Robus Dynamc Implemenaon Anono Pena Unversy of Wsconsn, Madson Dep. of Economcs March 10, 2015 Absrac Ths paper exends he belef-free approach o robus mechansm desgn o dynamc envronmens, n whch agens oban nformaon over me. A socal choce funcon (SCF) s robusly parally mplemened f s perfec Bayesan ncenve compable for all possble belefs. I s shown ha hs s possble f and only f he SCF s ex-pos ncenve compable. Robus full Implemenaon mposes he sronger condon ha, for all possble belefs, all he Perfec Bayesan Equlbra nduce oucomes conssen wh he SCF. Characerzng he se of such equlbra s a key d culy for sudyng hs problem. Ths paper shows ha, for a weaker noon of equlbrum, he se of all such equlbra can be compued by means of a recursve procedure whch combnes he logc of raonalzably and backward nducon reasonng. These resuls are hen used o show ha, n envronmens wh sngle crossng preferences and well-behaved neremporal e ecs, src ex-pos ncenve compably and a condon whch lms he srengh of preference nerdependences are su cen o guaranee robus full mplemenaon. Keywords: backward nducon reasonng dynamc mechansm desgn mplemenaon raonalzably robusness JEL Codes: C72; C73; D82. 1 Inroducon A common crcsm o classcal heory of mechansm desgn s ha reles on srong common knowledge assumpons ha are unlkely o be sas ed n realy. Ths vewpon, ofen referred o as he Wlson docrne, has recenly been revved by a seres of papers by Bergemann and Morrs (2005, 2009a,b, 2011), whch spurred a growng leraure on robus mechansm desgn. Bergemann and Morrs semnal work pursued a belef free approach, nvesgang condons under whch (full, paral or vrual) mplemenaon can be acheved ndependen [emal: apena@ssc.wsc.edu]. Prevous versons of hs paper crculaed under he le Robus Dynamc Mechansm Desgn. I am ndebed o my advsor, George Malah, for hs unque dedcaon and for consanly challengng me hroughou hs projec. I hank Andrew Poslewae, Qngmn Lu, Drk Bergemann, Sephen Morrs, Alessandro Pavan, Marzena Rosek, Bll Sandholm and he anonymous referees for her helpful commens. Fnancal suppor from he UW-Madson Graduae School s graefully acknowledged. 1
of he agens belefs. More recenly, alernave and less demandng approaches o robusness have also been proposed. 1 Ths leraure, however, has only consdered sac sengs. In conras, many suaons of economc neres presen problems of mechansm desgn ha are nherenly dynamc. For nsance, consder he problem of a socal planner who wans o assgn lcenses for he provson of a publc good o he mos producve rm n each perod. Frms producvy s her prvae nformaon and may change over me. Furhermore, producvy n earler perods may be nformave abou laer producvy, and laer producvy may depend on earler allocave choces as well (for example, f here s learnng-by-dong). hese suaons, sac mechansms may no su ce o guaranee a socally desrable oucome, and s mporan for he planner o ake no accoun he rms neremporal ncenves. Moreover, problems of hs knd canno be cas whn he framework receved by he leraure on robus mechansm desgn, whch assumes ha agens oban all he relevan nformaon before he mechansm s se n place. The presen paper exends he belef-free approach o sudy paral and full mplemenaon n problems of dynamc mechansm desgn. Dynamc mechansm desgn can be undersood n wo ways: rs, as he sudy of dynamc mechansms (e.g., an ascendng aucon) n sandard (sac) envronmens; second, as he sudy of mechansm desgn problems n envronmens ha are nherenly dynamc, such as he example above. Our analyss apples o boh cases and nnovaes on he prevous leraure along boh dmensons. 2 The belef-free approach s ofen crczed for beng excessvely demandng. From a heorecal vewpon, however, hs approach represens an mporan benchmark, parcularly o analyze he mehodologcal aspecs of robus mechansm desgn. The nsghs receved from he sac leraure, for nsance, have been adaped o less demandng noons of robusness, whch may be more appealng from an appled vewpon (cf. Secon 6). To wha exen he mehodology developed for sac problems can be exended o dynamc sengs, however, s no clear, parcularly for he sudy of he full mplemenaon problem. The belef-free approach herefore s a naural sarng pon o exend he heory of robus mechansm desgn o dynamc sengs. Tha s boh o undersand he possble lmaons of hs mporan benchmark, and o address he fundamenal mehodologcal quesons, whch may prove useful for fuure research based on more realsc assumpons. Unl recenly, dynamc mechansm desgn problems were surprsngly negleced by he leraure. In recen years, a growng leraure has developed o ll hs mporan gap. 3 The presen paper depars from hs leraure n wo man respecs. In Frs, exsng papers on dynamc mechansm desgn ypcally assume ha he sochasc process ha generaes payo s and sgnals s common knowledge among he agens and known o he desgner. As such, he 1 For envronmens wh paral resrcons on belefs, see Aremov e al. (2013), Lopomo e al. (2013), Km and Pena (2013) and Ollár and Pena (2014). For a d eren, second bes approach, see Börgers and Smh (2012a,b) and Yamasha (2012, 2013a,b). Ths leraure s dscussed more exensvely n Secon 6. 2 Müller (2012a,b) also exends he belef-free approach o sudy dynamc mechansms, bu he consders vrual mplemenaon and only sac envronmens. Ths work s furher dscussed n Secon 6. 3 Among ohers, see Ahey and Segal (2014), Bergemann and Valmak (2010), Pavan, Segal and Tokka (2013) and he references heren. 2
approach su ers from he non-robusness problem dscussed above. Second, he leraure hus far has focused solely on problems of paral mplemenaon. Tha s, he desgn of Perfec Bayesan Incenve Compable (PBIC) mechansms, n whch agens ruhfully reveal her nformaon n a Perfec Bayesan Equlbrum (PBE) of he game. PBIC, however, does no rule ou he possbly ha oher, undesrable PBE exs. The more demandng requremen ha no such equlbra exs s referred o as full mplemenaon. Ths s an mporan queson, especally f he dynamc neracon provdes agens wh more opporunes o collude. A he curren sae of he leraure, however, very lle s known on he problem. Ths s he rs paper o address he queson of dynamc full mplemenaon, le alone he robusness requremen dscussed above. 