whch may not algn entrely wth the ncentves of the organzaton. For example, each town may want the hosptal and put lttle or no value on havng the hosptal located n a dfferent town. Each unt n the frm may want the resources and prestge assocated wth headng up the new project wthout regard to whether t s the unt whch wll best carry out the project. The venture captal frm wshes to maxmze ts profts, but each of the competng startups desres fundng ndependently of whch startup wll yeld the hghest return. It s also natural to assume that the agents have prvate nformaton relevant to the choce by the head of the organzaton. The researchers know much more about the lkelhood of a breakthrough than the fundng agency. The ndvdual towns know more about the lkely level of use of a hosptal than the central government. The departments know more about the characterstcs of the people they would hre than does the dean. In many of these stuatons, the head of the organzaton can, at a cost, obtan and process some or all of ths nformaton. The fundng agency can nvestgate the research areas and progress to date of some or all of the competng researchers. The government can carry out a careful study of the towns. The frm can audt past performance of a unt and ts current capabltes n detal. The head of personnel can verfy some of the job applcants clams. Fnally, monetary transfers are not practcal or at least not used n many of these 2

5 cases. Frms allocate budgets to ndvdual unts based on what actvtes they want these unts to carry out t would be self defeatng to have unts bd these resources for the rght to head up the new project. Smlarly, t would be odd for a fundng agency to ask researchers to pay n order to receve grants. Governments may ask towns to share n the cost of a hosptal but f part of the purpose of the project s to serve the poor, such transfers would be undermne ths goal. 2 Lterature revew. Townsend (1979) ntated the lterature on the prncpal agent model wth costly state verfcaton. See also Gale and Hellwg (1985), Border and Sobel (1987), and Mookherjee and Png (1989). These models dffer from what we consder n that they nclude only one agent and allow monetary transfers. In ths sense, one can see our work as extendng the costly state verfcaton framework to multple agents when monetary transfers are not possble. Our work s also related to Glazer and Rubnsten (2004, 2006), partcularly the former whch can be nterpreted as model of a prncpal and one agent wth lmted but costless verfcaton and no monetary transfers. Fnally, t s related to the lterature on mechansm desgn and mplementaton wth evdence see Green and Laffont (1986), Bull and Watson (2007), Deneckere and Severnov (2008), Ben-Porath and Lpman (2012), Kartk and Terceux (2012), and Sher and Vohra (2011). Wth the excepton of Sher and Vohra, these papers focus general ssues, rather than on specfc mechansms and allocaton problems. Sher and Vohra do consder a specfc allocaton queston, but t s a barganng problem between a seller and a buyer, very dfferent from what s consdered here. There s a somewhat less related lterature on allocatons wthout transfers but wth costly sgnals (McAfee and McMllan (1992), Hartlne and Roughgarden (2008), Yoon (2011), Condorell (2012), and Chakravarty and Kaplan (2013)). 3 In these papers, agents can waste resources to sgnal ther values and the prncpal s payoff s the value of the type recevng the good less the cost of the wasted resources. The papers dffer n ther assumptons about the cost, the number of goods to allocate, and so on, but the common feature s that wastng resources can be useful n allocatng effcently and that the prncpal may partally gve up on allocatve effcency to save on these resources. See also Ambrus and Egorov (2012) who allow both monetary transfers and wastng of resources n a delegaton model. The remander of the paper s organzed as follows. In the next secton, we present the model. Secton 3 shows that all optmal mechansms are essentally randomzatons 2 In a smlar ven, Banerjee, Hanna, and Mullanathan (2011) gve the example of a government that wshes to allocate free hosptal beds. Ther focus s the possblty that corrupton may emerge n such mechansms where t becomes mpossble for the government to entrely exclude wllngness to pay from playng a role n the allocaton. We do not consder such possbltes here. 3 There s also a large lterature on allocatons wthout transfers, namely the matchng lterature; see, e.g., Roth and Sotomayor (1990) for a classc survey and Abdulkadroglu and Sonmez (2013) for a more recent one. 3

6 over optmal favored agent mechansms. In Secton 4, we characterze the set of best favored agent mechansms. In Secton 5, we gve comparatve statcs and dscuss varous propertes of the optmal mechansm. In Secton 6, we sketch the proof of our unqueness result, whle Secton 7 dscusses some smple extensons. Secton 8 concludes. Proofs not contaned n the text are ether n the Appendx or the Onlne Appendx. 2 Model The set of agents s I = {1,..., I}. There s a sngle ndvsble good to allocate among them. The value to the prncpal of assgnng the object to agent depends on nformaton whch s known only to. Formally, the value to the prncpal of allocatng the good to agent s t where t s prvate nformaton of agent. We normalze so that types are always non negatve and the value to the prncpal of assgnng the object to no one s zero. As we explan n Secton 7, the assumpton that the prncpal always prefers allocatng the object to the agents s used only to smplfy some statements the results easly extend to the case where the prncpal sometmes prefers to keep the object. We assume that the t s are ndependently dstrbuted. The dstrbuton of t has a strctly postve densty f over the nterval 4 T [t, t ] where 0 t < t <. We use F to denote the correspondng dstrbuton functon. Let T = T. The prncpal can check the type of agent at a cost c > 0. We nterpret checkng as obtanng nformaton (e.g., by requestng documentaton, ntervewng the agent, or hrng outsde evaluators) whch perfectly reveals the type of the agent beng checked. The cost to the agent of provdng nformaton s assumed to be zero. We dscuss these assumptons and the extent to whch they can be relaxed n Secton 7. We assume that every agent strctly prefers recevng the object to not recevng t. Consequently, we can take the payoff to an agent to be the probablty he receves the good. The ntensty of the agents preferences plays no role n the analyss, so these ntenstes may or may not be related to the types. 5 We also assume that each agent s 4 It s straghtforward to drop the assumpton of a fnte upper bound for the support as long as all expectatons are fnte. Also, when n Secton 7 we allow the prncpal to prefer keepng the object, t s smlarly straghtforward to drop the assumpton that the support s bounded below. 5 Suppose we let the payoff of agent from recevng the good be ū (t ) and let hs utlty from not recevng t be u (t ) where ū (t ) > u (t ) for all and all t. Then t s smply a renormalzaton to let ū (t ) = 1 and u (t ) = 0 for all t. To see ths, note that each of the ncentve constrants wll take the form pū (t ) + (1 p)u (t ) p ū (t ) + (1 p )u (t ) where p s the probablty player gets the good when he s type t and tells the truth and p s the probablty he gets the good f he les n some partcular fashon. It s easy to rearrange ths equaton as p[ū (t ) u (t )] p [ū (t ) u (t )]. Snce ū (t ) > u (t ), ths holds ff p p, exactly the ncentve 4

