Continuous Implementation

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1 Contnuous Implementaton Maron Oury HEC Pars Olver Terceux Pars School of Economcs and CNRS Abstract It s well-known that mechansm desgn lterature makes many smplfyng nformatonal assumptons n partcular n terms of common knowledge of the envronment among players. In ths paper, we ntroduce a noton of contnuous mplementaton and characterze when a socal choce functon s contnuously mplementable. More specfcally, we say that a socal choce functon s contnuously (partally) mplementable f t s (partally) mplementable for types n the model under study and t contnues to be (partally) mplementable for types "close" to ths ntal model. We rst show that f the model s of complete nformaton a socal choce functon s contnuously (partally) mplementable only f t sats es Maskn s monotoncty. We then extend ths result to general ncomplete nformaton settngs and show that a socal choce functon s contnuously (partally) mplementable only f t s fully mplementable n teratve domnance. For nte mechansms, ths condton s also su cent. We also dscuss mplcatons of ths characterzaton for the vrtual mplementaton approach. Journal of Economc Lterature Class caton Numbers: C79, D82 Keywords: Hgh order belefs, Robust mplementaton Frst verson: February 15, Ths verson: May 11, We wsh to thank Drk Bergemann, Phlppe Jehel, Atsush Kaj, Erc Maskn, Ncolas Velle and semnar partcpants at Pars School of Economcs, the EEA meetng n Budapest, Prnceton Unversty, the Insttute for Advanced Study n Prnceton, Rutgers Unversty, Mc Gul Unversty, the world congress of game theory n Northwestern, the workshop on new topcs n mechansm desgn n Madrd, GRIPS workshop on Global Games n Tokyo and the PSE-Northwestern jont workshop. We are especally grateful to Stephen Morrs for helpful remarks and dscussons at varous stages of ths research. Olver Terceux thanks the Insttute for Advanced Study at Prnceton for nancal support through the Deutsche Bank membershp. He s also grateful to the Insttute for ts hosptalty. 1

2 1 Introducton The noton of partal as opposed to full mplementaton conssts n desgnng games under whch some equlbrum but not necessarly all yelds the outcome desred by the socal planner. Despte the fact that undesrable equlbra may potentally exst, partal mplementaton s wdely used both n theoretcal and appled works. One of the man reasons for ts success s the celebrated revelaton prncple: f the desred outcomes can arse as an equlbrum n some mechansm, then t wll arse n a truth-tellng equlbrum of the drect mechansm. It s then nformally argued that truth-tellng s a focal pont and that agents should coordnate on ths equlbrum when t exsts. If the socal planner s not perfectly sure that the nformaton structure of the model he has n mnd corresponds exactly to the true stuaton, the very noton of truth-tellng strategy becomes problematc. Nevertheless, the focal pont argument used to defend the partal approach extends to ths (more realstc) context: f an agent s type t s very "close" to a type t of the ntal model, then reportng the message that corresponds to type t may reasonably be seen as a focal pont for type t. Followng ths lne of thought and takng nto account the doubts a socal planer may have about hs model, we characterze when a socal choce functon can be partally contnuously mplemented. More spec cally, we requre that n any perturbaton of the ntal model, there exsts an equlbrum that yelds the desred outcome, not only at all types of the ntal model but also at all types "close" to ntal types. Our man results state that ths contnuty requrement leads to necessary (and su cent) condtons that are tghtly lnked to full mplementaton. Otherwse stated, ths paper shows that the partal mplementaton paradgm s very fragle when slght mod catons of the nformaton structures are allowed. In a rst step, we focus on the smple case n whch the ntal model s of complete nformaton. In ths spec c settng, wdely used n mechansm desgn, the approach of partal mplementaton s very permssve. For nstance, when there are at least three agents, any socal choce functon can be partally mplemented 1. Let us be more spec c and descrbe our contnuty requrement n ths settng. A pro le of complete nformaton types may be seen as a pro le of degenerate herarches of belefs where every player knows the realzed state of nature, every player knows that everyone knows and so on... Put n another way, the modeler studes a set of (degenerate) herarches of belefs where 1 The mechansm allowng such a permssve result s drect. Just assume that whenever at least n 1 players out of n send the same state of nature, the mechansm assgns the outcome desred by the planer at ths state. In any other case, the mechansm assgns some arbtrary outcome. 2

3 some gven state of nature s commonly known. In our settng, a pro le of ncomplete nformaton types t s consdered to be close to a pro le t of complete nformaton types (where t s common knowledge that the real state of nature s ) f, t nduces herarches of belefs where each player beleves wth a hgh probablty that payo s are gven by ; each player beleves wth a hgh probablty that each player beleves wth a hgh probablty that payo s are gven by ; etc... up to hgh but nte order. In our rst result, we show that a socal choce functon s contnuously mplementable only f t sats es Maskn s monotoncty. Many socal choce functons are not monotonc and hence not contnuously mplementable. Snce Maskn s monotoncty s necessary and (almost) su cent for full Nash mplementaton n complete nformaton settngs (Maskn (1999)), ths result bulds a rst brdge between partal and full mplementaton. In other words, a lack of full mplementaton can be problematc even f the socal planer s only wllng to partally mplement the socal choce functon. Our contnuty requrement naturally extends to the case where the ntal model s of ncomplete nformaton. To formalze ths, we use the method ntroduced by Harsany (1967) and developed n Mertens and Zamr (1985). Each type n the ntal model s mapped nto a herarchy of belefs. Then, followng the nterm approach due to Wensten and Yldz (2007), we de ne a noton of nearby types. As already underlned n the complete nformaton settng, ths noton, formally descrbed by the product topology n the unversal type space, captures the restrctons on the modeler s ablty to observe the players (hgh order) belefs. In ths general settng, we provde our man result: f a socal choce functon s contnuously mplementable, then t must also be fully mplementable n ratonalzable messages. More precsely, we show that f some mechansm contnuously mplements a socal choce functon f, then we can extract from the ntal mechansm a "smaller" mechansm that fully mplements f n ratonalzable messages. For nte mechansms, ths condton s also su cent 2. Borgers (1995) shows that full mplementaton n ratonalzable messages s a demandng noton when consderng large preference domans. However, under complete nformaton, Bergemann and Morrs (2009a) establshes a tght connecton between ths noton and full mplementaton n Nash equlbrum. Bergemann and Morrs (2009b,c) provde necessary and su cent condtons for full mplementaton n ratonalzable messages whle Bergemann and Morrs (2007) studes an applcaton to ascendng auctons. Vrtual mplementaton corresponds to the requrement that the outcomes spec ed by the socal choce functon arse wth probablty arbtrarly close to but not necessarly 2 As wll be dscussed further, for n nte mechansms, the exstence of an equlbrum s not ensured and so ths condton need not be su cent. 3

