Commitment and Robustness in Mechanisms with Evidence 1

Transcription

1 Commtment and Robustness n Mechansms wth Evdence 1 Elchanan Ben-Porath 2 Edde Dekel 3 Barton L. Lpman 4 Frst Draft June We thank the Natonal Scence Foundaton, grant SES (Dekel), and the US Israel Bnatonal Scence Foundaton for support for ths research. 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 We show that n a class of I agent mechansm desgn problems wth evdence, commtment has no value for the prncpal, randomzaton has no value for the prncpal, and robust ncentve compatblty has no cost. In partcular, there s an equlbrum wth a relatvely smple structure n whch the prncpal obtans the same outcome wthout commtment as the best he can acheve wth commtment.

3 1 Introducton We show that n a class of I agent mechansm desgn problems wth evdence, randomzaton has no value for the prncpal and robust ncentve compatblty has no cost. Also, commtment has no value for the prncpal n the sense that there s an equlbrum of the game when the prncpal s not commtted to the mechansm wth the same outcome as n the optmal mechansm wth commtment. We also show that ths equlbrum has a relatvely smple structure. To understand the class of mechansm desgn problems our result apples to, consder the followng examples. Example 1. The smple allocaton problem. The prncpal has a sngle unt of an ndvsble good whch he can allocate to one of a set of I agents. Each agent has a type whch affects the value to the prncpal of allocatng the good to that agent. Each agent prefers gettng the good to not gettng t, regardless of her type. Types are ndependent across agents and monetary transfers are not possble. Each agent may have concrete evdence whch proves to the prncpal some facts about her type. For example, the prncpal may be a state government whch needs to choose a cty n whch to locate a publc hosptal. Each cty wants the hosptal. The state wants to place the hosptal where t wll be most effcently utlzed, but each cty has prvate nformaton on local needs. The state could ask the cty to bear the cost of the hosptal, but that would mply dvertng the cty s funds from other projects that the goverement consders mportant. Example 2. The publc goods problem. The prncpal has to choose whether or not to provde a publc good whch affects the utlty of I agents. If the prncpal provdes the good, the cost must be evenly dvded among the agents. Each agent has a type whch determnes her wllngness to pay for the good. If the wllngness to pay exceeds her share of the cost, she wants the good provded and otherwse prefers that t not be provded. Types are ndependent across agents and monetary transfers other than the cost sharng are not possble. Each agent may have evdence whch enables her to prove some facts to the prncpal about the value of the publc good to her. The prncpal wshes to maxmze the sum of the agents utltes. For example, the prncpal may be a government agency decdng whether or not to buld a hosptal n a partcular cty and the agents may be resdents of that cty who wll be taxed to pay for the hosptal f t s bult. Then an agent mght show documentaton of a health condton or past emergency room vsts to prove to the prncpal that she has a hgh value for a nearby hosptal. Example 3. The allocaton problem wth externaltes. Ths problem s the same as Example 1, but now agent cares about whch other agent receves the good f she does not. For example, suppose agent s most preferred outcome s to receve the good, her second best opton s that agent + 1 receves t (mod I), and all other allocatons 1

4 are ted for worst. In Example 3a, the payoff to the prncpal to allocatng the good to agent s stll just a functon of s type. Alternatvely, n Example 3b, we assume that the prncpal s payoff s a weghted sum of the payoffs to the agents where the weght on agent s utlty (whch could be negatve) depends on s type. Example 4. The publc goods problem wth nterdependence. Ths problem s the same as Example 2 except now the payoff to agent from the provson of the publc good depends on both her type and the types of the other agents. We wll show that optmal mechansms for Examples 1, 2, and 3b share several sgnfcant features. Frst, there s no value to commtment. In other words, f the prncpal s not commtted to the mechansm, there s stll an equlbrum of the game wth the same outcome as n the optmal mechansm. Second, the optmal mechansm s determnstc the prncpal does not need to randomze. Thrd, the optmal mechansm s not just ncentve compatble but s also what we wll call robustly ncentve compatble. We defne ths precsely later, but for now smply note that t s a strengthenng of domnant strategy ncentve compatblty. Thus the robustness of domnant strategy ncentve compatblty comes at no cost to the prncpal. Ths strong robustness of the mechansm n turn mples strong robustness propertes of the equlbrum whch acheves the same outcome. By contrast, the optmal mechansm for Examples 3a and 4 satsfes none of these propertes, n general. One useful mplcaton of ths result s that we can compute optmal mechansms by consderng equlbra. In partcular, we gve a relatvely smple characterzaton of the optmal equlbrum for the prncpal whch does not rely on much nformaton regardng the prncpal s preferences or the structure of the set of actons. Ths makes determnng the optmal mechansm straghtforward n some cases. To llustrate, we consder optmal mechansms when the evdence technology s the one orgnally proposed by Dye (1985). In Dye s model, each agent has some probablty of havng evdence that would enable her to exactly prove her true type and otherwse has no evdence at all. When we apply ths approach to the smple allocaton problem descrbed n Example 1 above, we fnd that the optmal mechansm has a favored agent structure remnscent of Ben-Porath, Dekel, and Lpman (2014), henceforth BDL. BDL consdered the smple allocaton problem descrbed above, but wth one dfference. BDL dd not consder evdence but nstead consdered what n some respects s the opposte assumpton, where the prncpal can verfy clams by the agents at a cost n a manner that the agents cannot control. BDL showed that the optmal mechansm was a favored agent mechansm. More precsely, a favored agent mechansm specfes an agent, say, and a threshold net value, say v. Call the net value of a type of an agent the value to the prncpal of gvng her the good when she has that type mnus the cost of verfyng ths clam. In a favored agent mechansm, f every agent other than the favored agent clams a net value below v, then 2

