Full surplus extraction and within-period ex post implementation in dynamic environments

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1 Full surplus exracion and wihin-period ex pos implemenaion in dynamic environmens Shunya Noda May 1, 2018 Absrac We sudy full surplus exracion and implemenaion in dynamic environmens. We exploi ineremporal correlaions of agens ypes o consruc wihin-period ex pos incenive compaible mechanisms. Firs, we formulae one-sho environmens, in which a single agen has a hidden ype and he planner observes a public signal abou he agen s ype afer a ype-coningen allocaion is chosen. We propose necessary and sufficien condiions for full surplus exracion (srong deecabiliy) and for implemenabiliy of he argeed allocaion rule (weak deecabiliy) in his one-sho problem. We decompose he general dynamic problem ino one-sho problems, and obain sufficien condiions for surplus exracion and implemenaion. JEL Classificaion: C73, D47, D82, D86 Keywords: dynamic mechanism design, wihin-period ex pos implemenaion, revenue maximizaion, full surplus exracion I have been suppored by he Funai Overseas Scholarship, he E. K. Poer fellowship, he JSPS research fellowship (DC1), and he JSPS KAKENHI Gran Number 26J I am hankful o Fuhio Kojima, Hioshi Masushima and Takuo Sugaya for heir guidance and encouragemen. I am also graeful o Alex Bloedel, Gabriel Carroll, Mahew Jackson, Michihiro Kandori, Toshihiro Masumura, Yuri Masumura, Delong Meng, Daisuke Oyama, Julia Salmi, Dan Sasaki, Ilya Segal, Waaru Tamura, Juuso Toikka, Yuichi Yamamoo, and all he seminar paricipans a he 2013 Economeric Sociey Summer School in Seoul, he Universiy of Tokyo, he spring meeing of he 2014 Japanese Economic Associaion, Sanford Universiy, he Universiy of Elecro-Communicaions, and he 2016 Norh American meeing and Asian meeing of he Economeric Sociey for helpful commens. My hanks also go o he co-edior, Johannes Hörner, and anonymous referees. All remaining errors are my own. Deparmen of Economics, Sanford Universiy, 579 Serra Mall, Sanford, CA , USA. shunya.noda[a]gmail.com 1

2 1 Inroducion This paper invesigaes he possibiliy of full surplus exracion and he implemenabiliy of general allocaion rules in dynamic environmens in which he agens may have inerdependen values and heir hidden ypes evolve over ime. For such environmens, we esablish a way o use correlaions among agens ypes o induce ruhful revelaion of ype realizaions. For saic problems, Crémer and McLean (1985, 1988) prove ha full surplus exracion is possible whenever beliefs are convex independen: for each agen i and for each agen i s ype (θ i Θ i ), his belief abou he oher agens ypes (θ ) ha is associaed wih θ i is no in he convex hull of he beliefs abou θ ha are associaed wih he oher ypes of agen i (Θ i \ {θ i }). Under his condiion, he planner can deec agens privae ypes wihou leaving informaion ren; herefore, full surplus exracion is achievable. Their convex-independence condiion is generically saisfied in saic environmens. However, many real-world problems are dynamic. We consider he following dynamic environmen. In each period, each agen i I privaely observes his ype, θ. i Hence, he sae in period is he profile of agens ypes in, θ (θ) i i I. The planner needs o collec informaion abou he sae θ in order o decide an allocaion. The sae in he nex period, θ +1, depends on he sae as well as on he allocaion decision in. The goal of his paper is o provide (reasonably igh) sufficien condiions for full surplus exracion and implemenaion in dynamic environmens, by exending he convexindependence condiion of Crémer and McLean. We allow inerdependen values, wih which implemenaion of an efficien allocaion rule iself is no rivial (see, e.g., Jehiel and Moldovanu (2001)). We require he mechanisms o be wihin-period ex pos incenive compaible (wp-epic); ha is, ruhelling would remain opimal if agens observed all he privae informaion up o he curren repors (including he oher agens curren ypes), as long as he oher agens make ruhful repors from his poin on. We require wp-epic for hree reasons. Firs, wp-epic is desirable because ruhelling consiues a perfec Bayesian equilibrium under every assumpion abou he observabiliy of he curren saes. Second, wp-epic seems he sronges incenive compaibiliy noion 2

3 ha we can hope for in our seing. 1 Third, wp-epic is saisfied by he privae-value benchmarks, proposed by Bergemann and Välimäki (2010), Cavallo, Parkes, and Singh (2010) and Ahey and Segal (2013). To saisfy wp-epic, we canno use he inraperiod correlaion of agens ypes (i.e., he correlaion beween θ i and θ ) because we mus incenivize agen i o make a ruhful repor even when he were o observe θ. However, we can insead use he fuure ypes as ex pos signals o consruc a paymen rule ha provides an incenive for ruhelling. No one knows he realizaion of fuure ypes a he iming of he repor. Therefore, incenive paymens coningen on fuure ypes are useful for consrucing wp-epic mechanisms. We sar by formulaing he one-sho problem, which concenraes on a reporing problem of a single agen (say, i) in a single period (say, ). The repored θ i deermines a ype-coningen allocaion. An ex pos signal, which may be correlaed wih he realizaion of θ i, is hen publicly observed. Realizaions of ex pos signals sand for realizaions of he sae profiles in he nex period (θ +1 ). For such a one-sho problem, we sudy he condiion on he correlaion beween privae ypes and ex pos signals ha enables he planner o consruc a ruhful (one-sho) mechanism. We propose wo necessary and sufficien condiions he srong-deecabiliy condiion and he weak-deecabiliy condiion. (The precise definiions and saemens are in Secion 4.) 1. Srong deecabiliy is he necessary and sufficien condiion for he following: (i) he argeed allocaion rule is implemenable wih arbirary valuaions over allocaions, and (ii) he planner can provide arbirary equilibrium payoffs for each ype (Lemma 1). Under srong deecabiliy, each agen s payoff can be se o zero, leaving him wih no informaion ren. 2. Weak deecabiliy is he necessary and sufficien condiion for he argeed allocaion rule o be implemenable wih arbirary valuaions over allocaions (Lemma 2). I is useful for implemening an efficien allocaion rule ha maximizes he social surplus. 2 1 For example, Ahey and Segal (2013) sae ha requiring he mechanism o be robus o observaion of fuure ypes would be oo srong for he dynamic seing, even wih a single agen (p. 2472). 2 For saic environmens, Aoyagi (1998) shows ha if each agen has a differen belief whenever his ype is differen, hen any allocaion rules can be implemened by a Bayesian incenive compaible mechanism. 3