7 reservaton utlty s less than or equal to hs utlty from not recevng the good. Snce monetary transfers are not allowed, ths s the worst payoff an agent could receve under a mechansm. Consequently, ndvdual ratonalty constrants do not bnd and so are dsregarded throughout. In ts most general form, a mechansm can be qute complex, havng multple stages of cheap talk statements by the agents and checkng by the prncpal, where who can speak and whch agents are checked depend on past statements and the results from past checks, fnally culmnatng n the allocaton of the good, perhaps to no one. However, t s not hard to show that one can use an argument smlar to the Revelaton Prncple to restrct attenton to a smple class of mechansms. 6 Specfcally, we show n part A of the Onlne Appendx that we can consder only drect mechansms (.e., mechansms whch ask agents to report ther types) for whch truthful revelaton s a Nash equlbrum and whch have the followng propertes. Frst, for any vector of reported types, the mechansm selects (perhaps va randomzaton) at most one agent who s checked. If an agent s checked and (as wll happen n equlbrum) found to have told the truth, then he receves the good. If no agent s checked, then the mechansm (agan, perhaps randomly) selects whch agent, f any, receves the good. Hence we can wrte a mechansm as specfyng for each vector of reports, two probabltes for each agent: the probablty he s awarded the object wthout beng checked and the probablty he s awarded the object condtonal on a successful check. Let p (t) denote the total probablty s assgned the good and q (t) the probablty s assgned the good and checked. Then these functons must satsfy p : T [0, 1], q : T [0, 1], p (t) 1 for all t T, and q (t) p (t) for all I and all t T. Henceforth, the word mechansm wll be used only to denote such a tuple of functons, generally denoted (p, q) for smplcty. The prncpal s objectve functon s E t [ The ncentve compatblty constrant for s (p (t)t q (t)c ) E t p (t) E t [p (t, t ) q (t, t )], t, t T, I. ]. Gven a mechansm (p, q), let ˆp (t ) = E t p (t) constrant we have f ū (t ) = 1 and u (t ) = 0. 6 The usual verson of the Revelaton Prncple does not apply to games wth verfcaton and hence cannot be used to obtan ths concluson. See Townsend (1988) for dscusson and an extenson to a class of verfcaton models whch does not nclude ours. 5

8 and ˆq (t ) = E t q (t). The 2I tuple of functons (ˆp, ˆq) I s the reduced form of the mechansm (p, q). We say that (p 1, q 1 ) and (p 2, q 2 ) are equvalent f ˆp 1 = ˆp 2 and ˆq 1 = ˆq 2 up to sets of measure zero. It s easy to see that we can wrte the ncentve compatblty constrants and the objectve functon of the prncpal as a functon only of the reduced form of the mechansm. Hence f (p 1, q 1 ) s an optmal ncentve compatble mechansm, (p 2, q 2 ) must be as well. Therefore, we can only dentfy the optmal mechansm up to equvalence. 3 The Suffcency of Favored Agent Mechansms Our man result n ths secton s that we can restrct attenton to a class of mechansms we call favored agent mechansms. To be specfc, we show that every optmal mechansm s equvalent to a randomzaton over favored agent mechansms. Hence to compute the set of optmal mechansms, we can smply optmze over the much smpler class of favored agent mechansms. In the next secton, we use ths result to characterze optmal mechansms n more detal. We say that (p, q) s a favored agent mechansm f there exsts a favored agent I and a threshold v R such that the followng holds up to sets of measure zero. Frst, f t c < v for all, then p (t) = 1 and q (t) = 0 for all. That s, f every agent other than the favored agent reports a value t c below the threshold, then the favored agent receves the object and no agent s checked. Second, f there exsts j such that t j c j > v and t c > max k (t k c k ), then p (t) = q (t) = 1 and p k (t) = q k (t) = 0 for all k. That s, f any agent other than the favored agent reports a value above the threshold, then the agent wth the hghest reported value (regardless of whether or not he s the favored agent) s checked and, f hs report s verfed, receves the good. 7 Note that ths s a very smple class of mechansms. Optmzng over ths set of mechansms smply requres us to pck one of the agents to favor and a number for the threshold, as opposed to probablty dstrbutons over checkng and allocaton decsons as a functon of the types. Obvously, any randomzaton over optmal mechansms s optmal. 8 Also, as noted n Secton 2, f a mechansm (p, q) s optmal, then any mechansm whch s essentally the same n the sense of havng the same reduced form up to sets of measure zero must also 7 If there are several agents that maxmze t c, then one of them s chosen arbtrarly. Ths event has probablty zero and so does not affect ncentves or the prncpal s payoff. 8 Randomzng refers to the pontwse convex combnaton of mechansms (p, q). 6

