Dynamic Mechanism Design:

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1 Dynamc Mechansm Desgn: Incentve Compatblty, Pro t Maxmzaton and Informaton Dsclosure Alessandro Pavan Northwestern Unversty Ilya Segal Stanford Unversty May 8, 2009 Juuso Tokka Stanford Unversty Abstract We examne the desgn of ncentve-compatble screenng mechansms for dynamc envronments n whch the agents types follow a (possbly non-markov) stochastc process, decsons may be made over tme and may a ect the type process, and payo s need not be tme-separable. We derve a formula for the dervatve of an agent s equlbrum payo wth respect to hs current type n an ncentve-compatble mechansm, whch summarzes all rst-order condtons for ncentve compatblty and generalzes Mrrlees s envelope formula of statc mechansm desgn. We provde condtons on the envronment under whch ths formula must hold n any ncentvecompatble mechansm. When specalzed to quas-lnear envronments, ths formula yelds a dynamc revenue-equvalence result and an expresson for dynamc vrtual surplus, whch s nstrumental for the desgn of optmal mechansms. We also provde some su cent condtons for ncentve compatblty, and for ts robustness to an agent s observaton of the other agents past and future types. We apply these results to a number of novel settngs, ncludng the desgn of pro t-maxmzng auctons and durable-good sellng mechansms for buyers whose values follow an AR(k) process. JEL Class caton Numbers: D82, C73, L. Keywords: asymmetrc nformaton, stochastc processes, ncentves For useful suggestons, we thank semnar partcpants at varous nsttutons where ths paper was presented. A specal thank s to L Hao and Narayana Kocherlakota for very detaled comments. Pavan also thanks the hosptalty of Collego Carlo Alberto where part of ths work was completed. Ths paper supersedes prevous workng papers Revenue Equvalence, Pro t Maxmzaton and Transparency n Dynamc Mechansms by Segal and Tokka and Long-Term Contractng n a Changng World by Pavan. Earler versons contaned a secton on the ndependentshock approach whch has now been ncorporated n a companon paper (stll work n progress). Whle watng for a draft, the reader nterested n ths part can look at Secton 3.3 n the November 2008 verson posted on our webpages.

2 Introducton We consder the desgn of ncentve-compatble mechansms n a dynamc envronment n whch agents receve prvate nformaton over tme and decsons may be made over tme. The model allows for seral correlaton of the agents prvate nformaton as well as the dependence of ths nformaton on past decsons. For example, t covers as specal cases such problems as the allocaton of resources to agents whose valuatons follow a stochastc process, the procedures for sellng new experence goods to consumers who re ne ther valuatons upon consumpton, or the desgn of mult-perod procurement auctons for bdders whose cost parameters evolve stochastcally over tme and may exhbt learnng-by-dong e ects. A fundamental d erence between dynamc and statc mechansm desgn s that n the former, an agent has access to a lot more potental devatons. Namely, nstead of a smple msrepresentaton of hs true type, the agent can make ths msrepresentaton condtonal on the nformaton he has observed n the mechansm, n partcular on hs past types, hs past reports (whch need not have been truthful), and past decsons (from whch he can make nferences about the other agents types). Despte the resultng complcatons, we delver some general necessary condtons for ncentve compatblty and some su cent condtons, and then use them to characterze optmal (e.g. pro t-maxmzng) mechansms n several applcatons. The cornerstone of our analyss s the dervaton of a formula for the dervatve of an agent s equlbrum expected payo n an ncentve-compatble mechansm wth respect to hs prvate nformaton. Smlarly to Mrrlees s rst-order approach for statc envronments (Mrrlees, 97), our formula (hereafter referred to as dynamc payo formula) provdes an envelope-theorem condton summarzng local ncentve compatblty constrants. In contrast to the statc model, however, the dervaton of ths formula reles on ncentve compatblty n all the future perods, not just n one gven perod. Furthermore, unlke some of the earler papers about dynamc mechansm desgn, we dentfy condtons on the prmtve envronment for whch the dynamc payo formula must hold n any ncentve-compatble mechansm (not just n well-behaved ones). In addton to carryng over the usual statc assumptons of smoothness of the agent s payo functon n hs type and connectedness of the type space (see, e.g., Mlgrom and Segal, 2002), the dynamc settng requres addtonal assumptons on the stochastc process governng the evoluton of each agent s nformaton. Intutvely, our dynamc payo formula represents the mpact of an (n ntesmal) change n the agent s current type on hs equlbrum expected payo. In addton to the famlar drect e ect of the current type on the agent s utlty, the formula also accounts for the current type s mpact on the type dstrbutons n each of the future perods, whch s both drect and ndrect through ts mpact on the dstrbuton of types n ntermedate perods. All these stochastc e ects are

3 summarzed wth a functon that can be nterpreted as a (nonlnear) mpulse response of the future type to the current type. Our dynamc payo formula adds up the utlty e ects of all the future types weghted by ther mpulse responses to the current type. As for the current type s e ects through the agent s messages to the mechansm, the formula gnores them, by the usual envelope theorem logc. For ease of exposton, n the rst part of the paper (Secton 3) we consder an envronment wth a sngle agent who observes all the relevant hstory of the mechansm and derve the dynamc payo formula for ths envronment. In Secton 4 we adapt the dynamc payo formula to a multagent envronment, n whch an agent may observe only a part of the entre hstory generated by the mechansm, and must therefore form belefs about the unobserved parts such as the types of the other agents as well as the unobserved past decsons made by the mechansm. We show that the sngle-agent analyss extends to mult-agent mechansms provded that the stochastc processes governng the evoluton of the agents types are ndependent of one another, except through ther e ect on the decsons observed by the agents. In other words, we show how the famlar Independent Types assumpton for statc mechansm desgn should be properly adjusted to a dynamc settng to guarantee that the agents equlbrum payo s can stll be pnned down by an envelope formula. For the specal case of quaslnear envronments, we use the dynamc payo formula to establsh a dynamc revenue equvalence theorem that lnks the payment rules n any two Bayesan ncentvecompatble mechansms that mplement the same allocaton rule. In partcular, for a sngle-agent determnstc mechansm, ths theorem pns down, n each state, the total payment that s necessary to mplement a gven allocaton rule, up to a state-ndependent constant. Wth many agents, or wth a stochastc mechansm, the theorem pns down the expected payments as functon of each agent s type hstory, where the expectaton s wth respect to the other agents types and/or the stochastc decsons taken by the mechansm. However, f one requres a strong form of robustness, whch we call Other-Ex Post Incentve Compatblty (OEP-IC) accordng to whch the mechansm must reman ncentve-compatble even f an agent s shown at the begnnng of the game all the other agents (future) types and randomzaton outcomes then the theorem agan pns down, for each agent and for each state, the total payment up to a state-ndependent constant (whch may depend on the other agents types and randomzaton outcomes). Next, we use the dynamc envelope formula to express the expected pro ts n an ncentvecompatble and ndvdually ratonal mechansm as the expected vrtual surplus, approprately de ned for the dynamc settng. Ths dervaton uses only the agents local ncentve constrants, and only the partcpaton constrants of the agents lowest types n the ntal perod. Ignorng all the other ncentve and partcpaton constrants yelds a dynamc Relaxed Program, whch 2

