Implementation in adaptive better-response dynamics: Towards a general theory of bounded rationality in mechanisms

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1 Implementaton n adaptve better-response dynamcs: Towards a general theory of bounded ratonalty n mechansms Antono Cabrales a,b,, Roberto Serrano c,d a Departamento de Economía, Unversdad Carlos III de Madrd, Madrd 126, Getafe, Span b CEPR, Unted Kngdom c Department of Economcs, Brown Unversty, Provdence, RI 02912, USA d IMDEA-Socal Scences Insttute, Madrd, Span abstract JEL classfcaton: C72 D70 D78 Keywords: Robust mplementaton Bounded ratonalty Evolutonary dynamcs Mechansms We study the classc mplementaton problem under the behavoral assumpton that agents myopcally adjust ther actons n the drecton of better-responses or bestresponses. Frst, we show that a necessary condton for recurrent mplementaton n better-response dynamcs (BRD) s a small varaton of Maskn monotoncty, whch we call quasmonotoncty. We also provde a mechansm for mplementaton n BRD f the rule s quasmonotonc and excludes worst alternatves no worst alternatve (NWA). Quasmonotoncty and NWA are both necessary and suffcent for absorbng mplementaton n BRD. Moreover, they characterze mplementaton n strct Nash equlbra. Under ncomplete nformaton, ncentve compatblty s necessary for any knd of stable mplementaton n our sense, whle Bayesan quasmonotoncty s necessary for recurrent mplementaton n nterm BRD. Both condtons are also essentally suffcent for recurrent mplementaton, together wth a Bayesan NWA. A characterzaton of mplementaton n strct Bayesan equlbra s also provded. Partal mplementaton results are also obtaned. 1. Introducton The correct desgn of nsttutons can be decsve for achevng economc systems wth good welfare propertes. But suppose that the correct desgn depends on the knowledge of key parameters n the envronment. Then, an mportant problem ensues f the bulder of the nsttutons does not have such knowledge. The theory of mplementaton looks n a systematc way at the desgn of rules for socal nteracton that do not assume a detaled knowledge of the fundamentals by those wth power to adjudcate socal outcomes. The last decades have seen mpressve advances n the theory of mplementaton. 1 As Sjöström (1994) ponted out, Wth enough ngenuty the planner can mplement anythng. On the other hand, several recent contrbutons 2 have hghlghted the fact that not all mechansms perform equally well, n terms of achevng the socally desrable outcomes. In partcular, some of the mechansms that are more permssve, leadng to a wder span of mplementable socal choce correspondences (SCCs) or socal choce functons (SCFs), may lead to dynamc nstablty or convergence to the wrong * Correspondng author at: Departamento de Economía, Unversdad Carlos III de Madrd, Madrd 126, Getafe, Span. E-mal addresses: antono.cabrales@uc3m.es (A. Cabrales), roberto_serrano@brown.edu (R. Serrano). URLs: (A. Cabrales), (R. Serrano). 1 See Jackson (2001), Maskn and Sjöström (2002), Palfrey (2002), Serrano (2004) or Corchón (2009) for recent surveys. 2 See e.g. Cabrales (1999) and the lterature revew at the end of ths ntroducton. 1

2 equlbrum when the players are boundedly ratonal. Ths should perhaps not be surprsng snce those mechansms were not desgned wth robustness to bounded ratonalty n mnd, and yet, somewhat surprsngly, the canoncal mechansm for Nash mplementaton turned out to be robust to some knds of bounded ratonalty (Cabrales, 1999). Gven these fndngs, t s natural to ask whether the dffculty wth permssve mechansms les n the partcular mechansms employed, or f t s a general problem. In other words, what are the necessary condtons on SCCs for ther evolutonary mplementaton? Are they close to beng suffcent as well? That s, can one characterze the set of rules that a planner could hope to decentralze n a socety populated by boundedly ratonal agents? The current paper answers these questons when one models bounded ratonalty by myopc behavor that tends to move n the drecton of better reples (or best reples) to a bounded sample of past hstory, whch conforms wth a wde class of evolutonary settngs. 3 Thus, we postulate a behavoral assumpton by whch agents (or generatons thereof) nteract myopcally wthn a gven nsttuton, and adjust ther actons n the drecton of better-responses wthn the mechansm. We remark that the wde class of dynamcs consdered here does not requre that all better-responses be played wth postve probablty, although all best-responses must be. Indeed, as we explan n the second remark after Theorem 2, our central results (n Secton 3) and ther proofs go through unchanged f we swtch from better-response dynamcs to (full support) best-response dynamcs. 4 Our crteron for successful mplementaton wll be the convergence of the better-response process to a rest pont or to a set of rest ponts. When all the outcomes of an SCC are the only lmts of the better-response dynamcs (BRD) of a mechansm for any allowed envronment, we shall say that the SCC s mplementable n recurrent strateges of BRD. If the only lmts of BRD le n the SCC, we speak of partal mplementablty n recurrent strateges. We come to descrbe our frst man fndng. A necessary condton for recurrent mplementaton s a small varaton of (Maskn) monotoncty (Maskn, 1999), whch we call quasmonotoncty. Quasmonotoncty prescrbes that the socal outcome reman n the SCC f the strctly lower contour sets of preferences at that socal outcome are nested for every agent across two envronments. In partcular, t s nether logcally stronger nor weaker than monotoncty, although both concde n many settngs (we dscuss the relatonshp n Secton 2). Furthermore, quasmonotoncty s also suffcent for recurrent mplementaton n BRD, f there are at least three agents n the envronment and the SCC satsfes the no worst alternatve (NWA) property. Ths property requres that, for every envronment, there s always a strctly worse outcome for every agent than each outcome prescrbed by the SCC. NWA becomes also necessary f the recurrent classes of the better-response dynamcs are sngletons (.e., absorbng states) n the mplementng mechansm. The typcal mechansm that we construct has good dynamc propertes. It mplements the socally desrable outcome accordng to the agents reports, f there s total agreement among them. If only one agent s report dsagrees wth the rest, other outcomes wll be mplemented. Those outcomes are meant to elct the rght behavor from agents. Fnally, f more than one agent dsagrees, a modulo game s played. We note that the modulo game s not an essental part of the mechansm. It can be replaced, for example, wth a unanmously bad outcome for all players. What really matters (and the modulo game delvers) s that outsde from a stuaton of total (or almost total) agreement, t does not hurt to tell the truth. Ths then mples that behavor can easly drft nto unanmous truth-tellng. 