4 Furhermore, he resuls are obaned under general assumpons on preferences, whch need no be quaslnear nor me separable. Ths s ye anoher nnovaon wh respec o he leraure on dynamc mechansm desgn. The rs resul of hs paper shows ha, as far as robus paral mplemenaon s concerned, he man nsghs from he sac leraure easly exend o dynamc envronmens: a Socal Choce Funcon (SCF) s PBIC for all possble belefs f and only f s ex-pos ncenve compable. Ths resul s bes seen as a necessary condon for robus full mplemenaon and enables us o focus on he novel ssues ha dynamc sengs rase for he full mplemenaon problem, whch s he man focus of he paper. These noveles nvolve boh he dynamc naure of agens neracon and he mehodology of he analyss. From a mehodologcal vewpon, even when he agens belefs are known o he desgner, characerzng he se of PBE of a gven mechansm can be very d cul. Ths may explan why he full mplemenaon queson has no been pursued n dynamc sengs. I may hus seem ha addng he robusness requremen o he already d cul full mplemenaon problem s doomed o make he problem nracable. Ths paper nroduces and provdes foundaons o a mehodology ha avods he d cules of compung he se of PBE for all possble belefs. The key ngreden s he noon of nerm perfec equlbrum (IPE). IPE weakens PBE allowng a larger se of belefs o he equlbrum pah. The advanage of weakenng PBE n hs conex s wofold: on he one hand, full mplemenaon resuls are sronger f obaned under a weaker soluon concep (f all he IPE nduce oucomes conssen wh he SCF, hen so do all he PBE, or any oher re nemen of IPE); on he oher hand, he weakness of IPE s crucal o makng he problem racable. In parcular, s shown ha he se of IPE-sraeges across models of belefs can be compued by means of a backwards procedure ha combnes he logc of raonalzably and backward nducon reasonng: For each hsory, compue he se of raonalzable connuaon-sraeges, reang prvae hsores as ypes, and proceed backwards from almos-ermnal hsores o he begnnng of he game. Re nemens of IPE would eher lack such a recursve srucure, or requre more complcaed backwards procedures. These resuls are hen appled o sudy condons for full mplemenaon n envronmens 4 The closes work o dynamc full mplemenaon s Lee and Sabouran (2011), who sudy repeaed full mplemenaon. The man d erence beween dynamc and repeaed mplemenaon s ha, n he laer, he dsrbuon of ypes, SCF and mechansm are he same n every perod, hence hey do no depend on he prevous hsory. 3
wh monoone aggregaors of nformaon: In hese envronmens, nformaon s revealed dynamcally, and whle agens preferences may depend on her opponens nformaon (nerdependen values) or on he sgnals receved n any perod, n each perod all he avalable nformaon (across agens and curren and prevous perods) can be summarzed by onedmensonal sascs. In envronmens wh sngle-crossng preferences, su cen condons for full mplemenaon n drec mechansms are suded: hese condons bound he srengh of preference nerdependence and requre ha he neremporal e ecs are well-behaved. The res of he paper s organzed as follows: Secon 2 nroduces he formalsm for he envronmens and agens belefs; Secon 3 nroduces mechansms, he noaon for he resulng dynamc games, and he key noons of mplemenaon. The analyss of paral and full mplemenaon s conaned n Secons 4 and 5, respecvely. Secon 6 concludes. 2 Seup Belef-Free Envronmens. Consder an envronmen wh n agens and T perods, T < 1. In each perod = 1; :::; T, each agen = 1; :::; n observes a sgnal ; 2 ; = [ l ; ; h ; ] R.5 For each, := T =1 ; s he se of s payo ypes: a payo -ype s a complee sequence of agen s sgnals n every perod. A sae of naure s a pro le of agens payo ypes, and he se of saes of naure s de ned as := 1 ::: n. As usual, we le ; := j6= j; and := j6= j. A smlar noaon wll be used for oher produc ses. In each perod, he socal planner chooses an allocaon from a non-empy subse of a nely dmensonal Eucldean space,. sequences of allocaons. 6 The se = T =1 denoes he se of feasble Agens are expeced uly maxmzers, wh preferences over sequences of allocaons ha depend on he realzaon of : for each = 1; :::; n, preferences are represened by uly funcons u :! R. Thus, he saes of naure characerze everybody s preferences over he ses of feasble allocaons. A (belef-free) envronmen herefore s de ned by a uple E = N; ; ; (u ) 2N, assumed common knowledge. Envronmen E hus represens agens nformaon and preferences, no her belefs. For each, le Y := =1 ; denoe he se of possble hsores of player s sgnals up o perod. For each and prvae sgnals y = ( ;1; :::; ; ) 2 Y, agen knows ha he rue sae of naure 2 belongs o he se y T =+1 ;. For any 2 and = 1; :::; T, we le y () = ( ;1; :::; ; ) denoe he hsory of s prvae sgnals realzed a sae, up o perod. We de ne y () and y () smlarly. Hsores of allocaons wll be denoed by x = ( 1 ; :::; ) 2 =1. 5 The ne horzon resrcon s mporan for he resuls on full mplemenaon, whch are based on a backwards procedure. The resrcon s also mananed by Müller (2012a,b), bu no by he (non-robus) dynamc mechansm desgn leraure, whch focuses on paral mplemenaon alone (e.g., Pavan e al., 2013). 6 In hs paper he socal choce funcon (SCF, nroduced below) s aken as a prmve. Hence, an explc represenaon of neremporal consrans s unnecessary. Possble neremporal consrans, as well as he desgner s objecve funcon, are accommodaed mplcly n he SCF, whch can be hough of as he argmax of a consraned opmzaon problem. The analyss ha follows herefore accommodaes he possbly of neremporal consrans n he desgner s opmzaon problem, of course provded ha he resulng SCF sas es he condons saed n he resuls. 4
Socal Choce Funcons. funcon (SCF), f :!. The descrpon of he prmves s compleed by a socal choce We assume ha he SCF s such ha perod- choces are measurable wh respec o he nformaon avalable n ha perod. Tha s, we assume ha here exs funcons f : Y!, = 1; :::; T, such ha f () = f y () T for each 2. =1 Models of Belefs. A any pon n me, agens have (subjecve) belefs abou he feaures of he envronmen hey do no know. These belefs are dsnc from nformaon, whch s encoded n he payo ypes and drecly a ecs he SCF. Such belefs are hus modeled separaely from players nformaon. A model of belefs for an envronmen E s a uple B = (B ; ) 2N such ha for each, B s he se of ypes, assumed Polsh, and : B! ( B ) s a connuous funcon. 