4 s n general a dynamc programmng problem. In partcular, the Relaxed Program gves us a smple ntuton for the optmal dstortons ntroduced by a pro t-maxmzng prncpal: Snce only the rst-perod partcpaton constrants bnd (due to the unlmted bondng possbltes n the quaslnear envronment wth unbounded transfers), the dstortons trade o e cency for extracton of the agents rst-perod nformaton rents. However, due to nformatonal lnkages n the stochastc type processes, the prncpal dstorts the agents consumptons not only n perod one, but also n any subsequent perod n whch hs type s responsve to the rst-perod type, as measured by our mpulse response functon. In partcular, we nd that when an agent s type n perod t > hts ts hghest or lowest possble value, the nformatonal lnkage dsappears and the prncpal mplements the e cent level of consumpton n that perod (provded that payo s are addtvely tme-separable). However, for ntermedate types n perod t, the optmal mechansm entals dstortons (for example, when types are postvely correlated over tme n the sense of Frst-Order Stochastc Domnance, and the agents payo s satsfy the sngle-crossng property, the optmal mechansm entals downward dstortons). Thus, n contrast to the statc model, wth a contnuous but bounded type space, dstortons n each perod t > are typcally non monotonc n the agents types. Ths s also n contrast wth the results of Battagln (2005) for the case of a Markov process wth only two types n each perod. Studyng the Relaxed Program s not satsfactory unless one ts solutons can be shown to satsfy all of the remanng ncentve and partcpaton constrants. We provde a few su cent condtons for these constrants to be sats ed. In partcular, we show that n the case where the agents types follow a Markov process and ther payo s are Markovan n ther types (so that t s enough to check one-stage devatons from truthtellng), a su cent condton for an allocaton rule to be mplementable s that the partal dervatve of the agent s expected utlty wth respect to hs current type when he msreports be nondecreasng n the report. One can then use the dynamc payo formula to calculate ths partal dervatve the condton s farly easy to check. (Unfortunately, ths condton s not necessary for ncentve-compatblty a tght characterzaton s evasve because of the multdmensonal decson space of the problem.) Ths su cent condton also turns useful when checkng ncentve compatblty s some non-markov settngs that are su cently separable. In some standard settngs we can actually state an even smpler su cent condton for ncentve compatblty, whch also ensures that ncentve compatblty s robust to an agent learnng n advance all of the other agents types (and therefore to any weaker form of nformaton leakage n the mechansm). Ths condton s that the transtons that descrbe the evoluton of the agents prvate nformaton are monotone n the sense of Frst-Order Domnance, the payo s satsfy the sngle-crossng property, and the allocaton rule s strongly monotonc n the sense that the 3

5 consumpton of a gven agent n any perod s nondecreasng n each of the agent s type reports, for any gven pro le of reports by the other agents. In Secton 5, we show how the aforementoned results can be put to work n a couple of smple, yet llumnatng, applcatons. The rst applcaton s a settng where the agents types follow an autoregressve stochastc process of degree k (AR(k)) and where each agent s payo s a ne n hs types (but not necessarly n hs consumpton). Ths spec caton can capture for example auctons wth ntertemporal capacty constrants, habt formaton, and learnng-by-dong. In ths case, the prncpal s Relaxed Program turns out to be very smlar to the expected socal surplus maxmzaton program, the only d erence beng that the agents true values n each perod are replaced by ther correspondng vrtual values. In the AR(k) case, the d erence between an agent s true value and hs vrtual value n perod t, whch can be called hs handcap n perod t, s determned by the agent s rst-perod type, the hazard rate of the rst perod type s dstrbuton, and the mpulse response functon, whch n the case of an AR(k) process s a constant that s ndependent of the realzatons of the process. Intutvely, the mpulse response constant determnes the nformatonal lnk between perod t and perod, whle the rst-perod hazard rate captures the mportance that the prncpal assgns to the trade-o between e cency and rent-extracton as perceved from perod one s perspectve (just as n the statc model). Snce the handcaps depend only on the rst-perod type reports, the Relaxed Program at any perod t 2 can be solved by runnng an e cent (.e., expected surplus-maxmzng) mechansm on the handcapped values. Thus, whle constructng an e cent mechansm may n general requre solvng an nvolved dynamc programmng problem (due to possble ntertemporal payo nteractons), once t s constructed t can be easly converted nto a soluton to the pro t-maxmzng Relaxed Program. We also use the fact that the soluton to the Relaxed Program looks quas-e cent from perod 2 onward to show that t can be mplemented n a mechansm that s ncentve compatble from perod 2 onward (followng truthtellng n perod one). Ths can be done for example usng the Team Mechansm payments proposed by Athey and Segal (2007) to mplement e cent allocaton rules. As for ncentve compatblty n perod one, we were only able to check t applcaton-by-applcaton, but we have been able to verfy t n a few specal settngs. The second applcaton s an envronment n whch the agents types contnue to follow an AR(k) process, but where all agents preferences are addtvely tme-separable, wth arbtrary ow payo s. Ths settng s partcularly smple because the Relaxed Program separates across perods and states and so we do not need to solve a dynamc programmng problem. Under the standard monotone hazard rate assumpton on the agents ntal type dstrbuton and the standard thrddervatve assumpton on ther utlty functons, the Relaxed Program s solved by a Strongly The term handcapped aucton was rst used n Eso and Szentes (2007), but n a more specal settng (see Secton 2). 4