4 equal to one. Movng to vrtual mplementaton may be seen as natural when consderng contnuty requrements. A corollary of our man result states that a socal choce functon s contnuously vrtually (partally) mplementable wth nte mechansms f and only f t s vrtually (fully) mplementable n ratonalzable messages wth nte mechansms. Whle apparently smlar to our man result, ths characterzaton s actually much less demandng. Indeed, under vrtual mplementaton, very permssve su cent condtons have been establshed by Abreu and Matsushma (1992a,b) for the soluton concept of ratonalzablty. More precsely, n complete nformaton settngs, Abreu and Matsushma (1992a) shows that under very weak doman restrctons, f there are more than three players, any socal choce functon s vrtually mplementable n ratonalzable messages. Abreu and Matsushma (1992b) extends ths result to ncomplete nformaton: they show that Bayesan Incentve Compatblty and a measurablty condton, whch seems weak and genercally sats ed 3, are both necessary and su cent. In other terms, under the vrtual approach, the gap between partal mplementaton (whch s equvalent to Bayesan Incentve Compatblty) and full mplementaton n ratonalzable messages s qute small. Snce mechansms used n ths lterature are nte, ths means that when movng to vrtual mplementaton, our contnuty requrement leads to much less severe restrctons than for exact mplementaton. We nterpret ths result as a new argument n favor of the vrtual approach: even f the socal planer s only nterested n partal mplementaton, consderng Abreu and Matsushma s mechansms makes sense. Snce the semnal paper by Rubnsten (1989) on the e-mal game, several approaches have been followed to analyze the connecton between hgh order belefs and strategc behavor; the so-called noton of robustness due to Kaj and Morrs (1997), the global games argument due to Carlsson and Van Damme (1993) and the nterm approach due to Wensten and Yldz (2007). These works share the common assumpton that n the perturbed models, some types may have preferences that are radcally d erent from those of types n the ntal model 4. Indeed, the behavor of these spec c types s used as a startng pont for contagon processes that drve results n these analyses. Note that the meanng of such an assumpton n the mechansm desgn context (where the socal planer xes the game form) would be problematc. However, n the present paper, we show that the logc of mplementaton makes ths assumpton unnecessary. Indeed, n mechansm desgn, several d erent states of natures are ex ante possble for the socal planer. Our 3 For nstance, as noted n Abreu and Matsushma (1992b) or Bergemann and Morrs (2009d), a smple su cent condton for all socal choce functons to satsfy the measurablty condton, s type dversty: every type has dstnct preferences over lotteres uncondtonal on others types. 4 Ths corresponds to the noton of "crazy types" n the robustness approach and to that of "domnance regons" n global games or the nterm approach. 4

5 argument n the proof uses ths multplcty and shows that ths settng s then rch enough: partal mplementaton n the ntal model s used as an (endogenous) startng pont for the contagon process at equlbrum. It s then enough to assume that sendng a message may nvolve an (arbtrarly) small cost. Snce n many real economc stuatons sendng a message s costly 5, ths techncal assumpton 6 s n the sprt of our local requrements: mechansms that are not robust to an arbtrarly small departure from the assumpton of costless messages are rather undesrable. Our results also contrbute to the lterature on the so-called "Wlson doctrne" 7. Bergemann and Morrs (2005) s one of the rst attempts to relax the mplct common knowledge assumptons made n the mechansm desgn lterature. 8 In ther settng, the modeler when choosng a mechansm has no nformaton on the real stuaton that wll nally preval among the agents 9. Consequently, ther noton of robust mplementaton follows a "global approach": a socal choce functon 10 s robustly (partally) mplementable f t s (partally) mplementable on all possble models. They show that a socal choce functon s robustly mplementable f and only f t s ex-post mplementable 11. On the contrary, we assume that the planer has some spec c model n mnd and s qute con dent about t. As a consequence, our requrement s only local: the socal choce functon must be mplemented only at types close to types n the ntal model. Ths s the reason why 5 For nstance, sendng a message n real lfe stuatons may consst n llng n a questonnare whch s tme-consumng and hence costly. It may sometmes requre to present costly physcal proofs such as observable characterstcs of products, endowments... See Bull and Watson (2007) or Kartk and Terceux (2009) for detals. 6 In case a player has several best responses aganst some belef, ths assumpton allows us to buld a small perturbaton of the envronment where ths player has a unque best response. 7 Wlson (1987) wrtes "I foresee the progress of game theory as dependng on successve reductons n the base of common knowledge requred to conduct useful analyses of practcal problems. Only by repeated weakenng of common knowledge assumptons wll the theory approxmate realty". 8 Another related paper s Chung and Ely (2001). They study full mplementaton n undomnated Nash equlbrum. They show that (under hedonc preferences), whle almost all socal choce functons are fully mplementable n undomnated Nash equlbrum, only monotonc socal choce functons can be fully mplemented n undomnated Nash f we also requre that no dscontnuty occurs at complete nformaton nformaton. There are two man d erences wth our work: rst we focus on partal mplementaton, second the topology behnd ther contnuty requrement s d erent from ours. See Kunmoto (2008) for addtonal detals on ther underlyng topology. 9 Alternatvely, Artemov, Kunmoto and Serrano (2007) consder that the planer knows the ( nte set of) rst-order belefs of the agents. 10 Bergemann and Morrs (2005) also consder socal choce correspondences. 11 Ex-post mplementaton requres that each agent s strategy be optmal for every possble realzaton of the types of other agents. The possblty of ex-post mplementaton has been recently studed (see Jehel et al (2006) and Bkhchandan (2006)). 5

