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 Draft August We thank Rcky Vohra and numerous semnar audences for helpful comments. We also thank the Natonal Scence Foundaton, grants 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 ths draft was n progress. 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 (dean) has an object (job slot) to allocate to one of I agents (departments). Each agent has a strctly postve value for recevng the object. Each agent also has prvate nformaton whch determnes the value to the prncpal of gvng the object to hm. There are no monetary transfers but the prncpal can check the value assocated wth any ndvdual at a cost whch may vary across ndvduals. We characterze the class of optmal Bayesan mechansms, that s, mechansms whch maxmze the expected value to the prncpal from hs assgnment of the good mnus the costs of checkng values. One partcularly smple mechansm n ths class, whch we call the favored agent mechansm, specfes a threshold value v and a favored 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 wth probablty 1 and receves the good ff her report s confrmed. We show that all optmal mechansms are essentally randomzatons over optmal favored agent mechansms.

3 1 Introducton Consder the problem of the head of an organzaton say, a dean who has an ndvsble resource or good (say, a job slot) that can be allocated to any of several dvsons (departments) wthn the organzaton (unversty). Naturally, the dean wshes to allocate ths slot to that department whch would fll the poston n the way whch best promotes the nterests of the unversty as a whole. Each department, on the other hand, would lke to hre n ts own department and puts less, perhaps no, value on hres n other departments. The problem faced by the dean s made more complex by the fact that each department has much more nformaton regardng the avalablty of promsng canddates and the lkelhood that these canddates wll produce valuable research, teach well, and more generally be of value to the unversty. The standard mechansm desgn approach to ths stuaton would construct a mechansm whereby each department would report ts type to the dean. Then the slot would be allocated and varous monetary transfers made as a functon of these reports. The drawback of ths approach s that the monetary transfers between the dean and the departments are assumed to have no effcency consequences. In realty, the monetary resources each department has s presumably chosen by the dean n order to ensure that the department can acheve certan goals the dean sees as mportant. To take back such funds as part of an allocaton of a job slot would undermne the approprate allocaton of these resources. In other words, such monetary transfers are part of the overall allocaton of all resources wthn the unversty and hence do have mportant effcency consequences. We focus on the admttedly extreme case where no monetary transfers are possble at all. Of course, wthout some means to ensure ncentve compatblty, the dean cannot extract any nformaton from the departments. In many stuatons, t s natural to assume that the head of the organzaton can demand to see documentaton whch proves that the dvson or department s clams are correct. Processng such nformaton s costly, though, to the dean and departments and so t s optmal to restrct such nformaton requests to the mnmum possble. Smlar problems arse n areas other than organzatonal economcs. For example, governments allocate varous goods or subsdes whch are ntended not for those wllng and able to pay the most but for the opposte group. Hence allocaton mechansms based on auctons or smlar approaches cannot acheve the government s goal, often leadng to the use of mechansms whch rely nstead on some form of verfcaton nstead. 1 1 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. 1

4 As another example, consder the problem of choosng whch of a set of job applcants to hre for a job wth a predetermned salary. Each applcant wants the job and presents clams about hs qualfcatons for the job. The person n charge of hrng can verfy these clams but dong so s costly. 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 documentaton s requred. If some agent other than reports a value above the threshold, then the agent who reports the hghest value s requred to document hs clams. 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 reduced form (see Secton 2 for defnton) as such a randomzaton up to sets of measure zero. 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, and the less rsky s hs dstrbuton of values. We also show that the mechansm s, n a sense, almost a domnant strategy mechansm and consequently s ex post ncentve compatble. Lterature revew. Townsend (1979) ntated the lterature on the prncpal agent model wth costly state verfcaton. 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. See also Gale and Hellwg (1985), Border and Sobel (1987), and Mookherjee and Png (1989). 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 (2011), Kartk and Terceux (2011), and Sher and Vohra (2011). Wth the excepton of Sher and Vohra, these papers focus more on general ssues of mechansm desgn and mplementaton n these envronments rather than on specfc mechansms and allocaton problems. Sher and Vohra do consder a specfc allocaton 2