6 Monotone allocaton rule, whch then mples that t s mplementable n an ncentve-compatble mechansm (and one that s robust to nformaton leakage). The optmal mechansm exhbts nterestng propertes: for example, an agent s consumpton n a gven perod depends only on hs ntal report and hs current report, but not on ntermedate reports. Ths can be nterpreted as a scheme where the agents make up-front payments that reduce ther future dstortons. The rest of the paper s organzed as follows. Secton 2 bre y dscusses some related lterature. Secton 3 presents the results for the sngle-agent case. Secton 4 extends the analyss to quas-lnear settngs wth multple agents. Secton 5 presents a few applcatons. All proofs are n the Appendx at the end of the manuscrpt. 2 Related Lterature The last few years have wtnessed a fast-growng lterature on dynamc mechansm desgn. A number of recent papers propose transfer schemes for mplementng e cent (expected surplusmaxmzng) mechansms that generalze statc VCG and expected-externalty mechansms (e.g., Bergemann and Välmäk (2007),Athey and Segal (2007), and references theren), but do not provde a general analyss of ncentve compatblty n dynamc settngs. Our analyss s more closely related to the poneerng work of Baron and Besanko (984) on regulaton of a natural monopoly and the more recent paper of Courty and L (2000) on advance tcket sales. Both papers consder a two-perod model wth one agent and use the rst-order approach to derve optmal mechansms. The agent s types n the two perods are serally correlated and ths correlaton determnes the dstortons n the optmal mechansm. Courty and L also provde some su cent condtons for the allocaton rule to be mplementable. Our paper bulds on the deas n these papers but extends the approach to allow for multple perods, multple agents, and more general payo s and stochastc structure. Contrary to these early papers, we also provde condtons on the prmtve envronment that valdate the rst-order approach. Battagln (2005) derves the optmal sellng mechansm for a monopolst facng a sngle consumer whose type follows a two-state Markov process. Our results for a model wth contnuous types ndcate that many of hs predctons are spec c to hs two-type settng (we dscuss ths n greater detal n subsecton 4.6). Gershkov and Moldovanu (2008a) and Gershkov and Moldovanu (2008b) consder both e cent and pro t maxmzng mechansms to allocate objects to buyers that arrve randomly over tme. Snce each agent n ther model lves only nstantaneously, ther ncentve constrants are statc. The papers payo -equvalence can be vewed as a statc result appled separately to each short-lved agent. 2 2 Other recent papers that study dynamc pro t-maxmzng mechansms nclude Bognar, Borgers, and Meyer-ter Vehn (2008) and Zhang (2008). The key d erence between these papers and ours s that these papers look at 5

7 Eso and Szentes (2007) consder a two-perod model wth many agents but wth a sngle decson n the second perod. They propose a novel approach to the characterzaton of optmal mechansms, whch uses the Probablty Integral Transform Theorem ( e.g., Angus, 994) to represent an agent s second-perod type as a functon of hs rst-perod type and a random shock that s ndependent of the rst-perod type. In Pavan, Segal, and Tokka (2009), we show how the ndependent-shock approach can be used to dentfy condtons for ncentve compatblty n n nte-horzon settngs. The results n that paper complement those n the current one n that, when appled to ntehorzon models, they permt one to valdate the dynamc payo formula dent ed n ths paper under a d erent (and not nested) set of assumptons. Eso and Szentes also derve a pro t-maxmzng aucton and con the term handcapped aucton to descrbe t. However, n ther two-perod AR() settng, t turns out that any ncentvecompatble mechansm, not just a pro t-maxmzng one, can be vewed as a handcapped aucton. What we nd more surprsng s that, under the specal assumptons of an AR(k) type process and a ne payo s, even wth many perods the optmal mechansm remans a handcapped mechansm. The dstngushng feature of such mechansms s that the allocaton n a gven perod depends only on that perod s reports and the rst-perod reports but not on any ntermedate reports. 3 The paper s also related to the macroeconomc lterature on dynamc optmal taxaton. Whle the early lterature typcally assumes..d. shocks (e.g. Green (987), Thomas and Worrall (990), Atkeson and Lucas (992)), the more recent lterature consders the case of persstent prvate nformaton (e.g. Fernandes and Phelan (2000), Golosov, Kocherlakota, and Tsyvnsk (2003), Kocherlakota (2005), Golosov and Tsyvnsk (2006), Kapcka (2006), Tchsty (2006), Bas, Marott, Plantn, and Rochet (2007), Zhang (2006), Wllams (2008)). Whle our work shares several modellng assumptons wth some of the papers n ths lterature, ts key dstnctve aspect s the general characterzaton of ncentve compatblty as opposed to some of the features of the optmal mechansm n the context of spec c applcatons. 4 Dynamc mechansm desgn s also related to the lterature on multdmensonal screenng, as noted, e.g., n Rochet and Stole (2003). Nevertheless, there s a sense n whch ncentve compatblty s much easer to ensure n a dynamc mechansm than n a statc multdmensonal mechansm. Ths s because n a dynamc envronment an agent s asked to report each dmenson of hs prvate nformaton before learnng the subsequent dmensons, and so he has fewer devatons partcular ssues that can emerge n dynamc envronments, such as costly partcpaton, whle our abstracts from some of these ssues but nstead provdes a more general characterzaton of ncentve-compatblty. 3 Also, whle Eso and Szentes use ther model to study prmarly the e ects of the seller s nformaton dsclosures on surplus extracton n a specal settng, here we focus on the characterzaton of ncentve compatble mechansms n general dynamc settngs. 4 Some of ths work lmts ts analyss to the characterzaton of rst-order condtons for ntetemporal consumpton smoothng (the nverse Euler equaton), ether leavng the dynamcs of the optmal mechansm unspec ed or solvng for t numercally. 6

