Mechanisms with Evidence: Commitment and Robustness 1

Transcription

1 Mechansms wth Evdence: Commtment and Robustness 1 Elchanan Ben-Porath 2 Edde Dekel 3 Barton L. Lpman 4 Frst Draft January 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 s unnecessary, randomzaton has no value, and robust ncentve compatblty has no cost. In partcular, for each agent, we construct a smple dsclosure game between the prncpal and agent where the equlbrum strateges of the agents n these dsclosure games gve ther equlbrum strateges n the game correspondng to the mechansm but where the prncpal s not commtted to hs response. In ths equlbrum, the prncpal obtans the same payoff as n the optmal mechansm wth commtment. As an applcaton, we show that certan costly verfcaton models can be characterzed usng equlbrum analyss of an assocated model of evdence.

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 s unnecessary 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 can be computed from a collecton of I auxlary games, where the th game s a smple dsclosure game between agent and the prncpal. As an applcaton, we show that certan mechansm desgn problems wth costly verfcaton nstead of evdence can be solved va an assocated evdence model. 1 To understand the class of mechansm desgn problems we consder, 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 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 dean wth one job slot to allocate to a department n the College. Each department wants the slot and each has prvate nformaton regardng the characterstcs of the person the department would lkely hre wth the slot, nformaton that s relevant to the value to the dean of assgnng the slot to the department. Alternatvely, the prncpal may be a state government whch needs to choose a cty n whch to locate a publc hosptal. The state wants to place the hosptal where t wll be most effcently utlzed, but each cty wants the hosptal and 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 government consders mportant. Extensons: More complex allocaton problems. A broader class of allocaton problems wll also ft n our framework. For example, consder agan the example of a dean gven above, but suppose the dean has several job slots to allocate where each department can have at most one and there are fewer slots than departments. A related problem s the allocaton of a budget across dvsons by the head of a frm. Suppose the organzaton has a fxed amount of money to allocate and that the value produced by a dvson s a functon of ts budget and ts prvately known productvty. Alternatvely, consder a task allocaton problem where the prncpal s a manager who must choose an 1 In a model wth costly verfcaton, the agents do not have evdence to present but the prncpal can learn the true type of an agent at a cost. 1

4 employee to carry out a partcular job. Suppose none of the employees wants to do the task and each has prvate nformaton about how well he would do t. Fnally, we could consder a task that some employees mght want to do and others would not want to do, where both the employee s ablty and desre to do the job are prvate nformaton. 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 to be 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. Alternatvely, the prncpal can maxmze a weghted sum of the agents utltes plus a utlty of her own for the publc good. We wll show that optmal mechansms for these examples share several sgnfcant features. Frst, commtment s not necessary. 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. The robustness of the mechansm n turn mples smlar robustness propertes of the equlbrum whch acheves the same outcome. 2 One useful mplcaton of ths result s that we can compute optmal mechansms by consderng equlbra of the game wthout commtment. In partcular, we gve a relatvely smple characterzaton of an 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. More specfcally, we construct a collecton of I auxlary games, one for each agent, where the game for agent s a smple dsclosure game between agent 2 Ths does not mean that the truth tellng strateges used n the mechansm are also used n the game wthout commtment. In general, agents may be mxng over reports and evdence n the equlbrum of the game. 2