9 be optmal. Hence gven any set of optmal mechansms, we know that all mechansms that are essentally equvalent to a randomzaton over mechansms n ths set must also be optmal. Theorem 1. A mechansm s optmal f and only f t s essentally a randomzaton over optmal favored agent mechansms. Hence we can restrct attenton to favored agent mechansms wthout loss of generalty. 9 Furthermore, f there s a unque optmal favored agent mechansm, then there s essentally a unque optmal mechansm. Secton 6 contans a sketch of the proof of ths result. A very ncomplete ntuton for ths result s the followng. For smplcty, suppose c = c for all and suppose T = [0, 1] for all. Clearly, the prncpal would deally gve the object to the agent wth the hghest t. Of course, ths sn t ncentve compatble as each agent would clam to have type 1. By always checkng the agent wth the hghest report, the prncpal can make ths allocaton of the good ncentve compatble. So suppose the prncpal uses ths mechansm. Consder what happens when the hghest reported type s below c. Obvously, t s better for the prncpal not to check n ths case snce t costs more to check than t could possbly be worth. Thus we can mprove on ths mechansm by only checkng the agent wth the hghest report when that report s above c, gvng the good to no one (and checkng no one) when the hghest report s below c. It s not hard to see that ths mechansm s ncentve compatble and, as noted, an mprovement over the prevous mechansm. However, we can mprove on ths mechansm as well. Obvously, the prncpal could select any agent at random f all the reports are below c and gve the good to that agent. Agan, ths s ncentve compatble. Snce all the types are postve, ths mechansm mproves on the prevous one. The prncpal can do stll better by further explotng hs selecton of the person to gve the good to when all the reports are below c. To see ths, suppose the prncpal gves the good to agent 1 f all reports are below c. Contnue to assume that f any agent reports a type above c, then the prncpal checks the hghest report and gves the good to ths agent f the report s true. Ths mechansm s clearly ncentve compatble. However, the prncpal can also acheve ncentve compatblty and the same allocaton of the good whle savng on checkng costs: he doesn t need to check 1 s report when he s the only agent to report a type above c. To see why ths cheaper mechansm s also ncentve 9 It s straghtforward to show that an optmal mechansm exsts, so Theorem 1 s not vacuous. 7

10 compatble, note that f everyone else s type s below c, 1 gets the good no matter what he says. Hence 1 only cares what happens f at least one other agent s report s above c. In ths case, he wll be checked f he has the hgh report and hence cannot obtan the good by lyng. Hence t s optmal for hm to tell the truth. Ths mechansm s the favored agent mechansm wth 1 as the favored agent and v = 0. Of course, f the prncpal chooses the favored agent and the threshold v optmally, he must mprove on ths payoff. Ths ntuton does not show that some more complex mechansm cannot be superor, so t does not establsh exstence of an optmal mechansm n the favored agent class, much less the unqueness part of Theorem 1. Indeed, the proof of ths theorem s rather complex. Remark 1. It s worth notng that the favored agent mechansm s ex post ncentve compatble. To see ths, note frst that any agent wth t c above the threshold has a domnant strategy to report honestly. Second, any nonfavored agent wth t c below the threshold gets the good wth zero probablty regardless of hs report. Fnally, f the favored agent has t c below the threshold, he obtans the good ff all the other agents report values below the threshold, ndependently of hs report. 10 Snce ex post ncentve compatblty s strcter than ncentve compatblty, ths mples that the favored agent mechansm s also the optmal ex post ncentve compatble mechansm. 4 Optmal Favored Agent Mechansms We complete the specfcaton of the optmal mechansm by characterzng the optmal threshold and the optmal favored agent. We show that condtonal on the selecton of the favored agent, the optmal favored agent mechansm s unque. After characterzng the optmal threshold gven the choce of the favored agent, we consder the optmal selecton of the favored agent. For each, defne t by E(t ) = E(max{t, t }) c. (1) 10 In other words, truth tellng s an almost domnant strategy n the sense that t s an optmal strategy for every type of every player regardless of the strateges of hs opponents. It s not domnant for an agent who s not favored and whose type s such that t c s below the threshold snce hs payoff s zero regardless of hs report. We can make truth tellng a domnant strategy by changng the mechansm off the equlbrum path. Specfcally, suppose we modfy our mechansm only by assumng that f an agent s checked and found to have led, we select another agent at random, check hm, and gve the good to hm ff he s found to have told the truth. It s easy to see that truth tellng s a domnant strategy n ths mechansm and that t generates the same allocaton as the orgnal mechansm. 8