6 ex-post mplementaton s not necessary n our settng. 12 Ths paper s organzed as follows. Secton 2 analyses the complete nformaton case. In Secton 3, we extend our noton of contnuous mplementaton to ncomplete nformaton and gve our man result. We conclude wth a dscusson of further ssues n Secton 4. 2 Complete Informaton Case We rst ntroduce the complete nformaton settng and the noton of mplementaton under complete nformaton. Then, we de ne our man noton of contnuous mplementaton. 2.1 Complete Informaton Implementaton We consder a nte set of players I = f1; :::; Ig: Each agent has a bounded utlty functon u : A! R where s the nte 13 set of states of nature and A s the set of outcomes endowed wth an arbtrary topology. A socal choce functon s a mappng f :! A. 14 If the true state of nature s ; the planner would lke the outcome to be f(): A mechansm spec es a message set M for each agent and a mappng g from message pro les to outcomes. More precsely, we wrte M as an abbrevaton for Q 2I M and for each player, M for Q j6= M j. 15 A mechansm M s a par (M; g) where the outcome functon g : M! A assgns to each message pro le m an alternatve g(m) 2 A. In what follows, we assume that message spaces are countable. 16 By a slght abuse of notatons, we wll sometmes note m for the degenerate dstrbuton n (M) assgnng probablty 1 to m. For each 2 ; a mechansm M = (M; g) nduces a complete nformaton game (M; ) = [I; fm g 2I ; fu (g(:); )g 2I ] where each agent s payo when message pro le m s sent s u (g(m); ). We also denote the set of pure Nash equlbra n (M; ) by 12 It s not su cent ether. Ths s due to the fact that Bergemann and Morrs (2005) use the so-called "known own payo type" unversal type space. To be more spec c, they de ne the set of states of nature by = 2I where s the set of player s payo types. Then, they assume that there s common knowledge that each player knows hs payo type. 13 The nteness assumpton s used to prove our man result (Theorem 2) but s not needed to establsh the necessty of Maskn monotoncty (Theorem 1). 14 In the paper, we restrct our attenton to socal choce functon for smplcty. Extensons to socal choce correspondences wll be dscussed further. 15 Smlar abbrevatons wll be used throughout the paper for analogous objects. 16 As wll become clear, ths assumpton wll allow us to prove our necessary condtons for contnuous mplementaton usng only models wth a countable set of types. When movng to su cent condtons, havng models wth countable set of types wll be useful to apply standard exstence theorems; see footnote 16. 6

7 NE(M; ) = fm 2 M : for each, u (g(m ; m ); ) u (g(m 0 ; m ); ) for all m 0 2 M g. Implementaton lterature (n partcular partal mplementaton) often focuses on the equlbrum concept of pure (and not mxed) Nash. We recall the de ntons of partal and full mplementaton under complete nformaton. De nton 1 A socal choce functon f :! A s partally mplementable f there exsts a mechansm M such that for each ; there exsts m 2 NE(M; ) such that g(m ) = f(). De nton 2 A socal choce functon f :! A s fully mplementable f there exsts a mechansm M such that for each ; NE(M; ) 6= ; and for any m 2 NE(M; ) we have: g(m ) = f(). 2.2 Contnuous Implementaton To de ne our noton of contnuous mplementaton,we embed the complete nformaton settng n a rcher settng that allows to perturb hgh order belefs. We also relax the assumpton that sendng a message s perfectly costless. Small costs of messages We assume that sendng a message may be slghtly costly. Indeed, sendng a message usually requres to ll n a questonnare, to wrte a letter and sometmes to present costly physcal proofs such as observable characterstcs of goods, endowments... A recent lterature n mplementaton takes nto account costs of messages 17. We beleve that a mechansm mplementng a socal choce functon should stll mplement t when we allow for slght departures from the assumpton of costless messages. In order to formalze ths dea, we proceed as follows. Gven a mechansm M = (M; g), for each player, we de ne a cost functon c : M ~! R + where ~ s the space of states of nature assocated wth costs of messages. We assume that the state space ~ s rch enough. More precsely, t s de ned by ~ = [ [ f ~ m g [ f ~ 0 g 2I m 2M where for each player and each message m ; we have c (m ; ~ 0 ) = 0, c (m ; ~ m ) = 0 and c (m 0 ; ~ m ) = for all m 0 6= m ; where s a strctly postve parameter that can be chosen arbtrarly close to 0. When no confuson arses, we wll omt the dependence wth respect to. Note that snce M has been assumed to be countable, ~ s also countable. Next, we wrte = ~ for the extended set of states of nature. For a gven state of 17 See for nstance Bull and Watson (2007), Deneckere and Severnov (2008), Matsushma (2008) and Kartk and Terceux (2009). 7

8 nature = (; ~ ) 2 ; the utlty functon of player for a gven message m and a gven outcome a s u (a; ) and (; ~ 0 ). c (m ; ~ ). For notatonal convenence, we wll sometmes dentfy Techncally, the above constructon s used to break tes. More precsely, f a type s nd erent between several messages, we can slghtly perturb hs nformaton so that ths type has a unque best reply. Our basc pont s that we want to allow a rch enough uncertanty such that for each message there exsts a state of nature where the cost of ths message s smaller than the other messages costs. Note that ths assumpton s remnscent of the rchness assumpton assumed n Wensten and Yldz (2007). However, t s much weaker. Indeed, the rchness assumpton states that for any player and any message m, there exsts a state nature where m s strctly domnant for player. Models There are two man classes of stuatons wth ncomplete nformaton. The rst one conssts n stuatons wth an ex ante stage durng whch each player observes a prvate sgnal about the payo s, and the jont dstrbuton of sgnals and payo s s commonly known. These stuatons are naturally modelled usng a standard type space. The second class, on whch we focus n ths paper, conssts n genune stuatons of ncomplete nformaton,.e. stuatons wth no ex ante stage: each player begns wth a herarchy of belefs. We follow the standard Harsany (1967) s approach and model these herarches of belefs by ntroducng a hypothetcal ex ante stage leadng to a standard type space. Ths allows us to study strategc behavor of players at types that are consdered to be close to a gven orgnal model. A model T s a par (T; ) where T = T 1 ::: T I s a countable 18 type space and t 2 ( T ) denotes the assocated belefs for each t 2 T. Gven a mechansm M and a model T ; we wrte U(M; T ) for the nduced ncomplete nformaton game. In ths game, a (behavoral) strategy of a player s any measurable functon : T! (M ): We wll note (m j t ) for the probablty that strategy assgns to message m when player s of type t. For each 2 I and for each belef 2 ( M ); set BR ( j M) = arg max m 2M X (; ~ ;m )2 M (; ~ ; m ) h u (g(m ; m ); ) c (m ; ~ ) : Gven any type t and any strategy pro le, we wrte ( j t ; ) 2 ( M ) for the jont dstrbuton on the underlyng uncertanty and the other players messages nduced by t and. 18 Ths assumpton s just made to ensure exstence of Bayes Nash equlbrum n nte games whch wll turn out to be useful when we deal wth su cent condtons for contnuous mplementaton. 8