5 and the prncpal. The equlbrum of the game wthout commtment between the I agents and the prncpal whch has the same outcome as the optmal mechansm can be constructed by assgnng to agent her equlbrum strategy n her auxlary game. 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 or to the publc good problem of Example 2, we fnd optmal mechansms remnscent of optmal mechansms n a dfferent context, namely, under costly verfcaton. We dscuss ths connecton to Ben-Porath, Dekel, and Lpman (2014) and to Erlanson and Klener (2015) n Secton 5 where we show that a class of costly verfcaton models can be solved usng our results for evdence models. The paper s organzed as follows. Secton 2 presents the formal model. In Secton 2.5, we state the man results sketched above, ncludng the characterzaton of the best equlbrum for the prncpal. The proof of ths theorem s sketched n Secton 4. In Secton 3, 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 also show that under some condtons, optmal mechansms for costly verfcaton nstead of evdence can be solved usng the optmal mechansms for Dye evdence. We offer concludng remarks n Secton 5. Proofs not contaned n the text are n the Appendx. Related lterature. Our work s related to the lterature on mechansm desgn wth evdence. The frst paper on ths topc was Green and Laffont (1986). We make use of results n Bull and Watson (2007) and Deneckere and Severnov (2008). 3 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) and extended by Sher (2011) and by Hart, Kremer, and Perry (2016, forthcomng). We dscuss these papers n more detal n Secton 5. Also, our result showng that commtment s not necessary 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. The frst papers on ths topc are Grossman (1981) and Mlgrom (1981). Our work makes partcular use of Dye (1985) and Jung and Kwon (1988). Fnally, the papers most closely related to our applcaton to 3 Other papers whch are less drectly related nclude Ben-Porath and Lpman (2012), Kartk and Terceux (2012), and Sher and Vohra (2015). 3

6 costly verfcaton models are Ben-Porath, Dekel, and Lpman (2014) and Erlanson and Klener (2015). 2 Model and Results 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. For example, n the smple allocaton problem, we have A = I where a = means that the good s allocated to. More generally, a can be nterpreted as an allocaton of money (where money s fntely dvsble) as well as other goods, publc or prvate. Each agent has prvate nformaton n the form of a type t where types are dstrbuted ndependently across agents. The fnte set of types of s denoted T and the (full support) pror s denoted ρ. 2.1 Preferences Gven acton a by the prncpal and type profle t, agent s utlty s ū (a, t ), ndependent of t. We need sgnfcantly more structure on the agents utlty functons than ths prvate values assumpton, as we dscuss n detal below. The prncpal s utlty s v(a, t) = u 0 (a) + ū (a, t ) v (t ). For notatonal convenence, we defne v 0 (t 0 ) = 1 so that we can wrte ths as ū (a, t ) v (t ) wth the conventon that the sum runs from = 0 to I. 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 v (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. 4 Of course, one can also nterpret the model as assumng both motvatons for the prncpal. 4 For example, consder the smple allocaton problem where the prncpal s the head of an organzaton who needs to choose one of the agents to promote. Assume that every agent wshes to be promoted, so s utlty s 1 f he s promoted and 0 otherwse. So n ths context, t s natural to assume that v (t ) measures the ablty of type t, not how much the prncpal cares about t s utlty. 4

7 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 ū (a, t ) to û (a, t ) = ū (a, t ) v (t ) and changng the prncpal s utlty functon to û (a, t ). Because û (a, t ) s a postve affne transformaton of ū (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 v (t ) nto the part that comes from how the prncpal evaluates s utlty and how agent evaluates outcomes. Also, note that we allow v (t ) 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 depends on the agents types. Turnng to the detals of our assumptons on the agents utlty functons, we go beyond the type ndependent preferences that the lterature has assumed, but requre that the type dependence takes a partcularly smple, multplcatvely separable, form. Specfcally, we say that ū (a, t ) satsfes smple type dependence f there exst functons u : A R and β : T R such that ū (a, t ) = u (a)β (t ) where β (t ) 0 for all t T. 5 After explanng how these preferences capture the examples above, we provde a renormalzaton that gves a more useful form for analyzng the model and whch also helps show how ths model generalzes the cases consdered n the lterature. To show that smple type dependence accommodates all the examples dscussed n the ntroducton, we llustrate wth two examples. Frst, consder the smple allocaton problem, Example 1. Let A = {1,..., I} where a = means the prncpal allocates the good to agent. Snce every agent desres the good regardless of t, we let β (t ) = 1 for all and let u () = 1 and u (j) = 0 for all j. Fnally, let u 0 (a) 0. Then we can nterpret v (t ) as the value to the prncpal of allocatng the good to agent when hs type s t. As another example, consder the publc goods problem, Example 2. Let A = {0, 1}, where 1 corresponds to provdng the good and 0 to not provdng t. Let β (t ) be the value of the publc good to type t net of s share of the cost of provson. Lettng u (a) = a, then the utlty of agent s ū (a, t ) = u (a)β (t ). If we take the utlty of the prncpal to be the sum of the utltes of the agents, then lettng v (t ) = 1 for every t and every, the utlty of the prncpal s v(a, t) = ū (a, t ) v (t ). 5 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. 5