5 the prncpal gves the good to the favored agent and does not verfy any reports. Otherwse, he verfes the report of the agent who makes the hghest net value clam and, f the report s truthful (as t wll be n equlbrum), he gves the good to that agent. Turnng to the model of the current paper, we defne a favored agent mechansm to specfy a favored agent and a threshold value v such that f every agent other than the favored agent ether provdes no evdence or proves a value below v, then the favored agent receves the good. Otherwse, the good goes to the agent who proves the hghest value. The fact that the prncpal does not need commtment makes ths result qute straghtforward to prove, by contrast to the complex proof n BDL. 1 We also show that the optmal mechansm for the publc good problem s very smlar to the optmal mechansm for the same problem wth costly verfcaton as derved by Erlanson and Klener (2015). The paper s organzed as follows. Secton 2 states the model formally. In Secton 3, we show the man results sketched above, ncludng the characterzaton of the best equlbrum for the prncpal. The proof of ths theorem s sketched n Secton 5. In Secton 4, we specalze to the Dye (1985) evdence structure and provde a characterzaton of optmal mechansms n ths settng. We then use ths characterzaton to gve optmal mechansms for a varety of more specfc settngs ncludng the smple allocaton problem and the publc goods problem. We offer concludng remarks n Secton??. Proofs not contaned n the text are n the Appendx. Related lterature. There are several lteratures related to ths paper. Frst, there s a lterature on mechansm desgn wth evdence see, for example, 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 (2015). A partcularly relevant subset of ths lterature s a set of papers on one agent mechansm desgn problems whch show that, under certan condtons, the prncpal does not need commtment to obtan the same outcome as under the optmal mechansm. Ths was frst shown by Glazer and Rubnsten (2004, 2006) for the case where the prncpal has two actons and was extended by Sher (2011) to concave utlty functons and by Hart, Kremer, and Perry (2015) to sngle peaked utlty. Our results extend these n two ways. Frst and most mportantly, we consder mult agent problems rather than sngle agent settngs. Of course, our result on robust ncentve compatblty cannot have any analog n the one agent settng. Second, unlke these prevous results, we allow the preferences of an agent to vary wth her type, albet n a specfc fashon. In partcular, n the two acton case consdered by Glazer and Rubnsten, we can allow arbtrary dependence of an agent s preference on her type. We dscuss the connectons to these papers n more detal n Secton 4. 1 Though see the smpler proof of that result n Lpman (2015) or Erlanson and Klener (

6 Second, our result showng that commtment s not valuable can be thought of as a characterzaton of equlbra n games wth evdence. Hence our work s also related to the lterature on communcaton games wth evdence. See, for example, Dye (1985), Jung and Kwon (1988), Shn (1994, 2003), Lpman and Sepp (1995), and Guttman, Kremer, and Skrzypacz (2014). As noted above, our work s also related to our prevous work on mechansm desgn wth costly verfcaton (Ben Porath, Dekel, and Lpman, 2014) and to Erlanson and Klener (2015). We dscuss ths connecton n Secton 3. 2 Model The set of agents s I = {1,..., I} where I 1. The prncpal has a fnte set of actons A and can randomze over these. Each agent has prvate nformaton n the form of a type t where types are dstrbuted ndependently across agents. The set of types of s denoted T and the (full support) pror s denoted ρ. T s fnte for all. Gven acton a by the prncpal and type profle t, agent s utlty s u (a, t ), ndependent of t. We add sgnfcantly more structure on the agents utlty functons below. The prncpal s utlty s v(a, t) = u 0 (a) + u (a, t )v (t ). In what follows, we often wrte ths smply as u (a)v (t ) wth the conventon that the sum runs from = 0 to I and v 0 (t 0 ) 1. There are two ways to nterpret the prncpal s utlty functon. The most obvous s a socal welfare nterpretaton where the prncpal maxmzes a weghted sum of the agent s utltes and t determnes how much he cares about agent s utlty. On the other hand, ths utlty functon does not requre the prncpal to care about the agents at all. A dfferent nterpretaton s to thnk of v (t ) as measurng the extent to whch the prncpal s nterests are algned wth those of agent. That s, a hgh value of v (t ) doesn t mean that the prncpal lkes agent but means that the prncpal lkes what agent lkes. For some settngs, both nterpretatons seem natural. For example, consder the smple allocaton problem, Example 1 n Secton 1. Here the prncpal has a sngle unt 4

7 of an ndvsble good that he can ether keep or allocate to one of the agents. Thus we can wrte the set of actons as {0, 1,..., I} where 0 s nterpreted as the prncpal keepng the good and 0 s nterpreted as allocatng t to agent. Assumng every type of every agent prefers havng the good to not havng t, we take the utlty functon for agent to be 1, f a = ; u (a, t ) = 0, otherwse. Lettng v (t ) denote the value to the prncpal of allocatng the good to agent when she s type t and lettng u 0 (0) denote the value to the prncpal of keepng t, we obtan the utlty functon for the prncpal of u 0 (a) + u (a, t )v (t ). Thus ths formulaton s consstent wth the second nterpretaton, but, of course, t s also consstent wth an nterpretaton that v (t ) measures how much the prncpal cares about type t s utlty. For other problems, the socal welfare nterpretaton s more natural. For example, suppose the prncpal has a set of objects to allocate. If each agent wants at most one object and s ndfferent across objects, then, just as above, we can nterpret v (t ) as the value to the prncpal of gvng an object to t. On the other hand, f agents may want multple objects and have nontrval preferences regardng bundles of such objects, t seems most natural to nterpret the prncpal s utlty functon as a socal welfare functon. Another mportant ssue for nterpretaton s that we cannot entrely separate assumptons about the prncpal s utlty functon and the agents utlty functons. For example, suppose v (t ) > 0 for all t and all. Then consder changng agent s utlty functon from u (a, t ) to û (a, t ) = u (a, t )v (t ) and changng the prncpal s utlty functon to û (a, t ). Becaue û (a, t ) s a postve affne transformaton of u (a, t ), we haven t changed best responses for the agents. Clearly, the prncpal s preferences have not changed snce ths s smply a dfferent way of wrtng the same functon. Hence we cannot separate the extent to whch v (t ) s part of the prncpal s utlty functon or a scalng factor for agent s utlty functon. Fnally, note that v (t ) s allowed to be zero or negatve. Thus the prncpal s nterests can be n conflct wth those of some or all agents n a way whch can depend on the agents types. Each agent may have evdence whch would prove some clams about her type. To model evdence, we assume that for every, there s a functon E : T 2 2T. In other words, E (t ) s a collecton of subsets of T, nterpreted as the set of events that t can prove. The dea s that f E E (t ), then type t has some set of documents or other tangble evdence whch she can present to the prncpal whch demonstrates conclusvely that her type s n the set E. We requre the followng propertes. Frst, proof s true. Formally, E E (t ) mples t E. Second, proof s consstent n the sense that 5