5 Once agents agree, the rules for almost unanmous agreement, whch make use of both quasmonotoncty and the punshments that are possble by NWA, avod any drftng out of generalzed agreement. Our agents are boundedly ratonal myopc better responders yet they are able to act n an apparently complex mechansm wth a modulo game. Whle one could see our mechansm as complex, t s fnte. The argument n the prevous paragraph already hnts at the fact that the nature of better responses n the complex mechansm s not unduly complcated. It requres to understand what s your most preferred outcome, what s the true profle and not much more. More mportantly, we beleve that the queston whether real agents are able to understand the game should be mostly emprcal. In ths respect, the results of Cabrales et al. (2003) are encouragng. In a laboratory experment, the lkelhood of the socally desred outcome beng mplemented was 0.80 n a treatment usng a mechansm smlar to the one we use for our suffcency condton. Nevertheless, as already acknowledged n footnote 3, more research n ths mportant queston s needed. Our results on recurrent mplementaton n BRD are obtaned for a general class of preferences and wll stand for any mutaton process. The latter means that, f one were to perturb the BRD va mutatons, an SCF that s mplementable n recurrent strateges would also be mplementable n stochastcally stable strateges of any perturbaton of BRD; the reader s referred to our companon paper for further detals on ths topc (Cabrales and Serrano, 2010). It follows that quasmonotoncty s dentfed as the key condton to essentally characterze very robust mplementaton wth respect to myopc BRD processes. 6 3 In dong so, we acknowledge that some of the canoncal mechansms we construct mght be too complex for boundedly ratonal agents, whch forces us to stress the word towards from our ttle even more strongly. References for evolutonary game theory n general are Webull (1995), Vega-Redondo (1996), Samuelson (1997), Fudenberg and Levne (1998) and Young (1998). 4 Fcttous play would requre some more dscusson, but an assumpton of lmted memory would lead to the same results. 5 Indeed, f agents were to have some arbtrarly small cost of msrepresentng ther prvate nformaton, as n Kartk (2009), ths drft nto unanmous truth-tellng could be qute fast. 6 It s mportant to stress how Maskn monotoncty has emerged n other works when robustness s requred n other senses: Chung and Ely (2003) f undomnated Nash mplementaton s to occur wth near-complete nformaton, Aghon et al. (2010) for subgame-perfect mplementaton also wth near-complete nformaton, and Bergemann et al. (2010) for ratonalzable mplementaton. 2

3 Gven that one can model bounded ratonalty n many dfferent ways, we also explore (n Secton 3.3) the robustness to other such approaches of the condtons dentfed for recurrent mplementaton under BRD. A common feature of many alternatve approaches s the stablty of strct Nash equlbra. Therefore we pose the queston of mplementaton n strct Nash equlbra and provde ts characterzaton. We fnd that we can dspense altogether wth the standard no-veto-power suffcent condton. In general envronments wth at least three agents both quasmonotoncty and NWA are necessary and suffcent for mplementaton n strct equlbra. Ths provdes a further strong endorsement of quasmonotoncty, and rases the stature of the NWA condton. Indeed, NWA also becomes necessary when we requre mplementaton n absorbng strateges of BRD (see Secton 3.3 for detals). 7 We vew the strct Nash mplementaton result as a confrmaton of the more general message sent by our fndngs for evolutonary mplementaton. 8 Next, we also provde an almost characterzaton of partal mplementablty n recurrent strateges of BRD. The key condton here s a verson of quasmonotoncty appled to the entre SCC. When the range of the correspondence s rch, ths condton s typcally trval and thus, we learn that convergence of BRD processes to a set s much easer to obtan. Our man nsghts already descrbed are confrmed n envronments wth ncomplete nformaton, and some others are obtaned theren (for smplcty, we present most of those results for SCFs and economc envronments). Frst, ncentve compatblty arses as a necessary condton for stable mplementaton n our sense, whatever the perturbaton one wshes to use, ncludng no perturbaton at all, of nterm BRD. 9 If one wshes to mplement n recurrent strateges, the condton of Bayesan quasmonotoncty s also necessary. The comparson between ths condton and Bayesan monotoncty, necessary for Bayesan mplementaton (e.g., Postlewate and Schmedler, 1986; Palfrey and Srvastava, 1989; Jackson, 1991), s smlar to that between quasmonotoncty and Maskn s condton. Moreover, ncentve compatblty, Bayesan quasmonotoncty and ε-securty are also suffcent for mplementaton n recurrent strateges of BRD processes when there are at least three agents. 10 Fnally, we provde a characterzaton of the SCFs that are mplementable n strct Bayesan equlbra n general envronments, a contrbuton n ts own rght, and also a vehcle to show the general mplcatons of our basc results for recurrent mplementaton of nterm BRD. That s, parallel to Secton 3.3, we also study n Secton 5.2 the robustness of the condtons dentfed for recurrent Bayesan mplementaton. Droppng the assumpton of economc envronments, we provde a full characterzaton of strct Bayesan mplementaton wth at least three agents. We remark that no such characterzaton s avalable for Bayesan mplementaton. 11 The condtons for strct Bayesan mplementaton are strct ncentve compatblty, Bayesan quasmonotoncty, and a verson of the NWA condton for Bayesan envronments Related lterature The study of mplementaton under bounded ratonalty has a long ntellectual hstory. Muench and Walker (1984) and de Trenqualye (1988) study the stablty of the Groves and Ledyard (1977) mechansm. 12 Walker (1984) descrbes a mechansm for whch almost Walrasan allocatons n large economes are stable. Vega-Redondo (1989) proposes a globally convergent mechansm (under best-response dynamcs) to the Lndahl equlbrum n an economy wth one prvate good, one publc good and lnear producton. Along the same lnes, de Trenqualye (1989) proposes a locally stable mechansm to mplement Lndahl equlbra n an economy wth multple prvate goods, one publc good, lnear producton and quas-lnear preferences. Wth respect to these early contrbutons, our paper does not focus on the stablty of partcular mechansms, SCFs or envronments, and we deal wth global, rather than local convergence. More recently, Cabrales (1999) studes suffcent condtons for global convergence of (a varaton of) the canoncal mechansm for Nash mplementaton. 