7 A perod 0 agens have no nformaon abou he envronmen. Ther (subjecve) prors abou he payo sae and he opponens belefs are mplcly represened by means of ypes b, as he belefs (b ) 2 ( B ). In perods = 1; :::; T, agens updae her belefs usng her prvae nformaon (he hsory of payo sgnals), and oher nformaon possbly dsclosed by he mechansm se n place. The man d erence wh respec o sandard (sac) ype spaces (as n Bergemann and Morrs (2005), for nsance), s ha players here do no know her own payo -ype a he ouse: payo -ypes are dsclosed over me, and known only a he end of perod T. Thus, an agen s ype a he begnnng of he game s compleely descrbed by a pror belef over he payo saes and he opponens ypes. Sandard models of mechansm desgn (e.g., references n foonoe 3) assume common knowledge of a spec c model of belefs, and assume a sngle common pror. Ths corresponds o he case n whch B = fb g s a sngleon for each, and belefs (b ) p 2 () are he same for all and have full suppor. The furher specal case of ndependen ypes (e.g., Pavan e al. (2013)) requres ha here exs ^p 2 ( ) for each such ha p = 2N ^p. A furher assumpon, common n he leraure, s ha such ^p are Markov processes (e.g., Bergemann and Valmak (2010)). In consras, here we wll be neresed on mplemenaon resuls for all possble models of belefs B = (B ; ) 2N. To summarze our ermnology, n an envronmen wh belefs (E; B) we dsngush he followng sages: n perod 0 (he nerm sage) agens have no nformaon, her (subjecve) pror s represened by ypes b, wh belefs (b ) 2 ( B ); T d eren perod- nerm sages, for each = 1; :::; T, when a ype s belefs afer a hsory of prvae sgnals y are concenraed on he se y T =+1 ; B : The erm ex-pos sage refers o he nal realzaon, when he sae of naure s revealed. 3 Mechansms, Incenve Compably and Implemenaon A mechansm s de ned by a se of messages M ; for every 2 N and = 1; :::; T, and by a collecon of oucome funcons (g ) =1;:::;T whch assgn allocaons o each hsory a 7 A Polsh space s a complee separable merc space. For any X, (X) denoes he se of probably measures on X, endowed wh he correspondng Borel sgma-algebra. 5
each sage. As usual, for each we de ne M = 2N M ;. I s assumed ha he repored messages are publcly observed a he end of each perod. Formally, le denoe he empy hsory, and de ne H 0 := fg. For each = 1; :::; T, he se of publc hsores of lengh s de ned as H := H 1 M, and he se of publc hsores s denoed by H := [ T =0 H. The perod- oucome funcon s a mappng g : H 1 M!. A mechansm herefore s a uple T M = h (M ; ) 2N ; g, assumed commond knowledge. We focus hroughou on mechansms =1 n whch he ses M ; are compac subses of nely dmensonal Eucldean spaces. A drec mechansm s such ha M ; = ;, and g = f for all 2 N and = 1; :::; T. Tha s, n a drec mechansm agens are asked o announce her sgnals a every perod. Based on he repors, he mechansm chooses he perod- allocaon as spec ed by he SCF, ha s accordng o he funcon f : Y!. To emphasze he dependence of he drec mechansm on he SCF f, we denoe by M f. Each mechansm nduces a dynamc game. If agens belefs B = (B ; ) 2N are spec ed, he resulng game s a sandard Bayesan game, whch can be analyzed usng sandard soluon conceps, such as Bayes-Nash (BNE) or Perfec Bayesan Equlbrum (PBE). For he analyss of robus mplemenaon, however, s useful o consder envronmens n whch agens belefs are no spec ed. For hs reason we also nroduce he noon of a belef-free game, whch obans mposng a mechansm M on a belef-free envronemn E. In hese games, soluon conceps such as BNE or PBE are no de ned. Ther analyss herefore requres novel soluon conceps, whch wll be nroduced n Secon 5.1. Belef-Free Games. An envronmen E and a mechansm M deermne a belef-free dynamc game, ha s a uple (E; M) = hn; (H ; ; u ) 2N. Ses N, and payo funcons u are as de ned n E. The ses H denoe he se of s prvae hsores, de ned as follows: for each and, le Y = =1 ;, H := H 1 Y and nally H := [ T =1 H. Tha s, for each and, H denoes he se of prvae hsores of lengh for player. Each prvae hsory h = h 1 ; y 2 H s made of wo componens: a publc componen, h 1, whch consss of he agens messages n perods 1 hrough 1; and a prvae componen, y, whch consss of agen s prvae sgnals from perod 1 hrough. I s convenen o nroduce noaon for he paral order represenng he precedence relaon on he ses H and H : h h ndcaes ha hsory h s a predecessor of h (smlarly for prvae hsores: h 1 ; y h 1 ; y f and only f h h and y y.) Agens sraeges n he belef-free game are measurable funcons s : H! S T =1 M ; such ha s h 2 M; for each h 2 H. The se of s pure sraeges s denoed by S. We also de ne he ses S = 2N S and S = j6= S j. For any sraegy pro le s 2 S, each realzaon of 2 nduces a ermnal allocaon g s () 2. Sraegc-form payo funcons, U : S! R, are such ha U (s; ) = u (g s () ; ) for each s and. For each publc hsory h and player, le S h denoe he se of player s sraeges ha are conssen wh hsory h beng observed. Snce s prvae hsores are only nformave of he opponens behavor 6
hrough he publc hsory, for each 2 N, h = h 1 ; y 2 H and j 6=, S j h = Sj h 1. 8 In a drec mechansm, he ruhellng sraeges are hose sraeges ha, condonally on havng repored ruhfully n he pas, repor each perod- sgnal ruhfully. Truhellng sraeges may d er n he behavor hey prescrbe a hsores followng pas msrepors, bu hey all are oucome equvalen and nduce ruhful revelaon n each perod on he pah. The se of such sraeges s denoed by S S, and le S := 2N S. For laer reference, s useful o nroduce he followng noon of ncenve compably: De non 1 (Ex-pos Incenve Compably) SCF f s ex pos ncenve compable (EPIC) f he ruhellng sraegy s an ex-pos equlbrum of he drec mechansm. Tha s, f for all 2 N, 2, s 2 S and s 0 2 S, U (s ; ) U s 0 ; s ;. SCF f s srcly EPIC f he nequaly holds srcly for all s 0 2 S ns. Bayesan Games. A uple (E; M; B) deermnes a dynamc Bayesan game. Sraeges n a Bayesan game are measurable mappngs : B! S. The se of sraeges n (E; M; B) s denoed by. Agen s nformaon ses n he Bayesan game are B (H [ fg), wh generc elemen (b ; h ). A perod 0, agens only know her own ype b. Perod-0 hsores herefore are of he form (b ; ) 2 B fg, and for each 1, perod- hsores are b ; h 2 B H. In he followng, we le h 0, so ha nformaon ses are wren as b ; h 2 B (H [ fg) for 0. From he pon of vew of each, for each b ; h 2 B (H [ fg) and sraegy pro le, he nduced ermnal hsory s a random varable ha depends on he realzaon of he sae of naure and of he opponens ypes. Ths random varable s denoed by g j(b ;h ) (; b ). We de ne he Bayesan game sraegc-form connuaon payo funcons as follows: U ; ; b ; b ; h = u g j(b ;h ) (; b ) ;. Snce (E; M; B) s a dynamc Bayesan game, we consder Perfec Bayesan Equlbrum (PBE) as soluon concep. Ths requres nroducng noaon for belef sysems, whch represen players belefs abou he payo sae and he opponens ypes a each nformaon se of he Bayesan game. Formally, a sysem of belefs for (E; M; B) s a pro le p = (p ) 2N where each p s a collecon of condonal belefs p b ; h 2 ( B ), one for each b ; h 2 B (H [ fg), such ha p (b ; ) = (b ) for all b 2 B. De non 2 Fx an envronmen E, a model of belefs B, and a SCF f. (Truhful PBE) The assessmen (; p) s a ruhful perfec Bayesan equlbrum of he Bayesan game E; M f ; B f sas es he followng condons: () (; p) s a PBE of E; M f ; B ; () (b ) 2 S for all and b 2 B ; () belefs p assgn probably one o he oher agens havng repored ruhfully a all hsores. (PBIC) The SCF f s perfec Bayesan ncenve compable (PBIC) on B, f here exss a ruhful PBE of E; M f ; B. 8 Ses H and S are endowed wh he sandard mercs derved from H T. See Appendx A.1 for deals. 7