4 Weak deecabiliy is weaker han srong deecabiliy: srong deecabiliy implies weak deecabiliy. In he online appendix, we also show ha weak deecabiliy is generically saisfied under a weaker condiion (abou dimensionaliy of he signal space) han srong deecabiliy is. Nex, we decompose he general dynamic problem ino one-sho problems o apply Lemmas 1 or 2 and obain one-sho mechanisms. Then, we combine he one-sho mechanisms o consruc a dynamic mechanism. Firs, we specify θ +1 as ex pos signals of θ i (Subsecion 5.1). The oher agens fuure ype θ +1 is a racable ex pos signal of θ i because an incenive paymen for θ i does no influence he reporing problem of his fuure ypes as long as i is independen of agen i s own fuure ype (θ i +1). We show ha (i) if srong deecabiliy is saisfied in he iniial period, we don have o leave any informaion ren (Proposiion 1); and (ii) if weak deecabiliy is saisfied for all periods, we can implemen a argeed allocaion rule (Proposiion 2). When boh condiions are saisfied, full surplus exracion is guaraneed. In Subsecion 5.2, we weaken he sufficien condiions furher. Even when he correlaion beween θ i and θ +1 does no saisfy eiher srong or weak deecabiliy, he correlaion beween θ i and θ +1 = (θ i +1, θ +1) may saisfy hese deecabiliy condiions. Recall ha srong deecabiliy guaranees ha we can provide arbirary coninuaion payoffs. Therefore, if srong deecabiliy in period + 1 is saisfied, he planner can use he coninuaion payoff a + 1 as a coningen incenive paymen o induce a ruhful repor of θ i. In his case, we can use θ +1 = (θ i +1, θ +1) raher han θ +1 as he ex pos signal. This yields our weakes assumpions, which are used in our main heorems (Theorems 1 and 2). 2 Relaed Lieraure Assuming privae values, Bergemann and Välimäki (2010), Cavallo e al. (2010), and Ahey and Segal (2013) consruc dynamic versions of Vickrey Clarke Groves (VCG) mechanisms, which implemen an efficien allocaion rule. They use disinc assumpions Weak deecabiliy is differen from his condiion for implemenaion since in our environmen, (i) he signal disribuion also depends on he seleced allocaion, and (ii) we do no have o give a srong incenive for ruhelling over misreporing ha does no change he resulan allocaion. 4

5 and heir mechanisms display disinc properies, bu all hree mechanisms are wp-epic under some assumpions, including privae values. Our formulaion of dynamic environmens is close o heirs. However, we consruc a dynamic version of Crémer-McLean mechanism; hus, we assume neiher privae values nor efficiency of he argeed allocaion rule, while we impose deecabiliy condiions on he sae ransiion. Furhermore, our mechanism saisfies wp-epic, same as dynamic VCG mechanisms hey esablish. Mezzei (2007) and Obara (2008) sudy full surplus exracion using ex pos signals. Mezzei (2004, 2007) examines a saic single-uni aucion problem, in which agens valuaions are inerdependen, while ypes are independen. He esablishes ha he planner can implemen an efficien allocaion (Mezzei (2004)) and exrac full surplus (Mezzei (2007)) under a wide variey of seings in which she can use a paymen rule ha depends on he agens realized payoffs. For hese saic problems wih ex pos signals (which correspond o realized payoffs), our Lemmas 1 and 2 provide necessary and sufficien condiions for surplus exracion and implemenabiliy when he ex pos signals need no be realized payoffs. 3 Obara (2008) sudies a wo-sage allocaion problem in which agens privaely choose acions before heir payoff-relevan ypes are realized, and derives he necessary and sufficien condiion for efficiency o be implemenable, wihou leaving informaion rens. Independen of our work, Liu (2017) also analyzes he implemenaion of an efficien allocaion rule in an inerdependen-value seing using he ineremporal correlaion of agens ypes. He provides a condiion, essenially equivalen o srong deecabiliy, under which he planner can align individual and collecive social incenives, as in he canonical VCG mechanism. In conras, we show ha weak deecabiliy is crucial for implemening an efficien allocaion rule, raher han srong deecabiliy; in his sense, our condiion for implemenabiliy is weaker han Liu s. Our conribuion relaive o Liu s is discussed 3 More recenly, Nah, Zoeer, Narahari, and Dance (2015) and He and Li (2016) exend Mezzei (2004) o implemen he efficien allocaion rule in dynamic environmens. In heir seings, he valuaions are inerdependen, ypes evolve independenly, and agens observe heir acual flow valuaions afer allocaions. In his environmen, Nah e al. (2015) develop an efficien dynamic mechanism in which ruhelling is sricly wp-epic. He and Li (2016) show ha a wihin-period budge balance can also be achieved wih inerim incenive compaibiliy (which requires ha ruhful sraegies consiue a perfec Bayesian equilibrium) in such environmens. To accommodae our model he assumpion ha he agens can recognize he realized flow valuaion (or signals abou ha) beween periods in our model, we can redefine he ime horizon as 2T, and le θ represen he rue payoff characerisics if is even and he valuaions realized if is odd. 5

6 furher in Remark 3 and Subsecion I is well known ha (i) full surplus exracion in saic mechanism design wih moneary ransfers, and (ii) folk heorems in repeaed games wihou moneary ransfers, are closely relaed; when he discoun facor is sufficienly large, we can rea coninuaion payoffs as moneary ransfers, as seen in Fudenberg and Levine (1994) and Fudenberg, Levine, and Maskin (1994). Recenly, Hörner, Takahashi, and Vieille (2015) show ha his relaionship is readily generalized o mechanism design and dynamic (sochasic) Bayesian games. Now ha his paper provides a dynamic mechanism o exrac he full surplus in a wide rage of environmens. Applying Hörner e al. (2015) s mehod o replace he moneary ransfer wih he coninuaion payoff would allow us o prove a new folk heorem in dynamic Bayesian games. In paricular, if srong deecabiliy holds for every T periods and weak deecabiliy holds for every period, hen by applying he mehod of Hörner e al. (2015) in every T periods, we can consruc an efficien equilibrium in dynamic Bayesian games wihou moneary ransfers. 5 3 Environmen: he Original Problem Consider an environmen wih a finie se of agens, indexed by i I = {1, 2,, I} where I 2. For now, we focus on a finie horizon, where ime is indexed by T = {0, 1,, T, T + 1} and where T Z +. We can exend he resuls o an infinie horizon, under some addiional assumpions (see Subsecion 6.3 and he online appendix). period, agen i observes his privae sae (or ype) θ i Θ i, and he planner can direcly observe θ 0 Θ 0, where Θ i is assumed o be finie for all i I {0} and T. 6 In Hence, he sae space in, Θ = I i=0θ i, is also finie. Afer θ is realized, he allocaion x X (where X is also assumed o be finie), and he ransfer (y 1,, y I ) R I are deermined based on he mechanism o which he planner commis ex ane. 4 In conras o his paper, which focuses on finie ype spaces, Liu also sudies a way o implemen he argeed allocaion rule when he ype space is infinie. 5 To be more precise, we need some addiional condiions for saisfying wihin-period budge balance, because his is a condiion required for he mechanism o be converible ino a dynamic Bayesian game. 6 Superscrips denoe he names of he agen and subscrips denoe ime periods. 6