8 avalable than n the correspondng statc envronment n whch he observes all the dmensons at once. Because of ths, the set of mplementable allocaton rules proves to be sgn cantly larger n a dynamc envronment than n the correspondng statc multdmensonal envronment. For example, the pro t-maxmzng dynamc allocaton rules obtaned n our applcatons would not be mplementable f the agents were to observe all of ther prvate nformaton at the outset of the mechansm. We also touch here upon the ssue of transparency n mechansms. Calzolar and Pavan (2006a) and Calzolar and Pavan (2006b) study ts role n envronments n whch downstream actons (e.g. resale o ers n secondary markets, or more generally contract o ers n sequental common agency) are not contractble upstream. Pancs (2007) also studes the role of transparency n envronments where agents take nonenforceable actons such as nvestment or nformaton acquston. 3 Sngle-agent case 3. General setup 3.. The Envronment We consder an envronment wth one agent and ntely many perods, ndexed by t = ; 2; : : : ; T. In each perod t there s a contractble decson y t 2 Y t, whose outcome s observed by the agent. (In the next secton we apply the model to a more general setup where y t s the part of the decson taken n perod t that s observed by the agent.) Each Y t s assumed to be a measurable space wth the sgma-algebra left mplct. The set of all possble hstores of feasble decsons s denoted by Y Q T = Y : That Y s a subset of Q T = Y captures the possblty that the decsons that are feasble n perod t may depend on the decsons made n prevous perods. Gven Y; for any t we then let Y t fy t 2 Q t = Y : (y t ; y t+ ; :::; y T ) 2 Y for some (y t+ ; :::; y T ) 2 Q T =t+ Y g denote the set of feasble perod-t hstores of decsons. 5 For the full hstores we drop the superscrpts so that y s an element of Y Y T. Before the perod-t decson s taken, the agent receves some prvate nformaton t 2 t R. We mplctly endow the set t wth the Borel sgma-algebra. We refer to t as the agent s current type. The set of all possble type hstores at perod t s then denoted by t Q t =. An element of T s referred to as the agent s type. The dstrbuton of the current type t may depend on the entre hstory of past types and on the hstory of past decsons ( t ; y t ) 2 t Y t. In partcular, we assume that the dstrbuton of t s governed by a hstory-dependent probablty measure ( kernel ) F t j t ; y t on t ; 5 By conventon, all products of measurable spaces encountered n the text are endowed wth the product sgmaalgebra. 7

9 such that F t (Aj) : t Y t! R s measurable for all measurable A t : 6 Note that the dstrbuton of t depends only on varables observed by the agent. We denote the collecton of kernels by F F t : t Y t! ( t ) T t= ; where for any measurable set A, (A) denotes the set of probablty measures on A. We abuse notaton by usng F t (j t ; y t ) to denote the cumulatve dstrbuton functon (c.d.f.) correspondng to the measure F t ( t ; y t ). The agent s a von Neumann-Morgenstern decson maker whose preferences over lotteres over Y are represented by the expectaton of a (measurable) Bernoull utlty functon U : Y! R: Although some form of tme separablty of U s typcally assumed n applcatons, ths s not needed for our results. What s essental s only that the agent s preferences be tme consstent, whch s captured here by the assumpton that the agent s an expected-utlty maxmzer, wth a Bernoull functon that s constant over tme. An often encountered specal case n applcatons s one where prvate nformaton evolves n a Markovan fashon, and where the agent s payo s Markovan n the followng sense. De nton The envronment s Markov f (M) for all t, and all ( t ; y t ) 2 t Y t, F t ( t ; y t ) does not depend on t 2, and (M2) there exsts functons A t : t Y t T! R ++ t= and B t : t Y t! R T such that for all t= (; y) 2 Y, U (; y) = TX t= Yt = A ( ; y )! B t t ; y t : Condton (M) guarantees that the stochastc process governng the evoluton of the agent s type s Markov, whle Condton (M2) ensures that n any gven perod t, after observng hstory t ; y t, the agent s von Neumann-Morgenstern preferences over future lotteres depend on hs type hstory t only through the current type t. In partcular, t encompasses the case of addtve separable preferences (A t preferences (B t t ; y t = 0 for all t < T ). t ; y t = for all t) as well as the case of multplcatve separable 6 Throughout, we adopt the conventon that for any set A, A 0 f?g. 8

10 3..2 Mechansms A mechansm n the above envronment assgns a set of possble messages to the agent n each perod. The agent sends a message from ths set and the mechansm responds wth a (possbly randomzed) decson that may depend on the entre hstory of messages sent up to perod t, and on past decsons. By the Revelaton Prncple (adapted from Myerson, 986), for any standard soluton concept, any dstrbuton on Y that can be nduced as an equlbrum outcome n any ndrect mechansm can also be nduced as an equlbrum outcome of a drect mechansm n whch the agent s asked to report the current type, and where, n each perod, he nds t optmal to report truthfully, condtonal on havng reported truthfully n the past. Let m t 2 t denote the agent s perod-t message, and m t (m ; : : : m t ) 2 t denote a collecton of messages, from perod one to perod t: De nton 2 A drect mechansm s a collecton t : t Y t! (Y t ) T t= such that () for all t, all measurable A Y t, t (Aj) : t Y t! [0; ] s measurable, and () for all t, all ( t ; y t ) 2 t Y t, y t 2 Supp[ t (j t ; y t )] =) (y t ; y t ) 2 Y t : The notaton t (Ajm t ; y t ) stands for the probablty that the mechansm generates y t 2 A Y t gven hstory (m t ; y t ) 2 t Y t, whle Supp[ t (j t ; y t )] denotes the support of the dstrbuton t (j t ; y t ) : The requrement that, for any y t 2 Supp[ t (j t ; y t )]; (y t ; y t ) 2 Y t guarantees that the decsons taken n perod t are feasble gven past decsons y t. Gven a drect mechansm, and a hstory ( t ; m t ; y t ) 2 t t Y t, the followng sequence of events takes place n each perod t:. The agent prvately observes hs current type t 2 t drawn from F t j t ; y t. 2. The agent sends a message m t 2 t. 3. The mechansm selects a decson y t 2 Y t accordng to t (jm t ; y t ). A (pure) strategy for the agent n a drect mechansm s thus a collecton of measurable functons t : t t Y t! t T t= : De nton 3 A strategy s truthful f for all t and all (( t ; t ); m t ; y t ) 2 t t Y t, t (( t ; t ); m t ; y t ) = t : 9