11 It s easy to show that t s well defned. 11 To nterpret t, suppose the prncpal s usng a favored agent mechansm wth as the favored agent. Note that the prncpal does not take nto account s report unless at least one of the other agents reports a value above the threshold. For ntuton, thnk of the prncpal as not even askng for a report unless ths happens. Suppose the profle of reports of the other agents s t. Let v = max j (t j c j ). Suppose the prncpal s not commtted to the threshold and consder hs decson at ths pont. He can choose a threshold above v or, equvalently, gve the object to agent wthout checkng hm. If he does so, hs expected payoff s E(t ). Alternatvely, he can choose a threshold below v or, equvalently, ask for a report from agent and gve the object to the agent wth the hghest value of t j c j after a check. In expectaton, ths yelds the prncpal E max{t c, v}. Hence equaton (1) says that when v = t c, the prncpal s ndfferent between these two optons. A slght extenson of ths reasonng yelds a proof of the followng result. Theorem 2. Wthn the set of favored agent mechansms wth as the favored agent, the unque optmal mechansm s obtaned by settng the threshold v equal to t c. Proof. For notatonal convenence, let the favored agent equal 1. Contrast the prncpal s payoff to thresholds t 1 c 1 and ˆv > t 1 c 1. Gven a profle of types for the agents other than 1, let x = max j 1 (t j c j ) that s, the hghest value of (and hence reported by) one of the other agents. Then the prncpal s payoff as a functon of the threshold and x s gven by x < t 1 c 1 < ˆv t 1 c 1 < x < ˆv t 1 c 1 < ˆv < x t 1 c 1 E(t 1 ) E max{t 1 c 1, x} E max{t 1 c 1, x} ˆv E(t 1 ) E(t 1 ) E max{t 1 c 1, x} To see ths, note that f x < t 1 c 1 < ˆv, then the prncpal gves the object to agent 1 wthout a check usng ether threshold. If t 1 c 1 < ˆv < x, then the prncpal gves the object to ether 1 or the hghest of the other agents wth a check and so receves a payoff of ether t 1 c 1 or x, whchever s larger. Fnally, f t 1 c 1 < x < ˆv, then wth threshold t 1 c 1, the prncpal s payoff s the larger of t 1 c 1 and x, whle wth threshold ˆv, she gves the object to agent 1 wthout a check and has payoff E(t 1 ). 11 To see ths, observe that the rght hand sde of equaton (1) s contnuous and strctly ncreasng n t for t t, below the left hand sde at t = t, and above t as t. Hence there s a unque soluton. Note that f we allowed t, then when c = 0, we would have t = t. (At c = 0, t s not unquely defned, but t s natural to take t to be the lmt value as c 0 whch gves t = t.) Ths fact together wth what we show below mples the unsurprsng observaton that f all the costs are zero, the prncpal always checks the agent who receves the object and gets the same payoff as under complete nformaton. Note also that f c s very large, we can have t > t. If so, then t = E(t ) + c > t. 9

12 Recall that t 1 > t 1. Hence t 1 < t 1 wth strctly postve probablty. Therefore, for x > t 1 c 1, we have E max{t 1 c 1, x} > E max{t 1 c 1, t 1 c 1 }. But the rght hand sde s E max{t 1, t 1} c 1 whch equals E(t 1 ) by defnton of t. Thus E max{t 1 c 1, x} > E(t 1 ). Hence gven that 1 s the favored agent, the threshold t 1 c 1 weakly domnates any larger threshold. A smlar argument shows that the threshold t 1 c 1 weakly domnates any smaller threshold, establshng that t s optmal. To see that the optmal mechansm n ths class s unque, note that the comparson of threshold t 1 c 1 to a larger threshold v s strct unless the mddle column of the table above has zero probablty. That s, the only stuaton n whch the prncpal s ndfferent between the threshold t 1 c 1 and the larger threshold v s when the allocaton of the good and checkng decsons are the same wth probablty 1 gven ether threshold. That s, ndfference occurs only when changes n the threshold do not change (p, q). Hence there s a unque best mechansm n F. Whle t resembles an ndex or reservaton value of the sort often seen n the search lterature, ths specfc defnton s not standard n that lterature. Indeed, t s not straghtforward to nterpret t n terms of search. To see the pont, compare t to the ndex dentfed n Wetzman (1979). In hs model (wth some mnor adjustments for easer comparablty), the value to learnng the payoff to opton s characterzed by a crtcal value t defned by t = E(max{t, t }) c where c s the cost of fndng out the payoff of opton and t s the random varable gvng the value of ths payoff. Note that ths s dentcal to our expresson, except that E(t ) appears on the left hand sde of equaton (1), not t. Wetzman s expresson s easly nterpreted. If the best alternatve found so far has value t, then the agent s ndfferent between stoppng hs search and choosng t versus checkng opton and then stoppng. Our expresson s not as obvously nterpreted n terms of search. 12 Now that the optmal threshold s characterzed gven the choce of the favored agent, t remans only to characterze the optmal favored agent. 12 Snce the frst draft of ths paper, Doval (2013) consdered a search model where ths defnton does appear. In Doval, the searcher could choose opton wthout checkng t, yeldng payoff E(t ). Thus t, computed accordng to our defnton, emerges as the cutoff for the best opton found so far wth the property that t leaves the agent ndfferent between takng the last opton wthout checkng t and checkng the last opton and then choosng between t and the outsde opton. To the best of our knowledge, ths s the frst tme ths defnton has appeared n the search lterature. 10