7 Agen i wans o maximize he expecaion of his payoff, T +1 δ [ v(x i, θ ) + y] i, =0 which is deermined by he sequence of sae profiles θ 0:T +1 (θ 0, θ 1,, θ T +1 ) Θ 0:T +1 T =0 +1 Θ, allocaions x 0:T +1, and agen i s moneary ransfers y0:t i +1. Throughou his paper, z :s (z,, z s ) (Z :s s k= Z k) denoes he sequence of variables (ses) z k (Z k ) from period o period s. The discoun facor δ (0, 1] is common and v i : X Θ R is agen i s flow valuaion funcion in. We assume ha he flow valuaion funcions are Markov, in he sense ha v i does no depend on θ 0: 1. We assume ha agens do no face an allocaion problem in period T + 1, bu ha hey receive addiional signals θ T +1 for he ype realizaions unil T, θ 0:T. Formally, we assume X T +1 = 1, and v i T +1 (x T +1, θ T +1 ) = 0 for all i I, θ T +1 Θ T The ype disribuion in period 0 is given by µ 0 (Θ 0 ), and subsequen saes evolve according o he ransiion probabiliy funcion µ : X 1 Θ 1 (Θ ). For simpliciy, we assume ha µ has full suppor, ha is, µ 0 (θ 0 ) > 0 for all θ 0 Θ 0, and µ (θ ; x 1, θ 1 ) > 0 for all (x 1, θ 1, θ ) X 1 Θ 1 Θ. We call (Θ, µ ) T +1 =0 he informaion srucure. The roles of hose assumpions are discussed in he online appendix. We focus on direc mechanisms, in which agen i repors his sae θ i in period. (χ, ψ ) T +1 =0 denoes he mechanism where χ : Θ X is he allocaion rule in period, and ψ = (ψ 1,, ψ I ) where ψ i : Θ 0: R is agen i s paymen rule in period. Slighly abusing erminology, we also call (χ, ψ i ) T +1 =0 a mechanism. We concenrae on Markov allocaion rules, i.e., we assume ha χ is deermined by he repor in period, θ, bu is no affeced by he repored ype profile unil 1, θ 0: 1. There exiss an efficien Markov allocaion rule since we assume ha neiher he flow valuaion funcion v i nor he ransiion probabiliy funcion µ is affeced by θ 0: 1. In he online appendix, we discuss 7 This assumpion simplifies he analysis for he las period ( = T ), in which we canno make use of he ex pos signals o induce ruhelling. Excep for he las period, our mechanism does no rely on his assumpion. To guaranee incenive compaibiliy in he las period, we can alernaively assume he exisence of a (saic) ex pos incenive compaible mechanism in T + 1 (which can leave arbirarily large informaion rens). For all he mechanisms presened in his paper, agen i s paymen in period T + 1 does no depend on his own ype in T + 1, θt i +1 ; hus, (saic) ex pos incenive compaibiliy in T + 1 is no affeced by he incenive scheme for = 0, 1,, T. 7

8 how our resuls are generalized o he case of non-markov allocaion rules. We define V i ( ; (χ k ) T +1 k=0 ) : Θ R as agen i s expeced presen value (hereafer, EPV) from valuaions by he following: [ T ] +1 V i (θ ; (χ k ) T +1 k=0 ) E δ s vs(χ i s (θ s ), θ s ) (χ k) T +1 k=0, θ. s= Recall ha once we specify (Θ s, µ s, χ s ) T +1 s=0 and θ, he probabiliy ha θ s realizes for s is deermined. Given allocaion rule (χ ) T +1 =0, he expeced social welfare is E [ i I V i 0 (θ 0 ; (χ ) T +1 =0 ) ]. An allocaion rule (χ ) T +1 =0 is efficien if i maximizes E [ i I V i 0 (θ 0 ; (χ ) T +1 =0 ) ]. Similarly, we define agen i s EPV from paymens Ψ i ( ; (χ k ) T +1 k=0 ) : Θ 0: R by [ T ] +1 Ψ i (ˆθ 0: 1, θ ; (χ k ) T +1 k=0 ) E δ s ψs(ˆθ i 0: 1, θ, θ +1:s ) (χ k) T +1 k=0, θ. s= Since he sae ransiion is Markov, condiional on θ, Ψ i is independen of realizaions of θ 0: 1. However, Ψ i depends on he repored ˆθ 0: 1 because we don assume ha ψ i is Markov. In his paper, we someimes decompose he ransfer rule ψ i ino several pars, e.g., ψ i (θ 0: ) = g i (θ 0: ) + φ i (θ 0: ). By analogy wih he relaionship beween ψ and Ψ, we represen he EPVs of he pars of paymen rules g, φ by he capial leers G, Φ, respecively. For noaional convenience, when we wrie EPV erms such as V i (θ ; (χ k ) T +1 k=0 ), Ψ i (θ 0: ; (χ k ) T +1 k=0 ), we drop (χ ) T +1 =0, and simply wrie V i (θ ), Ψ i (θ 0: ). We require a dynamic version of ex pos incenive compaibiliy. In dynamic environmens, here are many ways o model wha agen i learns abou he pas repors and realized ype profiles of he oher agens, (ˆθ 0: 1, θ 0:). Here, we ake a conservaive approach: we consruc mechanisms in which ruhful reporing of agen i s ype, θ i, is opimal even if he observed all of he pas repors ˆθ 0: 1 as well as he curren ype profile θ (including he ypes of he oher agens). 8 We do no exploi an agen s informaion abou he oher agens ypes. Insead, we consruc mechanisms ha are robus agains he leakage of he oher agens privae informaion. We also require ruhelling on and 8 Noe ha condiional on he realizaion of he curren ype profile θ, agen i s expeced payoff from period is independen of he realizaions of θ 0: 1 and he pas allocaion x 0: 1. 8

9 off he equilibrium pah, i.e., ruhful reporing maximizes each agen s payoff as long as oher agens ell he ruh from his poin on. This noion of incenive compaibiliy is called wihin-period ex pos incenive compaibiliy (wp-epic). Wp-EPIC is he incenivecompaibiliy noion ha he dynamic versions of VCG mechanisms (Bergemann and Välimäki (2010), Cavallo e al. (2010), and Ahey and Segal (2013)) saisfy. Definiion 1 (wp-epic). (χ, ψ i ) T +1 =0 is wihin-period ex pos incenive compaible (wp- EPIC) for agen i a (θ 0: 1, θ i, θ ) Θ 0: if, for all ˆθ i Θ i, V i (θ, i θ ) + Ψ i (θ 0: 1, θ, i θ ) v(χ i (ˆθ, i θ ), θ, i θ ) + ψ(θ i 0: 1, ˆθ, i θ ) (1) [ + δ E V+1(θ i +1 ) + Ψ i +1(θ 0: 1, ˆθ ], i θ, θ +1 ) χ (ˆθ, i θ ), θ, i θ. (χ, ψ i ) T +1 =0 is wp-epic for agen i if i is wp-epic for agen i for every and (θ 0: 1, θ ) Θ 0:. A mechanism (χ, ψ ) T +1 =0 is wp-epic if for all i I, (χ, ψ i ) T +1 =0 is wp-epic for i. We define he no-informaion-ren propery and full surplus exracion as follows: Definiion 2 (No Informaion Ren). (χ, ψ i ) T +1 =0 leaves no informaion ren for agen i if for all θ 0 Θ 0. V i 0 (θ 0 ; (χ ) T +1 =0 ) + Ψ i 0(θ 0 ; (χ ) T +1 =0 ) = 0, Definiion 3 (Full Surplus Exracion). (χ, ψ ) T +1 =0 exracs he full surplus if (i) he allocaion rule (χ ) T +1 =0 is efficien, and (ii) for each i I, (χ, ψ i ) T +1 =0 leaves no informaion ren. Here, we assume ha each agen s ouside opion in period 0 is zero for all θ 0 Θ 0. 9 Hence, Ψ i 0(θ 0 ; (χ ) T +1 =0 ) = V i 0 (θ 0 ; (χ ) T +1 =0 ) is he larges period-0 expeced revenue colleced from agen i when he allocaion rule (χ ) T +1 =0 is implemened. I is naural o 9 We se he ouside opion o zero for simpliciy s sake. We can sill achieve full surplus exracion even if he ouside opion depends on he sae profile, θ. 9