8 We now renormalze the agents utlty functons. Let u (a, t ) = ū(a, t ) β (t ). Clearly, u (, t ) represents the same preferences over (A) as ū (, t ) for every t and hence the model s strategcally equvalent f we use u for s utlty functon. Note that where u (a), f t u (a, t ) = T + u (a), f t T T + = {t T β (t ) > 0} and T = T \ T +. We refer to T + as the postve types of and T as the negatve types of agent. Thus all types have ndfference curves over (A) defned by u (a), though types may dffer n terms of the drecton of ncrease n utlty. Also, types can dffer n terms of preference ntensty as measured by β (t ). Ths ntensty factor does not have mplcatons for s preferences over (A) but does for the prncpal s n the sense that a change n β (t ), all else equal, changes the prncpal s preferences over (A) condtonal on t. 6,7 Wth ths rewrtng of the agents utltes, we can rewrte the prncpal s utlty as v(a, t) = u (a)β (t ) v (t ) = u (a)v (t ), where v (t ) = β (t ) v (t ) (wth β 0 (t 0 ) defned to be 1). We wll typcally wrte the utlty functons n ths form henceforth. Whle the assumpton of smple type dependence s restrctve n general, t obvously has type ndepdendent preferences as a specal case. If we set T =, we have the assumpton used n most of the lterature on mechansm desgn wth evdence and, n 6 It may seem odd to have a part of the agent s utlty functon whch s rrelevant to her preferences. We can thnk of β as measurng the ntensty of s preferences over A relatve to some other actons whch are not under the prncpal s control and do not affect the prncpal s choces. In other words, f agent s utlty functon s u (a)β (t ) + g (w ) where w s a bundle of prvate goods chosen by the agent, then β s relevant to s preferences overall, but not for s preferences wth respect to those choces whch are ncluded n the model. For example, n the publc goods problem dscussed above, our nterpretaton of β (t ) as the monetary value of the publc good to t mnus her share of the costs mplctly treats w as money and assumes g (w ) = w. 7 Ths does not requre us to assume that the prncpal cares about the ntensty of the agents preferences. If β (t ) > β (t ), we could have v (t ) < v (t ) to an extent whch offsets ths, leavng the prncpal s preferences condtonal on t the same as hs preferences condtonal on t. On the other hand, ths formulaton allows the prncpal to respond to dfferences n ntenstes. The publc goods problem dscussed above s one where the ntensty of agent preferences naturally matters to the prncpal. For another example, consder the smple allocaton problem descrbed above where the prncpal s a utltaran. Then the prncpal wshes to allocate the good to the agent whose preference for the good s the most ntense. ; 6

9 partcular, the papers on the value of commtment. More broadly, n many settngs, the agent has only two type ndependent ndfference curves over A and n ths case, smple type dependence s wthout loss of generalty. For example, f the prncpal has only two actons, then, obvously, there can only be two ndfference curves (at most). The type ndependent verson of ths settng s 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 others prefer not to. 8 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. 2.2 Evdence Each agent may have evdence whch would prove some clams about herself. 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 T. For example, f agent presents a house deed wth her name on t, t proves that she s one of the types who owns a house. We requre the followng propertes. Frst, proof s true. Formally, e E (t ) mples t e. Second, proof s consstent n the sense that 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 a convenent smplfyng assumpton used n much of the lterature. Ths property was ntroduced as the full reports condton by Lpman and Sepp (1995), but s more commonly referred to as normalty, followng Bull and Watson (2007). 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 ). 8 For example, f the good s a task assgnment as dscussed n the extensons of Example 1 n the ntroducton, ths formulaton s natural. 7