5 queston, but t s a barganng problem between a seller and a buyer, very dfferent from what s consdered here. The remander of the paper s organzed as follows. In the next secton, we present the model. Secton 3 contans the characterzaton of the class of optmal compatble mechansms, showng all optmal mechansms are essentally randomzatons over optmal favored agent mechansms. Snce these results show that we can restrct attenton to favored agent mechansms, we turn n Secton 4 to characterzng the set of best mechansms n ths class. In Secton 5, we gve comparatve statcs and some examples. In Secton 6, we sketch the proof of our unqueness result and dscuss several other ssues. Secton 7 concludes. Proofs not contaned n the text are n the 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. The value to the prncpal of assgnng the object to no one s normalzed to zero. We assume that the t s are ndependently dstrbuted. The dstrbuton of t has a strctly postve densty f over the nterval T [t, t ] where 0 t < t <. (All results extend to allowng the support to be unbounded above.) 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 requrng documentaton by agent to demonstrate what hs type s. If the prncpal checks some agent, she learns that agent s type. The cost c s nterpreted as the drect cost to the prncpal of revewng the nformaton provded plus the costs to the prncpal assocated wth the resource cost to the agent of provdng ths documentaton. The cost to the agent of provdng documentaton s zero. To understand ths, thnk of the agent s resources as allocated to actvtes whch are ether drectly productve for the prncpal or whch provde nformaton for checkng clams. The agent s ndfferent over how these resources are used snce they wll all be used regardless. Thus by drectng the agent to spend resources on provdng nformaton, the prncpal loses some output the agent would have produced wth the resources otherwse whle the agent s utlty s unaffected. 2 In Secton 6, we show one way to generalze our model to allow agents to bear some costs 2 One reason ths assumpton s a convenent smplfcaton s that droppng t allows a back door for transfers. If agents bear costs of provdng documentaton, then the prncpal can use threats to requre documentaton as a way of fnng agents and thus to help acheve ncentve compatblty. Ths both complcates the analyss and ndrectly ntroduces a form of the transfers we wsh to exclude. 3

6 of provdng documentaton whch does not change our results qualtatvely. 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. 3 We also assume that each agent s 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, allowng the prncpal to decde whch agents to check as a functon of the outcome of prevous checks and cheap talk statements for multple stages before fnally allocatng the good or decdng to not allocate t at all. Wthout loss of generalty, we can restrct attenton to truth tellng equlbra of mechansms where each agent sends a report of hs type to the prncpal who s commtted to (1) a probablty dstrbuton over whch agents (f any) are checked as a functon of the reports and (2) a probablty dstrbuton over whch agent (f any) receves the good as a functon of the reports and the outcome of checkng. Whle ths does not follow from the usual Revelaton Prncple drectly (as the usual verson does not apply to games wth verfcaton), the argument s smlar. Fx a dynamc mechansm and any equlbrum. The equlbrum defnes a functon from type profles nto probablty dstrbutons over outcomes. More specfcally, an outcome s a sequence of checks and an allocaton of the good (perhaps to no one). Replace ths mechansm wth a drect mechansm where agents report types and the outcome (or dstrbuton over outcomes) gven a vector of type reports s what would happen n the equlbrum f ths report were true. Clearly, just as n the usual Revelaton Prncple, truth tellng s an equlbrum of ths mechansm and ths equlbrum yelds the same outcome as the orgnal equlbrum n the dynamc mechansm. We can replace any outcome whch s a sequence of checks wth an outcome where exactly these checks are done smultaneously. All agents and the prncpal are ndfferent between these two outcomes. Hence the altered form of the mechansm where we change outcomes n ths way also has a truth tellng equlbrum and yelds an outcome whch s just as good for the prncpal as the orgnal equlbrum of the dynamc mechansm. Gven that we focus on truth tellng equlbra, all stuatons n whch agent s report s checked and found to be false are off the equlbrum path. The specfcaton of the mechansm for such a stuaton cannot affect the ncentves of any agent j snce agent j wll expect s report to be truthful. Thus the specfcaton only affects agent 3 In other words, suppose we let the payoff of 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. 4

7 s ncentves to be truthful. Snce we want to have the strongest possble ncentves to report truthfully, we may as well assume that f s report s checked and found to be false, then the good s gven to agent wth probablty 0. Hence we can further reduce the complexty of a mechansm to specfy whch agents are checked and whch agent receves the good as a functon of the reports, where the latter apples only when the checked reports are accurate. Fnally, t s not hard to see that any agent s ncentve to reveal hs type s unaffected by the possblty of beng checked n stuatons where he does not receve the object regardless of the outcome of the check. That s, f an agent s report s checked even when he would not receve the object f found to have told the truth, hs ncentves to report honestly are not affected. Snce checkng s costly for the prncpal, ths means that f the prncpal checks an agent, then (f he s found to have been honest), he must receve the object wth probablty 1. Therefore, we can thnk of the mechansm as specfyng 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. So a mechansm s a 2I tuple of functons, (p, q ) I where 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 [ (p (t)t q (t)c ) The ncentve compatblty constrant for s then E t p (t) E t î p (ˆt, t ) q (ˆt, t ) ó, ˆt, t T, I. ]. and Gven a mechansm (p, q), let ˆp (t ) = E t p (t) 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. 5

8 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 more specfc, frst we show that there s always a favored agent mechansm whch s an optmal mechansm. Second, we show that every Bayesan 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. To be more precse, 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 he s the favored agent or not) s checked and, f hs report s verfed, receves the good. 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 2I probablty dstrbutons. Theorem 1. There always exsts a Bayesan optmal mechansm whch s a favored agent mechansm. 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 ths s a feasble mechansm. Suppose the hghest reported type s below c. Obvously, t s better for the prncpal to 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. 6