8 s E E(t ) mples E E (s ). In other words, f there s a pece of evdence that some type can present whch does not rule out s, then t must be true that s could present that evdence. Clearly, f s could not present t, the evdence actually refutes the possblty of s. Puttng these two propertes together, we have t E f and only f E E (t ). The last property we assume s not necessary for the model to be nternally consstent but s an addtonal restrcton used n much of the lterature. Ths property s called normalty by Bull and Watson (2007) and the full reports condton by Lpman and Sepp (1995). The condton says that there s one event that t can present whch summarzes all the evdence she has avalable. Intutvely, ths condton means that there are no tme or other restrctons on the evdence an agent can present, so that she can present everythng she has. Formally, the statement s that for every t, we have E E (t ) E E (t ). That s, the event proved by showng all of t s evdence s tself an event that t can prove. Henceforth, we let M (t ) denote the maxmally nformatve event t can prove. I.e., we defne M (t ) = E. E E (t ) We sometmes refer to t presentng M (t ) as presentng maxmal evdence. Before formally defnng a mechansm, we note that gven our assumptons, t s wthout loss of generalty to focus on mechansms where the agents smultaneously make cheap talk reports of types and present evdence and where each agent truthfully reveals her type and presents maxmal evdence. Ths s not the standard Revelaton Prncple but has been shown by, among others, Bull and Watson (2007) and Deneckere and Severnov (2008). Formally, let E = t T E (t ) and E = E. A mechansm s then a functon P : T E (A). For notatonal brevty, gven a mechansm P, t (t, e ) T E, let T, (s, e ) T E (t ), and ũ (s, e, t, e t, P ) = a P (a s, e, t, e )u (a, t ) and û (s, e t, P ) = E t ũ (s, e, t, M (t ) t, P ). In words, ũ (s, e, t, e t, P ) s agent s expected utlty under mechansm P when her type s t but she reports s, presents evdence e, and expects all other agents to clam types t and report evdence e. Then û (s, e t, P ) s s expected utlty from 6

9 reportng (s, e ) when her type s t and she expects the other agents to report ther types truthfully and to provde maxmal evdence. A mechansm P s ncentve compatble f for every agent, û (t, M (t ) t, P ) û (s, e t, P ), for all s, t T and all e E (t ). The prncpal s expected payoff from an ncentve compatble mechansm P s E t P (a t, M(t))v(a, t). a Our man result s that f the type dependence of the agents utlty s suffcently smple, then for the prncpal, there s no value to commtment, no cost to robust ncentve compatblty, and no need to randomze. We now make ths statement more precse. Before defnng our noton of robust ncentve compatblty, we begn wth more standard notons. A mechansm s ex post ncentve compatble f for every agent, ũ (t, M (t ), t, M (t ) t, P ) ũ (s, e, t, M (t ) t, P ), for all s, t T, all t T, and all e E (t ). In other words, a mechansm s ex post ncentve compatble f each agent has an ncentve to report honestly and present maxmal evdence even f she knows all the other agents types and that they are reportng truthfully. Say that a reportng strategy σ j : T j T j E j s feasble f whenever σ j (t j ) = (s j, e j ), we have e j E j (t j ). A mechansm s domnant strategy ncentve compatble f for every agent, E t ũ (t, M (t ), σ (t ) t, P ) E t ũ (s, e, σ (t ) t, P ) for all s, t T, all feasble σ : T T E, and all e E (t ). In mechansms wth evdence, nether of these notons of ncentve compatblty mples the other. A mechansm could be ex post ncentve compatble, but an agent mght want to devate f she knew another agent were gong to report (s, e ) where e M (s ). That s, an agent mght want to devate from truth tellng and maxmal evdence f she knew another agent was gong to devate from truth tellng and maxmal evdence n a detectable way. Smlarly, a mechansm could be domnant strategy ncentve compatble but an agent could wsh to devate f she knew the specfc types of her opponents. The robustness noton we wll use combnes both the ex post and domnant strategy features of the above defntons. 7

10 We say that a mechansm s robustly ncentve compatble f for every agent, ũ (t, M (t ), t, e t, P ) ũ (s, e, t, e t, P ), for all s, t T, all (t, e ) T E, and all e E (t ). In other words, even f knew the exact type and evdence reports of all other agents, t would be optmal to report truthfully and provde maxmal evdence regardless of what those reports are. In mechansms wthout evdence but wth ndependent prvate values, robust ncentve compatblty, ex post ncentve compatblty, and domnant strategy ncentve compatblty are all equvalent. Whle we have ndependent prvate values (.e., the t s are ndependent across and u s not a functon of t ), these concepts are not equvalent here because of the evdence structure. Obvously, robust ncentve compatblty mples ncentve compatblty, but the converse s not true. Hence the best robustly ncentve compatble mechansm for the prncpal yelds her a weakly lower expected payoff than the best ncentve compatble mechansm, typcally strctly lower. Our result wll show that under our assumptons, the best ncentve compatble mechansm for the prncpal s always robustly ncentve compatble. We say a mechansm P s determnstc f for every (t, e) T E, P (t, e) s a degenerate dstrbuton. In other words, for every report and presentaton of evdence, whether or not t nvolves truth tellng and maxmal evdence, the prncpal chooses an a A wthout randomzng. Of course, randomzaton s an mportant feature of optmal mechansms n some settngs. We wll show that under our assumptons, there s an optmal mechansm whch s determnstc. Fnally, to state what t means that there s no value to commtment, we must defne what the prncpal can accomplsh n the absence of commtment. Wthout commtment, we assume that there s a game n whch, just as n the revelaton mechansm, agents smultaneously make type reports and present evdence, perhaps wth randomzaton. The prncpal observes these choces and then chooses some allocaton a, agan perhaps wth randomzaton. More formally, the set of strateges for agent, Σ, s the set of functons σ : T (T E ) such that σ(s, e t ) > 0 mples e E (t ). That s, f agent s type t and puts postve probablty on provdng evdence e, then ths evdence must be feasble for t n the sense that e E (t ). The prncpal s set of feasble strateges, Σ P, s the set of functons σ P : T E (A). We consder the set of perfect Bayesan equlbra of ths game. More precsely, a belef by the prncpal s a functon µ : T E (T ), gvng the prncpal s belefs about t gven a profle of reports and evdence presentaton. For notatonal convenence, gven σ Σ, σ P Σ P, a A, and (s, e ) T E, let Q (a s, e, σ, σ P ) = E t σ P (a s, e) σ j (s j, e j t j ). (s,e ) j 8