13 The suffcent condtons used there are very smlar to ours. However, Cabrales (1999) does not study necessary condtons for convergence, but rather the bad dynamc propertes of other (already exstng) mechansms. And unlke Cabrales and Serrano (2010), that paper does not provde ether addtonal condtons on dynamcs or preferences to delver ergodc dynamc propertes. 14 Sandholm (2005) provdes a suffcent concavty condton under whch smple prce schemes would be globally stable when mplementng effcent SCFs n economc envronments. 15 Mathevet (2007) studes suffcent condtons for mplemen- 7 Cabrales et al. (2003) show expermentally that n the absence of NWA, the performance of Maskn s canoncal mechansm for Nash mplementaton s substantally worse. 8 The characterzaton of strct Nash mplementaton has nterest n ts own rght. In partcular, t s noteworthy how one can dspense wth the no-veto condton n non-random settngs, even outsde of economc envronments. 9 Our comment above regardng better-responses versus best-responses apples here as well. 10 Under a weak dversty of preferences n the envronment, the condton of Bayesan quasmonotoncty can be entrely dropped. Ths can be done f the planner s satsfed wth mplementaton n stochastcally stable strateges under unform mutatons and at least fve agents; see agan Cabrales and Serrano (2010). 11 Theorem 2 n Jackson (1991) for general envronments s a suffcency result. 12 Chen and Plott (1996) and Chen and Tang (1998) confrm expermentally these fndngs. 13 Cabrales et al. (2003) confrm expermentally these fndngs. 14 Cabrales and Pont (2000) study the convergence and stablty propertes of Sjöström s (1994) mechansm assumng that the dynamcs are monotonc n the sense of Samuelson and Zhang (1992). Cabrales (1999) also dscusses some drawbacks, from a dynamc pont of vew, of the mechansms n Abreu and Matsushma (1992, 1994). 15 Sandholm (2002) provdes a concrete mechansm achevng effcency and stablty n a road congeston prcng problem. 3

4 taton n supermodular games, motvated by ther good learnng propertes. In contrast, we do not take advantage of the good convergence propertes of dynamc processes when posng our general questons and constructng our mechansms, and therefore, the SCFs mplemented n those papers must stll meet our necessary condtons f the processes follow the BRD drecton. Another mportant lne of research has studed mplementaton that s statcally robust to bounded ratonalty. Elaz (2002) shows that (a strengthenng of) monotoncty and no-veto power are suffcent to mplement an SCF even when agents are error-prone, provded a majorty of players are not. Tumennasan (2008) shows that (once agan) quasmonotoncty s necessary for mplementaton when all agents are error-prone n the style of Logt Quantal Response Equlbrum (LQRE), as defned n McKelvey and Palfrey (1995). 16 Snce the condtons used n the latter paper are closer to ours, we dscuss t n more detal later on. The papers we have dscussed so far typcally focus on envronments wth complete nformaton, that s, the agents know each others preferences. Notable exceptons are the prevously mentoned paper of Mathevet (2007) and the mpossblty result of Jordan (1986). We extend the lterature by also studyng ncomplete nformaton envronments. In ths respect, there s also a small related lterature on the stablty of equlbra for dfferent aucton and double aucton formats (ncludng Hon-Snr et al., 1998; Dawd, 1999; Saran and Serrano, 2010) Plan of the paper Secton 2 descrbes the model and the dynamcs we use. Secton 3 provdes necessary and suffcent condtons for recurrent mplementaton under complete nformaton. Secton 4 presents more permssve results.e., concernng nonquasmonotonc rules f one reles on partal mplementablty of SCCs. Secton 5 collects the extensons of our results to ncomplete nformaton envronments. Secton 6 concludes. 2. Prelmnares Let N ={1,...,n} be a set of agents and let Z be a fnte set of alternatves. Let θ denote agent s preference orderng over the set Z. Letθ = (θ ) N be a preference profle, and Θ be the (fnte) set of allowable preference profles. 17 We assume that Θ 3, wthout loss of generalty. 18 The symbol θ stands for s weak preference when the preference profle s θ, whle θ denotes s strct preference at θ and θ denotes ndfference. We refer to the set of agents, the set of alternatves and agents preferences over alternatves as an envronment. To the extent that the sets N and Z wll reman fxed throughout, two envronments dffer only n that the preferences of some agents change across them. (In Secton 5 we shall consder also ncomplete nformaton envronments, n whch agents nformaton may also change.) A socal choce correspondence (SCC) specfes a set of alternatves for each preference profle θ. Thus, an SCC F : Θ 2 Z. A socal choce functon (SCF) s an SCC that always assgns a sngleton to each θ Θ. We shall denote an SCF by f, and thus, f : Θ Z. A mechansm G = ((M ) N, g), where M s agent s message set, and g : N M Z s the outcome functon. A Nash equlbrum of the mechansm n state θ s a profle of messages m such that for every N, g(m ) θ g(m,m ) for all m m. A strct Nash equlbrum s a Nash equlbrum n whch all these nequaltes are strct. Gven a profle m N M,agent j s (weak) better-response to m s any m such that g(m j j,m j) θ g(m). j For our frst set of results, n the next secton, we begn by consderng a specfc class of SCCs. Most mportantly, we shall use the followng condton, whch turns out to be central to the theory we develop here: An SCC F satsfes quasmonotoncty whenever, for every a F (θ), f t s true that for every N and for all z A wth a θ z we have that a φ z, ths mples that a F (φ) for all θ,φ Θ. Note how quasmonotoncty resembles closely the condton of monotoncty uncovered n Maskn (1999). Indeed, the only dfference s that, whle Maskn s condton mposes that the lower contour sets of preferences be nested across two envronments, quasmonotoncty reles on the ncluson of the strctly lower contour sets. It follows from the defntons of both condtons that quasmonotoncty s nether stronger nor weaker than Maskn monotoncty. 19 The next two examples present two mportant correspondences that make ths pont: Example 1. Consder envronments n whch for no par of alternatves z, z Z t s the case that z θ z for all N. Defne the strong Pareto correspondence F SP : Θ 2 Z as follows: F SP (θ) { z Z: z Z, N, z θ z and j N, z θ j z}. 16 The noton of LQRE s tself closely assocated wth the stochastc choce models of Luce (1959) and McFadden (1973). 17 The fnteness of Θ guarantees that we can study the dynamcs for our mechansms wth fnte Markov chan tools. 18 If Θ =2, one can always add an extra envronment artfcally by replcatng one of the two preference profles and use a thrd name for the new one. 19 Ths contrasts wth other condtons prevously found n the lterature (e.g., the weaker almost monotoncty n Sanver, 2006), used to expand the scope of Nash mplementabllty by means of awards. 4