9 De nton 3 A pro le of strateges = ( 1 ; :::; I ) s a Bayes Nash equlbrum n U(M; T ) f for each 2 I and for each t 2 T ; m 2 Supp( (t )) ) m 2 BR ( ( j t ; ) j M). As spec ed earler, types close to complete nformaton are types where t s mutually beleved (wth arbtrarly large probablty) up to an arbtrarly large order that a gven complete nformaton stuaton occurred. More formally, gven a model (T; ) and any type t n type space T, we can compute the belef of t on (.e. hs rst-order belef ) by h 1 (t ) = marg t We can compute the second-order belef of t ;.e. hs belefs about ( ; h 1 1 (t 1); :::; h 1 I (t I)); by settng h 2 (t ) = t ( ( ; t ) j ( ; h 1 1(t 1 ); :::; h 1 I(t I )) 2 F ) for each measurable F ( ) I : We can compute an entre herarchy of belefs by proceedng n ths way. Hence, a type of a player nduces an n nte herarchy of belefs (h 1 (t ); h 2 (t ); :::; h k (t ); :::) where h 1 (t ) 2 ( ) s a probablty dstrbuton on ; representng the belefs of about ; h 2 (t ) 2 ( ( ) I ) s a probablty dstrbuton representng the belefs of about and the other rst order belefs. Let us wrte h (t ) for the resultng herarchy and h k (t ) for the kth-order belefs of type t. The set of all belef herarches for whch t s common knowledge that the belefs are coherent (.e., each player knows hs belefs and hs belefs at d erent orders are consstent wth each other) s the unversal type space (see Mertens and Zamr (1985) and Brandenburger and Dekel (1993)). We denote by T belef n ths space and wrte T = Y T. 2I the set of player 0 s herarches of In our formulaton, two types t and t are close f there exsts a su cently large k such that for each l k; the lth-order belefs h l (t) and h l (t) are close n the topology of convergence of measures. To be more precse, each T s endowed wth the product topology, so that a sequence of types ft [n]g 1 n=0 converges to a type t ; f, for each k : h k (t [n])! h k (t ) (.e. h k (t [n]) converges toward h k (t ) n the topology of weak convergence of measures 19 ). In such a case, we wrte t [n]! P t. We wll sometmes use the metrc d k (:; :) on the kth level belefs 20 that metrzes the topology of weak convergence of measures. 19 Recall that h k (t [n]) 2 (X k 1 ) where X 0 = and X k = [(X k 1 )] I X k I.e. on (X k 1 ) see the prevous footnote. One such metrc s the Prokhorov metrc; see Secton

10 We now ntroduce our man noton of contnuous mplementaton. We rst de ne the complete nformaton model T CI = (T CI ; ) as follows. For each player ; T CI = S ft ; g and t; = (; ~ 0 ;t ; ) where x denotes the probablty dstrbuton that puts probablty 1 on fxg. It s easly checked that h (t ; ) s the herarchy of player s belefs correspondng to common knowledge 21 of (; ~ 0 ). We wll henceforth call the type t ; a complete nformaton type. The modeler s nterested n strategc behavor of types where players mutually beleve (wth hgh probablty) up to a hgh (but nte) level that the state of nature s some 2. Our contnuty requrement wll ensure that the socal choce functon s mplemented not only at complete nformaton types but also at types that are so close to the complete nformaton stuaton that they cannot be ruled out by the modeler. In what follows, for two models T = (T; ) and T 0 = (T 0 ; 0 ) we wll note T T 0 f T T 0 and for all ; t 2 T 0 : t = 0 t. De nton 4 Fx a mechansm M and a model T T CI. We say that an equlbrum n U(M; T ) contnuously mplements f f for each t 2 T CI, () (t ) s pure and () for any sequence t[n]! P t where for each n : t[n] 2 T; we have g (t[n])! f(). Notce that pont () mantans the requrement of pure strategy behavor usually assumed n mplementaton theory; ths wll allow for smple comparsons wth exstng results 22. We now state a formal de nton of contnuous mplementaton. De nton 5 A socal choce functon f :! A s contnuously mplementable f there exsts a mechansm M, such that for any model T T CI, there s a Bayes Nash equlbrum n the nduced game U(M; T ) whch contnuously mplements f Monotoncty as a Necessary Condton In ths secton, we show that any socal choce functon that s contnuously mplementable sats es the well-known monotoncty condton as de ned n Maskn (1999). Ths result, whch s a rst step toward our man result, reduces the gap between partal and full mplementaton snce as proved by Maskn ths monotoncty condton s necessary and "almost" su cent for full mplementaton 23. Let us rst recall the de nton of monotoncty for socal choce functons. 21 In ths paper, we do not dstngush common knowledge and common belef. 22 Ths assumpton s dspensble for our man result (Theorem 2). In addton, provded that the de nton of monotoncty s extended to lotteres, Theorem 1 also extends. 23 Maskn (1999) showed that wth more than three players together wth the assumpton that f sats es the weak condton of No Veto Power, monotoncty actually mples full mplementaton. 10

11 De nton 6 A socal choce functon f s monotonc f for every par of states and 0 such that for each player and for each a 2 A; u (a; ) u (f(); ) ) u (a; 0 ) u (f(); 0 ); (?) we have f() = f( 0 ): We now state the man theorem of ths secton. Theorem 1 A socal choce functon s contnuously mplementable only f t s monotonc. Proof of Theorem 1. Assume that there exsts a mechansm M = (M; g) that contnuously mplements f: Pck ; 0 2 such that for each player and for each a 2 A; the relaton (?) s sats ed. We want to show that f() = f( 0 ). We show that there exsts a model T = (T; ) such that for any equlbrum that contnuously mplements f, there s a sequence of types ft[n]g 1 n=1 n T such that t[n]! P t 0 and g (t[n])! f(). By pont () of De nton 4: g (t[n])! f( 0 ); whch mples f() = f( 0 ). For ths purpose, we buld the desred model (T; ) n whch for each player, each set T sats es: T = T CI [ 1[ [ k=1 m2m t (k; m) where t (k; m) and are de ned recursvely as follows. For each m = (m 1 ; :::; m I ) 2 M : t (1; m) s such that marg T t (1;m)(t ; ) = 1; marg t (1;m)( 0 ) = 1;! and marg ~ t (1;m)( ~ m ) = 1. In addton, for each k 2; t (k; m) s de ned by marg ~ t (k;m)( ~ 0 ) = 1 marg T t (k;m)(t (k 1; m)) = 1; marg t (k;m)( 0 ) = 1; 1 k ; and, marg ~ t (k;m)( ~ m ) = 1 k : Observe that snce M has been assumed to be countable, each T s countable, and so s T. 11