10 assume ha he planner maximizes he ex ane expeced revenue from he agens, since he planner commis o a mechanism (χ, ψ ) T +1 =0 ex ane. 10 We do no impose paricipaion consrains for 1. Since we consider a finiehorizon problem wih finie ypes, for all (χ, ψ i ) T +1 =0 here exiss he wors-case EPV for agen i, namely, M i min,θ0: [V i (θ ) + Ψ i (θ 0: )]. When his is negaive, agen i leaves he mechanism once such θ 0: realizes. However, consider a modified mechanism (χ, ψ i ) T +1 =0 defined by ψ i 0(θ 0 ) ψ i 0(θ 0 ) + M i, ψ i (θ 0: ) ψ i (θ 0: ) for all {1,, T }, and ψ i T +1(θ 0:T +1 ) ψ i T +1(θ 0:T +1 ) δ (T +1) M i. Then, Ψ i 0(θ 0 ) = Ψ i 0(θ 0 ) holds in period 0. Furhermore, Ψ i (θ 0: ) = Ψ i (θ 0: ) δ M i holds for all 1; hus, V i (θ ) + Ψ i (θ 0: ) 0 for all 1. Inuiively, he planner addiionally requires a deposi o make sure ha agens do no leave he mechanism unil i erminaes. The deposi changes neiher he agens EPV in he iniial period nor he planner s revenue because he deposi will be paid back wih appropriae ineres in he las period, as long as agens say in. Using his deposi scheme, we can saisfy paricipaion consrains for 1 wihou increasing he ren. 4 Necessary and Sufficien Condiions for he One- Sho Problem 4.1 Formulaion To explain our main resuls for he original problem, we inroduce he following one-sho problem, which consiss of wo sages, and is characerized by (u i, δ) and (X, Θ i, χ, S, π). 1. A single agen, say, agen i, observes his privae ype θ i Θ i. He makes a ype repor o he planner, ˆθ i Θ i. The planner chooses an allocaion, x = χ(ˆθ i ) according o 10 The impossibiliy of saisfying he paricipaion consrains wih equaliy a every node θ 0: (insead of only in period 0) is explained in he online appendix. 10

11 a commied allocaion rule, χ : Θ i X. 2. An ex pos signal s S (where S is assumed o be finie) is drawn according o π : X Θ i (S), which depends on he allocaion and agen i s rue ype. According o paymen rule, p i : Θ i S R, agen i receives a moneary ransfer. Agen i s payoff is u i (χ(ˆθ i ), θ i ) + δp i (ˆθ i, s), where u i : X Θ i R denoes he agen s valuaion. We call (X, Θ i, χ, S, π) he signal srucure. Imporanly, he planner has o choose an allocaion when he agen repors ˆθ i, while he paymen can also depend on he realizaion of he ex pos signal s. Jus like Crémer and McLean (1988), we exploi he correlaion beween θ i and s o induce ruhelling. However, in conras o Crémer and McLean (1988), in our model, (i) he signal is observable only afer he allocaion is deermined, and (ii) is disribuion also depends on he allocaion. There are wo ways o inerpre his one-sho problem. 1. When T = 0, he original problem can be decomposed ino i I Θ 0 one-sho problems. Each one-sho problem is idenical o he reporing problem of θ i 0, given for each (i, θ 0 ) I Θ 0. The oher agens ypes θ 0 canno be used as a signal for achieving wp-epic, because agen i mus ell he ruh even when he observes θ 0. The only available ex pos signal o induce he ruhelling of θ i 0 is θ 1. Thus, by choosing X X 0, Θ i Θ i 0, χ χ 0, S Θ 1, π µ 1 (,, θ 0 ), u i v i 0(,, θ 0 ) and p i ψ i 1(, θ 0 ), he one-sho problem becomes equivalen o agen i s reporing problem of θ i 0, given ha θ 0 is realized. 2. For he general original problem, we can sill use θ +1 as an ex pos signal o solve he reporing problem of θ i (given an agen i and a paricular sequence of ype repors ˆθ 0: 1 and a ype profile of he oher agens θ ). Hence, defining S Θ +1 and π µ +1(,, θ ) and applying he resuls for one-sho problems, we can derive a (loose) sufficien condiion for full surplus exracion and implemenaion of an allocaion rule (Subsecion 5.1). Furhermore, under a cerain condiion (described laer), we can also use θ i +1 as an ex pos signal o induce he ruhelling of θ i. As an exreme case, we can even ake S Θ +1 and π µ +1 (,, θ ), which yields a 11

12 weaker sufficien condiion han he case of S Θ +1 and π µ +1(,, θ ). See Subsecion 5.2. Remark 1. While θ is dropped from he noaion, we are no assuming privae values. When we apply he resul from he one-sho problem he original problem, we can choose a differen u i for each θ, which allows us o model inerdependency of he valuaion funcion. Similarly, since we can choose a differen paymen rule p i for each hisory (θ 0: 1, θ ), he paymen rule does no need o be Markov eiher. 4.2 Exracion Firs, we sudy he condiion on (X, Θ i, χ, S, π) ha guaranees ha for all u i, here exiss p i such ha ruhelling is induced wih arbirary expeced payoffs. Then, in paricular, we can provide a zero expeced payoff o each agen for all θ i ; i.e., here are no informaion ren lef in he one-sho problem. Definiion 4 (Srong Deecabiliy). Θ i is srongly deecable wih (X, Θ i, χ, S, π) if, for all θ i Θ i, ( { π(χ(θ i ), θ i ) / co π(χ(θ i ), ˆθ } ) i ). (2) ˆθ i Θ i \{θ i } Parallel o Crémer and McLean (1988) s convex-independence condiion, srong deecabiliy is he necessary and sufficien condiion for he exisence of a loery λ : Θ i S R ha provides (i) a zero expeced payoff when he agen ells he ruh, and (ii) a negaive expeced payoff when he agen misrepors. The consrucion of λ is described in he online appendix. 11 Using his loery, we can punish any misrepor. The exisence of such loeries is necessary and sufficien for ruhelling, while providing arbirary expeced payoffs. Lemma 1. The following are equivalen: 1. Θ i is srongly deecable wih (X, Θ i, χ, S, π). 2. For all δ (0, 1], u i : X Θ i R, and U i : Θ i R, here exiss p i : Θ i S R such ha 11 See he proof of Lemma 3. U i (θ i ) = u i (χ(θ i ), θ i ) + δ E [ p i (θ i, s) χ(θ i ), θ i], (3) 12

13 for all θ i Θ i, and [ ] U i (θ i ) u i (χ(ˆθ i ), θ i ) + δ E p i (ˆθ i, s) χ(ˆθ i ), θ i, (4) for all (θ i, ˆθ i ) Θ i Θ i. All proofs are in Appendix. As shown in he proof, when srong deecabiliy is violaed, we can always find (u i, δ) such ha U i (θ i ) = 0 for all θ i canno be achieved when (3) and (4) are saisfied. 4.3 Implemenaion Nex, we consider he condiion on (X, Θ i, χ, S, π) ha guaranees ha for all u i, he planner can induce ruhelling for some payoffs. Definiion 5 (Weak deecabiliy). Θ i is weakly deecable wih (X, Θ i, χ, S, π) if, for all non-empy Θ i Θ i, here exiss θ i Θ i such ha ( { π(χ( θ i ), θ i ) / co π(χ( θ i ), ˆθ } ) i ). (5) ˆθ i Θ i s.. χ(ˆθ i ) χ( θ i ) Since we do no have o achieve arbirary payoffs, weak deecabiliy is weaker han srong deecabiliy. More precisely, if Θ i is srongly deecable wih (X, Θ i, χ, S, π), hen Θ i is also weakly deecable wih (X, Θ i, χ, S, π). This is because (i) he convex hull ha appears in he definiion of srong deecabiliy includes he convex hull of weak deecabiliy as a subse; and (ii) while srong deecabiliy requires ha, for all θ i Θ i, π(χ(θ i ), θ i ) is no in he (larger) convex hull, weak deecabiliy only requires ha (for every Θ i Θ i ) here exiss θ i Θ i Θ i such ha π(χ( θ i ), θ i ) is no in he (smaller) convex hull. In he online appendix, we furher (i) prove ha weak deecabiliy is generic under a weaker condiion han srong deecabiliy, and (ii) show some numerical simulaions which help undersand o wha exen weak deecabiliy is more likely o be saisfied han srong deecabiliy. Weak deecabiliy is necessary and sufficien o implemen χ wih arbirary (u i, δ). Lemma 2. The following are equivalen: 13