11 In other words, ths s the probablty the prncpal chooses allocaton a gven that she uses strategy σ P, agents other than use strateges σ j, j, and agent reports s and presents evdence e. We say that (σ 1,..., σ I, σ P, µ) s a perfect Bayesan equlbrum 2 f the followng condtons hold. Frst, for every and every t T, σ (s, e t ) > 0 mples (s, e ) arg max Q (a s, e, σ, σ P )u (a, t ). s T,e E (t ) a A Second, for every (s, e) T E, σ P (a s, e) > 0 mples a arg max a A µ(t s, e)v(a, t). t T Thrd, for every (s, e), µ( s, e) respects ndependence across agents. That s, s report (s, e ) only affects the prncpal s belefs about t and hs belefs about t and t j respect ndependence for all j. Formally, we have functons µ : T E (T ) such that for all t T and all (s, e) T E, µ(t s, e) = µ (t s, e ). Fourth, for all (s, e), µ( s, e) respects feasblty. That s, the prncpal s belefs must put zero probablty on any type whch s nfeasble gven (s, e). Formally, for every t T and (s, e ) T E, we have µ (t s, e ) = 0 f e / E (t ). Fnally, the prncpal s belefs are consstent wth Bayes rule whenever possble n the sense that for every (s, e ) T E such that there exsts t wth σ (s, e t ) > 0, we have σ (s, e t )ρ (t ) µ (t s, e ) = t T σ (s, e t )ρ (t ). (Recall that ρ s the prncpal s pror over t.) The equlbra whch wll gve the prncpal the same payoff as n the optmal mechansm wll satsfy a certan robustness property that, for lack of a better phrase, we smply call robustness. Specfcally, a perfect Bayesan equlbrum (σ, µ) s robust f for every and every t T, σ (s, e t ) > 0 mples (s, e ) arg max σ P (a s, e, s, e )u (a, t ), (s, e ) T E. s T,e E (t ) a A 2 Our defnton s the natural adaptaton of Fudenberg and Trole s (1991) defnton of perfect Bayesan equlbrum for games wth observed actons and ndependent types to allow type dependent sets of feasble actons. 9

12 In other words, σ (t ) s optmal for t regardless of the actons played by the other agents, gven the strategy of the prncpal. Gven a perfect Bayesan equlbrum (σ, µ), the prncpal s expected utlty s E t σ (s, e t )σ P (a s, e)v(a, t). a We wll show that there s a robust perfect Bayesan equlbrum of ths game whch gves the prncpal the same expected utlty as the optmal mechansm. In ths sense, the prncpal does not need the commtment assumed n characterzng the optmal mechansm. 3 Commtment, Determnsm, and Robust Incentve Compatblty Our result assumes that the type dependence of the agents utlty functons takes a partcularly smple form. Formally, we say that u (a, t ) satsfes smple type dependence f there exst functons u : A R and β : T R such that u (a, t ) = u (a)β (t ) where β (t ) 0 for all t T. 3 Ths multplcatve separablty s more restrctve than t may appear. Under smple type dependence, we effectvely have a model where all types have the same ndfference curves n (A) space, those defned by utlty functon u (a), but the drecton of mprovement may vary across types. Obvously, ths s qute restrctve n general. To see ths, renormalze u (a, t ) by dvdng through by β (t ). Hence t s strategcally equvalent to defne ū (a, t ) = u (a, t ) β (t ). Wth ths renormalzaton, we have where u (a), f t ū (a, t ) = T + u (a), f t T T + = {t T β (t ) > 0} and T = T \ T +. We can also rewrte the prncpal s utlty functon n terms of the 3 If β (t ) = 0 for some t, then that type s ndfferent over all actons by the prncpal and so wll always truthfully reveal. Hence we may as well dsregard such types. ; 10

13 u (a) s. Specfcally, note that v(a, t) = u 0 (a) + = u 0 (a) + = u 0 (a) + v (t )u (a, t ) v (t )β (t )u (a) v (t )u (a), where v (t ) = v (t )β (t ). Wth some abuse of notaton, we can then redefne v and wrte the prncpal s utlty functon as I I v(a, t) = u 0 (a) + u (a)v (t ) = u (a)v (t ). =1 =0 Because t s more convenent for analyss, we wll wrte the prncpal and agents utlty functons n the form above hereafter, referrng to T + as the postve types of and T as the negatve types. Whle ths assumpton s restrctve n general, there s one case where t s not restrctve at all, namely, where each agent has only two type ndependent ndfference curves over A. For example, ths obvously must hold n the case when the prncpal has only two pure actons, the case orgnally consdered by Glazer and Rubnsten (2004, 2006). Smlarly, consder a type dependent verson of the smple allocaton problem where each agent cares only about whether she receves the good or not, but some types prefer to get the good and other prefer not to. Here the prncpal has as many actons as there are agents (more f she can keep the good), but each agent has only two ndfference curves over A. In ths case, there are only two (nontrval) preferences over (A), so ths formulaton s not restrctve n that context. Of course, smple type dependence also nests type ndependence where β (t ) = 1 for all t or, equvalently, T + = T and T =. Ths s the case consdered n most of the lterature on mechansm desgn wth evdence, partcularly the papers on the value of commtment. For example, t s easy to see that the allocaton problems Example 1 and Example 3b dscussed n Secton 1 satsfy smple type dependence because of the type ndependence assumed. For an example whch satsfes smple type dependence but not type ndependence, consder the publc goods problem, Example 2, dscussed n the Introducton. Suppose A = {0, 1} where 0 corresponds to not provdng the publc good and 1 corresponds to provson. Let u (0, t ) = 0 and u (1, t ) = β (t ). The nterpretaton s that β (t ) s the value of the publc good to type t net of s share of the cost of provson. For types who do not value the publc good very much, we wll have β (t ) < 0, whle there may be other types who value t substantally and therefore have β (t ) > 0. Assume the prncpal s 11