9 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 selectng the best person to gve the good to when all the reports are below c. To thnk more about 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 compatble, note that f everyone else s type s below c, 1 gets the good no matter what he says. Hence he 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 s far from a proof. Indeed, we prove ths result as a corollary to the next theorem, a result whose proof s rather complex. Let F denote the set of favored agent mechansms and let F denote the set of optmal favored agent mechansms. By Theorem 1, f a favored agent mechansm s better for the prncpal than every other favored agent mechansm, then t must be better for the prncpal than every other ncentve compatble mechansm, whether n the favored agent class or not. Hence every mechansm n F s an optmal mechansm even wthout the restrcton to favored agent mechansms. Gven two mechansms, (p 1, q 1 ) and (p 2, q 2 ) and a number λ (0, 1), we can construct a new mechansm, say (p λ, q λ ), by (p λ, q λ ) = λ(p 1, q 1 ) + (1 λ)(p 2, q 2 ), where the rght hand sde refers to the pontwse convex combnaton of these functons. The mechansm (p λ, q λ ) s naturally nterpreted as a random choce by the prncpal between the mechansms (p 1, q 1 ) and (p 2, q 2 ). It s easy to see that f (p k, q k ) s ncentve compatble for k = 1, 2, then (p λ, q λ ) s ncentve compatble. Also, the prncpal s payoff s lnear n (p, q), so f both (p 1, q 1 ) and (p 2, q 2 ) are optmal for the prncpal, t must be true that (p λ, q λ ) s optmal for the prncpal. It s easy to see that ths mples that any mechansm n the convex hull of F, denoted conv(f ), s optmal. 7

10 Fnally, as noted n Secton 2, f a mechansm (p, q) s optmal, then any mechansm equvalent to t n the sense of havng the same reduced form up to sets of measure zero must also be optmal. Hence Theorem 1 mples that any mechansm equvalent to a mechansm n conv(f ) must be optmal. The followng theorem shows the stronger result that ths s precsely the set of optmal mechansms. Theorem 2. A mechansm s optmal f and only f t s equvalent to some mechansm n conv(f ). Secton 6 contans a sketch of the proof of ths result. Theorem 2 says that all optmal mechansms are, essentally, favored agent mechansms or randomzaton over such mechansms. Hence we can restrct attenton to favored agent mechansms wthout loss of generalty. Ths result also mples that f there s a unque optmal favored agent mechansm, then conv(f ) s a sngleton so that there s essentally a unque optmal 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 and showng ths result, we consder the optmal selecton of the favored agent. For each, defne t by It s easy to show that t s well defned. To see ths, let E(t ) = E(max{t, t }) c. (1) ψ (t ) = E(max{t, t }) c. Clearly, ψ (t ) = E(t ) c < E(t ). For t t, ψ s strctly ncreasng n t and goes to nfnty as t. Hence there s a unque t > t. 4 It wll prove useful to gve two alternatve defntons of t. Note that we can rearrange the defnton above as t t f (t ) dt = t F (t ) c t 4 Note that f we allowed c = 0, we would have 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. 8

11 or t = E[t t t ] + Fnally, note that we could rearrange the next to last equaton as c F (t ). (2) c = t F (t ) t t t f (t ) dt = t t F (τ) dτ. So a fnal defnton of t s t t F (τ) dτ = c. (3) Gven any, let F denote the set of favored agent mechansms wth as the favored agent. Theorem 3. The unque best mechansm n F 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. Let t 1 denote the profle of types of j 1 and 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 gve 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 ). 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 our frst defnton of t. Hence gven that 1 s the favored agent, the threshold t 1 c 1 weakly domnates than any larger threshold. A smlar argument shows that the threshold t 1 c 1 weakly domnates any smaller threshold, establshng that t s optmal. 9

12 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. Gven that the best mechansm n each F s unque, t remans only to characterze the optmal choce or choces of. Theorem 4. The optmal choce of the favored agent s any wth t c = max j (t j c j ). Proof. For notatonal convenence, number the agents so that 1 s any wth t c = max j t j c j and let 2 denote any other agent so t 1 c 1 t 2 c 2. Frst, we show that the prncpal must weakly prefer havng 1 as the favored agent at a threshold of t 2 c 2 to havng 2 as the favored agent at ths threshold. If t 1 c 1 = t 2 c 2, ths argument mples that the prncpal s ndfferent between havng 1 and 2 as the favored agents, so we then turn to the case where t 1 c 1 > t 2 c 2 and show that t must always be the case that the prncpal strctly prefers havng 1 as the favored agent at threshold t 1 c 1 to favorng 2 wth threshold t 2 c 2, establshng the clam. So frst let us show that t s weakly better to favor 1 at threshold t 2 c 2 than to favor 2 at the same threshold. Frst, note that f any agent other than 1 or 2 reports a value above t 2 c 2, the desgnaton of the favored agent s rrelevant snce the good wll be assgned to the agent wth the hghest reported value and ths report wll be checked. Hence we may as well condton on the event that all agents other than 1 and 2 report values below t 2 c 2. If ths event has zero probablty, we are done, so we may as well assume ths probablty s strctly postve. Smlarly, f both agents 1 and 2 report values above t 2 c 2, the object wll go to whchever reports a hgher value and the report wll be checked, so agan the desgnaton of the favored agent s rrelevant. Hence we can focus on stuatons where at most one of these two agents reports a value above t 2 c 2 and, agan, we may as well assume the probablty of ths event s strctly postve. If both agents 1 and 2 report values below t 2 c 2, then no one s checked under ether mechansm. In ths case, the good goes to the agent who s favored under the mechansm. So suppose 1 s reported value s above t 2 c 2 and 2 s s below. If 1 s the favored agent, he gets the good wthout beng checked, whle he receves the good wth a check f 2 were favored. The case where 2 s reported value s above t 2 c 2 and 1 s s below s symmetrc. For brevty, let ˆt 1 = t 2 c 2 + c 1. Note that 1 s report s below the threshold ff t 1 c 1 < t 2 c 2 or, equvalently, t 1 < ˆt 1. Gven the reasonng above, we see 10