Condon () s self-explanaory. Condon () requres he equlbrum pro le o be ruhful, and corresponds o he on-pah ruhful condon of Pavan e al. (2013). Condon () says ha agen always beleves ha he opponens have been followng her equlbrum sraeges. I does no necessarly follow from Bayesan updang because he general seup accommodaes models of belefs over a connuum of sgnals. In hese cases, every hsory has a zero probably. Ths condon s sandard n he leraure on dynamc mechansm desgn (e.g., Pavan e al. (2013), Bergemann and Valmak (2010)). Robus Implemenaon n drec mechansms. In he followng we focus on paral and full mplemenaon n drec mechansms, dscussed respecvely n Secons 4 and 5. The resrcon o drec mechansms s sandard n he leraure on dynamc mechansm desgn, whch has only focused on paral mplemenaon. The resrcon s known o be wh loss of generaly for he full mplemenaon problem, n ha more complcaed mechansms may make full mplemenaon easer o acheve. 9 The smplcy of drec mechansms, however, s an mporan desderaum from he vewpon of he Wlson docrne, and has he furher advanage of makng he comparson beween paral and full mplemenaon easer. Fnally, snce resrcng he class of mechansms makes full mplemenaon resuls harder o oban, he resrcon o drec mechansms srenghens he posve resuls obaned n Secons 5. Sandard mechansm desgn assumes ha agens belefs are known o he desgner, and herefore mplemenaon s de ned for a gven model B. To address he ssue of robusness, we requre ha paral and full mplemenaon are acheved for all possble belefs: De non 3 (Robus Paral Implemenaon) A SCF f s robusly parally mplemenable n he drec mechansm f s PBIC on all models of belefs. De non 4 (Robus Full Implemenaon) SCF f s robusly fully mplemenable n he drec mechansm f for every B = (B ; ) 2N, every PBE-sraegy pro le of he Bayesan game E; M f ; B s such ha (b) 2 S for all b 2 B. De nons 3 and 4 exend o dynamc sengs he noons nroduced, respecvely, by Bergemann and Morrs (2005, BM05) and Bergemann and Morrs (2009a, BM09). As dscussed n he nroducon, he belef-free approach o robusness s clearly very demandng. From a heorecal vewpon, however, s an mporan benchmark, parcularly o analyze he mehodologcal aspecs of robus mechansm desgn (cf. Secon 6). The belef-free approach herefore s he naural sarng pon o exend robus mechansm desgn o dynamc sengs. 9 For hs reason, he classcal leraure on Bayesan Implemenaon ypcally adops complex mechansms n whch agens repor more han her own ype (e.g., Maskn (1999) Poslewae and Schmedler (1988), Palfrey and Srvasava (1989) and Jackson (1991)). Full mplemenaon resuls va smple desgn of ransfers are provded by Ollár and Pena (2014). 8
4 Robus Paral Implemenaon BM05 de ne robus (paral) mplemenaon as nerm ncenve compably (IIC) on all ype spaces. De non 3 essenally adaps he underlyng noon of ncenve compably, replacng IIC wh he sandard n he leraure on dynamc mechansm desgn, PBIC. I can hus be seen as he dynamc counerpar of BM05, as well as he belef-free counerpar of he dynamc mechansm desgn leraure. BM05 show ha a SCF s IIC on all ype spaces f and only f s ex-pos ncenve compable. Snce, for any model of belefs, PBIC mples IIC, an mmedae mplcaon of her resul s ha ex-pos ncenve compably (Def. 1) s a necessary condon for robus mplemenaon n dynamc sengs (Def. 3). Bu snce PBIC s n general more demandng han IIC, achevng PBIC for a gven model of belefs n general requres sronger condons han IIC (e.g., Pavan e al. (2013)). As he nex resul shows, however, PBIC has no exra be once s requred for all models of belefs. EPIC herefore s boh necessary and su cen for robus paral mplemenaon (he proof s n Appendx D.1). Proposon 1 (Paral Implemenaon) SCF f s PBIC on all models of belefs f and only f s ex pos ncenve compable. 10 Thus, from he vewpon of paral mplemenaon, he belef-free approach enals he same ncenve compably condons n sac as n dynamc sengs. Ths resul, however, does no mean ha hs s an nherenly sac approach: beyond ncenve compably, dynamcs has an mporan role even whn he belef-free approach. The nex secon on full mplemenaon provdes one nsance of hs general pon (for anoher nsance of he same pon, see also Müller (2012a,b)). 5 Robus Full Implemenaon Ths secon focuses on robus full mplemenaon n drec mechansms (Def. 4). I can hus be seen as he dynamc counerpar of BM09. In sac sengs, BM09 de ne robus full mplemenaon by requrng ha, for all ype spaces, all he BNE nduce ruhful revelaon. Snce he se of all such equlbra can be compued applyng raonalzably o he belef-free (sac) game, BM09 sudy condons o ensure ha ruhful revelaon s he only raonalzable sraegy n he drec mechansm. Besdes (src) ex-pos ncenve compably of he SCF, hese condons requre he preference nerdependences o be no oo srong. Inuvely, he reason s ha n an EPIC mechansm, srong preference nerdependence deermnes srong sraegc exernales, whch are a source of mulplcy and may hus undermne he possbly of full mplemenaon. Dynamc sengs presen wo dsnc orders of problems. Frs, 10 Noe ha Proposon 1 concerns he properes of a SCF and of he assocaed drec mechansm. I does no sae ha, n general games, he se of ex-pos equlbra and he se of PBE for all models of belefs concde. See Borgers and McQuade (2007) for a dscusson of he relaons beween soluon conceps based on sequenal raonaly and ex-pos equlbra n dynamc games. 9