5 As s well known, F SP does not satsfy Maskn monotoncty: for nstance, let N ={1, 2} and Z ={z, z }, θ be such that z θ 1 z and z θ 2 z, so that F SP (θ) ={z, z }.Andletφ dffer from θ only n that z φ 1 z, mplyng that z / F SP (φ). However, t s easy to see that F SP wll satsfy quasmonotoncty n any such envronment. Indeed, suppose the hypothess of quasmonotoncty s satsfed (the ncluson of the strctly lower contour sets between θ and φ), and that z F SP (θ), but z / F SP(φ). The latter mples that there exsts z such that z φ z for all N and z φ z for some j N. Recallng j that t s not the case that z θ z for all N, the ncluson of the strctly lower contour sets would mply that z θ z for all N and that there exsts some j N for whom z θ j z, contradctng that z F SP (θ). 20 Example 2. Consder any envronment. Defne the weak Pareto correspondence F WP : Θ 2 Z as follows: F WP (θ) { z Z: z Z, N, z θ z }. As s well known, F WP always satsfes Maskn monotoncty. However, t may volate quasmonotoncty: for nstance, let N ={1, 2} and Z ={z, z }, θ be such that z θ 1 z and z θ 2 z, so that F WP (θ) ={z, z }.Andletφ dffer from θ only n that z φ 1 z, mplyng that z / F WP (φ). The above examples underscore the dfferences between Maskn monotoncty and quasmonotoncty. Nonetheless, there are many envronments n whch the two condtons concde. For example, consder a problem of assgnment of ndvsble goods n whch all preferences are strct (as n Shapley and Scarf, 1974; Roth and Postlewate, 1977), or envronments n whch agents preferences are contnuous (n ths case, the ncluson of lower contour sets and strctly lower contour sets s equvalent). In addton to quasmonotoncty, we shall use the followng condton: An SCC F satsfes no-worst-alternatve (NWA) whenever for every agent, every preference profle θ and every a F (θ), there exsts an outcome z a,θ such that a θ z a,θ. In partcular, NWA allows the possblty of punshng an agent at every outcome of the SCC. In Secton 5, n whch we concentrate on economc envronments and where the socal choce rule allocates a bundle to each agent, we shall use a verson of NWA for SCFs. An SCF f s sad to be ε-secure f there exsts ε > 0 such that for each θ, and for each N, f (θ) (ε,...,ε) 0. The condton of ε-securty amounts to establshng a mnmum threshold of lvng standards n the consumpton of all commodtes. We shall thnk of ε as beng a farly small number. Then, one could easly justfy t on normatve grounds. Next, we turn to dynamcs, the central approach n our paper. The mechansm wll be played smultaneously each perod by myopc agents. Or, n an nterpretaton closer to the evolutonary tradton, the mechansm wll be played successvely each perod by generatons of agents who lve and care for that perod only. Gven a mechansm, we take the set N M of message profles as the state space (whch, to avod further measure theoretc techncaltes, we assume to be countable). We shall begn by specfyng a Markov process on ths state space,.e., a matrx lstng down the transton probabltes from any state to any other n a sngle perod. 21 Such a process wll typcally have multple long-run predctons, whch we call recurrent classes. A recurrent class s a set of states that, f ever reached, wll never be abandoned by the process, and that does not contan any other set wth the same property. A sngleton recurrent class s called an absorbng state. The Markov process that we shall mpose on the play of the mechansm over tme s the followng better-response dynamcs (BRD). In each perod t, each of the agents s gven the chance, wth postve, ndependent and fxed probablty, to revse hs message or strategy. Smultaneous revson opportuntes for dfferent agents are allowed. Let m(t) be the strategy profle used n perod t, and say agent s chosen n perod t. Then, denotng by θ agent s true preferences, agent swtches wth postve probablty to any m such that g(m,m (t)) θ g(m(t)). 22 Thus, the planner, who has a long run perspectve on the socal choce problem, wshes to desgn an nsttuton or mechansm such that, when played by myopc agents who keep adjustng ther actons n the drecton of better-responses, wll eventually converge to the socally desrable outcomes as specfed by the SCC. Ths logc suggests the followng noton of mplementablty. An SCC F s mplementable n recurrent strateges (of BRD) f there exsts a mechansm G such that, for every θ Θ and every a F (θ), there s a recurrent class of the BRD process (appled to the nduced game under preferences θ) such that the class has a unque outcome and the outcome s equal to a. Furthermore, the outcome of any recurrent class of that BRD process must be some a F (θ). The dependence of long-run predctons of unperturbed Markov processes on ntal condtons s sometmes perceved as a drawback of ths analyss. One way out s to perturb the Markov process, and ths would lead to the concept of mplementablty n stochastcally stable strateges (the nterested reader should see Cabrales and Serrano, 2010). 20 In envronments smlar to these where ndvdual property rghts are defned, other examples of correspondences that volate Maskn monotoncty, but that satsfy quasmonotoncty, nclude the strct ndvdual ratonalty correspondence and ts ntersecton wth the strong Pareto correspondence. 21 For complete formal defntons of Markov chans, and related terms (recurrent classes, absorbng states, etc., see e.g. Karln and Taylor, 1975, Chapter 2). 22 As we dscuss n the second remark after Theorem 2, all the results can be extended to the case n whch agents swtch to play a subset of betterresponses (n partcular, best-responses) to the mxed-strategy of the others gven by the emprcal dstrbuton of play over some bounded sample of past hstory. 5