13 that under threshold t 2 c 2, t s weakly better to have 1 as the favored agent f F 1 (ˆt 1 )F 2 (t 2)E[t 1 t 1 ˆt 1 ] + [1 F 1 (ˆt 1 )]F 2 (t 2)E[t 1 t 1 > ˆt 1 ] + F 1 (ˆt 1 )[1 F 2 (t 2)] {E[t 2 t 2 > t 2] c 2 } F 1 (ˆt 1 )F 2 (t 2)E[t 2 t 2 t 2] + [1 F 1 (ˆt 1 )]F 2 (t 2) E[t 1 t 1 > ˆt 1 ] c 1 + F 1 (ˆt 1 )[1 F 2 (t 2)]E[t 2 t 2 > t 2]. If F 1 (ˆt 1 ) = 0, then ths equaton reduces to F 2 (t 2)E[t 1 t 1 > ˆt 1 ] F 2 (t 2) E[t 1 t 1 > ˆt 1 ] c 1, whch must hold. If F 1 (ˆt 1 ) > 0, then we can rewrte the equaton as E[t 1 t 1 ˆt 1 ] + c 1 F 1 (ˆt 1 ) c 1 E[t 2 t 2 t 2] + c 2 F 2 (t 2) c 2. From equaton (2), the rght hand sde equaton (4) s t 2 c 2. Hence we need to show E[t 1 t 1 ˆt 1 ] + c 1 F 1 (ˆt 1 ) c 1 t 2 c 2. (4) Recall that t 2 c 2 t 1 c 1 or, equvalently, ˆt 1 t 1. Hence from equaton (1), we have E(t 1 ) E[max{t 1, ˆt 1 }] c 1. A smlar rearrangement to our dervaton of equaton (2) yelds Hence E[t 1 t 1 ˆt 1] + c 1 F 1 (ˆt 1) ˆt 1. E[t 1 t 1 ˆt 1] + c 1 F 1 (ˆt 1) c 1 ˆt 1 c 1 = t 2 c 2 + c 1 c 1 = t 2 c 2, mplyng equaton (4). Hence as asserted, t s weakly better to have 1 as the favored agent wth threshold t 2 c 2 than to have 2 as the favored agent wth ths threshold. Suppose that t 1 c 1 = t 2 c 2. In ths case, an argument symmetrc to the one above shows that the prncpal weakly prefers favorng 2 at threshold t 1 c 1 to favorng 1 at the same threshold. Hence the prncpal must be ndfferent between favorng 1 or 2 at threshold t 1 c 1 = t 2 c 2. We now turn to the case where t 1 c 1 > t 2 c 2. The argument above s easly adapted to show that favorng 1 at threshold t 2 c 2 s strctly better than favorng 2 at ths threshold f the event that t j c j < t 2 c 2 for every j 1, 2 has strctly postve 11