10 That s, the event proved by showng all of t s evdence s tself an event that t can prove. Henceforth, we denote ths maxmally nformatve event by M (t ) = e. e E (t ) We sometmes refer to t presentng M (t ) as presentng maxmal evdence. 2.3 Mechansms 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 verson of the Revelaton Prncple 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 and ũ (s, e, t, e t, P ) = a T, (s, e ) T E (t ), and P (a s, e, t, e )u (a, t ) 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 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 ). In words, just as stated above, the agent fnds t optmal to report her type truthfully and present maxmal evdence gven that every other agent does the same. 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 agents preferences satsfy smple type dependence, then for the prncpal, commtment s not necessary, there s no cost to robust ncentve compatblty, and randomzaton has no value. We now make ths more precse. 8

11 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 ). That s, a mechansm s domnant strategy ncentve compatble f each agent has an ncentve to report honestly and present maxmal evdence gven any feasble strateges for her opponents. 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. 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 T, all e 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. As noted above, ex post ncentve compatblty and domnant strategy ncentve compatblty are not equvalent n mechansms wth evdence even wth ndependent prvate values. Robust ncentve compatblty mples both ex post ncentve compatblty and domnant strategy ncentve compatblty, but s not mpled by ether. We gve an example n Appendx A to llustrate. If a mechansm s robustly ncentve compatble, then t has several desrable propertes. Frst, the mechansm does not rely on the prncpal knowng the belefs of the 9

12 agents about each other s types or strateges. Second, the outcome of the mechansm need not change f the agents report publcly and sequentally, rather than smultaneously, regardless of the order n whch they report. 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 states assumptons under whch there s no dfference that s, the best ncentve compatble mechansm for the prncpal s 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. 2.4 Games Fnally, to state what t means that commtment s not necessary, 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. For clarty, we refer to ths as the game wthout commtment. 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 ). 9 The prncpal s set of feasble strateges, Σ P, s the set of functons σ P : T E (A). A belef by the prncpal s a functon µ : T E (T ) gvng the prncpal s belefs about t as a functon of the 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 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 9 We do not requre t to report truthfully and do not requre hs clam of a type to be consstent wth the evdence he presents. That s, we could have σ (s, e t ) > 0 even though s t and e / E (s ). 10

13 e. We study perfect Bayesan equlbra of ths game. 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. See Appendx B for detals. 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 In other words, σ (t ) s optmal for t regardless of the actons played by the other agents, gven the strategy of the prncpal. Note that s strategy s robust wth respect to the strateges of the other agents, but not wth respect to the prncpal s strategy. Gven a perfect Bayesan equlbrum (σ, µ), the prncpal s expected utlty s σ (s, e t )σ P (a s, e)v(a, t). E t (s,e) T E 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. When we show that commtment s unnecessary, we wll construct an equlbrum wth the same outcome as n the optmal mechansm. The equlbrum constructon 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 auxlary 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 orgnal game defned above, 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. 11