14 utlty functon s the sum of the agent s utltes. Lettng T + denote the set of types wth β (t ) > 0, we can rewrte ths model n the form of smple type dependence wth 0, f a = 0; u (a) = 1, f a = 1 and v (t ) = β (t ) for all t T and all. When we show that commtment has no value, we wll construct an equlbrum wth the same outcome as n the optmal mechansm. The equlbrum we construct s partcularly smple n that t can be constructed from a set of I one agent games whch do not depend on A or preferences over A. Specfcally, we defne the artfcal game for agent as follows. Ths s a game wth two players, the prncpal and agent. Agent has type set T. Type t has acton set T E (t ). The prncpal has acton set X R where X s the compact nterval [mn j mn tj T j v j (t j ), max j max tj T j v j (t j )]. Agent s payoff as a functon of t and the prncpal s choce of x s x, f t T + x, ; otherwse. The prncpal s utlty n ths stuaton s (x v (t )) 2. In other words, the artfcal game s a persuason game where postve types want the prncpal to beleve that v (t ) s large and negatve types want hm to beleve t s small. The structure of A and u (a) play no role. As n the real game defned earler, a strategy for agent s a functon σ : T (T E ) wth the property that σ (s, e t ) > 0 mples e E (t ). We denote a strategy for the prncpal as X : T E X. Theorem 1. If every u exhbts smple type dependence, then commtment and randomzaton have no value for the prncpal, whle robust ncentve compatblty has no cost. That s, there s an optmal ncentve compatble mechansm for the prncpal whch s determnstc and robustly ncentve compatble. In addton, there s a robust perfect Bayesan equlbrum wth the same outcome as n ths optmal mechansm. In ths equlbrum, agent s strategy s a perfect Bayesan equlbrum strategy n the artfcal game for agent. As mentoned n Secton 1, there are earler results for the one agent settng showng that commtment s not valuable to the prncpal. Our result extends these n several ways. Frst, we consder multple agents. Second, because we have multple agents, we can consder robust ncentve compatblty that s, the queston of robustness wth respect to agents belefs about other agents, an ssue absent n the one agent settng. Thrd, our characterzaton of these equlbrum strateges s novel. Even when we restrct our analyss to the case of I = 1 so that we also only have one agent, our results are not nested by the prevous lterature. For the remander of ths 12

15 secton, we dscuss the one agent case, so t refers to the type of the sngle agent, T her set of types, and u her utlty functon. The frst papers to show no value to commtment n a mechansm desgn problem wth evdence were Glazer and Rubnsten (2004, 2006). These papers used weaker assumptons on evdence than we use as they do not requre normalty. However, they assumed that the prncpal only had two actons avalable and the agent s preference over these actons was ndependent of her type. Our assumptons on preferences, restrcted to the one agent case where the prncpal has only two actons, are completely general, though. Unlke our model, randomness may be mportant for optmal mechansm n Glazer Rubnsten because they allow for volatons of normalty. Sher (2011) generalzes the Glazer Rubnsten result va a concavty assumpton. In our notaton, hs assumptons are as follows. Frst, the utlty functon of the agent u s ndependent of her type t. Sher assumes that the agent s utlty s strctly ncreasng wth respect to an order over the prncpal s actons. The concavty assumpton s that for every type t, there exsts a concave functon ϕ t such that v(a, t) = ϕ t (u(a)). That s, gven the agent s type, the prncpal s utlty functon over A s a concave transformaton of the agent s utlty functon. It s not hard to see that ths mples that the prncpal does not need to randomze n the optmal mechansm. In the one agent verson of our model, the prncpal s utlty functon s v(a, t) = u 0 (a) + v(t)u(a). Snce the prncpal s utlty over A gven t depends on more than just the agent s utlty, ths s not nested by Sher s assumptons, even n the type ndependent verson of our model. Fnally, Hart, Kremer, and Perry (2015) gve a verson of the Glazer Rubnsten result whch, lke our result, assumes normalty of evdence. Lke Glazer Rubnsten and Sher, they assume that the agent s utlty functon s ndependent of her type. Lke Sher, they assume that the agent s utlty s ncreasng n a. In addton, they dscuss two other preference assumptons. Frst, they gve a no value to commtment result whch assumes that the prncpal cannot randomze. For ths result, they assume that for each t T, the utlty functon of the prncpal over A can be wrtten as v(a, t) = ϕ t (u(a)) where gven any µ (T ), t µ(t)ϕ t s a sngle peaked functon. That s, there exsts u µ such that t µ(t)ϕ t (u) s strctly ncreasng n u for u < u µ and strctly decreasng n u for u > u µ. Agan, because we allow the prncpal s utlty to depend on a drectly as well as through u(a) n the form v(a, t) = u 0 (a) + u(a)v(t), our model volates ths assumpton n general, even n the type ndependent verson of our model. Also, our assumptons mply that the prncpal does not need to randomze, whle Hart, Kremer, and Perry s frst theorem assumes that he cannot. Hart, Kremer, and Perry s second no value to commtment result allows the prncpal to randomze. Here they use an assumpton called PUB or Prncpal s Unform Best. To state ths, say that ū R s a feasble utlty level for the agent f there exsts 13