14 probablty. To see ths, note that f ths event has strctly postve probablty, then the clam follows ff F 1 (ˆt 1 )F 2 (t 2)E[t 1 t 1 ˆt 1 ] + [1 F 1 (ˆt 1 )]F 2 (t 2)E[t 1 t 1 > ˆt 1 ] + F 1 (ˆt 1 )[1 F 2 (t 2)] {E[t 2 t 2 > t 2] c 2 } > F 1 (ˆt 1 )F 2 (t 2)E[t 2 t 2 t 2] + [1 F 1 (ˆt 1 )]F 2 (t 2) E[t 1 t 1 > ˆt 1 ] c 1 + F 1 (ˆt 1 )[1 F 2 (t 2)]E[t 2 t 2 > t 2]. If F 1 (ˆt 1 ) = 0, ths holds ff F 2 (t 2)c 1 > 0. By assumpton, c > 0 for all. Also, t 2 < t 2, so F 2 (t 2) > 0. Hence ths must hold f F 1 (ˆt 1 ) = 0. If F 1 (ˆt 1 ) > 0, then ths holds f equaton (4) holds strctly. It s easy to use the argument above and t 1 c 1 > t 2 c 2 to show that ths holds. So f the event that t j c j < t 2 c 2 for every j 1, 2 has strctly postve probablty, the prncpal strctly prefers havng 1 as the favored agent to havng 2. Suppose, then, that ths event has zero probablty. That s, there s some j 1, 2 such that t j c j t 2 c 2 wth probablty 1. In ths case, the prncpal s ndfferent between havng 1 as the favored agent at threshold t 2 c 2 versus favorng 2 at ths threshold. However, we now show that the prncpal must strctly prefer favorng 1 wth threshold t 1 c 1 to ether opton and thus strctly prefers havng 1 as the favored agent. To see ths, recall from the proof of Theorem 3 that the prncpal strctly prefers favorng 1 at threshold t 1 c 1 to favorng hm at a lower threshold v f there s a postve probablty that v < t j c j < t 1 c 1 for some j 1. Thus, n partcular, the prncpal strctly prefers favorng 1 at threshold t 1 c 1 to favorng hm at t 2 c 2 f there s a j 1, 2 such that the event t 2 c 2 < t j c j < t 1 c 1 has strctly postve probablty. By hypothess, there s a j 1, 2 such that t 2 c 2 < t j c j wth probablty 1, so we only have to establsh that for ths j, we have a postve probablty of t j c j < t 1 c 1. Recall that t j c j < t j c j by defnton of t j. By hypothess, t j c j < t 1 c 1. Hence we have t j c j < t 1 c 1 wth strctly postve probablty, completng the proof. 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 t c = max 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 wth t c equal to ths threshold to favor. Loosely speakng, for generc dstrbutons and checkng costs, there wll be a unque wth t c = max j t j c j and hence a unque optmal mechansm. 12

15 5 Comparatve Statcs and Examples Our characterzaton of the optmal favored agent and threshold makes t easy to gve comparatve statcs. Recall our thrd expresson for t whch s t t F (τ) dτ = c. (5) Hence 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 favored agent s the one the prncpal checks least often, ths makes t more desrable to make the favored agent. 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 (5) for fxed t, so to mantan the equalty, t must ncrease. Therefore, such a shft makes t more lkely than 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 value, he s a better choce to be the favored agent. The ntuton for why less rsky agents are favored s not as mmedate. One way to see the dea s to suppose that there s one agent whose type s completely rskless.e., s known to the prncpal. Obvously, there s no reason for the prncpal to check ths agent snce hs type s known. Thus settng hm as the favored agent the least lkely agent to be checked seems natural. We llustrate wth two examples. Frst, suppose we have two agents where t 1 U[0, 1], t 2 U[0, 2] and c 1 = c 2 = c. It s easy to calculate t. From equaton (1), we have For = 1, f t 1 < 1, t must solve E(t ) = E max{t, t } c. or 1 2 = t 1 t 1 ds t 1 s ds c 1 2 = (t 1) (t 1) 2 c 2 13

16 so t 1 = 2c. Ths holds only f c 1/2 so that t 1 1. Otherwse, E max{t 1, t 1} = t 1, so t 1 = (1/2)+c. Hence 2c, f c 1/2 t 1 = (1/2) + c, otherwse. A smlar calculaton shows that t 2 = 2 c, f c c, otherwse. It s easy to see that t 2 > t 1 for all c > 0, so 2 s the favored agent. The optmal threshold value s 2 c c, f c 1 t 2 c = 1, otherwse. Note that f 2 c 1 (or c 1/4), then the threshold value 2 c c 1 c. In ths case, the fact that t 1 s always less than 1 mples that t 1 c 1 v wth probablty 1. In ths case, the favored agent mechansm corresponds to smply gvng the good to agent 2 ndependently of the reports. If c (0, 1/4), then there are type profles for whch agent 1 receves the good, specfcally those wth t 1 > max{2 c, t 2 }. For a second example, suppose agan we have two agents, but now t U[0, 1] for = 1, 2. Assume c 2 > c 1 > 0. In ths case, calculatons smlar to those above show that so t = t c = 2c, f c 1/2 (1/2) + c, otherwse 2c c, f c 1/2 (1/2), otherwse. It s easy to see that 2c c s an ncreasng functon for c (0, 1/2). Thus f c 1 < 1/2, we must have t 2 c 2 > t 1 c 1, so that 2 s the favored agent. If c 1 1/2, then t 1 c 1 = t 2 c 2 = 1/2, so the prncpal s ndfferent over whch agent should be favored. Note that n ths case, the cost of checkng s so hgh that the prncpal never checks, so that the favored agent smply receves the good ndependent of the reports. Snce the dstrbutons of t 1 and t 2 are the same, t s not surprsng that the prncpal s ndfferent over who should be favored n ths case. It s not hard to show that when c 1 < 1/2 so that 2 s the favored agent, 2 s payoff s hgher than 1 s. That s, t s advantageous to be favored. Note that ths mples that agents may have ncentves to ncrease the cost of beng checked n order to become favored, an ncentve whch s costly for the prncpal. 14