14 1. Θ i is weakly deecable wih (X, Θ i, χ, S, π). 2. For all δ (0, 1] and u i : X Θ i R, here exis U i : Θ i R and p i : Θ i S R such ha U i (θ i ) = u i (χ(θ i ), θ i ) + δ E [ p i (θ i, s) χ(θ i ), θ i] (3, revisied) for all θ i Θ i, and [ ] U i (θ i ) u i (χ(ˆθ i ), θ i ) + δ E p i (ˆθ i, s) χ(ˆθ i ), θ i (4, revisied) for all (θ i, ˆθ i ) Θ i Θ i. Whereas Lemma 1 says ha we can induce ruhelling while providing for all on-pah payoff funcions U i : Θ i R (wih srong deecabiliy), Lemma 2 only says ha here exiss U i : Θ i R. Hence, while weak deecabiliy implies wp-epic, i guaranees neiher full surplus exracion nor flexible conrol of on-pah expeced payoffs. The following wo examples illusrae he sufficiency and necessiy of weak deecabiliy. Example 1 (Sufficiency). Assume ha X = {l, r}; Θ i = {L, R 1, R 2 }; χ(l) = l; χ(r 1 ) = χ(r 2 ) = r; and π(l, L) = π(l, R 1 ) = π(l, R 2 ); bu π(r, L) π(r, R 1 ) = π(r, R 2 ). In his example, Θ i is no srongly deecable wih (X, Θ i, χ, S, π), for wo reasons: (i) π(r, R 1 ) = π(r, R 2 ) implies violaions of (2) wih θ i = R 1, R 2 ; and (ii) π(l, L) = π(l, R 1 ) = π(l, R 2 ) implies a violaion of (2) wih θ i = L. However, weak deecabiliy is saisfied. To see his, if we ake Θ i such ha {R 1, R 2 } Θ i, hen we can choose eiher θ i = R 1 or R 2 o show (5) (he convex hull becomes eiher {π(r, L)} or ). Oherwise, Θ i = {L} and (5) is rivially saisfied. How can we induce ruhelling? Firs, recall ha in order o induce ruhelling, we do no have o disinguish he repors of R 1 and R 2 because hese repors lead o he same allocaion, r. If we se p i (R 1, s) = p i (R 2, s) for all s, hen he repors of R 1 and R 2 resul in an idenical allocaion and paymen; hus, he agen becomes indifferen beween reporing R 1 and R 2. Accordingly, he has a (weak) incenive for ruhelling. Hereafer, 14

15 we regard he ype repors of R 1 and R 2 as he idenical repor, say, R. 12 Even afer R 1 and R 2 are clusered, srong deecabiliy is sill no saisfied because χ(l) = l and π(l, L) = π(l, R) imply a violaion of (2). On he oher hand, if agen i repors ˆθ i = R, he allocaion r is chosen, and π(r, L) π(r, R). Therefore, R can preend o be L: If he agen repors L when his rue ype is R, he signal disribuion is π(l, R) = π(l, L). The planner canno saisically idenify he agen s rue ype. L canno preend o be R: When L repors R, he resulan signal disribuion is differen from he one generaed when R repors R i.e., π(r, L) π(r, R). Hence, he planner can saisically idenify his deviaion. Formally, we can consruc a loery such ha, when R is repored (and χ(r) = r is chosen), i follows ha (i) if i s rue ype is R (i.e., he expecaion is aken wih respec o π(r, R)), he loery s expeced value is zero, and (ii) if i s rue ype is L (i.e., he expecaion is aken wih respec o π(r, L)), he loery s expeced value is negaive. Using his loery as a par of he paymen rule when R is repored, we can provide arbirarily srong punishmen o preven L from reporing R. Since L canno preend o be R, we can induce R s ruhful repor by giving a consan subsidy (independen of s) when he agen repors R. Here, he planner needs o disribue a subsidy (so weak deecabiliy does no guaranee ha U i can be conrolled arbirarily), bu ruhelling can be induced wih arbirary valuaion funcions. More generally, when weak deecabiliy is saisfied, he planner can consruc a weak order of he agen s ypes and a se of loeries ha enable he planner o punish he agen s upward misrepor (i.e., he agen would be punished if he preended o be of a higher ype) wihou changing each agen s on-pah payoffs. Furhermore, according o he consruced order, ypes are equivalen only if hey lead o an idenical allocaion and 12 To be more precise, we mus sill require agen i o disinguish beween R 1 and R 2 because he paymens of he oher agens may be differen. However, he repors of R 1 and R 2 lead o he same allocaion and paymen for agen i; as a resul, when we consider agen i s problem, we do no have o disinguish beween hem. Once agen i has an incenive o repor R {R 1, R 2 }, agen i is indifferen beween reporing R 1 and R 2, i.e., he has a weak incenive for ruhelling. 15

16 loery (e.g. R 1 and R 2 of Example 1 are equivalen and lead o he same allocaion and paymen). When ex pos signals are absen, an allocaion rule χ is implemenable if and only if along wih he endowed valuaion funcion u i, χ saisfies he cycle-monooniciy condiion of Roche (1987): for all finie cycles θ(0) i, θi (1),, θi (K), θi (K+1) = θi (0) in Θi, we have K+1 k=1 { u i (χ(θ i (k)), θ i (k)) u i (χ(θ i (k)), θ i (k 1)) } 0. (6) Weak deecabiliy guaranees ha we can arificially generae cycle monooniciy from he signal srucure. Formally, weak deecabiliy ensures he exisence of a loery λ : Θ i S R such ha for all finie cycles θ i (0),, θi (K), θi (K+1) = θi (0) in Θi, we have K+1 k=1 [ ]) (u i (χ(θ i(k) ), θi(k) ) + E λ(θ(k) i, s) χ(θ i (k) ), θ(k) i [ ]) (u i (χ(θ i(k) ), θi(k 1) ) + E λ(θ(k) i, s) χ(θ i (k) ), θ(k 1) i 0. (7) If all ypes in a cycle are equivalen (wih respec o he consruced order), hen hey lead o he same allocaion and loery; hus, (7) is rivially saisfied wih equaliy. Oherwise, he cycle conains a leas one upward misrepor (i.e., here exiss k such ha θ i (k) is a higher ype han θ i (k 1) ). Weak deecabiliy enables he planner o punish such an upward misrepor o saisfy (7). Accordingly, we can implemen χ as if i saisfies cycle monooniciy. Weak deecabiliy is no only sufficien bu also necessary for signal srucures o generae such a loery. Accordingly, if weak deecabiliy is no saisfied and χ does no saisfy cycle monooniciy wih respec o he valuaion funcion u i, ruhelling may no be induced. Example 2 illusraes his fac. Example 2 (Necessiy). We assume X = {a, b, c}, Θ i = {A, B, C}, χ(a) = a, χ(b) = b, 16