16 p (A) such that a p(a)u(a) = ū. Gven a feasble ū, let P (ū) denote the set of p (A) such that a p(a)u(a) = ū. The assumpton then s that for every feasble ū, there exsts p P (ū) such that a p(a)v(a, t) a ˆp(a)v(a, t) for every ˆp P (ū) for every t T. In other words, f the prncpal s constraned to gve the agent utlty ū, then there s a utlty maxmzng way for hm to do ths whch s ndependent of the agent s type. In the one agent case, our model does satsfy ths assumpton. Clearly, f v(a, t) = u 0 (a) + u(a)v(t), then for any t, the prncpal s preferred p P (ū) s any p maxmzng a p(a)u 0 (a). So our model satsfes ther PUB assumpton. Thus the type ndependent one agent verson of our model s nested n ther second result. 4 Optmal Mechansms wth Dye Evdence In lght of Theorem 1, we can compute the outcomes of optmal mechansms by dentfyng the best robust equlbrum for the prncpal. In partcular, we can compute these equlbra by consderng the artfcal game for each agent. In some cases, these equlbra are very easy to characterze. In ths secton, we llustrate by consderng optmal mechansms wth a partcular evdence structure ntroduced by Dye (1985) and studed extensvely n both the economcs and accountng lteratures. We say that the model has Dye evdence f for every, for all t T, ether E (t ) = {T } or E (t ) = {{t }, T }. In other words, any gven type ether has no evdence n the sense that she can only prove the trval event T or can choose between provng nothng and provng exactly her type. Let T 0 denote the set of t T wth E (t ) = {T }. Our artfcal games dffer n one respect from the usual persuason games n the lterature. In our artfcal game, agent both presents evdence and makes a cheap talk clam regardng her type. Of course, f these cheap talk clams convey nformaton, we can always permute agent s use of these clams and the prncpal s nterpretaton of them to obtan another equlbrum. There s also another form of multplcty whch s more standard n the lterature on games wth evdence. In some cases, we may have an equlbrum where the prncpal has the same belefs about the agent whether she presents evdence e or evdence e. In these cases, we can construct an equlbrum where the agent presents evdence e and another where she presents evdence e. Note that n both of these cases, the prncpal s belefs about the agent along the equlbrum path are the same across these varous equlbra. That s, f the agent s type t, the belef the prncpal wll have about t s the same across these equlbra. Wth ths ssue n mnd, we say that an equlbrum n the artfcal game for agent s 14

17 essentally unque f all equlbra have the same outcome n ths sense. To be precse, gven equlbra (σ, x ) and (ˆσ, ˆx ) of the artfcal game for, we say these equlbra are essentally equvalent f for every x X and every t T, we have σ ({(s, e ) T E (t ) x (s, e ) = x} t ) = ˆσ ({(s, e ) T E (t ) ˆx (s, e ) = x} t ). If there s an equlbrum wth the property that every other equlbrum s essentally equvalent to t, we say the equlbrum s essentally unque. The smplest case to consder wth Dye evdence s where the utlty functons are not type dependent at all. We say that the model exhbts type ndependent utlty f u (a, t ) s ndependent of t for all and a. In ths case, we abuse notaton and wrte u (a, t ) = u (a). Note that ths s equvalent to assumng T = (or redefnng u and takng T + = ). The followng results buld on well known characterzatons of equlbra n evdence games usng the Dye evdence structure. Theorem 2. In any model wth Dye evdence, for every, there exsts a unque v such that v = E î v (t ) t T 0 or v (t ) v ó. If T =, the essentally unque equlbrum n the artfcal game for s a pure strategy equlbrum where every type makes the same cheap talk clam, say s, and only types wth evdence wth v (t ) v present (nontrval) evdence. That s, type t sends (s, e (t )) wth probablty 1 where e (t ) = T, f t T 0 or v (t ) < v ; {t }, otherwse. To see the ntuton, note frst that cheap talk cannot be credble n ths game snce every type wants the prncpal to beleve that v s large. So f has no evdence (.e., can only prove the trval event T ), then she has no ablty to convey any nformaton to the prncpal she can only send an unnformatve cheap talk message and prove nothng. If can prove her type s t, she wants to do so only f v (t ) s at least as large as what the prncpal would beleve f she showed no evdence. Thus types wth evdence but lower values of v (t ) wll pool wth the types who have no evdence, leadng to an expectaton of v (t ) equal to v. In ths equlbrum, the prncpal s expectaton of v (t ) wll be v gven a type wth no evdence or wth v (t ) < v and wll equal the true value otherwse. More formally, 15

18 let v ˆv (t ) =, f t T 0 or v (t ) < v ; v (t ), otherwse. For every ˆv = (ˆv 1,..., ˆv I ), let ˆp( ˆv) denote any p (A) maxmzng [ p(a) u 0 (a) + ] u (a)ˆv. a A The followng s a corollary to Theorems 1 and 2. Corollary 1. In any model wth type ndependent utlty and Dye evdence, there s an optmal mechansm P wth P ( t, M(t)) = ˆp( ˆv(t)). In other words, the outcome selected by the prncpal when the profle of types s t s ˆp( ˆv(t)). We can use Corollary 1 to gve smple characterzatons of optmal mechansms n many cases of nterest. Example 1. The smple allocaton problem wth Dye evdence. In ths case, ˆp( t) > 0 ff ˆv (t ) = max j ˆv j (t j ). We can break ndfferences n a partcularly smple way and recast ths characterzaton n the form of a favored agent mechansm. More specfcally, say that P s a favored agent mechansm f there s a threshold v R and an agent, the favored agent, such that the followng holds. Frst, f no agent j proves that v j (t j ) > v, then receves the good. Second, f some agent j does prove that v j (t j ) > v, then the good s gven to the agent who proves the hghest v j (t j ) (where ths may be agent ). More specfcally, a favored agent mechansm where the favored agent s any satsfyng v = max j vj and the threshold v s gven by v s an optmal mechansm. To see ths, fx any t. By defnton, ˆv j (t j ) vj for all j. Hence f v vj for all j, then ˆv (t ) vj for all j. Hence for any j such that E j (t j ) = {T j } or v j (t j ) < vj, we have ˆv (t ) v vj = ˆv j (t j ). So f every j satsfes ths, t s optmal for the prncpal to gve the good to. Otherwse, t s optmal for hm to gve t to any agent who proves the hghest value. As noted n Secton 1, ths mechansm s very smlar to the favored agent mechansm dscussed by BDL for the allocaton problem wth costly verfcaton, a pont we return to below. Example 2. The mult unt allocaton problem wth Dye evdence. It s not hard to extend the above analyss to the case where the prncpal has multple dentcal unts of the good to allocate. Suppose he has K < I unts and, for smplcty, assume he must allocate all of them. Suppose each agent can only have ether 0 or 1 unt. Then 16