17 6 Dscusson 6.1 Proof Sketch In ths secton, we sketch the proof of Theorem 2. It s easy to see that Theorem 1 s a corollary. Frst, t s useful to rewrte the optmzaton problem as follows. 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. Clearly, ths holds f and only f 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. Hence we can rewrte the objectve functon 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 ] ] = E t [ [p (t)(t c ) + ϕ c ]. Some of the arguments below wll use the reduced form probabltes and hence rely on the frst expresson for the payoff functon, whle others focus on the nonreduced mechansm and so rely on the second expresson. Summarzng, we can replace the choce of p and q functons for each wth the choce of a number ϕ [0, 1] for each and a functon p : T [0, 1] satsfyng p (t) 1 and E t p (t) ϕ 0. Note that ths last constrant mples E t p (t) ϕ, so ϕ E t p (t) = E t p (t) 1. Hence the constrant that ϕ 1 cannot bnd and so can be gnored. 15

18 The remander of the proof sketch s more complex and so we ntroduce several smplfcatons for smplcty. Frst, the proof sketch assumes I = 2 and c = c for all. The equal costs assumpton mples that the threshold value v can be thought of as defnng a threshold type t to whch we compare the t reports. Second, we wll consder the case of fnte type spaces and dsregard certan boundary ssues. To explan, the statement of Theorem 2 s made cleaner by our use of a contnuum of types. Wthout ths, we would have boundary types where there s some arbtrarness to the optmal mechansm, makng a statement of unqueness more complex. On the other hand, the contnuum of types greatly complcates the proof of our characterzaton of optmal mechansms. In ths proof sketch, we explan how the proof would work f we focused on fnte type spaces, gnorng what happens at boundares. The proof n the appendx can be seen as a generalzaton of these deas to contnuous type spaces. The proof sketch has fve steps. Frst, we observe that every optmal mechansm s monotonc n the sense that hgher types are more lkely to receve the object. That s, for all, t > t mples ˆp (t ) ˆp (t ). To see the ntuton, suppose we have an optmal mechansm whch volates ths monotoncty property so that we have types t and t such that ˆp (t ) < ˆp (t ) even though t > t. To smplfy further, suppose that these two types have the same probablty. Then consder the mechansm p whch s the same as ths one except we flp the roles of t and t. That s, for any type profle ˆt where ˆt / {t, t }, we let p (ˆt) = p (ˆt). For any type profle of the form (t, t ) we assgn the p s the orgnal mechansm assgned to (t, t ) and conversely. Snce the probabltes of these types are the same, our ndependence assumpton mples that for every j, agent j s unaffected by the change n the sense that ˆp j = ˆp j. Obvously, ˆp (t ) ˆp (t ) = ˆp (t ). Snce the orgnal mechansm was feasble, we must have ˆp (t ) ϕ, so ths mechansm must be feasble. It s easy to see that ths change mproves the objectve functon, so the orgnal mechansm could not have been optmal. Ths monotoncty property mples that any optmal mechansm has the property that there s a cutoff type, say ˆt [t, t ], such that ˆp (t ) = ϕ for t < ˆt and ˆp (t ) > ϕ for t > ˆt. The second step shows that f we have a type profle t = (t 1, t 2 ) such that t 2 > t 1 > ˆt 1, then the optmal mechansm has p 2 (t) = 1. To see ths, suppose to the contrary that p 2 (t) < 1. Then we can change the mechansm by ncreasng ths probablty slghtly and lowerng the probablty of gvng the good to 1 (or, f the probablty of gvng t to 1 was 0, lowerng the probablty that the good s not gven to ether agent). Snce t 1 > ˆt 1, we have ˆp 1 (t 1 ) > ϕ 1 before the change, so f the change s small enough, we stll satsfy ths constrant. Snce t 2 > t 1, the value of the objectve functon ncreases, so the orgnal mechansm could not have been optmal. The thrd step s to show that for a type profle t = (t 1, t 2 ) such that t 1 > ˆt 1 and 16

19 t 2 < ˆt 2, we must have p 1 (t) = 1. To see ths, consder the pont labeled α = ( t 1, t 2 ) n Fgure 1 below where t 1 > ˆt 1 whle t 2 < ˆt 2. Suppose that at α, player 1 receves the good wth probablty strctly less than 1. It s not hard to see that at any pont drectly below α but above ˆt 1, such as the one labeled β = (t 1, t 2 ), player 1 must receve the good wth probablty zero. To see ths, note that f 1 dd receve the good wth strctly postve probablty here, we could change the mechansm by lowerng ths probablty slghtly, gvng the good to 2 at β wth hgher probablty, and ncreasng the probablty wth whch 1 receves the good at α. By choosng these probabltes approprately, we do not affect ˆp 2 ( t 2 ) so ths remans at ϕ 2. Also, by makng the reducton n p 1 small enough, ˆp 1 (t 1) wll reman above ϕ 1. Hence ths new mechansm would be feasble. Snce t would swtch probablty from one type of player 1 to a hgher type, the new mechansm would be better than the old one, mplyng the orgnal one was not optmal. 5 Smlar reasonng mples that for every t 1 t 1, we must have p (t 1, t 2 ) = 1. Otherwse, the prncpal would be strctly better off ncreasng p 2 (t 1, t 2 ), decreasng p 2 ( t 1, t 2 ), and ncreasng p 1 ( t 1, t 2 ). Agan, f we choose the szes of these changes approprately, ˆp 2 ( t 2 ) s unchanged but ˆp 1 ( t 1 ) s ncreased, an mprovement. 5 Snce ˆp 2 ( t 2 ) s unchanged, the ex ante probablty of type t 1 gettng the good goes up by the same amount that the ex ante probablty of the lower type t 1 gettng t goes down. 17