6 3. Necessary and suffcent condtons for recurrent mplementaton: Complete nformaton In ths secton we dentfy necessary and suffcent condtons for mplementablty n recurrent strateges of BRD. The secton also dscusses more general mplcatons of these results, and n partcular, the connectons wth mplementaton n strct Nash equlbra Necessty We seek mplementaton n recurrent strateges of the BRD. We wsh to show now that quasmonotoncty of F s a necessary condton for ts mplementablty n recurrent strateges. Theorem 1. If F s mplementable n recurrent strateges of a BRD process, F s quasmonotonc. Proof. Let the true preference profle be θ. BecauseF s mplementable n recurrent strateges of BRD, the only outcomes that correspond to strategy profles n recurrent classes of the dynamcs are a F (θ). Let a F (θ), and now consder a preference profle φ such that for all a θ z mples that a φ z. Sncea s the only outcome compatble wth a recurrent class of the dynamcs when preferences are θ, ths means that agent s unlateral devatons from recurrent strategy profles n that class must yeld ether a agan, or outcomes z such that a θ z. But then, the same recurrent class s also a recurrent class under φ,.e., a s also supported by recurrent profles of BRD when preferences are φ. SnceF s mplementable n recurrent strateges of BRD, ths mples that a F (φ). That s, F must be quasmonotonc Suffcency We now present our next result. Together wth Theorem 1, t provdes almost a characterzaton of the SCCs that are mplementable n recurrent strateges of BRD. Theorem 2. Let n 3. If an SCC F satsfes quasmonotoncty and NWA, t s mplementable n recurrent strateges of BRD. Proof. Consder the mechansm G = ((M ) N, g), where agent s message set s M = Θ Z Z N. Denoteagent s message m = (θ, a, z,n ), and the agents message profle by m. Let the outcome functon g be defned by the followng rules: () If for all N, m = (θ, a, z,n ) and a F (θ), g(m) = a. () If for all j, m j = (θ, a, z j,n j ) wth a F (θ), and m = (φ, a, z,n ), (φ, a ) (θ, a), one can have two cases: (.a) If z θ a, g(m) = z a,θ recall the defnton of the NWA property. (.b) If a θ z, g(m) = z. () In all other cases, g(m) = z t, where t = ( N n )(mod n + 1). Note how, because F satsfes NWA, rule (.a) s well defned. Now, we begn by argung n the next two steps that all recurrent classes of the BRD process yeld outcomes under rule (). Let θ be the true preference profle. Step 1: No message profle n rule () s part of a recurrent class. Argung by contradcton, from any profle m n (), one can construct a path as follows. There are two possble cases to consder. Case A. If fewer than n 2 agents make a common announcement of the preference profle and of the alternatve n the SCC, no sngle agent can change hs strategy and move the game to a profle outsde of rule (). Then, t s a best-response for any agent to choose a strategy (θ, a, a,n ) wth a F (θ), n sutably chosen to wn the modulo game and a one of the best possble outcomes for under θ. If all agents get a smultaneous revson opportunty, the game goes to rule () and yelds a F (θ), from whch one can never ext under BRD. Case B. If, on the other hand, exactly n 2 agents make a common announcement of the preference profle and of the alternatve n the SCC, say (φ, a) wth a F (φ), there s an agent who can change hs strategy and move the game to a profle n (). Then, let any agent announcng (φ, a) get a revson opportunty and change to announcng (φ, a, a,n ) wth a F (φ ) and φ dfferent from any preference profle announced n m f n > 3 (recall that Θ 3; on the other hand, f n = 3, there s no need for ths change so that φ = φ), n sutably chosen to wn the modulo game and a one of the best possble outcomes for under θ. Ths s possble under BRD, and now we are n Case A. Step 2: No message profle under rule () s part of a recurrent class of BRD. We argue by contradcton. Recall that the true preference profle s θ, and let the message profle under rule () (.a) or (.b) n queston be the followng: all 6

7 agents j announce m j = (φ, a, z j,n j ), whereas agent s message s (φ, a, z,n ). Then, the outcome s ether z a,φ under rule (.a) or z under rule (.b). But n ether case, from here, each of the other agents j can swtch to (φ j, a j, a j,n j ) (φ, a, z j,n j ),wthn j sutably chosen to wn the modulo game and a j one of the best possble outcomes for j under θ. Thus, we fnd ourselves under rule (), whch s a contradcton. Therefore, from steps 1 and 2, all recurrent classes contan only profles under rule (). Moreover, each recurrent class must yeld a unque outcome, say a F (φ). To see ths, notce that a class whch ncludes strategy profles wth dfferent outcomes, must nclude profles under rule () or (), whch cannot happen by steps 1 and 2. In addton, each strategy profle n a recurrent class must be a Nash equlbrum of the game nduced by the mechansm when the true preferences are θ. A non-equlbrum profle would produce wth postve probablty a devaton from rule (), so the recurrent class would nclude profles under rules () or (), whch cannot happen by steps 1 and 2. There s one famly of recurrent classes of Nash equlbra that exsts for all games nduced by the mechansm. It s composed of all the recurrent classes contanng the truthful profles (θ, a, z,n ) for every a F (θ) reported by every agent N. A unlateral devaton from any such profle to a profle outsde the recurrent class would produce a profle ether under rule (.a) or under rule (.b). In ether case, no such swtch can happen under BRD (recall that F satsfes NWA). But some games nduced by the mechansm may have other (non-truthful) recurrent classes of Nash equlbra under rule (). Let us call one such class, the class (φ, a) and let (φ, a, z,n ) N, a F (φ), be one Nash equlbrum n class (φ, a). Snce ths profle s a Nash equlbrum, t must be true that for all N, a φ z mples that a θ z. If that were not the case (.e. f z θ a), gven that can obtan z by devatng to rule (.b) because a φ z, he would choose to devate and (φ, a, z,n ) N would not be a Nash equlbrum. But n fact, we know even more. Any z a that any agent can obtan by changng hs message would yeld a message profle outsde the class (φ, a). So f there s no devaton to such messages that yelds z a n a BRD t must be because a devaton would leave the devant strctly worse off. That s, z cannot be ndfferent to a for under θ. Hence,wemust have that for all N, a φ z mples that a θ z. But, because F s quasmonotonc, ths mples that a F (θ) for any arbtrary profle (φ, a, z,n ) N n any recurrent class of the BRD. Therefore, F s mplementable n recurrent strateges of BRD. Remark. The reader wll have notced the smlartes between the mechansm used n ths proof and the one n Maskn (1999). There are some notceable dfferences. Our rule (.a) s mportant because all better-responses (or best-responses) could be used wth postve probablty by agents who get a revson opportunty. If an agent chooses to announce some preference profle/alternatve dfferent from the consensus and he s not punshed, he wll thus do so wth postve probablty. But, from there, t s a best-response for someone else to move to the modulo game, and the mechansm would unravel. Ths justfes the use of the punshments n rule (.a). In the parallel rule of Maskn s (1999) mechansm, devatons from rule () are allowed and are not a problem because they do not upset a Nash equlbrum. Our rule (.b), on the other hand, has to be modfed to complement the descrbed change n rule (.a). Rule () s a modulo game, nstead of an nteger game. Fnte games are usually vewed as more natural. Fnally, our strategy set allows two reported alternatves, as opposed to only one n Maskn (1999). Ths s done to prevent recurrent classes of the dynamcs wthn rule (), creatng a path back to rule (), as explaned n step 1 of the proof. Remark. Although we defne the stochastc process to be BRD, the proof of Theorem 2 uses only best-responses. Ths mmedately shows that our results hold under best-reply dynamcs (provded all best-responses are chosen wth postve probablty). Gven that players do not swtch strateges wth postve probablty (and hence there can be perods of arbtrary fnte length wthout any changes), they would also apply to dynamcs that best reply to the emprcal dstrbuton of play over some bounded sample of past hstory (a fnte memory verson of fcttous play 23 ) Connectons wth other approaches To the extent that one can model bounded ratonalty n many dfferent ways, the reader may be wonderng whether the condtons dentfed for recurrent mplementaton of BRD processes mght have some bearng wth other such approaches. Ths subsecton answers ths queston n the affrmatve. In partcular, a common thread found by several learnng, evoluton and bounded ratonalty models n game theory s ther support to certan classes of strct Nash equlbra as beng robust to many of these exercses. Ths ratonale perhaps justfes the queston of the dentfcaton of the condtons behnd strct Nash mplementaton, a queston that the lterature so far has overlooked. We turn to t next. An SCC F s mplementable n strct Nash equlbrum f there exsts a mechansm G such that, for every θ Θ, thesetof all strct Nash equlbrum outcomes of ts nduced game when the preference profle s θ s F (θ). A mechansm s non-mposng f for each agent N, thereexstm,m M, such that for some m M g(m,m ) g(m,m ). (In partcular, non-mposng mechansms allow all agents to meanngfully partcpate.) 23 Where we would agan requre all best-responses to be chosen wth postve probablty. 7

8 We frst state a result that dentfes necessary condtons for strct Nash mplementablty: Theorem 3. If F s mplementable n strct Nash equlbrum, F satsfes quasmonotoncty. Furthermore, f F s mplementable n strct Nash equlbrum by means of a non-mposng mechansm, F satsfes NWA. Proof. The smple proof s left to the reader. We move rght away to the suffcency result: Theorem 4. Let n 3. If an SCC F satsfes quasmonotoncty and NWA, t s mplementable n strct Nash equlbrum. Proof. We sketch the proof by specfyng a canoncal mechansm that wll serve to prove the statement. The mechansm s a small varant of the canoncal mechansm for Nash mplementaton, proposed by Repullo (1987) and used n Maskn (1999). Consder the mechansm G = ((M ) N, g), where agent s message set s M = Θ Z N. Denoteagent s message m = (θ, z,n ), and the agents message profle by m. Let the outcome functon g be defned by the followng rules: () If for all N, m = (θ, a, 1) wth a F (θ), g(m) = a. () If for all j, m j = (θ, a, 1) wth a F (θ), and m = (φ, z,n ) (θ, a, 1), one can have two cases: (.a) If z θ a, g(m) = z a,θ recall the defnton of NWA. (.b) If a θ z, g(m) = z. () In all other cases, g(m) = z t, where t = ( N n )(mod n + 1). Agan, NWA mples that rule (.a) s well defned. Then, one can frst establsh that unanmous truthful announcements under rule () are strct Nash equlbra that mplement each of the desred outcomes. It s easy to see that one cannot have strct equlbra ether under rule () or rule (). Thus, all strct Nash equlbra must happen under rule (), and quasmonotoncty of F and the constructon of rule () mply that any such strct equlbrum must yeld an outcome n F (θ). Remark. Note how the soluton concept employed strct Nash equlbrum allows one to dspense completely wth the no-veto condton, one of the suffcent condtons for Nash mplementaton outsde of economc envronments (see Maskn, 1999). 24 Therefore, we have establshed that for any envronment (economc or otherwse), quasmonotoncty and NWA provde a complete characterzaton of strct Nash mplementablty f there are at least three agents. The comparson wth the condtons for recurrent mplementablty n BRD s apparent: only NWA ceases to be necessary then. Next, we nvestgate how one can nal down the connecton even further. An SCC F s mplementable n absorbng strateges (of BRD) f t s mplementable n recurrent strateges and all recurrent classes of the BRD process appled to ts nduced game are sngletons. Then, one can show the followng connectng proposton. Its smple proof s also left to the reader: Proposton 1. If an SCC F s mplementable n absorbng strateges of BRD, F s mplementable n strct Nash equlbrum. Conversely, f an SCC F s mplementable n strct Nash equlbrum and there exsts an mplementng mechansm whose recurrent classes of BRD are all sngletons, F s mplementable n absorbng strateges of BRD. Note how the second hypothess of the converse clause n the proposton rules out non-sngleton recurrent classes of BRD, such as those nvolved n the modulo game of the mechansm we used to prove Theorem 4. Indeed, n some envronments, each agent s best alternatve may stll be better than the outcomes of the SCC for all agents, and n ths case, we would have a recurrent class of BRD n whch we cycle among each agent s top-ranked outcomes. Followng a dfferent approach to bounded ratonalty, Tumennasan (2008) consders mplementaton n lmtng logt quantal response equlbra (LLQRE), as defned n McKelvey and Palfrey (1995), where the lmt s taken as nose n random payoff maxmzaton s removed,.e., as behavor approaches full ratonalty. Notng some dffcultes n characterzng the structure of non-strct LLQRE, Tumennasan (2008) confnes hs attenton to strct LLQRE and calls hs mplementablty noton restrcted LLQRE mplementaton. Interestngly, he shows that both quasmonotoncty and a stronger verson of NWA are necessary for restrcted LLQRE mplementablty, and n economc envronments wth at least three agents, he shows that these two condtons, together wth no-veto, are also suffcent. In other words, to the extent that strct Nash equlbra 24 See also Bochet (2007) and Benot and Ok (2008), who escape no-veto wthn Nash mplementaton by usng stochastc mechansms. 8