17 and χ(c) = c. Furhermore, we assume: π(a, A) co ({π(a, C), π(a, B)}), π(b, B) co ({π(b, A), π(b, C)}), π(c, C) co ({π(c, B), π(c, A)}). Clearly, aking Θ i = Θ i produces a violaion of weak deecabiliy. We assume ha δ = 1 and u i (χ(θ i ), θ i ) = 0 and u i (x, θ i ) = 1 for x χ(θ i ). Noe ha χ does no saisfy cycle monooniciy wih respec o u i. 13 Towards a conradicion, suppose ha here exiss p i ha saisfies (3) and (4). Since π(a, A) co ({π(a, B), π(a, C)}), here exiss α [0, 1] such ha π(a, A) = απ(a, B) + (1 α)π(a, C). Regarding p i (A) : S R as a S -dimensional vecor, and muliplying i from he lef, we have p i (A) π(a, A) = αp i (A) π(a, B) + (1 α)p i (A) π(a, C); or, equivalenly, E [ p i (A, s) a, A ] = αe [ p i (A, s) a, B ] + (1 α)e [ p i (A, s) a, C ]. (8) (8) indicaes ha eiher E [p i (A, s) a, B] E [p i (A, s) a, A] holds or E [p i (A, s) a, C] E [p i (A, s) a, A] does. Oherwise, αe [p i (A, s) a, B]+(1 α)e [p i (A, s) a, C] < E [p i (A, s) a, A], which conradics (8). Wihou loss of generaliy, we assume E [p i (A, s) a, B] E [p i (A, s) a, A]. For B o make a ruhful repor agains misreporing A, he following mus hold: U i (B) = 0 + E [ p i (B, s) b, B ] 1 + E [ p i (A, s) a, B ]. 13 Such an allocaion rule canno be efficien under privae values. However, wihou he assumpion of privae values, he flow valuaion funcion of he oher agens could be affeced by θ i. In ha case, if he oher agens srongly preferred such an allocaion rule, hen his allocaion rule would maximize social welfare. See he online appendix. 17

18 When his is aken ogeher wih he fac ha U i (A) = 0 + E [p i (A, s) a, A], we obain ha U i (B) > U i (A) is necessary. Applying he above argumen o B, we obain eiher E [p i (B, s) b, A] E [p i (B, s) b, B] or E [p i (B, s) b, C] E [p i (B, s) b, B]. If he former inequaliy holds, we obain U i (A) > U i (B), which conradics U i (A) < U i (B). If he laer inequaliy holds, we have U i (C) > U i (B) (> U i (A)). However, applying he above argumen o C, we obain eiher U i (A) > U i (C) or U i (B) > U i (C). In every case, here is a conradicion. Hence, here is no p i and U i ha saisfy (3) and (4). Generalizing he argumen in Examples 1 and 2, we can prove ha weak deecabiliy is he necessary and sufficien condiion for he implemenabiliy of he argeed allocaion rule χ of he one-sho problem. 5 Sufficien Condiions for he Original Problem 5.1 Wihou Backup: a Basic bu Loose Sufficien Condiion We now consruc a dynamic mechanism for he original problem (defined in Secion 3). I is insrucive o begin by consrucing simpler mechanisms from sronger condiions. To consider he reporing problem of θ i, we can always use θ +1 as an ex pos signal for he realizaion of θ i because agen i s incenive for reporing afer period + 1 is no disurbed by such paymens. 14 In his subsecion, we describe a sufficien condiion ha relies only on he correlaion beween θ i and θ +1. Firs, we formulae he one-sho problem for deecing θ i (for each θ ). The planner chooses an allocaion in period from X. The ype space of agen i is rivially Θ i. The allocaion rule in he one-sho problem is χ ( ; θ ) : Θ i X. In his subsecion, we specify he se of ex pos signals as Θ +1. Given θ, θ+1 s (marginal) disribuion, condiional on (x, θ), i is µ +1( ;, θ ) : X Θ i (Θ +1), where µ +1(θ +1; x, θ ) = θ i +1 Θi +1 µ +1 (θ i +1, θ +1; x, θ ). 14 Noe ha his propery is no always guaraneed when we also use θ i +1 as an ex pos signal for θ i. 18

19 Hence, he signal srucure for deecing θ i given θ is: Γ i (θ ) (X, Θ i, χ ( ; θ ), Θ +1, µ +1(, ; θ )). Given ha here exiss a wp-epic mechanism (χ, g ) T +1 =0, under wha condiions can we modify i o saisfy he no-informaion-ren propery? Since we consider period-0 full surplus exracion (i.e., o exploi all he expeced payoffs from paricipaion in period 0), i suffices o incenivize ruhful reporing of ypes in he iniial period. Proposiion 1. Given an allocaion rule (χ ) T +1 =0, suppose ha for all i I and θ 0 Θ 0, Θ i 0 is srongly deecable wih Γ i 0(θ 0 ). Suppose also ha here exiss a paymen rule (g ) T +1 =0 such ha he mechanism (χ, g ) T +1 =0 is wp-epic. Then, here exiss a mechanism (χ, ψ ) T +1 =0 ha is wp-epic and leaves no informaion ren. To consruc (χ, ψ ) T +1 =0, (i) for = 0, we define ψ 0 0; and (ii) for = 2,, T + 1, we fix some θ 0 Θ 0 arbirarily, and define ψ i (θ 1: ) g i ( θ 0, θ 1: ) for all (i, θ 1: ) I Θ 1:. This makes ψ i for 2 independen of he repor in period 0. To obain ψ i 1, for each (i, θ 0 ) I Θ 0, we apply Lemma 1 where we se u i (x 0, θ i 0; θ 0 ) v i 0(x 0, θ 0 ) + δe [ V i 1 (θ 1 ) + G i 1( θ 0, θ 1 ) x0, θ 0 ] (9) U i (θ i 0; θ 0 ) 0 o obain p i (, ; θ 0 ) : Θ i 0 Θ 1 R ha saisfies (3) and (4). Define ψ i 1(θ i 0, θ 0, θ 1 ) p i (θ i 0, θ 1 ; θ 0 ) + g i 1( θ 0, θ 1 ). Imporanly, p i (, ; θ 0 ) is independen of agen i s own repor in period 1, θ i 1. This mechanism, (χ, ψ ) T +1 =0, leaves no informaion ren. From 0 U i (θ 0 ) = V i 0 (θ 0 ) + δ E [ G i 1( θ 0, θ 1 ) + p i (θ i 0, θ 1 ; θ 0 ) x0, θ 0 ] and 19

20 Ψ i 0(θ 0 ) = δ E [ Ψ i 1(θ 0, θ 1 ) x0, θ 0 ], = δ E [ G i 1( θ 0, θ 1 ) + p i (θ i 0, θ 1 ; θ 0 ) x0, θ 0 ] i follows ha for all θ 0 Θ 0, U i (θ 0 ) = V i 0 (θ 0 ) + Ψ i 0(θ 0 ) = 0. (10) Furhermore, his mechanism, (χ, ψ ) T +1 =0, saisfies wp-epic. For = 1,, T + 1, he fac ha (χ, ψ ) T +1 =0 saisfies wp-epic a θ 0: immediaely follows from he fac ha (χ, g ) T +1 =0 saisfies wp-epic a ( θ 0, θ 1: ). In addiion, for = 0, we subsiue (10) and u i (χ 0 (ˆθ i 0; θ 0 ), θ i 0; θ 0 ) [ =v0(χ i 0 (ˆθ 0; i θ0 ), θ 0 ) + δe =v i 0(χ 0 (ˆθ i 0; θ 0 ), θ 0 ) + δe V i 1 (θ 1 ) + G i 1( θ 0, θ 1 ) χ0 (ˆθ i 0; θ 0 ), θ 0 ] [ V i 1 (θ 1 ) + Ψ i 1(ˆθ i 0, θ 0, θ 1 ) p i (ˆθ i 0, θ 1 ; θ 0 ) ] χ 0 (ˆθ 0; i θ0 ), θ 0 for (4) o verify (1). Accordingly, we also have wp-epic for i in period 0. Remark 2. Assuming privae values, Ahey and Segal (2013) esablish an efficien mechanism ha saisfies wp-epic, irrespecive of he ransiion probabiliy funcions, (µ ) T +1 =0. This is an example of efficien mechanisms (χ, g ) T +1 =0, whose surplus is exraced by srong deecabiliy in he iniial period. Nex, we consider a condiion for a argeed allocaion rule o be implemenable. As we discussed in Secion 4, weak deecabiliy is crucial. Proposiion 2. Given an allocaion rule (χ ) T +1 =0, suppose ha for all i I, {0,, T }, and θ Θ, Θ i is weakly deecable wih Γ i (θ ). Then, here exiss a mechanism (χ, ψ ) T +1 =0 ha saisfies wp-epic. We can consruc (χ, ψ ) T +1 =0 by applying Lemma 2 backward. Since we apply Lemma 2 muliple imes, we add subscrips o (u i, p i, U i ) o denoe periods. For = T, and for 20