13 Theorem 3. The optmal choce of the favored agent s any wth t c = max j (t j c j ). A rough ntuton for ths result s that t c can be thought of as the standard that agents j must satsfy to persuade the prncpal to not gve the good to wthout checkng anyone. Intutvely, the hgher s ths threshold, the more nclned the prncpal s to gve the good to wthout checkng anyone. It s not surprsng that the agent towards whom the prncpal s most nclned n ths sense s the prncpal s choce for the favored agent. We sketch the man part of the proof here (see part C of the Onlne Appendx for the omtted detals). Fx any two agents, and j, and assume t c t j c j. We wll show that ths nequalty mples that gven a threshold of t j c j, t s weakly better for the prncpal to favor agent than agent j. By Theorem 2, favorng wth a threshold of t c s better stll, establshng the result. If any agent k other than or j reports t k c k above the threshold, then the agent wth the hghest report s checked and receves the object, ndependent of whch agent was favored. Hence we may as well condton on the event that all agents other than and j report values below the threshold. Also, f both and j report values above the threshold, agan, t does not matter to the prncpal whch agent was favored. Hence we only need to consder realzatons such that at least one of these agents reports a value below the threshold. Note that j s report s above the threshold f t j c j t j c j.e., t j t j. On the other hand, s report s above the threshold f t c t j c j or t ˆt t j c j + c. Gven ths, we see that the prncpal s better off favorng than j f Rewrtng, F (ˆt )F j (t j)e[t t ˆt ] + [1 F (ˆt )]F j (t j)e[t t > ˆt ] + F (ˆt )[1 F j (t j)] { E[t j t j > t j] c j } F (ˆt )F j (t j)e[t j t j t j] + [1 F (ˆt )]F j (t j) { E[t t > ˆt ] c } + F (ˆt )[1 F j (t j)]e[t j t j > t j]. F (ˆt )F j (t j) ( E[t t ˆt ] E[t j t j t j] ) + [1 F (ˆt )]F j (t j)c F (ˆt )[1 F j (t j)]c j 0. Ths equaton summarzes the change n the prncpal s payoff from swtchng from j as the favored agent to. The frst term s the change n the expected payoff condtonal on gvng the object to the favored agent wthout a check, whle the last two terms gve the change n the expected costs of checkng. It s easy to show that ths must hold f F (ˆt ) = 0, so assume F (ˆt ) > 0. It follows 11

14 from the defnton of t j that F j (t j) > 0, so we can rewrte ths as E[t t ˆt ] + We now show that ths s mpled by t c t j c j. Recall that t s defned by c F (ˆt ) c E[t j t j t j] + c j F j (t j ) c j. (2) E(t ) = E max{t, t } c, so t c t j c j or, equvalently, t ˆt mples E(t ) = E max{t, t } c E max{t, ˆt } c or ˆt or Hence t E[t t ˆt ] + E[t t ˆt ] + t f (t ) dt F (ˆt )ˆt c c F (ˆt ) ˆt = t j c j + c. c F (ˆt ) c t j c j. But the same rearrangng of the defnton of t j shows that t j c j = E[t j t j t j] + c j F j (t j ) c j. Combnng the last two nequaltes yelds equaton (2) as asserted. Summarzng, we see that the set of optmal favored agent mechansms s easly characterzed. A favored agent mechansm s optmal f and only f the favored agent satsfes argmax j (t j c j ) and the threshold v satsfes v = max j (t j c j ). Thus the set of optmal mechansms s equvalent to pckng a favored agent mechansm wth threshold v = max j (t j c j ) and randomzng over whch of the agents n argmax j (t j c j ) to favor. Clearly for generc checkng costs, there wll be a unque j maxmzng t j c j and hence a unque optmal mechansm. Moreover, fxng c and c j, the set of (F, F j ) such that t c = t j c j s nowhere dense n the product weak* topology. Hence n ether sense, such tes are non generc We thank Y-Chun Chen and Syang Xong for showng us a proof of ths result. 12

15 5 Propertes of Optmal Mechansms Gven that optmal mechansms are favored agent mechansms, t s easy to compare outcomes under the optmal mechansm to the frst best. For smplcty, suppose there are two agents and that the optmal mechansm favors agent 1 and has threshold v. Then there are exactly three ways the outcome can be nferor for the prncpal to the frst best. Frst, f t 2 c 2 < v but t 2 > t 1, then agent 1 receves the object, even though the prncpal would be better off gvng t to agent 2. Second, f t 2 c 2 > v, then the prncpal ends up checkng one of the agents, a cost he would avod n the frst best. Fnally, f t 2 c 2 > v, the good could stll go to the wrong agent relatve to the frst best. In partcular, f t 1 > t 2 but t 2 c 2 > t 1 c 1, then agent 2 wll receve the good even though the prncpal prefers agent 1 to have t and smlarly f we reverse the roles of 1 and 2. Also, our characterzaton of the optmal favored agent and threshold makes t easy to compute optmal mechansms and analyze ther propertes. Consder the followng example. There are two agents. Agent 1 s cost of beng checked s large n the sense that c 1 > t 1 E(t 1 ). As dscussed n footnote 11, ths mples t 1 = E(t 1 ) + c 1. For concreteness, assume E(t 1 ) = 1. Suppose c 2 = ε where ε > 0 but very small. Fnally, assume t 2 s unformly dstrbuted over [.99, 1.99]. It s easy to see that as ε 0, we have t 2 c 2 t 2 =.99 < 1 = t 1 c 1. Hence for ε suffcently small, 1 wll be the favored agent and the threshold v wll equal 1. However, suppose 2 reports t 2 c 2 > v = 1. Note that t 1 c 1 < t 1 c 1 < E(t 1 ) = 1. Thus f 2 s above the threshold, he receves the good for sure. Therefore, 1, even though he s favored, only receves the good f t 2 c 2 < 1, or t 2 < 1 + ε. Recall t 2 U[.99, 1.99], so for ε small, 1 receves the good slghtly more than 1% of the tme, even though he s favored. Note that we have assumed very lttle about the dstrbuton of t 1, so t could well be true that 1 s very lkely to have the hgher type. 14 Ths example hghlghts the fact that the favored agent s not necessarly the agent wth the hghest probablty of recevng the good, even condtonal on hs type. So n what sense s the favored agent favored? Compare a favored agent mechansm wth favored agent and threshold v to a favored agent mechansm wth j favored and threshold vj. Then for any v, vj, and j, agent (at least weakly) prefers the former mechansm to the latter. That s, t s always better to be favored than not. To see ths, smply note that n the mechansm where j s favored, a necessary condton for to receve the good s that t c t k c k for all k. Ths s true because can only receve the good f he or some other agent s above the threshold and s value s the hghest. However, n the mechansm where s 14 We thank an anonymous referee for rasng ths ssue wth a smlar example. 13