19 the prncpal s acton can be thought of as selectng a subset of {1,..., I} of cardnalty K. The prncpal s utlty gven the set Î s Î v (t ). As before, agent s utlty s 0 f she does not get a unt, 1 f she does. In ths case, t s easy to see that the prncpal allocates unts to the K agents wth the hghest values of ˆv (t ). Ths can be computed recursvely as a knd of favored agent mechansm. In other words, we allocate the frst unt to the agent wth the hghest value of v f no other agent proves a hgher value and to the agent wth the hghest proven value otherwse. Once removng ths agent and unt, we follow the same procedure for the second unt, and so on. It s easy to see that the agent wth the hghest value of v s the most favored agent n the sense that at least K agents must prove a value above her v for her to not get a unt. Smlarly, the agent wth the second hghest value of v s the second most favored agent n the sense that at least K 1 of the lower ranked agents must prove a value above her v for her not to get a unt, etc. Example 3. Allocatng a bad. Another settng of nterest s where the prncpal has to choose one agent to carry out an unpleasant task (e.g., serve as department char). It s easy to see that ths problem s effectvely dentcal to havng I 1 goods to allocate snce not recevng the assgnment s the same as recevng a good. Thus we can treat the prncpal s set of feasble actons as the set of subsets of {1,..., I} of cardnalty I 1, nterpreted as the set of agents who are not assgned the task. The one aspect of ths example that may seem odd s that the prncpal s utlty f he assgns the task to agent s then j v j (t j ). On the other hand, t s an nnocuous renormalzaton of the prncpal s utlty functon to subtract the allocaton ndependent term j v j (t j ) from her utlty. In ths case, we see that the prncpal s payoff to assgnng the task to agent s v (t ), so v (t ) s naturally nterpreted as t s level of ncompetence n carryng out the task. One can apply the analyss of the prevous example for the specal case of K = I 1 to characterze the optmal mechansm for ths example. Whle the case of type ndependent utlty wth Dye evdence s partcularly tractable, the case of smple type dependence s not much more dffcult. To see the ntuton, agan consder the artfcal game for where some types wsh to persuade the prncpal that v (t ) s large and other types want to convnce hm v (t ) s small. Suppose that when the agent doesn t prove her type, she makes a cheap talk clam regardng whether her type s postve (.e., she wants the prncpal to thnk v (t ) s large) or negatve (.e., the reverse). Let v + denote the prncpal s belef about v f does not prove her type but says t s postve and let v be the analog for the case where clams her type s negatve. If v + > v, then every postve type wthout evdence prefers to truthfully report that her type s postve, whle every negatve type wthout evdence wll honestly reveal that her type s negatve snce ths leads to the best possble belef from s pont of vew. If s a postve type wth evdence, she wll want to prove her type only f v (t ) > v +, whle a negatve type wth evdence wll prove her type only f v (t ) < v. Hence for ths to 17

20 be an equlbrum, we must have and v + = E î v (t ) (t T + T 0 ) or (t T + \ T 0 and v (t ) v + ) ó v = E î v (t ) (t T T 0 ) or (t T \ T 0 and v (t ) v ) ó. Suppose ths gves a unque value for v + and v. If these values do not satsfy v + v, then ths doesn t work as the postve types wll prefer to act lke negatve types and vce versa. In ths case, we must pool all types. Ths motvates the followng result. Theorem 3. In any model wth Dye evdence, for every, there exsts a unque trple v +, v, and v such that v + = E î v (t ) (t T + T 0 ) or (t T + \ T 0 and v (t ) v + ) ó, and v = E î v (t ) (t T T 0 ) or (t T \ T 0 and v (t ) v ) ó, v = E î v (t ) (t T 0 ) or (t T \ T 0 and v (t ) v ) or (t T + \ T 0 and v (t ) v ) ó. If v + v, then there s an essentally unque equlbrum n the artfcal game for. In ths pure strategy equlbrum, there s a fxed type ŝ such that t reports (ŝ, e (t )) where e (t ) = T, f t T 0 or (t T + and v (t ) < v ) or (t T and v (t ) > v ); {t }, otherwse. If v + > v, there are two equlbra that are not essentally equvalent to one another and every other equlbrum s essentally equvalent to one of the two. One of these s the equlbrum descrbed above. The other s another equlbrum n pure strateges. In ths second equlbrum, there are types ŝ + and ŝ wth ŝ + ŝ such that t T k sends (ŝ k, e k (t )), k =, +, where e + (t ) = T, f t T 0 or v (t ) < v + ; {t }, otherwse, and e (t ) = T, f t T 0 or v (t ) > v ; {t }, otherwse. If v + > v, then there are (essentally) two equlbra n the artfcal game. As the result below wll show, we can always compare these equlbra for the prncpal and the better one s the one whch separates the postve and negatve types. Thus ths s the 18