14 2.5 Results: Commtment, Determnsm, and Robust Incentve Compatblty Our man results are stated n the followng theorem. Theorem 1. If every u exhbts smple type dependence, then 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 of the game wthout commtment wth the same outcome as n ths optmal mechansm. In ths equlbrum, agent s strategy s also a perfect Bayesan equlbrum strategy n the auxlary game for agent. 3 Optmal Mechansms wth Dye Evdence 3.1 Characterzng the Optmal Mechansm In lght of Theorem 1, we can compute the outcomes of optmal mechansms by dentfyng the best perfect Bayesan equlbrum for the prncpal. In partcular, we can compute these equlbra by consderng the auxlary 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. After characterzng optmal mechansms wth Dye evdence, we show that these results can also be used to characterze optmal mechansms n a dfferent settng. Specfcally, n certan models wthout evdence but where the prncpal can verfy the type of an agent at a cost, we show that the optmal mechansm can be computed from the optmal mechansm for an assocated Dye evdence model. 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 has access to perfect evdence and can then choose between provng nothng (.e., provng T ) and provng exactly her type. Let T 0 denote the set of t T wth E (t ) = {T }. In what follows, we sometmes refer to types who have only trval evdence as havng no evdence and types wth E (t ) = {T, {t }} as havng evdence. A small complcaton n statng our results s that there s an essentally rrelevant but unavodable multplcty of equlbrum n our auxlary games. To understand ths, note that our auxlary games dffer n one respect from the usual persuason games n the 12

15 lterature n that agent both presents evdence and makes a cheap talk clam regardng her type n the former. 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 auxlary game for agent s essentally unque f all equlbra have the same outcome n ths sense. To be precse, gven equlbra (σ, X ) and (ˆσ, ˆX ) of the auxlary 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 Ä (s, e ) T E (t ) ˆX (s, e ) = x ä 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 other words, T =, so u (a, t ) = u (a) for all 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 auxlary game for s a pure strategy equlbrum where every type makes the same cheap talk clam, say s, and only types wth evdence and wth v (t ) > v present (nontrval) evdence. That s, type t sends (s, e (t )) wth probablty 1 where e T, f t (t ) = T 0 or v (t ) v ; {t }, otherwse. 13

16 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, 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, wth or wthout commtment, 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 3. The smple allocaton problem (Example 1) wth Dye evdence. In ths case, ˆp( t) > 0 ff ˆv (t ) = max j ˆv j (t j ). That s, the good s gven to one of the agents wth the hghest ˆv j (t j ) or, equvalently, who s beleved to have the hghest v 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 ). Then a favored agent mechansm where the favored agent s any satsfyng v = max j v j and the threshold v s gven by v s an optmal mechansm. To see ths, fx any 14

17 t. By defnton, ˆv j (t j ) v j for all j. Hence f v v j for all j, then ˆv (t ) v j for all j. Hence for any j such that E j (t j ) = {T j } or v j (t j ) v j, we have ˆv (t ) v v j = 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 we dscuss further below, ths mechansm s remnscent of the favored agent mechansm dscussed by Ben-Porath, Dekel, and Lpman (2014) for the allocaton problem wth costly verfcaton. We now extend the smple allocaton problem as dscussed n the ntroducton. Example 4. 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 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 and 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 ) as computed above. It s not dffcult to show that ths can be nterpreted as a knd of recursve favored agent mechansm. 10 Example 5. 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, 10 More specfcally, 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. 15

18 the case of smple type dependence s not much more dffcult. To see the ntuton, agan consder the auxlary 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 and 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 be an equlbrum, we must have 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. Suppose ths gves a unque value for v + and v. If these values do not satsfy v + v, then we can t have an equlbrum of ths knd snce the postve types who don t present evdence wll prefer to act lke the negatve type and vce versa. In ths case, we must pool all types. If we do have v + v, then these strateges do gve an equlbrum. In the case where v + = v, the cheap talk does not convey any extra nformaton. When v + > v, cheap talk s useful, but there s another equlbrum as well where cheap talk s dsregarded or treated as babblng, as n all models wth cheap talk. The followng lemma provdes the background for the equlbrum characterzaton. Lemma 1. In any model wth Dye evdence, for every, there exsts a unque trple v +, v, and v such that 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, 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. Usng these cutoffs, we can characterze the equlbra of the auxlary game for. Theorem 3. If v + v, then there s an essentally unque equlbrum n the auxlary game for. In ths pure strategy equlbrum, there s a fxed type ŝ such that t reports (ŝ, e (t )) where e T, f t (t ) = T 0 or (t T + and v (t ) v ) or (t T and v (t ) v ); {t }, otherwse. 16