12 Now pck any equlbrum of the nduced game U(M; T ) that contnuously mplements f. By pont () n De nton 4; (t ) s a pure Nash equlbrum n the complete nformaton game (M; ). In addton, pont () n De nton 4 mples that g (t ) = f(). In the sequel, we note m := (t ; ) and m := (t ). We have for any player and m 0 2 M : u (g(m ); ) u (g(m 0 ; m ); ); and so, u (f(); ) u (g(m 0 ; m ); ). By (?); ths mples that u (f(); 0 ) u (g(m 0 ; m ); 0 ); whch n turn mples u (g(m ); 0 ) u (g(m 0 ; m ); 0 );.e. m s a pure Nash equlbrum n (M; 0 ). Otherwse stated, for each player : m 2 BR ( ( 0 ; ~ 0 ;m ) j M); n addton, t s easly checked that ( j t (1; m ); ) = ( 0 ; ~ m ;m ). Consequently, fm g = BR ( ( 0 ; ~ m j M); and snce s an equlbrum: ;m ) (t (1; m )) = m. Usng a smlar reasonng, t s easy to show nductvely that for all k 2 (t (k; m )) = m. Ths means that for each k 1; g (t(k; m )) = g(m ) = f() and so obvously, g (t(k; m ))! f() (as k! 1) whch completes the proof snce t(k; m )! P t 0 (as k! 1). One may thnk that our strong result stems from lack of a common pror n the model we buld. However, t s possble to slghtly perturb the condtonal belefs of types n our model so that usng an argument due to Lpman (2003, 2005), these types could be pcked from models where players share a common pror. Our paper focuses on socal choce functons; for socal choce correspondences, two de ntons of partal mplementaton coexst. To be more spec c, n the rst de nton, whch s "weak", a socal choce correspondence F : A s partally mplementable f there exsts a mechansm M and a selecton f of F such that the mechansm M partally mplements f. In the second de nton, whch s "strong", a socal choce correspondence F : A s partally mplementable f there exsts a mechansm M that partally mplements each selecton f of F. Maskn (1999) gves a de nton of Maskn monotoncty 12

13 for socal choce correspondences 24 and shows that under the same condtons as for socal choce functons, ths noton mples "strong" full mplementaton,.e. that there exsts a mechansm M such that for each : F () = g(ne(m; )); note that the mechansm M partally mplements each selecton of F. Usng a "strong" de nton of contnuous partal mplementaton, Theorem 1 can easly be extended to socal choce correspondences. It s clear that f a mechansm contnuously mplements a socal choce functon, then t must partally mplement n NE ths socal choce functon. If we add the requrement that t must partally mplement n strct NE, then we may dspense wth the assumpton on cost of messages. In ths case, a necessary condton would be the strct monotoncty condton (.e. where the nequaltes n the de nton of monotoncty are replaced by strct nequaltes). 3 Incomplete Informaton So far, we focused our attenton on stuatons where the planer has a complete nformaton settng n mnd. We now relax ths assumpton and consder the general case where the ntal model of the planer s (potentally) an ncomplete nformaton one. As before, the modeler wants to see how strategc behavor s a ected under hs mechansm when the assumpton that hs model s common knowledge s relaxed. We now move to our man result whch establshes a tght connecton between our noton of contnuous mplementaton and full mplementaton n ratonalzable messages. 3.1 De ntons We rst extend the de nton of contnuous mplementaton to an ncomplete nformaton settng. In the sequel, we x a nte 25 model T = ( T ; ) whch s the model the planer has n mnd. De nton 7 Fx a mechansm M and a model T T, we say that an equlbrum n U(M; T ) contnuously mplements f f for each t 2 T, () (t) s pure and () for any sequence t[n]! P t where for each n : t[n] 2 T; we have g (t[n])! f(t). 24 A socal choce correspondence s monotonc f for all ; b 2 F () and any 0 such that for each player and a 2 A u (a; ) u (b; ) ) u (a; 0 ) u (b; 0 ) we have b 2 F ( 0 ). 25 The nteness assumpton allows to prove Theorem 2 usng only models that are countable whch agan wll be useful when movng to su cency results. 13

14 De nton 8 A socal choce functon f : T! A s contnuously mplementable f there exsts a mechansm M, such that for each model T T, there s a Bayes Nash equlbrum n the nduced game U(M; T ) whch contnuously mplements f. 3.2 Necessary Condton Our characterzaton result reles on the noton of full mplementaton n ratonalzable messages. Frst, let us recall the de nton of (nterm correlated) ratonalzablty gven n Dekel, Fudenberg and Morrs (2006, 2007). Pck any pro le of types t drawn from some arbtrary model T = (T; ). For each and t ; set R 0 (t j M; T ) = M ; and de ne for any nteger k > 0 the sets R k(t j M; T ) teratvely, by 8 >< m 2 BR (marg R k M j M) for some 2 ( T M ) [t j M; T ] = m 2 M where marg >: T = t and ( ; t ; m ) > 0 =) m 2 R k 1 (t j M; T ) 9 >=. >; where R k 1 (t j M; T ) stands for Q j6= Rk j 1 (t j j M; T ). The set of all ratonalzable messages for player (wth type t ) s 1\ IY R 1 (t j M; T ) = R k (t j M; T ) and R 1 (t j M; T ) = R 1 (t j M; T ). k=0 =1 Remark 1 Lpman (1994) gves an alternatve de nton of ratonalzablty for the case of countable acton sets. Whle hs de nton s consstent wth common knowledge of ratonalty, the one we use n ths paper s a coarser soluton concept. Usng a coarser soluton concept strengthens our necessary condton; ths condton wll reman vald under any ner noton of ratonalzablty. Our su cency results wll be proved for nte mechansms where both concepts concde. We say that a socal choce functon s fully mplementable n ratonalzable messages, or smply fully ratonalzable mplementable, f there s a mechansm M so that at each pro le of types t 2 T : m 2 R 1 (t j M; T ) ) g(m) = f(t). In the sequel, for two mechansms M = (M; g) and M 0 = (M 0 ; g 0 ), we wrte M 0 M f M 0 M and g jm 0 = g 0 where g jm 0 denotes the restrcton of g to M 0. Our man theorem s stated as follows. Theorem 2 A socal choce functon f : T! A s contnuously mplementable wth a mechansm M only f t s fully ratonalzable mplementable by some mechansm M 0 M. The above result states a necessary condton for contnuous mplementaton. When movng to su cency, exstence of equlbra becomes an ssue. Gven that the set of 14