21 equlbrum that corresponds to the optmal mechansm. Wth ths n mnd, now defne ˆv (t ) as follows. If v + > v, we let ˆv (t ) = v (t ), v +, f t T 0 T + or t T + \ T 0 v, f t T 0 otherwse. T or t T \ T 0 and v (t ) < v ; and v (t ) > v ; If v + v, let ˆv (t ) = v (t ), f (t T + \ T 0 v, otherwse. For each t T, let ˆp( ˆv) denote any p (A) maxmzng and v (t ) > v ) or (t T \ T 0 and v (t ) < v ); [ p(a) u 0 (a) + a A u (a, t )ˆv ]. (1) The followng result s a corollary to Theorems 1 and 3. Corollary 2. In any model wth smple type dependence and Dye evdence, there s an optmal mechansm P wth P ( t, M(t)) = ˆp( ˆv(t)). In other words, the outcome selected by the prncpal when the profle of types s t s ˆp( ˆv(t)). The only part of ths result that requres proof s the clam above that when v + > v, the equlbrum that s better for the prncpal s the one that separates the postve and negatve types. Ths s shown n Appendx D. Example 4. The publc goods problem. As an applcaton, consder the publc goods model dscussed n Secton 3. For smplcty, we wrte out the optmal mechansm only for the case where v + > v for all, but smlar comments apply more generally. We know that n equlbrum, gven a profle of types t, the prncpal s expectaton of v wll be gven by ˆv (t ) defned n equaton (1) above. Then the prncpal wll provde the publc good ff ˆv (t ) > 0. Ths s the analog of the optmal mechansm for costly verfcaton dentfed by Erlanson and Klener (2015). Specfcally, n ther model, we compute what they call an adjusted report for each agent gven t. Exactly as above for the types wth evdence, the adjusted report for a postve type wth evdence s max{v +, v (t )}, whle the adjusted report for a negatve type wth evdence s mn{v, v (t )} for certan cutoffs v + and v. These reports are adjusted by the verfcaton cost and then summed to determne the optmal acton by the prncpal. There s a smple reason why the optmal mechansm here for the smple allocaton problem parallels the optmal mechansm under costly verfcaton dentfed by BDL and why the optmal mechansm here for the publc goods problem parallels the optmal 19

22 mechansm under costly verfcaton dentfed by Erlanson and Klener. To explan, frst note that n both of the costly verfcaton models, the assumpton s that the prncpal can pay a cost c to learn the realzaton of agent s type, t. The agent cannot affect ths verfcaton process. So consder the followng hybrd model. Assume the Dye structure for evdence as above. However, change our assumptons so that when an agent provdes evdence, ths does not mmedately prove anythng to the prncpal. Instead, ths s a verfable message, but one that s costly to verfy. Equvalently, the prncpal has to pay to read the evdence to see that t does ndeed prove what the agent has sad that t proves. Ths adds verfcaton costs to the current model and adds nonverfable types to the earler costly verfcaton models, where the prncpal cannot verfy the valdty (or lack thereof) of a clam to be an unverfable type. It s not hard to show that ths latter change alters the optmal mechansm n BDL only n small and relatvely obvous ways. We conjecture that a smlar result occurs n the Erlanson and Klener model. As we take the verfcaton costs n those models to zero, the optmal mechansms converge contnuously to the optmal mechansms n the model of ths paper. Thus t s not surprsng that for small verfcaton costs, the optmal mechansms look smlar. The more ntrgung observaton s that ths s true for large costs as well. Whle the optmal mechansms look smlar n the two models, ths does not mean that all propertes are the same. Whle BDL note that ther mechansm s ex post and domnant strategy ncentve compatble, the optmal mechansm n Erlanson and Klener s ex post ncentve compatble but not domnant strategy ncentve compatble. Whle these results have been shown only for the model wthout unverfable types, t s easy to show the same holds for the hybrd verson of BDL and we conjecture t holds for the extenson of Erlanson Klener. Also, we beleve that commtment s valuable for the prncpal n both models under costly verfcaton, whether or not there are unverfable types. 5 Proof Sketch In ths secton, we sketch the proof of Theorem 1. For smplcty, we sketch the proof n the context of a specal case, namely, the smple allocaton problem. So assume for ths secton that the prncpal has one unt of an ndvsble good to allocate to some agent. All agents desre the good and the prncpal must gve t to one of the agents. So A = {1,..., I} where a = means that the prncpal allocates the good to agent. The 20

23 utlty functons of the agents are type ndependent wth u (a) = 1, f a = 0, otherwse. The payoff to the prncpal to allocatng the good to agent gven type profle t s v (t ) whch we assume s strctly postve for all and all t. One convenent smplfcaton n the type ndependent case s that we can wrte a mechansm as a functon only from type reports nto choces by the prncpal, where t s understood that f clams type t, she also reports maxmal evdence for t, namely, M (t ). Ths works n the type ndependent case because f clams type t but does not show evdence M (t ), the prncpal knows how to punsh namely, he can gve the good wth zero probablty, the worst possble outcome for. Ths wll deter any such obvous devaton. Of course, the mechansm must stll deter the more subtle devatons to reportng some s t and provdng evdence M (s ). So for ths proof sketch, we wll wrte a mechansm as a functon P : T (A). Fx an optmal mechansm P. Gven ths mechansm, we can construct the probablty that any gven type of any gven agent receves the good. Let ˆp (t ) = E t P ( t, t ). Ths s type t s probablty of beng allocated the good n mechansm P. Partton each T accordng to equalty under ˆp. In other words, for each α [0, 1], let T α = {t T ˆp (t ) = α}. Of course, snce T s fnte, there are only fntely many values of α such that T α. Unless stated otherwse, any reference below to a T α set assumes that ths set s nonempty. Let T denote the partton of T so defned and T the nduced (product) partton of T. It s easy to see that ncentve compatblty s equvalent to the statement that M (s ) E (t ) mples ˆp (t ) ˆp (s ). In other words, f t can report s credbly n the sense that t has avalable the maxmal evdence of s, then the mechansm must gve the good to t at least as often as s. The frst key observaton s that wthout loss of generalty, we can take the mechansm to be measurable wth respect to T. Whle ths property may seem techncal, t s the key property behnd our results and s not generally true for models wth more type dependence. To see why ths property holds n the smple allocaton problem, suppose t s volated. In other words, suppose we have some par of types s, s T such that ˆp (s ) = ˆp (s ) but 21