19 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. The frst s exactly the same strategy profle as above. 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. Thus f v + > v, then there are (essentally) two equlbra n the auxlary 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 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 does not follow mmedately from Theorems 1 and 3 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 equlbrum s better snce t provdes more nformaton for the prncpal. Ths s shown n Appendx F. 17

20 Example 8. The publc goods problem. As an applcaton, consder the publc goods model dscussed n Secton 1. 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. Just as the analyss of Example 3 above was remnscent of Ben-Porath, Dekel, and Lpman s (2014) analyss of allocaton wth costly verfcaton, the optmal mechansm n ths example s very remnscent of the optmal mechansm under costly verfcaton dentfed by Erlanson and Klener (2015). 3.2 Costly Verfcaton In Ben-Porath, Dekel, and Lpman (2014) and Erlanson and Klener (2015), costly verfcaton s modeled by assumng the prncpal can pay a cost c to check or learn the realzaton of agent s type, t. The agent cannot affect ths verfcaton process. By contrast, n the evdence model we consder here, the prncpal cannot acqure nformaton about an agent wthout somehow nducng the agent to reveal t. Despte ths dfference, the optmal mechansms look very smlar. Ben-Porath, Dekel, and Lpman dentfy an optmal mechansm n the costly verfcaton verson of the smple allocaton problem whch s very smlar to the optmal mechansm here. In both cases, there s a favored agent and a threshold. In both cases, f no non favored agent reports above the threshold, the favored agent receves the object regardless of hs report. Here, reportng above the threshold means to prove a value of v (t ) above the threshold. In Ben-Porath, Dekel, and Lpman, t means to make a cheap talk report of a type such that the type mnus the checkng cost s above the threshold. In both cases, f some non favored agent reports above the threshold, the good goes to the agent wth the hghest such report. In the costly verfcaton model, ths s after checkng ths type. Smlarly, Erlanson and Klener consder the publc goods model under costly verfcaton. In ther mechansm and n the optmal mechansm here, we compute adjusted reports for each agent gven t. In both cases, the adjusted report for a postve type s max{v +, v (t )}, whle the adjusted report for a negatve type s mn{v, v (t )} for certan cutoffs v + and v. Just as wth the allocaton problem, the dfference between these two scenaros s that the report s proven n the evdence model and s a cheap talk clam of a type adjusted by the verfcaton cost n the costly verfcaton model. In both the allocaton and publc goods problems, these reports are adjusted by the verfcaton cost and then summed to determne the optmal acton by the prncpal. Agan, ths ncludes some checkng n the costly verfcaton model. 18