15 messages may be n nte, there may exst models where no equlbrum exsts even f f s fully ratonalzable mplementable (n the model T ). However, ths condton wll turn to be su cent when consderng nte mechansms. Note that f f s contnuously mplementable, then t s partally mplementable. Hence, the prevous result shows that full mplementaton n (Bayes) Nash equlbrum s a necessary condton for contnuous mplementaton. Jackson (1991) has extended Maskn s monotoncty to ncomplete nformaton settngs. He de nes Bayesan monotoncty and shows that ths noton s a necessary condton for full mplementaton n Nash equlbra n ncomplete nformaton settngs. Hence, as a corollary of the above result, we get that Bayesan monotoncty s also necessary for contnuous mplementaton whch generalzes our Theorem Fnally, usng the "weak" de nton of partal mplementaton for socal choce correspondences gven n Secton 2.3, t s possble to extend Theorem 2 and to establsh that a necessary condton for F to be (weakly) contnuously mplementable s that F must be (weakly) fully mplementable n ratonalzable messages (.e. there s a mechansm M and a selecton f of F such that M full mplements f n ratonalzable messages.) 27 Let us move now to the proof of Theorem 2. Snce f s contnuously mplementable, there exsts a mechansm M = (M; g), such that for any model T = (T; ), there s a Bayes Nash equlbrum n the nduced game U(M; T ) where for each t 2 T, () (t) s pure and () for any sequence t[n]! P t where for each n : t[n] 2 T; we have g (t[n])! f(t). We let be the set of pure Bayesan Nash equlbra of U(M; T ). Note that because T s nte and M s countable, s countable. For each 2 ; we buld the set of message pro les M() n the followng way. For each player and each postve nteger k, we de ne nductvely M k (). Frst, we set M 0 () = ( T ). Then, for each k 1 : M k+1 () = BR (( f ~ 0 g M k ()) j M): Recall that n the model T = ( T ; ); marg ~ t ( ~ 0 ) = 1, for each and t 2 T. Snce s an equlbrum n U(M; T ), M 0() = ( T ) BR (( f ~ 0 g M 0 ()) j M) = M 1(). Consequently, t s clear that for each k : M k k+1 () M (). Fnally, set M () = lm k!1 M k() = S M k(). In the sequel, for each 2, we wll note M() k2n the mechansm (M(); g jm() ). 26 Bergemann and Morrs (2009b) de ne the noton of nterm ratonalzable monotoncty whch s necessary for full ratonalzable mplementaton. Clearly, Theorem 1 mples that nterm ratonalzable monotoncty s necessary for contnuous mplementaton. 27 For contnuous "strong" partal mplementaton, we beleve that Bayesan monotoncty as de ned by Jackson (1991) s necessary. 15

16 Notce that gven any model T = (T; ) such that T T, T s a belef closed subspace n T,.e., for any and t 2 T : marg T t ( T ) = 1. Hence, for any model T T and any equlbrum n U(M; T ); the restrcton of to T denoted jt s an equlbrum n U(M; T ). A rst nterestng property of the famly of sets fm()g 2 s as follows: there s a model T T for whch any equlbrum n U(M; T ) has full range n M( jt ).e. each message pro le n M( jt ) s played under at some pro le of types n the model T. More precsely, Proposton 1 s the rst step of the proof of Theorem 2. Proposton 1 There exsts a model T =(T; ) such that for any 2 and m 2 M(), there exsts t[; m] 2 T s.t. (t[; m]) = m for any equlbrum n U(M; T ) s.t. jt =. Proof. We buld the model T =(T; ) as follows. For each equlbrum 2, player and nteger k, we de ne nductvely t [; k; m ] for each m 2 M k () and set T = [ 2 1[ k=1 [ m 2M k () t [; k; m ] [ T Note that T s countable. In the sequel, we x an arbtrary 2. Ths equlbrum s sometmes omtted n our notatons. M k 1 For each k 1 and m 2 M k(), we know that there exsts k;m ()) such that m 2 BR ( k;m ). Thus we can buld ^ k;m such that whle marg ~^ k;m marg M k 1 () ^k;m = marg M k 1 = ~ m. Note that BR (^ k;m j M) = fm g. 2 ( f ~ 0 g 2 ( ~ M k 1 ()) () k;m In the sequel, for each player and m 2 M 0 (), we pck one type denoted t [; 0; m ] n T satsfyng (t [; 0; m ]) = m. Ths s well-de ned because by constructon, M 0 () = ( T ). Now, for each m 2 M 1 (), we let t [; 1; m ] be de ned by 28 t [;1;m ](; ~ ; t ) = ( 0 f t 6= t [; 0; m ] for each m 2 M 0 () ^ 1;m (; ~ ; m ) f t = t [; 0; m ] for some m 2 M 0 (). Ths probablty measure s well-de ned snce ^ 1;m ( ~ M 0 ()) = 1. In the same way, for each k > 1 and m 2 M k(), we de ne nductvely t [; k; m ] by: t [;k;m ](; ~ ; t ) = ( 0 f t 6= t [; k 1; m ] for each m 2 M k 1 () ^ k;m (; ~ ; m ) f t = t [; k 1; m ] for some m 2 M k 1 () : Agan, ths probablty measure s well-de ned snce ^ k;m ( ~ M k 1 ()) = Here agan, we abuse notatons and wrte t [; 0; m ] for (t j[; 0; m j]) j6=. Smlarly, t[; 0; m] stands for (t [; 0; m ]) 2I. Smlar abuse wll be used along ths proof. 16

17 To complete the proof, we show that for any equlbrum of U(M; T ) such that jt =, we have: (t [; k; m ]) = m ; (1) for each player, nteger k and message m 2 M k (): The proof proceeds by nducton on k. Frst note that, by constructon, of t [; 0; m ], we must have for any equlbrum of U(M; T ) such that jt = : (t [; 0; m ]) = m ; for each player and message m 2 M 0 (): Now, assume that Equaton (1) s sats ed at rank k 1 and let us prove t s also sats ed at rank k. Fx any m 2 M k () and any equlbrum of U(M; T ) such that jt =. Note that (t [; k; m ]) 2 BR ( j M) where 2 ( ~ M ) s such that (; ~ ; m ) = X t [;k;m ](; ~ ; t ) (m j t ). t In addton, by the nductve hypothess and the fact that s an equlbrum of U(M; T ) satsfyng jt =, we have (m j t [; k 1; m ]) = 1 for any m 2 M k 1 (): Hence, by constructon of t [;k;m ], we have (; ~ ; m ) = X t t [;k;m ](; ~ ; t ) (m j t ). = t [;k;m ](; ~ ; t [; k 1; m ]) = ^ k;m (; ~ ; m ) We get that (t [; k; m ]) 2 BR ( j M) = BR (^ m j M) = fm g as clamed. We now gve a rst nsght on the second step of the proof of our man result. Frst notce that, by constructon, each M() sats es the followng closure property: takng any belef 2 ( f ~ 0 g M ()) such that BR ( j M) 6= ;, we must have BR ( j M) M () and hence, BR ( j M) = BR ( j M()). Now pck a type t 2 T and a message m 2 R 1 ( t jm(); T ), t s possble to add a type t m to the model T de ned n Proposton 1 satsfyng the followng two propertes. Frst, h 1 (tm ) s arbtrarly close to h 1 ( t ); second, for any equlbrum wth jt =, (t m ) = m. Indeed, by de nton of R 1( t jm(); T ), there exsts a belef m 2 ( T M ()) where marg m = marg t and such that m 2 BR (marg M () m j M()). Usng our assumpton on cost of messages, we can slghtly perturb m so that m becomes a unque best reply. So let us assume for smplcty that fm g = BR (marg M () m 17 j M()). Hence, we can de ne the type t m