11 Ths de nton dent es a unque strategy; such a strategy has the property that the agent reports hs current type truthfully after any hstory, ncludng non-truthful ones. Note that we are not clamng t s safe to restrct attenton to mechansms wth the property that the truthful strategy (as de ned above) s optmal at all hstores. As explaned above, the Revelaton Prncple n fact smply guarantees that t s safe to restrct attenton to mechansms n whch the agent nds t optmal to report truthfully condtonal on havng reported truthfully n the past; ths s equvalent to requrng that the truthful strategy (as de ned above) be optmal at all truthful hstores. 7 In order to descrbe expected payo s, t s convenent to develop some more notaton. Frst we de ne hstores. For all t = 0; ; : : : ; T, let H t t t Y t [ t t Y t [ t t Y t ; where by conventon H 0 = f?g, and H = [ ( ) [ ( Y ). Then H t s the set of all hstores termnatng wthn perod t, and, accordngly, any h 2 H t s referred to as a perod-t hstory. We let H T[ t=0 denote the set of all hstores. A hstory ( s ; m t ; y u ) 2 H s a successor to hstory (^ j ; ^m k ; ^y l ) 2 H f () (s; t; u) (j; k; l), and (2) ( j ; m k ; y l ) = (^ j ; ^m k ; ^y l ). A hstory h = ( s ; m t ; y u ) 2 H s a truthful hstory f m t = t. Fx a drect mechansm, a strategy, and a hstory h 2 H. Let [; ]jh denote the (unque) probablty measure on Y the product space of types, messages, and decsons nduced by assumng that followng hstory h n mechansm, the agent follows strategy n the future. More precsely, let h = ( s ; m t ; y u ). Then [; ]jh assgns probablty one to (^; ^m; ^y) such that (^ s ; ^m t ; ^y u ) = ( s ; m t ; y u ). Its behavor on Y s otherwse nduced by the stochastc process that starts n perod s wth hstory h, and whose transtons are determned by the strategy, mechansm, and kernels F. If h s the null hstory we then smply wrte [; ]. We also adopt the conventon of omttng from the arguments of when s the truthful strategy. Thus [] s the ex-ante measure nduced by truthtellng whle []jh s the measure nduced by the truthful strategy followng hstory h. 7 One can safely restrct attenton to mechansms n whch the agent nds t optmal to report truthfully at any hstory, provded n each perod t, the agent s asked to report hs complete hstory t as opposed to the new nformaton t. Ths alternatve class of drect mechansms was proposed by Doepke and Townsend (2006). Whle these mechansms permt one to restrct attenton to one-stage devatons from truthtellng, the devatons that one must consder are multdmensonal and contngent on possbly nconsstent reportng hstores. Whether ths alternatve class of mechansms facltates the characterzaton of mplementable outcomes s thus unclear. H t 0

12 , 9 E [] [U( ~ ; ~y)] E [;] [U( ~ ; ~y)]: Gven ths notaton, E [;]jh [U( ~ ; ~y)] denotes the agent s expected payo at the hstory h when he plays accordng to the strategy n the future. 8 For most of the results we use ex-ante ratonalty as our soluton concept. That s, we requre the agent s strategy to be optmal when evaluated at date zero, before learnng. In a drect mechansm ths corresponds to ex-ante ncentve compatblty de ned as follows. De nton 4 A drect mechansm s ex-ante ncentve compatble (ex-ante IC) f for all strateges Ths noton of IC s arguably the weakest for a dynamc envronment. Thus dervng necessary condtons for ths noton gves the strongest results. However, for certan results t s convenent to de ne IC at a gven hstory. De nton 5 Gven a drect mechansm, the agent s value functon s a mappng V : H! R such that for all h 2 H, V (h) = sup E [;]jh [U( ~ ; ~y)]: De nton 6 Let h 2 H. A drect mechansm s ncentve compatble at hstory h (IC at h) f E []jh [U( ~ ; ~y)] = V (h): In words, s IC at h f truthful reportng n the future maxmzes the agent s expected payo followng hstory h. Ths de nton s exble n that t allows us to generate d erent notons of IC as specal cases by requrng IC at all hstores n a partcular subset. For example, ex-ante IC s equvalent to requrng IC only at the null hstory. Or n a statc model (.e., f T = ), the standard de nton of nterm ncentve compatblty obtans by requrng to be IC at all hstores where the agent knows only hs type. In a dynamc model a natural alternatve s to requre that f the agent has been truthful n the past, he nds t optmal to contnue to report truthfully. Ths s obtaned by requrng to be IC at all truthful hstores. The Prncple of Optmalty mples the followng lemma. Lemma If s IC at h, then for []jh-almost all successors h 0 to h, s IC at h 0. 8 Throughout we use tldes to denote random varables wth the same symbol wthout the tlde correspondng to a partcular realzaton. 9 Restrctng attenton to pure strateges s wthout loss: By the Revelaton Prncple the agent can be assumed to report truthfully on the equlbrum path. As for devatons, a mxed strategy (or a collecton of behavoral strateges) nduces a lottery over payo s from pure strateges. Thus, f there s a pro table devaton to a mxed strategy, then there s also a pro table devaton to a pure strategy n the support of the mxed strategy.

13 In partcular, f s ex-ante IC, then truthtellng s also sequentally optmal at truthful future hstores h wth probablty one, and the agent s equlbrum payo at such hstores s gven by V (h) wth probablty one. We wll sometmes nd t convenent to focus on such hstores, and they are the only ones that matter for ex-ante expectatons. 3.2 Necessary Condtons for IC We now set out to derve a recursve formula for (the dervatve of) the agent s expected payo n an ncentve compatble mechansm. Ths formula extends to dynamc models the standard use of the envelope theorem n statc models to pn down the dependence of the agent s equlbrum utlty on hs true type (see, e.g., Mlgrom and Segal, 2002). We begn wth a heurstc dervaton of the result. Frst recall the standard approach wth T =, whch expresses the dervatve of the agent s equlbrum payo n an IC mechansm wth respect to hs type as the partal dervatve of hs utlty functon wth respect to the true type holdng the truthful equlbrum message xed: dv Z " ( ( ; y ) = d (y j ) = E ~ # ; ~y ) : For the moment we gnore the precse condtons for the argument to be vald. Wth T >, we may be nterested n evaluatng the equlbrum payo startng from any perod t. h = ( t ; t ; y t ) s In general, the agent s expected utlty from truthtellng followng a truthful hstory h E []jh U( ~ ; ~y) = Z U (; y) df T + T + j T ; y T d T y T jm T ; y T df t+ t+ j t ; y t d t y t jm t ; y t m= ; where df T + ( T + j T ; y T ). Assume for the moment that ths expresson s su cently wellbehaved so that the dervatves encountered below exst. Suppose one then replcates the argument from the statc case. That s, consder the agent s problem of choosng a contnuaton strategy gven the truthful hstory ( t ; t ; y t ). Assumng that an envelope argument apples, one would then d erentate wth respect to the agent s current type t holdng the agent s truthful future messages xed. The current type drectly enters the payo n two ways. Frst, t enters the agent s utlty functon U. Ths gves the term E []jh [@U( ~ ; ~y)=@ t ]. Second, t enters the kernels F. Ths gves (after ntegratng by parts and d erentatng wthn the ntegral) for each > t the term " Z E ( j ~ ; ~y (( ~ ; ); ~ # ; ~y ) d 2