21 These parallels are specal cases of a more general result that certan costly verfcaton models can be rewrtten as a Dye evdence model, so that the optmal mechansm can be computed drectly from our results about optmal mechansms wth evdence. In the subsequent text, we explan ths for the smple allocaton problem. We gve the more general result and explan the connecton to Erlanson and Klener n Appendx G. So consder the followng alternatve model. Agan, the prncpal has a sngle unt of an ndvsble good to allocate, each agent prefers to receve the good, and v (t ) s the payoff to the prncpal to allocatng the good to agent. For smplcty, we assume here that v (t ) > 0 for all t and all and that no two types have the same value of v (t ). Instead of assumng that agents may have evdence to present, assume that the prncpal can pay a cost c > 0 to learn the type of agent, whch we refer to as checkng. As shown n Ben-Porath, Dekel, and Lpman, we can characterze an optmal mechansm as specfyng functons p : T ({1,..., I}) and q : T [0, 1] where p(t) gves the probablty dstrbuton over whch agent the prncpal gves the good to as a functon of type reports t and q (t) gves the probablty that the prncpal checks gven reports t. The prncpal s objectve functon then s E t [ p (t)v (t ) q (t)c ] where p(t) = (p 1 (t),..., p I (t)). The ncentve compatblty constrants are ˆp (t ) ˆp (t ) ˆq (t ), t, t T, where ˆp (t ) = E t p (t) and ˆq (t ) = E t q (t). To see ths, note that f type t reports truthfully, then whether he s checked or not, he wll receve the good wth expected probablty ˆp (t ). On the other hand, f he msreports and clams to be type t, he wll be checked wth expected probablty ˆq (t ). In ths case, the prncpal wll learn that he has led and wll not gve hm the good. Thus hs probablty of recevng the good s the same as t s probablty, mnus the probablty of beng checked. It s not hard to show that the soluton s monotonc n the sense that ˆp (t ) ˆp (t ) f v (t ) v (t ). For each, let t 0 be the type wth the smallest value of v (t ). The monotoncty of the soluton mples that f ncentve compatblty holds for type t 0, then t holds for every other type of agent. Hence we can rewrte the ncentve compatblty constrants as ˆq (t ) ˆp (t ) ˆp (t 0 ), t T,. It s easy to see that the optmal soluton must set ˆq as small as possble snce checkng s costly. Hence ˆq (t ) = ˆp (t ) ˆp (t 0 ) for all t. We can then rewrte the objectve functon as E t [ˆp (t )v (t ) ˆq (t )c ] = E t îˆp (t )(v (t ) c ) + ˆp (t 0 ó )c. 19

22 Thus we can solve the prncpal s problem by choosng p to maxmze the above subject to the constrant that ˆp (t ) ˆp (t 0 ) for all t T and all. Rewrtng the objectve functon once more, we can wrte t as [ ] E t [ˆp (t )ṽ (t )] = E t p (t)ṽ (t ) where { v (t ) c, f t t 0 ṽ (t ) = v (t 0 ) c + c ρ (t 0), f t = t 0. (Recall that ρ s the prncpal s pror over T.) Now consder the smple allocaton problem wth Dye evdence where the value to the prncpal of allocatng the good to agent s ṽ (t ). Assume that E (t 0 ) = {T } and E (t ) = {{t }, T } for all t t 0. In ths case, the objectve functon s the same as the one above. The ncentve compatblty constrant s smply that no type who can prove hs type wshes to mtate the type who cannot. That s, ˆp (t ) ˆp (t 0 ), the same ncentve compatblty constrant as n the costly verfcaton model. Thus we can drectly apply our characterzaton of optmal mechansms wth Dye evdence to derve the soluton to ths problem. It s straghtforward to use ths to gve a characterzaton for the orgnal costly verfcaton model by nvertng the ṽ s and wrtng the soluton n terms of the orgnal v s. In partcular, we obtan the optmal mechansm dentfed by Ben-Porath, Dekel, and Lpman. To see ths, for each, defne the cutoffs ṽ from the ṽ functons the same way we defned v from the v functons. That s, ṽ s the expectaton of ṽ condtonal on t not havng evdence (here beng the type t 0 ) or havng ṽ (t ) ṽ. As shown above, the optmal mechansm for ths allocaton problem wth evdence s to select a favored agent who has ṽ ṽj for all j and to set threshold ṽ. Ths mples that t s optmal to gve the good to f ṽ j (t j ) = v j (t j ) c j ṽ for all j and to gve the good to that agent j who maxmzes v j (t j ) c j otherwse. Ths s exactly the mechansm dscussed by Ben-Porath, Dekel, and Lpman. One can use ths approach to characterze optmal mechansms wth costly verfcaton for less smple allocaton problems such as the extensons of Example 1 n Secton 1 and for the model of Erlanson and Klener, as dscussed n Appendx G. 4 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 20