ndependen of he robusness requremen, characerzng he se of equlbra s d cul n a dynamc seng. Second, he dynamc srucure enrches he possbles of preference nerdependences as well as sraegc exernales, whch can boh exhb neremporal e ecs. The nex secon addresses he rs problem. Frs, we nroduce a weakenng of PBE, whch we call nerm perfec equlbrum (IPE), and show ha he se of IPE-sraeges for all models of belefs s characerzed by a backwards procedure whch combnes he logc of raonalzably and backward nducon. Ths resul s convenen because allows a recursve analyss of he full mplemenaon problem. Endowed wh hese resuls, we urn o he second order of problems n Secon 5.3, where we provde su cen condons for robus full mplemenaon. Snce IPE s weaker han PBE, achevng full mplemenaon hrough he backwards procedure su ces for robus full mplemenaon, as de ned n De non 4. 11 5.1 IPE and he backwards procedure Fx a Bayesan game, (E; M; B). We say ha a sraegy pro le s sequenally raonal wh respec o belef sysem p, f for every 2 N and every (b ; h ) 2 B H, 2 arg max 0 2 RB U (; ; b ; b ; h ) dp (b ; h ). For each agen and for each nformaon se b ; h 1, a sraegy pro le and condonal belefs p (b ; h 1 ) nduce a probably measure P ;p (b ; h 1 ) 2 (H 1 Y ) over he prvae hsores of lengh. De non 5 (IPE) An assessmen (; p) s an Inerm Perfec Equlbrum (IPE) f s sequenally raonal wh respec o p and f p sas es he followng condons: (B-1) for each h = (y ; h 1 ) 2 H and b 2 B, p (b ; h ) 2 (fy g (T =+1 ; ) B ), and (B-2) for each h 1 such ha h h, for every measurable E B, p (b ; h 1 ) [E] = p (b ; h ) [E] P ;p (b ; h 1 ) h. Condon (B-1) requres condonal belefs a each nformaon se o be conssen wh he player s prvae nformaon. Condon (B-2) requres ha belef sysem p s conssen wh Bayesan updang whenever possble, boh on- and o -he-pah. The laer condon n parcular s no requred by weak PBE (cf. Mas-Colell e al. (1995, p.285)). Condons (B-1) and (B-2) mpose essenally no resrcons on he belefs held a hsores ha receve zero probably a he precedng node. 12 IPE herefore s weaker han PBE. Also noe ha any player s devaon s a zero probably even, and reaed he same way: n parcular, f hsory h s precluded by (b ; h 1 ) alone, hen P ;p (b ; h 1 )[h ] = 0, and agen s belefs a (b ; h ) are unresrced he same way hey would be afer an unexpeced move of he opponens. As wll be dscussed shorly, hs feaure of IPE s key o obanng a recursve and racable characerzaon of he se of equlbra. Belef-Free Backwards Raonalzably. We nroduce nex a soluon concep for beleffree dynamc games, whch wll be shown o characerze he se of all IPE-sraeges over all 11 Noe ha, conrary o paral mplemenaon, full mplemenaon resuls n general are sronger f obaned for weaker soluon conceps. 12 Hence, condons B(), B() and B(v) n Fudenberg and Trole (1991, p.332) need no hold n an IPE. 10
models of belefs. The formal de non s noaonally cumbersome, and lef o Appendx B, bu he dea s sraghforward. Fx a belef-free game, (E; M), and a publc hsory of lengh T 1, h T 1. For each payo -ype y T 2 of each agen, he connuaon game s a (belef-free) sac game wh sraeges s jh T 1 2 S ht 1. We apply belef-free raonalzably (e.g., BM09) o hs game, and le R (h T 1 ) denoe he se of pars (y T ; s jh T 1 ) such ha connuaon sraegy s jh T 1 s raonalzable n he connuaon game from h T 1 for ype y T. We do hs for all publc hsores of lengh T 1. We hen proceed backwards: for each publc hsory of lengh T 2, h T 2, we apply raonalzably o he connuaon game from h T 2, resrcng connuaon sraeges s jh T 2 2 S ht 2 o be raonalzable n he connuaon games from hsores of lengh h T 1. We le R (h T 2 ) denoe he se of pars (y T 1 ; s jh T 2 ) such ha connuaon sraegy s jh T 2 s raonalzable n he connuaon game from h T 2 for ype y T 1. Inducvely, hs s done for each h 1, unl he nal node s reached, for whch he se of Belef-Free Backwards Raonalzable (BFBR) sraeges, R, s compued. Proposon 2 (Characerzaon of he se of IPE.) Fx a belef-free game (E; M). For each : ^s 2 R f and only f 9B = (B ; ) 2N s.. 9^b 2 B and (^; ^p) such ha (^; ^p) s an IPE of (E; M; B) and ^s = ^ (^b ): The proof of hs resul can be found n Appendx C. To grasp he basc nuon, noce ha an mplcaon of hs proposon s ha, for each publc hsory h, he se of IPE sraeges n he connuaon game from h concdes wh he se of IPE sraeges of he connuaon game consdered n solaon. Hence, he se of IPE sraeges for all models of belefs has a recursve srucure analogous o ha of he se of subgame perfec equlbra n complee nformaon games. Such se of equlbra can hus be compued backwards, analyzng each connuaon game n solaon. The propery of IPE hghlghed above, ha own-devaons are reaed he same as he opponens, s key o he possbly of consderng connuaon games n solaon, whch s needed for hs resul. Snce IPE s weaker han PBE, Proposons 1 and 2 mply ha an EPIC SCF s fully robusly mplemened by a drec mechansm f all he BFBR-sraeges are ruhful: Corollary 1 Le f be EPIC, and consder he belef-free game nduced by drec mechansm assocaed o SCF f, E; M f. If for all, R 6= ; and R S, hen f s fully robusly mplemenable (Def. 4). By ransformng he orgnal problem no an ar cal sequence of sac problems, he backwards procedure enables us o buld on he nsghs of he sac leraure o oban su cen condons ha guaranee ha all he sraeges conssen wh hs soluon concep are ruhful. Robus full mplemenaon hen follows from Corollary 1. The nex secon conans an llusrave example. Secon 5.3 presens he general resuls. 11
5.2 Example: A Dynamc Publc Good Problem Consder an envronmen wh wo agens (n = 2) and wo perods (T = 2), and le ; = [0; 1] for each and. In each perod, he planner chooses some quany q 2 R + of publc good. Agen s margnal uly for he publc good n perod s a funcon ; () of he realzed sae: for any = ( ;1 ; ;2 ; j;1 ; j;2 ) 2, ;1 ( 1 ) = ;1 + j;1 and ;2 ( 1 ; 2 ) = ' ( ;1 ; ;2 ) + ' ( j;1 ; j;2 ) ; where 0 and ' : [0; 1] 2! R s connuously d erenable and srcly ncreasng n boh argumens (hence, rs perod sgnals a ec second perod valuaons). Furhermore, f = 0, hs s a prvae-values seng; for any > 0, agens have nerdependen values. Fnally, we assume me-separably and ransferable uly. Agen s uly funcon herefore s: u (q 1 ; q 2 ; ;1 ; ;2 ; ) = ;1 ( 1 ) q 1 + ;1 (1) + [ ;2 ( 1 ; 2 ) q 2 + j;2 ], where ; denoes he ransfer a perod = 1; 2. Noaon ; s mnemonc for aggregaor : funcons ( ; ) =1;2 aggregae he nformaon avalable o all he agens up o perod no real numbers whch unquely pn down s preferences. Assumng a cos of producon equal o c (q ) = 1 2 q2, he opmal levels of publc good n he wo perods are such ha, for any 2 : q1 () = ;1 ( 1 ) + j;1 ( 1 ) and (2) q2 () = ;2 ( 1 ; 2 ) + j;2 ( 1 ; 2 ). (3) We consder here robus mplemenaon of he e cen rule, (q ) =1;2. By Proposon 1, ex-pos ncenve compably s a necessary condon for robus mplemenaon. We herefore le (; ; j; ) =1;2 denoe he ex-pos ncenve compable ransfers, and consder he SCF f = (q ; ; ; j; ) =1;2. 13 To see ha f s ndeed EPIC, for any (; m), le (; m) := ;2 (m) ;2 (). Gven a pro le of rs perod repors ^m 1 = ( ^m ;1 ; ^m j;1 ) and prvae sgnals (^ ;1 ; ^ ;2 ), s bes response m ;2 o pon belefs ( j;1; j;2 ; m j;2 ) a he second perod sas es: 14 ^;1 ; ^ ;2 ; j;1 ; j;2 ; ^m 1 ; m ;2; m j;2 = 0. (4) 13 The EPIC ransfers are de ned as follows: for any 2, ;1 ( ;1; j;1) = (1 + ) ;1 j;1 + 1 2 2 ;1 and ;2 ( ;1; j;1) = (1 + ) ' ( ;1; ;2) ' ( j;1 j;2) + 12 ' (;1;2)2. 14 For he sake of llusraon, we gnore here he possbly of corner soluons, whch do no a ec he fundamenal nsgh. Corner soluons wll be dscussed n Secon 5.3. 12