18 that assgns probablty marg M () m ( ; m ) to ( ; t [; m ]) where t [; m ] s de ned as n Proposton 1 (.e. t [; m ] plays m under any equlbrum n U(M; T ) such that j T = ). Now pck any equlbrum n U(M; T ) such that j T =. By constructon, Supp( (t m )) BR (marg M m j M) and so BR (marg M () m j M) 6= ;. By the closure property descrbed above, BR (marg M m j M) = BR (marg M () m j M()) and so we get that type t m plays m under the equlbrum and wll satsfy the desred property. Usng a smlar reasonng, we show nductvely the followng "contagon" result. Proposton 2 There exsts a model ^T = ( ^T ; ^) such that for each equlbrum 2 and each player the followng holds. For all t 2 T and m 2 R 1 (t j M(); T ), there exsts a sequence of types f^t [n]g 1 n=0 n ^T such that (1) ^t [n]! P t and (2) (^t [n]) = m for each equlbrum of U(M; ^T ) satsfyng j T =. Proof. See Appendx. We are now n a poston to complete the proof of our man Theorem. Proof of Theorem 2. Pck ^T = ( ^T ; ^) as de ned n Proposton 2. By de nton of contnuous mplementaton, there exsts an equlbrum n U(M; ^T ) that contnuously mplements f and pont () n De nton 7 ensures that j T s a pure equlbrum. Now pck any t 2 T and m 2 R 1 (t j M( j T ); T ), we show that g jm(j T )(m) = f(t) provng that the mechansm M( j T ) full mplements f n ratonalzable messages. Applyng Proposton 2, we know that there exsts a sequence of types f^t[n]g 1 n=0 n ^T such that (1) ^t[n]! P t and (2) (^t[n]) = m for all n. By (1) and the fact that contnuously mplements f, we have g (^t[n])! f(t) whle by (2) we have g (^t[n]) = g(m) for all n. Hence, we must have g(m) = f(t) and so g jm(j T )(m) = f(t) as clamed. 3.3 A Characterzaton Our man Theorem provdes a necessary condton for contnuous mplementaton. Now, we show that f we restrct our attenton to nte mechansms, ths condton s actually su cent. Theorem 3 A socal choce functon f s contnuously mplementable by a nte mechansm f and only f t s fully ratonalzable mplementable by a nte mechansm. Proof of Theorem 3. The only f part s proved by Theorem 2. Let us prove the f part. Assume that f : T! A s fully ratonalzable-mplementable by a nte mechansm M = (M; g).e. for all t 2 T, m 2 R 1 (t j M; T ) =) g(m) = f(t). 18

19 Lemma 1 (Dekel, Fudenberg and Morrs (2006)) Fx any model T = (T; ) such that T T and any nte mechansm M. (1) For any t 2 T and any sequence ft[n]g 1 n=0 n T, f t[n]! P t then, for n large enough, we have R 1 (t[n] j M; T ) R 1 (t j M; T ). (2) For any type t 2 T : R 1 (t j M; T ) s non-empty. Now pck any model T = (T; ) such that T T, we show that there exsts an equlbrum that contnuously mplements f. Because M s nte and T s countable, standard arguments show that there exsts a Bayes Nash equlbrum n U(M; T ). Pck any sequence ft[n]g 1 n=0 n T, such that t[n]! P t. It s clear that for each n : (t[n]) 2 R 1 (t[n] j M; T ). In addton, for n large enough, we know by Lemma 1 that R 1 (t[n] j M; T ) R 1 (t j M; T ). Then, for n large enough, (t[n]) 2 R 1 (t j M; T ) and so g (t[n]) 2 g (R 1 (t j M; T )) = ff(t)g as clamed. Theorem 3 allows to gve a new ratonale for the noton of vrtual mplementaton where nte mechansms are usually used. In the sequel, we assume that A s a metrc space and note d the assocated metrc. Gven a socal choce functon f; for each > 0, we note B (f) = ff 0 : T! A : d(f 0 (t); f(t)) < for all t 2 T g. A socal choce functon f s sad to be partally vrtually mplementable by nte mechansms f for each > 0; there exsts a socal choce functon f 0 2 B (f) that s partally mplementable by a nte mechansm (that may depend on ). In the same way, we can extend the de nton of contnuous mplementaton. De nton 9 A socal choce functon f s vrtually contnuously mplementable by nte mechansms f for all > 0, there exsts a socal choce functon f 0 contnuously mplementable by a nte mechansm. 2 B (f) that s We also say that a socal choce functon f s vrtually fully ratonalzable mplementable by nte mechansms f for all > 0, there exsts a socal choce functon f 0 2 B (f) that s fully ratonalzable mplementable by a nte mechansm. Usng Theorem 3 above we can extend our characterzaton result to vrtual mplementaton. Proposton 3 A socal choce functon f s vrtually contnuously mplementable by - nte mechansms f and only f t s vrtually fully ratonalzable mplementable by nte mechansms. Whle the formulatons n Proposton 3 and Theorem 3 are smlar, ther mplcatons are qute d erent. Indeed, n Abreu and Matsushma (1992b) settng 29, Bayesan Incentve 29 In ths settng, the ( nte) set of outcomes s extended to the set of lotteres over outcomes and the natural metrc s used over ths set. 19

20 Compatblty and a measurablty condton are both necessary and su cent for vrtual mplementaton n ratonalzable messages. The measurablty condton seems weak and s genercally sats ed 30. Hence, any socal choce functon that s partally mplementable (whch s equvalent to Bayesan Incentve Compatblty) s vrtually mplementable n ratonalzable messages. Snce mechansms used n these papers are nte, we know by Proposton 3 that they also ensures vrtual contnuous mplementaton. Hence, we beleve that Proposton 3 provdes a new foundaton to the approach of vrtual mplementaton n ratonalzable messages. 4 Dscusson 4.1 Falure of the Revelaton Prncple We present a varant of the well-known Solomon s predcament and establsh that the revelaton prncple does not hold for contnuous mplementaton. Each of two agents, 1 and 2, clams an object. There are two payo types: at 1 (resp. 2 ), player 1 (resp. player 2) s the legtmate owner. The set of outcomes s A = f(x; p 1 ; p 2 ) j x 2 f0; 1; 2; 3g and p 1 ; p 2 2 R + g where p s the level of the ne mposed on player and the varable x correspond to the followng stuatons. If x = 0, the object s not gven to ether player; f x 2 f1; 2g, t s attrbuted to player x, and f x = 3, both players are punshed and the none of them receve the object. The socal planner wshes to gve the good to the true owner,.e. he wants to mplement contnuously the socal choce functon f : f 1 ; 2 g! A for whch f( 1 ) = (1; 0; 0) and f( 2 ) = (2; 0; 0). Utlty functons are assumed to be quas-lnear and the object to have a monetary value for each player. More precsely, ths value for player s v H f he s the legtmate owner of the object and v L f he s not, wth v H > v L > 0. Fnally, the punshment outcome (x = 3) corresponds to a ne f L for player f he s the legtmate owner and to a ne f H f he s not, wth f H > f L > 0. For nstance when the payo type s 1 ; the utlty of player 1 when the outcome s (3; p 1 ; p 2 ) s: u 1 ((3; p 1 ; p 2 ); 1 ) = f L p 1 and when outcome (1; p 1 ; p 2 ) s gven: u 1 ((1; p 1 ; p 2 ); 1 ) = v H p 1. The followng two clams establsh the falure of the revelaton prncple when a contnuty requrement s taken nto account. Clam 1 f s not contnuously mplementable wth a drect mechansm.e. a mechansm M = (M; g) n whch for each 2 f1; 2g, M = f 1 ; 2 g. 30 As noted n the ntroducton, type dversty (whch states that every type has dstnct preferences over lotteres uncondtonal on others types) s su cent for all socal choce functons to be measurable n the sense of Abreu and Matsushma (1992b). 20