16 the favored agent, a suffcent condton for to receve the good s that t c t k c k for all k. Thus must weakly prefer beng favored and typcally wll strctly prefer t. For comparatve statcs, t s useful to gve an equvalent defnton of t. Our orgnal defnton can be rewrtten as or t t c = t F (t ) So an equvalent defnton of t s t f (t ) dt = t F (t ) c t t t f (t ) dt = t t F (τ) dτ. t t F (τ) dτ = c. (3) From (3), t s easy to see that an ncrease n c ncreases t. Also, from our frst defnton of t, note that t c s that value of v solvng E(t ) = E max{t c, v }. Obvously for fxed v, the rght hand sde s decreasng n c, so t c must be ncreasng n c. Hence, all else equal, the hgher s c, the more lkely s to be selected as the favored agent. To see the ntuton, note that f c s larger, then the prncpal s less wllng to check agent s report. Snce the agent who s favored s the one the prncpal checks least often, ths makes t more desrable to favor. It s also easy to see that a frst order or second order stochastc domnance shft upward n F reduces the left hand sde of equaton (3) for fxed t, so to mantan the equalty, t must ncrease. Therefore, such a shft makes t more lkely that s the favored agent and ncreases the threshold n ths case. Hence both better (FOSD) and less rsky (SOSD) agents are more lkely to be favored. The ntuton for the effect of a frst order stochastc domnance ncrease n t s clear. If agent s more lkely to have hgh type, he s a better choce to be the favored agent. The ntuton for why less rsky agents are favored s that there s less beneft from checkng f there s less uncertanty about hs type. Fnally, equaton (3) shows that f we change F only at values of t larger than t, then t s unaffected. In ths sense, the optmal favored agent and the optmal threshold are ndependent of the upper tals of the dstrbutons of the t s. Intutvely, ths s because the choce of threshold condtonal on favorng agent s based on comparng E(t ) versus E(max{t c, v}) where v = max j (t j c j ), for the reasons explaned n Secton 4. Changng the probabltes for very hgh values of t affects both parts of ths comparson symmetrcally and hence are rrelevant. 14

17 Now that we have shown how changes n the parameters affect the optmal mechansm, we turn to how these changes affect the payoffs of the prncpal and agents. Frst, consder changes n the realzed type vector. Obvously, an ncrease n t ncreases agent s probablty of recevng the good and thus hs ex post payoff. Therefore, hs ex ante payoff ncreases wth an FOSD shft upward n F. Smlarly, the ex post payoffs of other agents are decreasng n t, so ther ex ante payoffs decrease wth an FOSD shft upward n F. However, the prncpal s ex post payoff does not necessarly ncrease as an agent s type ncreases: f at some profle, the favored agent s recevng the good wthout beng checked, an ncrease n another agent s type mght result n the same allocaton but wth costly verfcaton. 15 Nevertheless, an FOSD ncrease n any F does ncrease the prncpal s ex ante payoff. To see ths, suppose t has dstrbuton functon F and ˆt has dstrbuton functon ˆF where ˆF domnates F n the sense of FOSD. It s not hard to see that F 1 ( ˆF (ˆt )) has the same dstrbuton as t. So suppose after changng the dstrbuton of s type from F to ˆF, the prncpal uses the same mechansm as he used before the shft, but converts s report to keep the dstrbuton of s reports unchanged. That s, f reports ˆt, the prncpal treats ths as a report of F 1 ( ˆF (ˆt )). Gven ths, we see that the prncpal s payoff s affected by the shft n the dstrbuton only f he gves the good to agent. In ths case, he would have receved F 1 ( ˆF (ˆt )) under the orgnal dstrbuton, but receves ˆt nstead. Snce F (ˆt ) ˆF (ˆt ) by FOSD, we see that ˆt F 1 ( ˆF (ˆt )), so the prncpal s better off. 16 Turnng to the effect of changes n c, t s obvous that a decrease n c makes the prncpal better off as he could use the same mechansm and save on costs. It s also easy to see that f agent s not favored, then ncreases n c make hm worse off and make all other agents better off, as long as the ncrease n c does not change the dentty of the favored agent. Ths s true smply because receves the good ff t c s large enough, so a hgher c makes less lkely to receve the good and other agents more lkely to do so. On the other hand, changes n the cost of checkng the favored agent have ambguous effects n general. Ths s true because t c s ncreasng n c. Hence f the cost of the favored agent ncreases, all other agents are less lkely to be above the threshold. Ths effect makes the favored agent better off and the other agents worse off. However, t s also true that f t j c j s above the threshold, then t s the comparson of t j c j to t c that matters. Clearly, an ncrease n c makes ths comparson worse for the favored agent and better for j. The total effect can be postve or negatve for the favored agent. For 15 For example, f 1 s the favored agent and t satsfes t 1 > t 1 and t c < t 1 c 1 for all 1, the payoff to the prncpal s t 1. If t 2, say, ncreases to t 2 such that t 1 c 1 < t 2 c 2 < t 1 c 1, then the prncpal s payoff falls to t 1 c We thank Andy Skrzypacz for suggestng ths approach. 15