Smlarly, gven prvae sgnal ^ ;1 and pon belefs abou he fuure own sgnal and message and abou he opponens sgnals and repors n boh perods, ( ;2 ; m ;2 ; ; m ), he rs perod bes-response m ;1 for agen sas es: m ;1 ^ ;1 = ( j;1 m j;1 ) @' m ;1 ; m ;2 + ^;1 ; ;2 ; m ;1; m ;2 ; ; m 1 @m ;1 (5) I s now easy o verfy ha f = (q ; ; ; j; ) =1;2 s EPIC: for any, f agen has repored ruhfully n he pas (m ;1 = ;1 ) and he expecs he opponens o repor ruhfully ( = m ), hen (4) s sas ed f and only f repors ruhfully n he second perod (m ;2 = ;2 ). Furhermore, f (; m) = 0, he rgh-hand sde of (5) s zero f he opponens repor ruhfully n he rs perod, and so s opmal o repor m ;1 = ^ ;1. Noce ha hs s he case for any 0, whch means ha f s robusly parally mplemenable for all 0. Ex-pos ncenve compably, however, does no su ce for full mplemenaon, as nonruhful equlbra may also exs. To address hs problem, we apply he backwards procedure R nroduced above. Frs, noce ha (4) mples ha, condonal on havng repored ruhfully n he rs perod (m ;1 = ;1 ), ruhful revelaon n he second perod s a bes-response o ruhful revelaon of he opponen, rrespecve of he realzaon of. If msrepored n he rs perod (m ;1 6= ;1 ), hen (mananng m j; = j; for = 1; 2) he opmal repor n he second perod s a furher msrepor (m ;2 6= ;2 ) such ha he mpled value of he aggregaor ;2 s equal o s rue value (.e., (; ^m 1 ; m 2 ) = 0). Ths s he noon of self-correcng sraegy, s c : a sraegy ha repors ruhfully a he begnnng of he game and a every ruhful hsory, bu n whch earler msrepors are followed by furher msrepors, o correc he mpac of he prevous msrepors on he value of he aggregaor ;2. 15 We show nex ha, f < 1, he self-correcng sraegy pro le s he only pro le survvng he backwards procedure nroduced above. Hence, gven he resul n Proposon 2, he selfcorrecng sraegy s he only IPE-sraegy, hence he only PBE-sraegy, for any model of belefs. Snce s c nduces ruhful revelaon, hs mples ha f s fully robusly mplemenable f < 1. To hs end, x he pro le of rs perod repors, ^m 1. Gven = ( ;1 ; ;2 ) and m ;2 2 M ;2, le w ( ^m ;1 ; m ;2 ; ) = [' ( ^m ;1 ; m ;2 ) ' ( ;1 ; ;2 )] denoe ype s mpled overrepor of he value of '. Then, equaon (4) can be nerpreed as sayng ha he opmal over-repor of ' ( ) s equal o mes he (expeced) under-repor of ' ( j ). Le 0 j w and w 0 j denoe, respecvely, he mnmum and maxmum possble values of w ( ^m ;1 ; m ;2 ; ) over (m ;2 ; ). Then, f s raonal, he opmal over-repor for ype a hsory ^m 1, w ( ; ^m 1 ), s bounded above and below, respecvely, by w 1 w 0 j and 1 w w0 j. Recursvely, de ne w k = w k 1 j and k w = wk j 1. Also, for each k and, le z k [ wk k ] denoe he dsance w beween he maxmum and lowes possble over-repor a sep k. Subsung, we oban a 15 In hs example, he self-correcng sraegy can be relaed o he srongly ruhful sraegy of Pavan e al. (2013), rede nng s second perod sgnals as ' ( ). Ths ransformaon, however, s no always possble n he general envronmens of Secon 5.3. Foonoe 17 dscusses hs pon n some deal. 13
sysem of d erence equaons, z k = z k 1, where:! " z k = zk zj k and = 0 0 # : (6) Noce ha he connuaon game from ^m 1 s domnance solvable f and only f z k! 0 as k! 1. In ha case, for each, w ( ; ^m 1 )! 0, and so he connuaon of he self-correcng sraegy s unquely raonalzable n he connuaon game. In hs example, hs s he case f and only f < 1. Hence, f < 1, he only raonalzable oucome n he connuaon from ^m 1 guaranees ha = 0. Gven hs, he rs perod bes response (5) smpl es o m ;1 ^ ;1 = ( j;1 m j;1 ) : The same argumen can be appled o show ha ruhful revelaon s he only raonalzable sraegy n he rs perod f and only f < 1. Then, f < 1, he self-correcng sraegy s he only sraegy survvng he backwards procedure R, hence he only sraegy played as par of PBE, for any model of belefs. I follows ha, f preference nerdependences are no oo srong, f s robusly fully mplemenable. Key properes and her generalzaons. The nex secon generalzes he key nsghs of hs example o envronmens wh monoone aggregaors of nformaon (EMA, Def. 6). As n he example above, hese envronmens have he propery ha for each agen and n each perod, all he avalable nformaon (across me and agens) can be summarzed by T one-dmensonal sascs (one for each perod), whch jonly pn down he agen s preferences. Preferences, however, need no be addvely separable over perods nor quaslnear. To accommodae he more general class of preferences, he noon of self-correcng sraegy wll be generalzed. A conracon propery wll be nroduced o formalze he dea of boundng preference nerdependences, whch generalzes he condon < 1 n he example. The man resuls show ha, f he SCF s srcly EPIC and preferences sasfy such a conracon propery and properly de ned sngle-crossng condons, hen s c s he only sraegy ha survves he backwards procedure. The SCF s herefore robusly fully mplemenable. As shown by he example, despe EPIC s sas ed here, he analyss sll presens non rval dynamcs due o he neremporal neracon and he full mplemenaon requremen: n he presence of sraegc uncerany, ruled ou n he paral mplemenaon approach, dependng on he agen s belefs abou fuure sgnals and ohers sraeges, msreporng n one perod may be a bes response even f he SCF s EPIC. I s only hanks o he mehodology based on BFBR ha we can ensure ha, f < 1, (; m) = 0 ndependen of he agen s curren choce, and hence we can analyze he problem as f s a sac one. Furher neremporal e ecs, whch do no arse n he example above, are possble n he general framework (e.g., an example wh pah dependen preferences s dscussed n Secons 5.3.1 and 5.3.2). 14