20 t 1 t 1 p 1 < 1 t 0 1 p 1 =0 p 1 > 0 ˆt 1 t 00 1 p 1 > 0 " p 1 < 1 t 2 ˆt 2 t 0 2 t 2 Fgure 1 Snce player 1 receves the good wth zero probablty at β but type t 1 does have a postve probablty overall of recevng the good, there must be some pont lke the one labeled γ = (t 1, t 2) where 1 receves the good wth strctly postve probablty. We do not know whether t 2 s above or below ˆt 2 the poston of γ relatve to ths cutoff plays no role n the argument to follow. Fnally, there must be a t 1 (not necessary below ˆt 1 ) correspondng to ponts δ and ε where p 1 s strctly postve at δ and strctly less than 1 at ε. To see that such a t 1 must exst, suppose not. Then for all t 1 t 1, ether p 1 (t 1, t 2 ) = 0 or p 1 (t 1, t 2) = 1. Snce p (t 1, t 2 ) = 1 for all t 1 t 1, ths mples that for all t 1 t 1, ether p 2 (t 1, t 2 ) = 1 18

21 or p 2 (t 1, t 2) = 0. Ether way, p 2 (t 1, t 2 ) p 2 (t 1, t 2) for all t 1 t 1. But we also have p 2 (t 1, t 2 ) = 1 > 1 p 1 (t 1, t 2) p 2 (t 1, t 2). So ˆp 2 ( t 2 ) > ˆp 2 (t 2). But ˆp 2 ( t 2 ) = ϕ 2, so ths mples ˆp 2 (t 2) < ϕ 2, whch volates the constrants on our optmzaton problem. From ths, we can derve a contradcton to the optmalty of the mechansm. At γ, lower p 1 and ncrease p 2 by the same amount. At ε, rase p 1 and lower p 2 n such a way that ˆp 2 (t 2) s unchanged. In dong so, keep the reducton of p 1 at γ small enough that ˆp 2 (t 1) remans above ϕ 1. Ths s clearly feasble. Now that we have ncreased p 1 at ε, we can lower t at δ whle ncreasng p 2 n such a way that ˆp 1 (t 1) remans unchanged. Fnally, snce we have lowered p 1 at δ, we can ncrease t at α, now lowerng p 2, n such a way that ˆp 2 ( t 2 ) s unchanged. Note the overall effect: ˆp 1 s unaffected at t 1 and lowered n a way whch retans feasblty at t 1. ˆp 2 s unchanged at t 2 and at t 2. Hence the resultng p s feasble. But we have shfted some of the probablty that 1 gets the object from γ to α. Snce 1 s type s hgher at α, ths s an mprovement, mplyng that the orgnal mechansm was not optmal. The fourth step s to show that ˆt 1 = ˆt 2. To see ths, suppose to the contrary that ˆt 2 > ˆt 1. Then consder a type profle t = (t 1, t 2 ) such that ˆt 2 > t 2 > t 1 > ˆt 1. From our second step, the fact that t 2 > t 1 > ˆt 1 mples p 2 (t) = 1. However, from our thrd step, t 1 > ˆt 1 and t 2 < ˆt 2 mples p 1 (t) = 1, a contradcton. Hence there cannot be any such profle of types, mplyng ˆt 2 ˆt 1. Reversng the roles of the players then mples ˆt 1 = ˆt 2. Let t = ˆt 1 = ˆt 2. Ths common value of these ndvdual thresholds wll yeld the threshold of our favored agent mechansm as we wll see shortly. To sum up the frst four steps, we can characterze any optmal mechansm by specfyng t, ϕ 1, and ϕ 2. From our second step, f we have t 2 > t 1 > t, then p 2 (t) = 1. That s, f both agents are above the threshold, the hgher type agent receves the object. From our thrd step, f t 1 > t > t 2, then p 1 (t) = 1. That s, f only one agent s above the threshold, ths agent receves the object. Ether way, then, f there s at least one agent whose type s above the threshold, the agent wth the hghest type receves the object. Also, by defnton, f t < t, then ˆp (t ) = ϕ = nf t ˆp (t ). Recall that we showed ˆq (t ) = ˆp (t ) ϕ, so ˆq (t ) = 0 whenever t < t. That s, f an agent s below the threshold, he receves the good wth the lowest possble probablty and s not checked. Ths mples that ˆp s completely pnned down as a functon of t, ϕ 1. and ϕ 2. If t > t, then ˆp (t ) must be the probablty t > t j. If t < t, then ˆp (t ) = ϕ. We already saw that ˆq s pnned down by ˆp and the ϕ s, so the reduced form s a functon only of t and the ϕ s. Snce we can wrte the prncpal s payoff can be wrtten as a functon only of the reduced form, ths mples that the prncpal s payoff s completely pnned down once we specfy t and the ϕ s. It s not hard to see that the prncpal s 19