21 each θ T Θ T, we apply Lemma 2 wih u i T (x T, θ i T ; θ T ) vi T (x T, θ T ) o obain p i T +1 and U i T ha saisfy (3) and (4). We se ψi T +1 (θ T, θ T +1 ) pi T +1 (θi T, θ T +1 ; θ T ). Here, ψt i +1 does no depend on he repors unil T 1. Afer consrucing (ψ i s) T +1 s=+2 such ha each ψ i s is independen of he repors unil s 2, we consruc ψ i +1 in he following manner. For each θ Θ, we apply Lemma 2 wih u i (x, θ; i θ ) v(x i, θ ) + δe [ V+1(θ i +1 ) ] x, θ + δ 2 E [ Ψ i +2(θ +1, θ +2 ) ] x, θ o obain p i +1 and U i ha saisfy (3) and (4). Noe ha Ψ i +2 is independen of he repor of θ i because (ψs) i T s=+2 +1 does no depend on he repors unil. We se ψ+1(θ i, i θ, θ p i +1(θ i, θ +1; θ ). Here, ψ i +1 does no depend on he repors unil 1 eiher. +1) Ieraing his process and seing ψ i 0 0, we obain a wp-epic mechanism (χ, ψ ) T +1 =0. Remark 3. Liu (2017) also sudies implemenabiliy of allocaion rules in dynamic environmens. His Theorem 3.1 claims ha when one assumes his Assumpion 2 (convex independence) which is essenially equivalen o srong deecabiliy wih Γ i (θ ) for all (i,, θ, χ ) we can implemen arbirary allocaion rules. Recall ha (i) srong deecabiliy implies weak deecabiliy, and (ii) Proposiion 2 relies only on weak deecabiliy. (2017). Hence, Proposiion 2 uses a weaker assumpion han Theorem 3.1 in Liu Liu (2017) also proves ha his Assumpion 2 is a sufficien condiion for full surplus exracion. Proposiions 1 and 2 also provide a sufficien condiion bu a weaker one. We need srong deecabiliy only in he iniial period (for = 0) o make he paricipaion consrain binding, and we implemen an efficien allocaion rule wih weak deecabiliy in laer periods (for = 1, 2,, T ). 5.2 Backup by Srong Deecabiliy in Laer Periods We can furher weaken he assumpions of Proposiions 1 and 2. In Subsecion 5.1, we have used only he correlaion beween θ i and θ +1 o induce ruhelling of θ i. From now 21

22 on, we also use agen i s own ype in he nex period, θ i +1, as an ex pos signal for he reporing problem of θ i o obain weaker condiions. Example 3 illusraes he idea. Example 3. Consider a hree-sage problem, in which Θ = Θ = Θ i 2 = 1; Θ i 0 = {L 0, R 0 }; Θ i 1 = {A 1, B 1, C 1, D 1 }; Θ 2 = {E 2, F 2, G 2 }; and X = 1 for = 0, 1, 2. The sae ransiion funcions µ 1 and µ 2 are described in Table E 2 F 2 G 2 A 1 B 1 C 1 D 1 µ 1 ( ; L 0 ) µ 1 ( ; R 0 ) µ 2 ( ; A 1 ) µ 2 ( ; B 1 ) µ 2 ( ; C 1 ) µ 2 ( ; D 1 ) Table 1: The sae ransiion of Example 3. Since he allocaion spaces are singleon, he argeed allocaion rule is rivially implemenable. We consider wheher full surplus exracion is guaraneed for his problem. If we consider only he correlaion beween θ i and θ +1, here are no ex pos signals in period 0 (because Θ 1 = 1). Hence, agen i s ype is no srongly deecable wih Γ i 0. However, here exiss a mechanism ha leaves no informaion ren. The following wo observaions are crucial. 1. Θ i 0 is srongly deecable wih (X 0, Θ i 0, χ 0, Θ 1, µ 1 ) (alhough Θ i 0 is no srongly deecable wih Γ i 0 = (X 0, Θ i 0, χ 0, Θ 1, µ 1 )). In words, if we regard agen i s own ype in period 1, θ i 1, as an ex pos signal of θ i 0, srong deecabiliy is saisfied in period Θ i 1 is srongly deecable wih Γ i 1 = (X 1, Θ i 1, χ 1, Θ 2, µ 2 ), indicaing ha we can achieve arbirary EPV in period 1. Using srong deecabiliy wih (X 0, Θ i 0, χ 0, Θ 1, µ 1 ) (raher han (X 0, Θ i 0, χ 0, Θ 1, µ 1 )), we apply Lemma 1 o he reporing problem of θ i 0, wih U i 0(θ i 0) 0 and u i 0(x 0, θ i 0) = v i 0(x 0, θ i 0). 22

23 Then, we obain p i 1 : Θ i 0 Θ i 1 R such ha 0 =v0(χ i 0 (θ0), i θ0) i + δe [ p i 1(θ0, i θ1) i χ0 (θ0), i θ0] i ; [ ] 0 v0(χ i 0 (ˆθ 0), i θ0) i + δe p i 1(ˆθ 0, i θ1) i χ 0 (ˆθ 0), i θ0 i for all ˆθ i 0 Θ i 0. Imporanly, unlike he analysis in he previous secion, p i 1 depends on agen i s own ype in he nex period. The equaion and inequaliy above imply ha if we can se each agen s EPV in period o he one specified by p i 1 (i.e., if we can se V i 1 (θ i 1) + Ψ 1 (θ i 0, θ i 1) = p i 1(θ i 0, θ i 1)), hen wp-epic in period 0 and he no-informaion-ren propery are saisfied. Γ i 1 In his case, i is indeed possible because Θ i 1 is srongly deecable in period 1 (wih = (X 1, Θ i 1, χ 1, Θ 2, µ 2 )). For each θ i 0 Θ i 0, we apply Lemma 1 o he reporing problem of θ i 1 wih U i 1(θ i 1; θ i 0) p i 1(θ i 0, θ i 1) and u i 1(x 1, θ i 1; θ i 0) = v i 1(x 1, θ i 1). Using p i 2(, ; θ i 0) obained from Lemma 1 as he paymen rule in period 2 (i.e., defining ψ i 2(θ i 0, θ i 1, θ 2 ) p i 2(θ i 1, θ 2 ; θ i 0)) we can saisfy wp-epic in period 1, achieving he EPV specified by p i 1. The consruced (χ, ψ i ) 2 =0 saisfies wp-epic for i and leaves no informaion ren for i. As illusraed in Example 3, if srong deecabiliy is saisfied in period + 1, we can use he EPV from + 1 iself as an incenive paymen for he period- repor because we can provide an arbirary EPV in period + 1 (wihou collapsing wp-epic in period + 1). In his case, we can use no only θ +1 bu also θ i +1 as he ex pos signal of θ i. As we can see in Proposiion 2, srong deecabiliy in laer periods is no a necessary condiion for implemening a argeed allocaion rule. However, if i is saisfied in laer periods, we can generae finer signal spaces, wih which srong and weak deecabiliy are more likely o be saisfied in earlier periods. In general, srong deecabiliy wih (X +1, Θ i +1, χ +1 ( ; θ +1), Θ +2, µ +2(, ; θ +1)) migh be saisfied only if θ +1 belongs o a paricular subse, say, B +1 Θ +1. In such a case, we can use a parial approach. If θ +1 B +1 is realized, hen we also use θ i +1 as an ex pos signal of θ i. Oherwise, we only use he even θ +1 is realized as he ex pos signal of θ i, and we do no disinguish beween he realizaion of (θ i +1, θ +1) and (ˆθ i +1, θ +1) for θ i +1 ˆθ i +1. To make he above argumen formally, we inroduce he following noaions: 23