5.3 Robus Full Implemenaon De non 6 (EMA) An Envronmen adms monoone aggregaors (EMA) f, for each, and for each = 1; :::; T, here exss an aggregaor funcon ; : Y! R and a valuaon funcon v : R T! R ha sasfy he followng condons: 1. For each ( ; ) 2, u ( ; ) = v ; ( (y ( ))) T =1 2. ; and v are connuous and ; s srcly ncreasng n ;. 3. For any y ; ^y 2 Y, f ; y ; y > ; ^y ; y for some y 2 Y, hen ; y ; ^y > ; ^y ; ^y for all ^y 2 Y. Assumng he exsence of he aggregaors and he valuaon funcons (condon 1), per se, enals lle loss of generaly. The be of he represenaon derves manly from he connuy and monooncy condons (2), and from condon (3), whch guaranees ha s prvae hsores of sgnals Y aggregaor ;. 16 can be ordered n erms of he nduced values of he perod- The nex de non generalzes he dea of self-correcng sraegy nroduced n he example. De non 7 (Self-correcng sraegy) The self-correcng sraegy, s c 2 S, s such ha for each = 1; :::; T and publc hsory h 1 = ~y 1, and for each h = h 1 ; y ;, s c h = arg mn m ; 2 ; ( max y 2Y ; y; y : ) ; ~y 1 ; m ; ; y : (7) In words, condonal on pas ruhful revelaon, sraegy s c ruhfully repors s perod- sgnal; a hsores ha come afer prevous msrepors of agen, s c enals a furher msrepor, o o se he mpac on he perod- aggregaor of he prevous msrespors. Sraegy s c herefore nduces ruhful reporng on s pah, hence s c 2 S. Gven prvae hsory h = h 1 ; y (and he nduced repor s c h ), le ^y h be s..: ^y h 2 arg max y 2Y ; y; y ; ~y 1 ; s c h ; y. (8) Then, he de non of s c and he properes of ; (Def. 6) mply ha, for any h = h 1 ; y ;, 16 To undersand he resrcon enaled by condon (3), suppose ha aggregaor funcon ;2 n he example of Secon 5.2 s replaced by he followng: ;2 () = ( ;2 + j;2)+( ;1+ j;1)1 f j;1 ;1g (where 1 fg denoes he ndcaor funcon). To see how hs volaes condon (3), consder y 2 = (3=4; 3=4) and ^y 2 = (1=2; 1=2). Then, ;2(y 2 ; yj 2 ) > ;2(^y 2 ; yj 2 ) f yj 2 = (1=2; 1=2), bu ;2(y 2 ; yj 2 ) < ;2(^y 2 ; yj 2 ) f yj 2 = (3=4; 3=4). Hence, y 2 does no mply an unambguously hgher aggregaed value han ^y 2 does: wheher one hsory of sgnals nduces a hgher aggregaed value han he oher (hence a hgher margnal uly for q 2 n he example) depends on he sgnals of he oher player. Ths s ruled ou by condon (3), whch requres one hsory of sgnals o be unambguously hgher han he oher, n erms of he nduced aggregaed value. 15
hree cases are possble: ; ; ; y; y = ; ~y 1 ; s c h ; y for all y 2 Y ; (9) y; ~y > ; ~y 1 ; s c h ; ^y h and s c h = h ;, (10) y; ~y < ; ~y 1 ; s c h ; ^y h and s c h = l ;. (11) Equaon (9) corresponds o he case n whch sraegy s c can compleely o se he prevous msrepors. Bu here may exs hsores a whch no curren repor can o se he prevous msrepors. In he example of Secon 5.2, suppose ha he rs perod under- (respecvely, over-) repor s so low (hgh), ha even reporng he hghes (lowes) possble message n he second perod s no enough o correc he mpled value of '. These hsores correspond, respecvely, o cases (10) and (11), assocaed wh he hghes and lowe perod- repors h ; and ; l, and n he example hey nduce corner soluons a he second perod.17 We nroduce nex a conracon propery whch bounds he srengh of preference nerdependence, and provdes a mul-perod exenson of he analogous condon for sac envronmens n BM09. To nroduce he condon formally, some exra noaon s needed: for each se of sraegy pro les D S, and for each prvae hsory h = (h 1 ; y ), le D (h ) := fm ; : 9 (s ; s ) 2 D s.. s (h ) = m ;g and D h 1 := S y 2Y D h 1 ; y. Le s D h 1 denoe he se of pars m ; ; y 2 M; Y such ha m ; 2 D (h 1 ; y ), and s c h 1 he se of m ; ; y such ha m; = s c h 1 ; y. De non 8 (Conracon Propery) An envronmen wh monoone aggregaors of nformaon sas es he Conracon Propery f, for each non-empy D S such ha D 6= fs c g and for each publc hsory h 1 = ~y 1 such ha s D h 1 6= s c h 1, here exss y and m 0 ; 2 D h 1 ; y, m 0 ; 6= s c h 1 ; y, such ha: sgn s c h 1 ; y m 0 ; = sgn ; y; y for all y = y 1 ; ; 2 Y and m 0 ; 2 D h 1 ; y. ; ~y 1 ; m 0 ;; m 0 ; ; (12) To nerpre he condon, and make more easly comparable o BM09 s, s useful o 17 Suppose ha he aggregaor funcons sasfy he followng reducon propery : 8; 8, 9' ; : Y! R and ^ ; : R Y s.. ; y; y = ^; ' ; y ; y for all y ; y (he example n Secon 5.2 has hs propery). Then, agens sgnals can be relabeled as follows: for each and, le he ransformed sgnal be ; 0 = ' ; y (). In he relabeled model, ; only depends on ; 0 (no on ; 0 for < ). Sraegy s c herefore nduces ruhful revelaon of he ransformed sgnal a every perod. The self-correcng sraegy s hus relaed o he srongly ruhful sraeges consdered, for nsance, by Pavan e al. (2013) o sudy PBIC n Markovan envronmens. Ths connecon, however, s no perfec: rs, he envronmens of Def. 6 need no sasfy he reducon propery, and he relabelng ha ransforms he self-correcng no a srongly ruhful sraegy may no be possble; second, s c s de ned n belef-free envronmens, whch do no nclude a sochasc process, hence no assumpons are made on he evoluon of sgnals over me. Thus, even f he reducon propery s sas ed, so ha he perod- aggregaor ; s compleely pnned down by he (relabeled) perod- sgnal 0 ; = g ; y, he envronmen need no be Markovan: s belefs abou fuure sgnals, for nsance, may depend on earler sgnals, even f ; does no. 16
rewre he argumen of he rgh-hand sde of (12) as follows: y; y ; ~y 1 ; m 0 ;; m 0 ; = ; ~y 1 ; s c h 1 ; y ; y ; ~y 1 ; m 0 ;; m 0 ; ; + h 1 ; y; y ; where h 1 ; y; y = ; y; y (13) ; ~y 1 ; s c h 1 ; y ; y : (14) Frs noe ha he erm represens he exen by whch he self-correcng sraegy s ncapable of o seng he prevous msrepors. To undersand how he conracon propery formalzes he dea ha preference nerdependences are no oo srong (resrcon < 1 n he example), consder rs a publc hsory ~y 1 along whch has repored ruhfully (~y 1 = y 1 ). A such a hsory, he self-correcng sraegy requres ha repors ; ruhfully, so ha (h 1 ; y ; y ) = 0. Then, he condon bols down o requrng ha, gven ~y 1 and s perod- sgnal ;, for any m 0 ; 6= ; and for all y and m0 ;, sgn [ ; m ; ] = h sgn ; (y 1 ; ; ); y ; ~y 1 ; m 0 ; ; m0 ;. ) Tha s, he drec mpac of s prvae sgnal ; on he aggregaor ; s always su cenly srong ha he d erence n he aggregaed value beween he rue sgnals and he repored sgnals always has he same sgn as he d erence beween he rue and repored sgnal of agen by self, regardless of ohers repors n hs perod (m 0 1 ; ), or wheher her earler repors were ruhful or no (~y s gven, bu he condon s requesed for all y ). The same logc apples o oher hsores wh he propery ha = 0 (whch, by equaons (9)-(11), occur whenever s c (h ) 2 (l ; ; h ; )), wh he only d erence ha he ruhful repor ; s now replaced by he self-correcng repor, s c h. To accoun for he possbly ha, a some hsores, he self-correcng sraegy s no su cen o o se he prevous msrepors (he corner soluons n he example of Secon 5.2), he conracon propery furher requres ha he sgn of he mpac on he aggregaor ; s no o se by he prevous msrepors, measured by > 0. 