14 Ths suggests that a margnal change n the current type a ects the equlbrum payo through two d erent channels. Frst, t changes the agent s payo from any allocaton. Second, t changes the dstrbuton of future types n all perods > t, and hence leads to a change n the perod- expected utlty captured by the dervatve of the value functon V evaluated at the approprate hstory. Whle the above heurstc dervaton solates the e ects of the current type on the agent s equlbrum payo, t does not address the techncal condtons for the dervaton to be vald. In fact, n general the d erentablty of the value functon at future hstores can not be taken for granted so that the actual formal argument s more nvolved. (See the dscusson after Proposton.) Furthermore, we do not want to mpose any restrcton on the mechansm to guarantee d erentablty of the value functon. Ths would clearly be restrctve, for example, for the purposes of dervng mplcatons for optmal mechansms. Instead, we seek to dentfy propertes of the envronment that guarantee that, n any IC mechansm, the value functon s su cently well behaved. Our dervaton makes use of the followng assumptons. Assumpton For all t, t = ( t ; t ) R for some t t +. Assumpton 2 For all t, and all ( t ; y t ) 2 t Y t, R j t jdf t ( t j t ; y t ) < +. Assumpton 3 For all t, and all ( t ; y t ) 2 t Y t, the c.d.f. F t (j t ; y t ) s strctly ncreasng on t. Assumpton 4 For all t, and all (; y) 2 y)=@ t exsts and s bounded unformly n (; y). Assumpton 5 For all t, all < t, and all ( t ; y t ) 2 t Y t ( t j t ; y t )=@ exsts. Furthermore, for all t, there exsts an ntegrable functon B t : t! R + [ f+g such that for all < t, and all ( t ; y t ) 2 t Y t ( t j t ; y t )=@ B t ( t ): Assumpton 6 For all t, and all y t 2 Y t, the probablty measure F t j t ; y t s contnuous n t n the total varaton metrc. 0 Assumptons and 4 are famlar from statc settngs (see, e.g., Mlgrom and Segal, 2002). The assumpton that each t s open permts us (a) to accommodate the possblty of an unbounded 0 See, e.g., Stokey and Lucas (989) for the de nton of the total varaton metrc. 3

15 support (e.g. t = R), and (b) to avod qual catons about left and rght dervatves at the boundares. All subsequent results extend to the case that t s closed. Assumptons 2 and 3 are also typcally made n statc models. Assumpton 2 s trvally sats ed f t s bounded. Assumpton 3 s a full support assumpton, whch s related to Assumpton. Whle Assumpton requres that the set of feasble types be connected, Assumpton 3 requres that the set of relevant types also be connected. The assumpton that the support t s hstory-ndependent smpl es some of the proofs but s not essental; one may for example accommodate the case that t follows an ARIMA process wth bounded nose, n whch case Assumpton 3 s volated (see e.g. Pavan, Segal, and Tokka (2009)). Assumpton 5 requres that the dstrbuton of the current types depends su cently smoothly on past types. The motvaton for t s essentally the same as for requrng that, even n statc settngs, utlty depends smoothly on types (.e., Assumpton 4). In a dynamc model the agent s expected payo depends on hs true type both through the utlty functon U and the kernels F. The combnaton of Assumptons 4 and 5 guarantees that the agent s expected payo depends smoothly on types. 2 Snce Assumpton 5 does not have a counterpart n the statc model, t s nstructve to consder what restrctons t mposes on the stochastc process for t. In partcular, t mples that the partal dervatve of the expected current type wth respect to any E[ ~ t j t ; y t ], exsts and s bounded unformly n ( t ; y t ) see Lemma A n the Appendx. It turns out that for non-markov settngs Assumpton 5 by tself does not mpose enough smoothness on the dependence of the kernels on past types, whch s why we mpose also Assumpton 6. Note that when the functons F t j t ; y t are absolutely contnuous, ths assumpton s equvalent to the contnuty of ther denstes n t We are now ready to state our rst man result. n the L metrc. Proposton Suppose Assumptons -6 hold. (If the envronment s Markov, then Assumpton 6 can be dspensed wth.) If s IC at the truthful hstory h t t ; t ; y t, then V ( t ; h t ) s Lpschtz contnuous n t, and for a.e. ( t ; h t ) t " E []j(t;ht ~ ; t TX =t+ ( j ~ ; ~y (( ~ ; ); ~ # ; ~y ) d () Dependng on the noton of IC, the assumpton of connectedness may also be dropped. Ths s the case, e.g., f one requres IC at all possble t. Wthout connectedness, the nterpretaton however becomes an ssue. For example, consder a statc model where = [0; ] but where F assgns probablty one to the set f0; g. Is ths a model wth a contnuum of types n whch IC s mposed for all 2 [0; ], or a model wth two types wth IC mposed only on 2 f0; g? 2 A weaker jont (or reduced form ) assumpton mposng restrctons drectly on the expected payo as opposed to U and F separately would also su ce, although need not be easer to verfy. 4