18 example, f I = 2 and F 1 = F 2 = F, then the favored agent benefts from an ncrease n hs cost of beng checked f the densty f s ncreasng and conversely f t s decreasng; see Onlne Appendx D. In short, every agent has an ncentve to ncrease hs cost of beng checked f ths can make hm the favored agent and, dependng on the densty, the favored agent may have ncentves to ncrease hs cost of beng checked even beyond that pont. Clearly, such an ncentve s potentally costly for the prncpal. 6 Proof Sketch for Theorem 1 Mechansm desgn problems wthout transfers are very dffcult to solve n general snce the usual ntegral characterzaton of feasble allocaton rules s unavalable. As we explan n ths secton, n our case, the structure of the problem provdes some useful smplfcatons whch allow for a complete characterzaton of optmal mechansms. The frst step s to rewrte the optmzaton problem. Recall that ˆp (t ) = E t p (t, t ) and ˆq (t ) = E t q (t, t ). We can wrte the ncentve compatblty constrant as ˆp (t ) ˆp (t ) ˆq (t ), t, t T. That s, the payoff to type t from tellng the truth exceeds the payoff to clamng to be type t. The unusual property of our ncentve compatblty constrant s that the payoff to falsely clamng to be type t does not depend on the true type t. We have ths structure because any le s caught ff the agent s checked and because we can normalze so that an agent s payoffs to recevng or not recevng the good do not depend on hs type. Normally, one rearranges the ncentve constrants to say that for each type, tellng the truth s better than the best possble le. Because the payoff to lyng does not depend on the truth, we can rearrange the ncentve constrant to say that the worst truth s better than any le. In other words, a mechansm s ncentve compatble f and only f t satsfes nf ˆp (t ) ˆp (t ) ˆq (t ), t T. t T Lettng ϕ = nf t T ˆp (t ), we can rewrte the ncentve compatblty constrant as ˆq (t ) ˆp (t ) ϕ, t T. Because the objectve functon s strctly decreasng n ˆq (t ), ths constrant must bnd, so ˆq (t ) = ˆp (t ) ϕ. (4) 16

19 We can substtute ths result nto the objectve functon and rewrte t as [ E t p (t)t ] c q (t) = E t [ˆp (t )t c ˆq (t )] = E t [ˆp (t )(t c ) + ϕ c ] (5) [ ] = E t [p (t)(t c ) + ϕ c ]. (6) Some of the arguments below wll use the reduced form probabltes and hence rely on the frst expresson, (5), for the payoff functon, whle others focus on the nonreduced mechansm and so rely on the second expresson, (6). From ths pont forward, we treat the prncpal s problem as choosng the ϕ s and p s subject to the constrants that the p s be well defned probabltes and the constrants mpled by ϕ = nf t ˆp (t ). Gven ths, equaton (6) expresses the key tradeoff n the model. Recall that ϕ s the mnmum probablty for any type of to receve the good. Snce the prncpal only needs to check t often enough that a fake clam of t succeeds wth probablty equal to ths mnmum, a hgher value of ϕ means that the prncpal does not have to check agent as often, as shown by equaton (4). Thus, as the payoff functon (6) shows, ths helps the prncpal n proporton to the cost c that he saves. However, ths puts a more severe constrant on ˆp, thus tendng to force a less effcent allocaton of the good. Next, we gve a partal characterzaton of the optmal mechansm takng the ϕ s as gven and optmzng over the p s. Ths partal characterzaton enables us to reduce the problem to choosng the ϕ s and one other varable whch wll turn out to be the threshold. Thus n what follows, we fx ϕ [0, 1] for each and characterze the soluton to what we wll call the relaxed problem of maxmzng (6) by the choce of functons p : T [0, 1] for I subject to p (t) 1 for all t and E t p (t) ϕ for all t and all. 17 Note that snce the ϕ s are fxed, we can take the objectve functon to be [ ] E t ˆp (t )(t c ) = E t p (t)(t c ). We show below that every optmal soluton to the relaxed problem s what we call a threshold mechansm. Specfcally, every soluton has a threshold v wth the followng 17 The constrant set s nonempty ff ϕ 1. It s not hard to show that the soluton to the overall problem wll necessarly satsfy ths and so we assume t n what follows. 17