24 Definiion 6. Given B +1 Θ +1, we define Θ +1 [B +1] as a pariion of Θ +1 such ha { (θ i +1, θ+1) } Θ +1 [B+1] { } (ˆθ +1, i θ+1) Θ +1 [B+1] ˆθ +1 i Θi +1 for all θ +1 B +1 and θ i +1 Θ i +1; and for all θ +1 / B +1. We define µ +1 [B +1]: X Θ (Θ +1 [B +1]) as he condiional probabiliy funcion such ha µ +1 [B +1](s; x, θ ) represens he probabiliy ha he even ha s Θ +1 [B +1] occurs afer (x, θ ). Formally, we define µ +1 [B +1](s; x, θ ) θ +1 s µ +1 (θ +1 ; x, θ ). We call B+1 a backup se. We also define he signal srucure generaed by (θ, B as follows: +1) Γ i (θ, B+1) (X, Θ i, χ ( ; θ ), Θ +1 [B+1], µ +1 [B+1](, ; θ )). Noe ha (Θ +1 [Θ +1], µ +1 [Θ +1]) is equivalen o (Θ +1, µ +1 ), and (Θ +1 [ ], µ +1 [ ]) is equivalen o (Θ +1, µ +1). Accordingly, Γ i (θ, ) = Γ i (θ ). Figure 1: An example of Θ +1 [B+1] and µ +1 [B+1]. Θ i +1 = {T, B}, Θ +1 = {L, R}, and B+1 = {R}. Example 4. Figure 1 illusraes Θ +1 [B +1] and µ +1 [B +1] given some (x, θ ) X Θ. In his example, Θ +1 = {L, R}, and B +1 = {R}. Since L / B +1, we do no disinguish beween (T, L) and (B, L). On he oher hand, since R B +1, he realizaions of (T, R) 24

25 and (B, R) can be used as disinc realizaions of ex pos signals. I follows ha: Θ +1[{R}] = {{(T, L), (B, L)}, {(T, R)}, {(B, R)}} and µ +1 [{R}]({(T, L), (B, L)}; x, θ ) = = 0.3; µ +1 [{R}]({(T, R)}; x, θ ) =0.3; µ +1 [{R}]({(B, R)}; x, θ ) =0.4. Hence, µ +1 [{R}]( ; x, θ ) = (0.3, 0.3, 0.4), and we can check he deecabiliy condiions wih such hree-dimensional vecors. When we apply Lemmas 1 and 2 for Γ i (θ, B+1), he paymen rule p i +1 which is hereby generaed saisfies p i +1(θ i, θ i +1, θ +1; θ ) = p i +1(θ i, ˆθ i +1, θ +1; θ ) for θ+1 / B+1 and for all θ+1, i ˆθ +1 i Θ i +1 because p i +1 : Θ i Θ +1 [B +1] R. As a resul, p i +1 can be expressed in he following way: ψ+1(θ i, θ+1) + 1 {θ +1 B +1 } (V +1(θ i +1 ) + Ψ i +1(θ, θ +1 ) ) =p i +1(θ i, θ +1 ; θ ). Hence, if he planner can se an arbirary on-pah EPV V i +1(θ +1 )+Ψ i +1(θ, θ +1 ) o every θ +1 B +1 (i.e., if srong deecabiliy is saisfied a every θ +1 B +1), hen (i) srong deecabiliy wih Γ i (θ, B+1) guaranee ha he planner can also choose an arbirary on-pah EPV a θ ; and (ii) weak deecabiliy wih Γ i (θ, B+1) guaranee ha he planner can implemen he argeed allocaion rule a θ. Checking srong deecabiliy sequenially, we can consruc a sequence of backup ses. Definiion 7. (B ) T =1 +1, where B Θ agen i along (χ ) T +1 =1 if boh he following hold: 25 for each, is a sequence of backup ses for

26 1. B T +1 =. 2. For = 1, 2,, T, θ B only if Θ i is srongly deecable wih Γ i (θ, B+1). When B +1 becomes larger, he generaed pariion, Θ +1 [B +1], becomes finer. Accordingly, for all ˆB +1 B+1, if Θ i is srongly (weakly) deecable wih Γ i (θ, B+1), Θ i is also srongly (weakly) deecable wih Γ i (θ, ˆB +1). Hence, we can obain he sequence of he larges backup ses by replacing Condiion 2 of Definiion 7 wih his revised condiion: 2. For = 1, 2,, T, θ B if and only if Θ i is srongly deecable wih Γ i (θ, B+1). The sequence of he larges backup ses forms an upper envelope of all sequences of backup ses. Tha is, if ( is a sequence of backup ses, hen ˆB ) T +1 =1 is he sequence of he larges backup se and (B ) T +1 =1 ˆB B holds for = 1, 2,, T + 1. If we wan o maximize he chance o saisfy srong or weak deecabiliy, we can concenrae on he sequence of he larges backup ses. If Θ i 0 is srongly deecable wih Γ i 0(θ 0, B 1 ) for every θ 0 Θ 0 where (B ) T +1 =1 is a sequence of backup ses, hen we can choose agen i s EPV in period 0, {V i 0 (θ 0 ) + Ψ i 0(θ 0 )} θ0 Θ 0, arbirarily. In paricular, we can selec V i 0 (θ 0 ) + Ψ i 0(θ 0 ) = 0 for all θ 0 Θ 0. We now generalize Proposiion 1. Theorem 1 (Exracion). Le (B ) T +1 =1 be a sequence of backup ses for agen i along (χ ) T +1 =0. Suppose ha (i) here exiss a paymen rule (g i ) T +1 =0 ha makes (χ, g i ) T +1 =0 saisfy wp-epic for i, and (ii) for all θ 0 Θ 0, Θ i 0 is srongly deecable wih Γ i 0(θ 0, B 1 ). Then here exiss (ψ i ) T +1 =0 such ha (χ, ψ i ) T +1 =0 saisfies wp-epic and leaves no informaion ren for i. Similarly, for a sequence of backup ses (B ) T +1 =1, if weak deecabiliy is saisfied for every period and a every θ Θ \ B, hen he implemenabiliy of he argeed allocaion rule (χ ) T +1 =0 is guaraneed. This generalizes Proposiion 2. Theorem 2 (Implemenaion). Le (B ) T +1 =1 be a sequence of backup ses for agen i along (χ ) T =0 +1. Suppose ha for all {0, 1,, T }, and for all θ Θ \ B, Θ i 26