16 The recursve formula () pns down how the agent s equlbrum utlty vares as a functon of the current type t. It s a dynamc generalzaton of Mrrlees s statc envelope theorem formula (Mrrlees, 97) (whch obtans as a specal case when T = t = ). As suggested n the heurstc dervaton precedng the result, an n ntesmal change n the current type has two knds of e ects n a dynamc model. Frst, there s a drect e ect on the nal utlty from decsons, whch s captured by the partal dervatve of U wth respect to t. Ths s the only e ect present n statc models. Wth more than one perod, there s a second, ndrect, e ect through the mpact of the current type on the dstrbuton of future types. Ths s captured by the sum wthn the expectaton. The e ect of the current type t on the dstrbuton of perod type s captured by the partal dervatve of F wth respect to t. The nduced change n utlty s evaluated by consderng the partal dervatve of the perod- value functon V wth respect to. The proof, whch s n the appendx, s by backward nducton, rollng backwards an envelope result at each stage. The argument s not trval because, contrary to the statc case, the contnuaton payo cannot be assumed to be d erentable n the current type. Ths s because the contnuaton payo s tself a recursve formulaton of the agent s future optmzatons. Hence the standard verson of the envelope theorem does not apply. However, we show that, at each hstory, the value functon s Lpschtz contnuous n the current type t, and thus has left- and rght-sde dervatves wth respect to t everywhere. We then prove an envelope theorem smlar to those n Mlgrom and Segal (2002) and n Ely (200) that relates the sde dervatves of the value functon to the sde dervatves of the objectve functon (the contnuaton payo ) evaluated at the optmum. The backward nducton n the proof uses ths envelope theorem to roll backward the sde dervatves. A nal complcaton arses from the fact that, as noted by Ely (200), an envelope theorem based on sde dervatves yelds payo equvalence only f the possble knks n the objectve functon are convex (.e., open upwards). However, t turns out that whenever a maxmzer exsts, the value functon has convex knks. Thus, along the equlbrum path, where truthtellng s optmal by ncentve compatblty (wth probablty one), any knks n the value functon must ndeed be convex. Remark Whle we have restrcted the agent to observe a one-dmensonal type n each perod, the same necessary condton () for ncentve compatblty can be appled to a model n whch the agent observes a multdmensonal type n each perod, by restrctng the agent to observe one dmenson of hs current type at a tme and report t before observng the subsequent dmensons. Indeed, snce ths restrcton only reduces the set of possble devatons, t preserves ncentve compatblty, and so condton () must stll hold. However, ncentve compatblty s harder to ensure when the agent observes several dmensons at once (see Remark 2 for more detal). 5

17 3.2. Closed-form expresson for expected payo dervatve The recursve formula for the partal dervatve of V wth respect to current type t n Proposton can be terated backwards to get a closed-form formula. Although ths can n prncple be done under the assumptons of the proposton, a more compact expresson obtans f we make the followng addtonal assumpton. Assumpton 7 For all t and all t ; y t 2 t Y t, the functon F t j t ; y t s absolutely contnuous wth densty f t t j t ; y t > 0 for a.e. t 2 t. The exstence of a strctly postve densty allows us to wrte the formula n () terms of expectaton operators rather than ntegrals. Usng terated expectatons then yelds the followng result. Proposton 2 Suppose Assumptons -7 hold. (If the envronment s Markov, then Assumpton 6 can be dspensed wth.) If s IC at the truthful hstory h t ( t ; t ; y t ), then V ( t ; h t ) s Lpschtz contnuous n t, and for a.e. t, ( t ; h t T # ) X = E []j(t;ht ) Jt ~ ; ~y ; ~y) ; =t (2) where J t t ( ~ t ; ~y t ) and J t ( ; y ) X K2N, l2n K+ : t=l 0 <:::<l K = KY k= I l k lk ( l k ; y l k ) for > t; wth Il m ( m ; y m m( m j m ; y m )=@ l f m ( m j m : ; y m ) The ntuton for the formula n (2) s as follows. The term Il m can be nterpreted as the drect response of sgnal l to a small change n sgnal m, m > l: The term Jt can then be nterpreted as the total mpulse response of t to, > t: It ncorporates all the ways n whch t can a ect the dstrbuton of, both drectly and through ts e ect on the ntermedate sgnals observed by the agent. The calculaton of Jt counts all possble chans of such e ects. For example, n the Markov case, Il m = 0 for l < m, hence we have Jt ; y Y = Ik k (k ; y k ) there s only one chan of e ects, whch goes through all the perods. k=t+ 6

18 Example Let t evolve accordng to an AR(k) process: kx t = j t j + " t, j= where t = 0 for any t 0; j 2 R for any j = ; :::; k, and where " t s the realzaton of the random varable ~" t dstrbuted accordng to some absolutely contnuous c.d.f. G t wth strctly postve densty over R f t 2 and over a connected set R f t =, wth (~" s ) T s= jontly ndependent. For convenence, hereafter we let j 0 for all j > k. Then 0 F j ; y = kx j j= j A ; so that for any > t, I t ; j ; y =@ t f ( j ; y ) = t ; and Jt ; y X MY = lm l m : M2N, l2n M+ :t=l 0 <:::<l M = m= Thus n ths case the mpulse response J t ; y s a constant that does not depend on the realzatons of (; y) (but may stll depend on t and ). In the specal case of an AR() process we have ( f = t + I t ; y = 0 otherwse, whch mples that J t ; y = ( ) t. 3.3 Su cent condtons for IC Whle the formula n (2) summarzes local ( rst-order) ncentve constrants, t does not mply the satsfacton of all (global) ncentve constrants. In ths secton we formulate some su cent condtons for ncentve compatblty. These condtons generalze the well-known monotoncty condton, whch together wth the rst-order condton characterzes ncentve-compatble mechansms n the statc model wth a one-dmensonal type space. The statc characterzaton cannot be extended to the dynamc model, whch could be vewed as an nstance of a multdmensonal mechansm desgn problem, for whch the characterzaton of ncentve compatblty s more d cult (see, e.g., Rochet and Stole, 2003). More precsely, there are two sources of d culty n ensurng 7