22 payoff s lnear n the ϕ s. Because of ths and the fact that the set of feasble ϕ vectors s convex, we see that gven v, there must be a soluton to the prncpal s problem at an extreme pont of the set of feasble (ϕ 1, ϕ 2 ). Furthermore, every optmal choce of the ϕ s s a randomzaton over optmal extreme ponts. The last step s to show that such extreme ponts correspond to favored agent mechansms. It s not hard to see that at an extreme pont, one of the ϕ s s set to zero and the other s as large as possble, where we clarfy the meanng of ths phrase below. 6 For notatonal convenence, consder the extreme pont where ϕ 2 = 0 and ϕ 1 s set as hgh as possble. We now show that ths corresponds to the favored agent mechansm where 1 s the favored agent and the threshold v s t c. To see ths, frst note that the reduced form for an arbtrary t, ϕ 1, and ϕ 2 s Pr[tj < t ˆp (t ) = ], f t > t, ϕ, otherwse and ˆq (t ) = ˆp (t ) ϕ. The reduced form of the favored agent mechansm where 1 s favored, say (p, q ), s gven by and ˆq 1(t 1 ) = ˆp Pr[t2 < t 1(t 1 ) = 1 ], f t 1 > t, Pr[t 2 < t ], otherwse, Pr[t2 < t 1 ] Pr[t 2 < t ], f t 1 > t, 0, otherwse, ˆq 2(t 2 ) = ˆp 2(t 2 ) = Pr[t1 < t 2 ], f t 2 > t, 0, otherwse, It s easy to see that these are equal f we set ϕ 1 = Pr[t 2 < t ] and ϕ 2 = 0. It turns out that Pr[t 2 < t ] s what as large as possble means n ths example. 6.2 Almost Domnance and Ex Post Incentve Compatblty One appealng property of the favored agent mechansm s that t s almost a domnant strategy mechansm. That s, for every agent, truth tellng s a best response to any strateges by the opponents. It s not always a domnant strategy, however, as the agent may be completely ndfferent between truth tellng and les. To see ths, consder any agent who s not favored and a type t such that t c > v. If t reports hs type truthfully, then receves the object wth strctly postve probablty 6 Under some condtons, ϕ = 0 for all s also an extreme pont. See the appendx for detals. 20

23 under a wde range of strategy profles for the opponents. Specfcally, any strategy profle for the opponents wth the property that t c s the hghest report for some type profles has ths property. On the other hand, f t les, then receves the object wth zero probablty gven any strategy profle for the opponents. Ths follows because s not favored and so cannot receve the object wthout beng checked. Hence for such a type, truth tellng weakly domnates any le for t. Contnung to assume s not favored, consder any t such that t c < v. For any profle of strateges by the opponents, t s probablty of recevng the object s zero regardless of hs report. To see ths, smply note that f reports truthfully, he cannot receve the good (snce t wll ether go to another nonfavored agent f one has the hghest t j c j and reports honestly or to the favored agent). Smlarly, f les, he cannot receve the object snce he wll be caught lyng when checked. Hence truth tellng s an optmal strategy for t, though t s not weakly domnant snce the agent s ndfferent over all reports gven any strateges by the other agents. A smlar argument apples to the favored agent. Agan, f hs type satsfes t c > v, truth tellng s domnant, whle f t c < v, he s completely ndfferent over all strateges. Ether way, truth tellng s an optmal strategy regardless of the strateges of the opponents. Because of ths property, the favored agent mechansm s ex post ncentve compatble. Formally, (p, q) s ex post ncentve compatble f p (t) p (ˆt, t ) q (ˆt, t ), ˆt, t T, t T, I. That s, t prefers reportng honestly to lyng even condtonal on knowng the types of the other agents. It s easy to see that the favored agent mechansm s almost domnance property mples ths. Of course, the ex post ncentve constrants are strcter than the Bayesan ncentve constrants, so ths mples that that favored agent mechansm s ex post optmal. Whle the almost domnance property mples a certan robustness of the mechansm, the complete ndfference for types below the threshold s troublng. There are smple modfcatons of the mechansm whch do not change ts equlbrum propertes but make truth tellng weakly domnant rather than just almost domnant. For example, suppose there are at least three agents and that every agent satsfes t c > v. 7 Suppose we modfy the favored agent mechansm as follows. If an agent s checked and found to have led, then one of the other agents s chosen at random and hs report s checked. If t s truthful, he receves the object. Otherwse, no agent receves t. It s easy to see 7 Note that f t c < v, then the favored agent mechansm never gves the object to, so s report s entrely rrelevant to the mechansm. Thus we cannot make truth tellng domnant for such an agent, but the report of such an agent s rrelevant anyway. 21