18 Thus, smlar o BM09 s analogous condon, he conracon propery lms he srengh of he preference nerdependence. The key d erence here s ha payo ypes are revealed over me, and he srengh of he preference nerdependence may vary from perod o perod. The condon above ensures ha such preference nerdependence remans small a any pon n me, for all possble repors ha may have already been revealed. The las assumpon o oban he full mplemenaon resul s a sngle-crossng condon. As usual, sngle-crossng condons allow o sor ypes wh respec o he mplemened allocaon. The key d erence n dynamc envronmens s ha hs sorng mus also ake no accoun he neremporal ncenves. The sngle-crossng condon herefore wll nvolve resrcons ha are boh whn and beween perods. To cleanly separae he wo, we rs consder aggregaor based SCFs (Secon 5.3.1), and show ha a smple whn perod 18 Appendx D.4 llusraes how hs complexy may be avoded by adopng smple mechansms wh exended message spaces, so ha any possble pas msrepor can be correced, nducng h 1 ; y ; y = 0 a all hsores. For a smlar rck, see Pavan (2008). 17
sngle-crossng condon su ces o oban full mplemenaon n hs case. We hen dscuss he resrcveness of he aggregaor-based assumpon, and he exra complcaons due o relaxng. The general resuls are provded n Secon 5.3.2. 5.3.1 Aggregaor-Based SCF Consder he SCF n he example of Secon 5.2: he SCF has he propery ha he allocaon chosen by he SCF n perod s only a funcon of he values of he aggregaors n perod. The noon of aggregaor-based SCF generalzes hs dea. De non 9 (AB-SCF) The SCF f = (f ) T =1 s aggregaor-based f for each, ; y = ; ~y for all mples f y = f ~y. We nex nroduce a sandard sngle-crossng condon (SCC), appled o every perod. De non 10 (SCC-1) An envronmen wh monoone aggregaors of nformaon sas es SCC-1 f, for each, valuaon funcon v s such ha: for each, and ; 0 2 s.. = 0 for all 6=, for each a ; 2 RT 1 and for each ; < ; 0 < 00 ;, v (; ; ; a ; ) > v ( 0 ; ; ; a ; ) and v (; ; 0 ; a ; ) = v ( 0 ; ; 0 ; a ; ) mples v (; ; 00 ; a ; ) < v (0 ; ; 00 ; a ; ). Equvalenly: for any wo allocaons and 0 ha only d er nher perod- componen, for any a ; 2 RT 1, he d erence ; (; 0 ; ; ) = v ; ; ; a ; v 0 ; ; ; a ;, as a funcon of ;, crosses zero a mos once (Fgure 1.a, p. 20). If T = 1, SCC-1 concdes wh BM09 s condon. Proposon 3 (Full Implemenaon: AB-SCF) In an envronmen wh monoone aggregaors (Def. 6) sasfyng SCC-1 (Def. 10) and he conracon propery (Def. 8), f an aggregaor-based socal choce funcon sas es Src EPIC (De non 1), hen R = fs c g. The argumen of he proof, whch can be found n Appendx D.2, s analogous o ha of he example n Secon 5.2. For each hsory of lengh T 1, s proved ha he conracon propery and SCC-1 mply ha agens play accordng o s c n he las sage. Then, he argumen proceeds by nducon: gven ha n perods +1; :::; T agens follow s c, a msrepor a perod only a ecs he perod- aggregaor. Snce he SCF s aggregaor-based, he allocaons a perods 6= are xed, hence he problem a perod- s essenally sac. The conracon propery and he whn perod SCC-1 herefore mply ha he self-correcng sraegy s followed a sage. And so on. Clearly, he enre argumen reles on he mehodologcal resuls of Secon 5.1, whch enable us o sudy connuaon games n solaon, and o apply he backwards procedure. An apprasal of he aggregaor-based assumpon. Consder he mporan specal case of me-separable preferences: for each and = 1; :::; T, here exs an aggregaor funcon 18
; : Y! R and a valuaon funcon v : R! R such ha for each ( ; ) 2, TX u ( ; ) = v ; ; y ( ) : 19 =1 In hs case, he condon ha he SCF s aggregaor-based (Def. 9) can be nerpreed as sayng ha he SCF only responds o changes n preferences. These preferences, however, canno accommodae phenomena of pah-dependence such as hab formaon or learnng-bydong. If preferences are pah-dependen, he aggregaor-based assumpon s oo resrcve. For nsance, suppose ha agens n he example of Secon 5.2 have he followng preferences: u (q 1 ; q 2 ; ;1 ; ;2 ; ) = ;1 ( 1 ) q 1 + ;1 (15) + [ ;2 ( 1 ; 2 ) F (q 1 ) q 2 + j;2 ]. Then, he margnal uly of q 2 also depends on he amoun of publc good provded n he rs perod. Then, he e cen rule for he second perod s q2 () = [ ;2 () + j;2 ()]F (q 1 ), whch s no aggregaor-based. To allow for pah-dependen preferences, herefore, s mporan o relax he aggregaor-based assumpon. 5.3.2 Relaxng he AB-assumpon The problem wh relaxng he aggregaor-based assumpon s ha a one-sho devaon from s c a perod- may nduce d eren allocaons n perod- and also n subsequen perods. Hence, he whn perod sngle-crossng condon (SCC-1) may no su ce o conclude he nducve sep n he proof of Proposon 3, and guaranee ha sraegy s c s played a perod- for all T. To avod hs problem, some bound s needed on he mpac ha a one-sho devaon has on he oucome of he SCF. The nex sngle-crossng condon guaranees ha such neremporal e ecs of one-sho devaons are no oo srong. De non 11 (SCC-2) An envronmen wh monoone aggregaors of nformaon sas es SCC-2 f, for each : for each ; 0 2 such ha 9 2 f1; :::; T g s.. y () = y ( 0 ) for all < and j () = j (0 ) for all > and for all j, for each a ; 2 RT 1 and for each ; < 0 ; < ; 00, v (f () ; ; ; a ; ) > v (f ( 0 ) ; ; ; a ; ) and v (f () ; ; 0 ; a ; ) = v (f ( 0 ) ; ; 0 ; a ; ) mples v (f ( 0 ) ; ; 00 ; a ; ) < v (f (0 ) ; ; 00 ; a ; ): SCC-2 compares he allocaons chosen for any wo smlar saes of naure, and 0. These saes are smlar n he sense ha hey are dencal up o perod 1, and mply he same value of he aggregaors a all perods oher han. Snce agens preferences are unquely deermned by he values of he aggregaors (Def. 6), he preferences nduced by saes and 0 only d er along he dmenson of he perod- aggregaor. The condon requres a 19 These preferences are me-separable n he sense ha each v does no depend on he allocaons chosen n perods oher han. Funcon v, however, may depend on prevous sgnals. 19