20 propertes. Frst, f t c < v, then ˆp (t ) = ϕ. Second, for any t such that some agent has t c > v, the agent wth the hghest t c receves the good wth probablty 1. As we show below, for an approprate selecton of the ϕ s, a threshold mechansm s equvalent to a favored agent mechansm. Theorem 4. Every soluton to the relaxed problem s a threshold mechansm. To see the ntuton for ths, frst note that the soluton to the relaxed problem would be trval f we dd not have the constrant that E t p (t) ϕ. Wthout ths constrant, the relaxed problem s equvalent to allocatng the good where the prncpal receves t c f he allocates the good to. Hence the soluton would be p (t) = 0 for all f t < c for all and otherwse p (t) = 1 for that such that t c = max k (t k c k ). Typcally, ths soluton wll volate the lower bound constrant on ˆp (t ). For example, f t < c, ths soluton would have ˆp (t ) = 0, so the constrant could only be satsfed f ϕ = 0. Thus the constrant forces the prncpal to sometmes allocate the good to an agent wth t < c or to an agent who does not have the hghest value of t c. When should the prncpal devate from gvng the good to the agent wth the hghest value of t c n order to satsfy the constrant? Intutvely, t s obvous that the prncpal should do ths only when max (t c ) s relatvely small. That s, t s natural that there should be a threshold for max (t c ) such that the prncpal allocates the good to the agent wth the hghest t c when ths threshold s met. It s also natural that f the prncpal does not want to allocate the good to a partcular type of a partcular agent because t c s relatvely small, then he gves t to that agent wth the smallest probablty allowed by the constrants. It s not as mmedately ntutve that the same threshold should apply to every for ths second statement or that ths threshold should also apply to the frst statement, but ths s the content of Theorem 4. We sketch the proof of Theorem 4 below, but frst explan how ths result enables us to complete the proof of Theorem 1. Theorem 4 mples that ˆp s completely pnned down as a functon of v and the ϕ s. Specfcally, f t c > v, then ˆp (t ) must be the probablty that t c > max j (t j c j ). If t < t, then ˆp (t ) = ϕ. By substtutng nto equaton (5), we can wrte the prncpal s payoff as a functon only of v and the ϕ s. Note that Theorem 4 does not say that any v s consstent wth gven ϕ s. To see why, note that the theorem pns down ˆp (t ) as a functon of v for any t > v + c. But f ths mpled value of ˆp (t ) s below ϕ, then the threshold v s not feasble gven the ϕ s. It turns out to be more convenent to fx the threshold v and ask whch ϕ s are consstent wth t. We show n Appendx A (Lemma 1) that for any v, the set of consstent ϕ s s convex. It s also easy to show that the objectve functon of the prncpal, holdng v fxed, s lnear n the ϕ s. Hence gven v, there must be a soluton 18

21 to the prncpal s optmzaton problem at an extreme pont of the set of consstent ϕ s. Furthermore, every optmal choce of the ϕ s s a randomzaton over optmal extreme ponts. The last step s to show that the optmal extreme ponts correspond to favored agent mechansms. Frst, we show that extreme ponts take on one of two forms. In one type of extreme pont, all but one of the ϕ s s set to zero and the remanng one s as large as possble. For convenence, consder the extreme pont where ϕ j = 0 for all j 1 and ϕ 1 s set as hgh as possble. In ths case, we have a favored agent mechansm where 1 s the favored agent and v s the threshold. To see ths, frst observe that the allocaton probabltes match ths. If every agent other than the favored agent has t j c j < v, then each of these agents receves the good wth probablty ϕ j = 0. It s not hard to show that makng ϕ 1 as large as possble entals gvng the good to agent 1 n ths stuaton, whether or not t 1 c 1 s above v. If some agent j 1 has t j c j > v, then he receves the good f and only f he has the hghest value of t j c j, just as requred by the favored agent mechansm. Equaton (4) can be used to show that the checkng probabltes, up to equvalence, are as requred for a favored agent mechansm. The other type of extreme pont s where ϕ = 0 for all. One way ths can come about s f there are two agents, say and j, wth t c > v and t j c j > v wth probablty 1. In ths case, we have the same mechansm as a favored agent mechansm wth threshold v and any agent selected as the favored agent. Snce we always have more than one agent above the threshold, the dentty of the favored agent s rrelevant as the mechansm always pcks the agent wth the largest t c, checks hm, and gves hm the good after a successful check. There s one other way to have an extreme pont wth ϕ = 0 for all. Ths occurs when every agent has a strctly postve probablty of beng below the threshold. Recall that Theorem 4 mples that f t c < v, then ˆp (t ) = ϕ. Hence ths mechansm has the property that for any t such that all agents have t c < v, no agent receves the good. It s easy to use our assumpton that the value of the good to the prncpal s zero and that types are postve wth probablty 1 to show that ths extreme pont cannot be optmal. 18 We now sketch the proof of Theorem 4. The proof of ths theorem has some techncal complcatons because of our use of a contnuum of types. The contnuum has the sgnfcant advantage that t makes the statement of the optmal mechansm much cleaner, whle we would have messy specfcatons of the mechansm at certan boundary types f we assumed fntely many types nstead. On the other hand, the contnuum of types assumpton ntroduces measurablty consderatons and makes the argument more com- 18 As we dscuss n Secton 7, f the prncpal has a strctly postve value for keepng the good, then ths extreme pont can be optmal. 19