9 are the common thread to these robustness results, the condtons dentfed n the current paper are also relevant for other approaches to bounded ratonalty The examples revsted To understand the dfferences between our approach and the one taken by the Nash mplementaton lterature, t wll be nstructve to revst the two examples of Secton 2. Consder the strong Pareto correspondence F SP and the followng two envronments, somewhat enhanced wth respect to Example 1 by the ntroducton of a unformly worst alternatve w and a thrd agent. Let N ={1, 2, 3} and Z ={z, z, w}. Let state θ be such that z θ 1 z θ 1 w, z θ 2 z θ 2 w and z θ 3 z θ 3 w, so that F SP (θ) ={z, z }.Andletφ dffer from θ only n that z φ 1 z, mplyng that z / F SP (φ). There s no mechansm to Nash-mplement F SP f the doman ncludes these two envronments. The reason s that, f one could fnd such a mechansm, t should support z by a Nash equlbrum when the state s θ. But those same messages would stll consttute a Nash equlbrum of the same mechansm when the state s φ, and thus, the mechansm would always nclude an undesrable Nash equlbrum n state φ. On the other hand, F SP satsfes quasmonotoncty and NWA over these two envronments. Ths mples that one can apply to t the canoncal mechansm of Theorem 4 to mplement F SP n strct Nash equlbra. Further, one can also apply to t the mechansm n the proof of Theorem 2, and make the correspondence concde wth the recurrent classes of BRD processes. Note, however, how n both cases these mechansms fal to mplement n Nash equlbrum. For example, n the mechansm of the proof of Theorem 4, the unanmous profle (θ, z, 1) s a (non-strct) Nash equlbrum when the state s φ: the outcome s z, but agent 1 can nduce z through rule (.b). Exactly the same happens n the mechansm of Theorem 2 wth the unanmous profle (θ, z, z, 1) n state φ. In addton, both mechansms may generally have other non-strct Nash equlbra n rules () or () when the modulo game, wthout the help of the non-veto condton, does not suffce to rule out undesred equlbra. We remark that all these addtonal equlbra that prevent Nash mplementaton of F SP are non-strct, and therefore, non-robust to our dynamc approaches. Consder now the weak Pareto correspondence F WP n a smlarly enhanced verson of Example 2. Let N ={1, 2, 3} and Z ={z, z }.Letθ be such that z θ 1 z, z θ 2 z and z θ 3 z, so that F WP (θ) ={z, z }.Andletφ dffer from θ only n that z φ 1 z, mplyng that z / F WP (φ). Observe that F WP satsfes monotoncty and no-veto, and thus, by Maskn s (1999) theorem, t s mplementable n Nash equlbrum. But note how, whatever the mplementng mechansm s, the Nash equlbrum supportng z n state θ must be non-strct. Snce F WP volates quasmonotoncty n any doman that ncludes these two envronments, t s not possble to construct a mechansm wth a dynamcally robust convergence to z n state θ, whch explans why the correspondence fals to be mplementable n recurrent strateges of BRD. 4. Non-quasmonotonc rules Thus far we have seen that quasmonotoncty s the key condton that s necessary for (full) mplementablty n recurrent strateges. Moreover, wth at least three agents, t s also suffcent f one mposes NWA. In ths secton, we explore the possbltes of mplementng non-quasmonotonc rules under more permssve mplementablty notons. We shall nvestgate the condtons for partal mplementablty, as opposed to full. 25 Suppose that the planner contents herself wth socety eventually reachng one of the socally desrable outcomes, but s not concerned wth constructng an nsttuton that over tme could get to all of them. That s, one can sometmes requre that the lmtng play of BRD over tme lead to outcomes n the SCC, wthout nsstng that, for each of them, there be a dynamc path that end up n t. Ths motvates the followng noton of weak or partal mplementablty: An SCC F s partally mplementable n recurrent strateges (of BRD) f there exsts a mechansm G such that, for every θ Θ, all the outcomes of all recurrent classes of the BRD process appled to ts nduced game when the preference profle s θ le n F (θ). When one consders SCCs as opposed to just SCFs, when both notons of partal and full mplementablty are the same, one would expect that the condtons that characterze partal mplementablty be substantally weaker than quasmonotoncty. And ndeed, we next show that ths s the case. An SCC F s weakly quasmonotonc whenever for all θ,φ Θ, f t s true that for every a F (θ) and for every N, a θ z mples that a φ z, we have that F (θ) F (φ). For SCCs, weak quasmonotoncty s weaker than quasmonotoncty, as t mposes a restrcton on the SCC only when the preferences n two envronments are such that the strctly lower contour sets of each agent are nested at every outcome of the SCC. If the range of the SCC s rch, ths wll often be a requrement that s hard to meet, and therefore, n these cases, weak quasmonotoncty wll be vacuously satsfed. Of course, for SCFs, weak quasmonotoncty reduces to quasmonotoncty. 25 A second approach s pursued n Cabrales and Serrano (2010). Whle nsstng on full mplementaton, n that paper we drop the requrement of convergng usng BRD no matter what mutaton process one mght use. One thus gets to mplement a larger class of SCFs, but to do so, we make extra specfc assumptons on envronments, preferences and mutatons. 9

10 We frst state the necessty result: Theorem 5. If F s partally mplementable n recurrent strateges of a BRD process, F s weakly quasmonotonc. Proof. Let the true preference profle be θ. Because F s partally mplementable n recurrent strateges of BRD, the only outcomes that correspond to strategy profles n recurrent classes of the dynamcs s some a F (θ). Let any a F (θ) be an outcome of a recurrent class of a BRD process. Let a preference profle φ be such that for all b F (θ), and for all, b θ z mples that b φ z. Sncea s an outcome compatble wth a recurrent class of the dynamcs when preferences are θ, ths means that agent s unlateral devatons from recurrent strategy profles n that class must yeld ether a agan, or outcomes z such that a θ z. But ths mples that a s also supported by recurrent profles of BRD when preferences are φ. SnceF s partally mplementable n recurrent strateges of BRD, ths mples that a F (φ). That s, F must be weakly quasmonotonc. To obtan the suffcency result, we need to strengthen the NWA condton. The strong NWA condton s stll compatble wth many SCCs of nterest, for example n standard exchange economc envronments n whch one can defne a worst outcome the zero bundle for all agents: An SCC F satsfes strong-no-worst-alternatve (SNWA) whenever for every agent, every preference profle θ there exsts an outcome z θ such that for every a F (θ), a θ z,θ. The suffcency result for partal mplementaton n recurrent classes of BRD follows: Theorem 6. Let n 3. If an SCC F satsfes weak quasmonotoncty and SNWA, t s partally mplementable n recurrent strateges of BRD. Proof. We sketch the proof as follows. For the canoncal mechansm we construct, we shall need the followng defnton: Let F (θ) ={a F (θ) φ Θ, N, b Z such that a θ b and b φ a}. Note that F (θ) = mples that F (θ) F (φ) for all φ Θ f F s weakly quasmonotonc. Consder the mechansm G = ((M ) N, g), where agent s message set s M = Θ Z Z N. Denoteagent s message m = (θ, a, z,n ), and the agents message profle by m. Let the outcome functon g be defned by the followng rules: () If for all N, m = (θ, a, z,n ): (.a) If F (θ) and a F (θ) for all, then g(m) = at, where t = ( N n )(mod n + 1). (.b) If F (θ) = and a F (θ) for all, then g(m) = a