19 ncentve compatblty n a dynamc settng: (a) n general one needs to consder multperod devatons, snce once the agent les n one perod, hs optmal contnuaton strategy may requre lyng n subsequent perods as well; 3 and (b) even f one focuses on sngle-perod devatons, n whch the agent msrepresents hs current one-dmensonal type, the decsons assgned by the mechansm from that perod onward form a multdmensonal decson space. Whle these problems make t hard to have a complete characterzaton of ncentve compatblty, we can stll formulate su cent condtons for IC that prove useful n a number of applcatons. Problem (a) s sdestepped by focusng on envronments n whch we can assure that truthtellng s an optmal contnuaton strategy even followng devatons, and so ncentve compatblty can be assured by checkng one-stage devatons. Whle ths focus s qute restrctve, t ncludes all Markov envronments, as well as some other nterestng cases see for example the applcaton to sequental auctons wth AR(k) values consdered n subsecton 5.2. Problem (b) s sdestepped by formulatng a monotoncty condton that, whle not always necessary for IC, s su cent and s easy to check n applcatons. Proposton 3 Suppose the envronment sats es the assumptons of Proposton 2. Fx any perod t and for any perod-t hstory h, let " X T D (h) E []jh Jt ( ~ ; ~y =t Suppose that for any truthful hstory t ; t ; y t, # ; ~y) () E []j((t ; t); t ;y t ) [U( ~ ; ~y)] s Lpschtz contnuous n t, and for a.e. t, d d t E []j((t ; t); t ;y t ) [U( ~ ; ~y)] = D ( t ; t ); ( t ; t ); y t : () For any m t, for a.e. t, D ( t ; t ); ( t ; t ); y t D ( t ; t ); ( t ; m t ); y t ( t m t ) 0; (3) () s IC at any (possbly non-truthful) perod t + hstory. Then s IC at any truthful perod-t hstory. Proposton 2 mples that condton () n Proposton 3 s a necessary condton for the mechansm to be IC at all truthful perod-t hstores (Recall that ths means that the agent s value 3 Note that the d culty of controllng for mult-stage devatons s somethng one must deal wth even f one consders the alternatve class of drect mechansms proposed by Doepke and Townsend (2006) n whch the agent reports hs complete hstory t n each perod t, as opposed to the new nformaton t: 8

20 functon at these hstores concdes wth the expected equlbrum payo ). The addton of condtons () and () s then su cent (but n general not necessary) for IC at all truthful perod-t hstores. The proof s based on a lemma n the appendx that extends to a dynamc settng a result by Garca (2005) for statc mechansm desgn wth one-dmensonal type and multdmensonal decsons. The assumpton that the mechansm s IC at all perod t + hstores, ncludng non-truthful ones, s rather strong, but t s sats ed n some applcatons. As a promnent example, n a Markov settng, the hstory t of the agent s true types does not a ect hs ncentves n perod t + after t+ s observed. Thus, any mechansm that s IC at all truthful perod t + hstores must also be IC at all perod t + hstores. In ths case, the Proposton can be terated backward startng from perod T + to establsh IC n all perods and at all hstores. Proposton 3 can be generalzed to establsh the optmalty of an arbtrary strategy at any arbtrary hstory h: To ths purpose, t su ces to consder a cttous mechansm ^ that responds to the agent s reports about hs type ^ t wth a recommendaton t (h t ; t ) about the message to send n the orgnal mechansm : Provded the support of t (h t ; ) concdes wth the entre set t whch guarantees that any devaton n perod t from t (h t ; t ) can be nterpreted as a msrepresentaton of t to the new mechansm ^ then the optmalty of at h t = (h t ; t ) can be ver ed by checkng a sngle-crossng condton smlar to the one that n (3) establshes optmalty of a truthful strategy at a truthful hstory. 4 Mult-agent quaslnear case We now ntroduce multple agents. The mult-agent model we consder features three mportant assumptons: () the envronment s quaslnear (.e., the decson taken n each perod can be decomposed nto an allocaton and a vector of monetary payments and the agents preferences are quaslnear n the payments), (2) the type dstrbutons are ndependent of past monetary payments (but they may stll depend on past allocatons), and (3) types are ndependent across agents. After settng up the model we show how from the perspectve of an ndvdual agent, the model reduces to the sngle-agent case studed n the prevous secton. 4. Quaslnear envronment There are N agents ndexed by = ; : : : ; N. shown a nonmonetary allocaton x t In each perod t = ; : : : ; T; each agent s 2 X t, where X t s a measurable space, and asked to make a payment p t 2 R. The set of all feasble hstores of jont allocatons s denoted by X Q T Q N t= = X t: Gven X, we then let X t Q N = X t denote the set of perod-t feasble 9

21 allocatons, X ;t Q j2f;:::;ng;j6= X jt the set of perod-t allocatons for all agents other than, and X t Q t s= X s, X t Q t Q N s= = X s, and X t Q t Q s= j2f;:::;ng;j6= X js the correspondng sets of perod-t hstores. Ths formulaton allows for the possblty that the set of feasble allocatons n each perod depend on the allocatons n the prevous perods, and/or that the set of feasble decsons wth each agent depends on the decsons taken wth the other agents. 4;5 Each agent observes hs own allocatons x T but not the other agents allocatons x T : The observablty of x t should be thought of as a technologcal restrcton: n each perod, a mechansm can reveal more nformaton to agent than x t, but t cannot conceal x t. As for the payments, because the results do not hnge on the spec c nformaton the agents receve about p, we leave the descrpton of the nformaton the agents receve about p unspec ed. As n the sngle-agent case, hstores are denoted usng the superscrpt notaton. For example, x t ; p t s an element of X t R Nt. In each perod t, each agent prvately observes hs current type t 2 t R. The current type pro le s then denoted by t ( t ; : : : ; Nt ) 2 t Q t. The dstrbuton of the type pro le 2 Q T t= t s descrbed n the followng de nton. We omt superscrpts for full hstores, wth the excepton of x T (x ; :::; x T ), p T (p ; :::; p T ), and T ( ; :::; T ) (and the sets they are elements of). Ths s to avod confuson between, e.g., x t (x t ; :::; x Nt ) and x (x ; :::; x T ). Agent s payo functon s denoted by U : X R T! R. We then de ne a quas-lnear envronment as follows. De nton 7 The envronment s quaslnear f the followng hold:. There s a sequence of decsons (x; p) 2 X R NT, where x = x T ; : : : ; xt N s an allocaton, p s a vector of payments, and for all, x T agent. s the mnmal nformaton about x receved by 2. The dstrbuton of the type pro le s governed by the kernels F t : t X t! ( t ) T t=. 3. For all, the payo functon of each agent, U : X R T! R, takes the quaslnear form U (; x; p T ) = u (; x) 4 For example, the (ntertemporal) allocaton of a prvate good n xed supply x can be modelled by lettng X = fx 2 R tn + : P t xt xg; whle the provson of a publc good whose perod-t producton s ndependent of the level of producton n any other perod can be modelled by lettng X = Q T t= Xt wth Xt = x t 2 R N + : x t = x 2t = ::: = x Nt. 5 Ths formulaton does not explctly allow for decsons that are not observed by any agent at the tme they are made; however, such decsons can easly be accomodated by ntroducng a cttous agent observng them. In ths case, one can also nterpret x t as the sgnal that agent receves n perod t about the unobservable decson. TX t= p t 20