21 Proof. We establsh that no mechansm M 0 M can fully mplement n NE the socal choce functon f. Theorem 2 completes the proof of Clam 1. Obvously, f cannot be mplemented f the set of message pro les s a sngleton. Now assume that the set of message pro les s a sngleton for one player, say player 1,.e. M1 0 = fg for some 2 f 1; 2 g and M2 0 = f 1; 2 g. In ths case, player 2 must have a message m 2 such that g(m 2 ; ) = (2; 0; 0). Then, m 2 strctly domnates any message that yelds outcome (1; 0; 0). Hence, (1; 0; 0) cannot be an equlbrum outcome at state 1. Fnally, we show that the drect mechansm M cannot fully mplement n NE the socal choce functon f. Proceed by contradcton and assume that there exst two message pro les m 1 = (m 1; 1 ; m 2; 1 ) 2 NE(M; 1 ) and m 2 = (m 1; 2 ; m 2; 2 ) 2 NE(M; 2 ) such that g(m 1 ) = f( 1 ) and g(m 2 ) = f( 2 ). It s easly checked that for each player : m ; 1 6= m ; 2, otherwse, at some state, one player would have an ncentve to devate from the equlbrum. Now for message pro le (m 1; 1 ; m 2; 2 ), there s (at least) one player who does not receve the object. Assume wthout loss of generalty that ths s player 1 (a smlar reasonng holds for player 2). Let us show that necessarly, m 2 = (m 1; 2 ; m 2; 2 ) s a pure Nash equlbrum at 1. Snce g(m 1; 2 ; m 2; 2 ) = (2; 0; 0) s the best outcome for player 2, he has no ncentve to devate. We also know by constructon that f player 1 devates, the outcome s g(m 1; 1 ; m 2; 2 ) = (x; p 1 ; p 2 ) where player 1 does not get the object. Hence, u 1 (g(m 2 ); 1 ) = u 1 ((2; 0; 0); 1 ) = 0 u 1 ((x; p 1 ; p 2 ); 1 ) = u 1 (g(m 1; 2 ; m 2; 2 ); 1 ) and so player 1 does not have any ncentve to devate ether. Thus f s not fully mplementable by M whch completes the proof. However, as we wll show n the followng lnes, when we expand the set of messages usng ndrect mechansms, f can be contnuously mplemented. Clam 2 There exsts an ndrect mechansm that contnuously mplements f. Proof. Consder the followng ndrect mechansm. Each player has three possble messages (Mne, Hs, and Mne+) and the outcome functon s gven by the matrx below, where v L < P < v H ; f L < p < f H, and > p. 31 Mne Hs Mne+ Mne (0; ; ) (1; 0; 0) (2; ; P ) Hs (2; 0; 0) (0; ; ) (0; p; 0) Mne+ (1; P; ) (0; 0; p) (3; 0; 0) At 1, acton Hs s strctly domnated by Mne+ for player 1. Consequently, n the second round of elmnaton, Mne and Mne+ are strctly domnated by Hs 31 Usng the usual conventon, player 1 s the row player whle player 2 s the column player. 21

22 for player 2 at 1. Fnally, n the thrd round, Mne s strctly better than Mne+ for player 1. Hence, (Mne; Hs) s the unque ratonalzable acton pro le at 1. A symmetrc reasonng apples at 2. By Theorem 3, we conclude that ths nte ndrect mechansm mplements contnuously the socal choce functon f. 4.2 Alternatve Topology: Unform Convergence In ths paper, we de ne the noton of contnuous mplementaton usng the topology of pont-wse convergence. Ths topology s standard when workng n the unversal type space and has a smple nterpretaton. However, other topologes are nterestng and so other notons of contnuous mplementaton are worth to be nvestgated. One natural canddate s the topology of unform convergence. 32 Whle nterestng n ts own rght, we show that all socal choce functons that are partally mplementable n strct Nash equlbra wth a nte mechansm are contnuously mplementable under ths topology. Ths condton s much weaker than the one obtaned under the topology of pont-wse convergence. In partcular, recall that n complete nformaton settngs, under mld condtons 33 and wth more than three players, any socal functon s partally mplementable n strct Nash equlbra wth a drect mechansm (and so wth a nte mechansm n our settng). To ntroduce the topology of unform convergence, we rst recall the de nton of the Prohorov dstance that metrzes the topology of weak convergence of measures. Gven a metrc space (X; ) the Prohorov dstance between any two ; 0 2 (X) s nff > 0 : 0 (A) (A ) + for every Borel set A Xg where A = fx 2 X : nf y2a (x; y) < g. Wrte X 0 = and for each k 1 : X k = [(X k 1 )] I X k 1. Now, let d 0 be the dscrete metrc on and d 1 the Prohorov dstance on 1st level belefs ( ). Then, recursvely, for any k 2, let d k be the Prohorov dstance on the kth level belefs (X k 1 ) when X k 1 s gven the product metrc nduced by d 0 ; d 1 ; :::; d k 1. We say that a sequence of types ft [n]g 1 n=0 converges unformly to a type t, f d U (t [n]; t ) sup k1 d k (h k (t [n]); h k (t ))! 0; n ths case we wrte t [n]! U t. We also wrte t[n]! U t; f, t [n]! U t ; for each 2 I. In ths topology, two types are close f they have very smlar rst-order belefs, second order-belefs and so on up to n nty where the degree of 32 Another topology n the unversal type space s the strategc topology as de ned n Dekel, Fudenberg and Morrs (2006). D Tllo and Fangold (2007) establshed the equvalence between unform topology and strategc topology around nte types. Snce T s nte, the result of ths secton s also true under the strategc topology. 33 For nstance n quas-lnear settngs wth arbtrary small transfers. 22