t, where t = ( N n )(mod n + 1). () If for all j, m j = (θ, a j, z j,n j ) wth a j as n rule (), and m = (φ, a, z,n ) wth ether φ θ, ora / F (θ) or a / F (θ), F (θ) =, then one can have two cases: (.a) If z θ a j for some j N, g(m) = z θ recall the defnton of the SNWA property. (.b) If a j θ z for all j N, g(m) = z. () In all other cases, g(m) = z t, where t = ( N n )(mod n + 1). Note how the mechansm makes use of two dstnct modulo games, one under rule () and a second one under rule (). The former s the real novelty of ths mechansm, as the recurrent classes of BRD wll all make use of t. Indeed, the steps to show that no recurrent classes are compatble wth ether rule () or rule () follow smlar arguments to those n the proof of Theorem 2 and we omt them. Let θ be the true preference profle. We can have two types of recurrent classes under rule (). One famly of recurrent classes that always exsts s the class that contans the truthful profles (θ, a, z,n ). Ths can happen both when all a F (θ) and F (θ), or when a F (θ) and F (θ) =. In ether case, no ndvdual wants to move out of the class, because extng the class nvolves gong through rules (.a) or (.b), whch strctly decrease hs payoff. But one may have other (non-truthful) recurrent classes under rule (). Let (φ, a, z,n ) N, be such an arbtrary class. Ths can happen both when all a F (φ) and F (φ), or when a F (φ) and F (φ) =. If F (φ) = recall that ths mples that F (φ) F (φ ) for all φ Θ, f F s weakly quasmonotonc. In partcular t s true that F (φ) F (θ), and then ths agrees wth our concept of mplementablty. On the other hand, f F (φ), we have two possble cases. One f the hypothess of weak quasmonotoncty s met (.e. for every a F (φ) and for every N, a φ z mples that a θ z). In ths case we have that F (φ) F (θ), and hence every a F (θ). In the second case the hypothess of weak quasmonotoncty does not hold. In that case, there exsts an ndvdual, alternatves a F (φ) and b such that a φ b and b θ a. Note that, f the consdered profle s part of a recurrent class, the same class contans a profle where all agents announce the same a (ths s so because an agent can always choose an nteger not to wn the modulo game so as not to change the exstng outcome, a weak better reply). Hence, he can weakly mprove hs payoff by nducng rule (.b) choosng (θ, a, b,n ). Ths would contradct that ths outcome belongs to a recurrent class. 10

11 Remark. Ths proof shows that one can converge to a set of outcomes, where the convergence can be to more than one outcome lterally: the typcal recurrent class makes use of the modulo game wthn rule (). Ths s the weak convergence noton used n Hart and Mas-Colell (2000), where regret matchng dynamcs s proposed as a way to converge to the set of correlated equlbrum dstrbutons. It s nterestng to note that, whle the entre correspondence of correlated equlbrum dstrbutons wll satsfy the weak quasmonotoncty condton over large classes of normal form games vewed as our envronments, certan specfc correlated equlbrum dstrbutons vewed as SCFs volate quasmonotoncty (see Kar et al., 2010). Taken together, these results show the dffcultes of BRD convergng to specfc correlated equlbrum dstrbutons, whereas convergence to the entre set s possble. 5. Incomplete nformaton Ths secton tackles the extenson of our results to ncomplete nformaton envronments. For ease of exposton, we shall present our results for SCFs. We shall begn wth the (almost) characterzaton of SCFs that are mplementable n recurrent strateges of nterm BRD. We consder economc envronments for our results n nterm BRD, although our last subsecton, amng to connect wth other approaches through mplementaton n strct Bayesan equlbra, dspenses wth ths assumpton Necessary and suffcent condtons We now descrbe an ncomplete nformaton envronment. Consder for smplcty economc envronments. More precsely, let agent s consumpton set be a fnte set, X R l +, where 0 X. One can specfy that each agent holds ntally the bundle ω X wth ω N = ω (prvate ownershp economes), or smply that there s an aggregate endowment of goods ω (dstrbuton economes). The set of alternatves s the set of allocatons: { Z = (x ) N } X : x ω. N Each agent knows hs type θ Θ, a fnte set of possble types. Let Θ = N Θ be the set of possble states of the world, let Θ = j Θ j of type profles θ of agents other than. We shall sometmes wrte a state θ = (θ,θ ).We assume that all states n Θ have postve ex-ante probablty. 26 Let q (θ θ ) be type θ s nterm probablty dstrbuton over the type profles θ of the other agents. An SCF (or state-contngent allocaton) s a mappng f : Θ Z that assgns to each state of the world a feasble allocaton. Let A denote the set of SCFs. We shall assume that uncertanty concernng the states of the world does not affect the economy s endowments, but only preferences and belefs. We shall wrte type θ s nterm expected utlty over an SCF f as follows: U ( f θ ) ( q (θ θ )u f (θ,θ ), (θ,θ ) ). θ Θ Note how the Bernoull (ex-post) utlty functon u may change wth the state. We also make the followng assumptons on preferences: (1) No consumpton externaltes: for each state θ, u : X R. That s, an agent s ex-post utlty functon n each state depends on the bundle of goods that he consumes, and not on other agents bundles. (2) Strctly ncreasng ex-post utlty functons: For all, forallθ and for all x X,fy x, u (y,θ)>u (x,θ). 27 Note how ths mples that 0 s the worst bundle for every agent. A mechansm G = ((M ) N, g), played smultaneously by myopc agents, conssts of agent s set M of messages (for each N, agent s message s a mappng from Θ to M ), and the outcome functon g : M Z. A Bayesan equlbrum s a message profle n whch each type chooses an nterm best-response to the other agents messages, and a strct Bayesan equlbrum s a Bayesan equlbrum n whch every type s nterm best-response s a strct best-response. To prevent any knd of learnng about the state, we shall assume that, after an outcome s observed, agents forget t (or, closer to the evolutonary tradton, agents are replaced by other agents who share the same preferences and pror belefs as ther predecessors, but are not aware of ther experence) We make ths assumpton for smplcty n the presentaton. Wth some mnor modfcatons n the arguments, one can prove smlar results f Θ Θ s the set of states wth postve probablty, accordng to every agent s pror belef. 27 For vectors x, y X, we use the standard conventons: x y whenever x l y l wth at least one strct nequalty; and x y whenever x l > y l for every commodty l. 28 There are a host of alternatve assumptons one could make, for example, that each agent receves hs type n each perod as a draw from the..d. underlyng dstrbuton; see Dekel et al. (2004) for an apprasal of such dfferent modelng choces. 11