ROBUST PREDICTIONS IN GAMES WITH INCOMPLETE INFORMATION By Dirk Bergemnn nd Stephen Morris September 2011 Revised December 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1821R COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Hven, Connecticut 06520-8281 http://cowles.econ.yle.edu/
Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn y Stephen Morris z December 10, 2011 Abstrct We nlyze gmes of incomplete informtion nd o er equilibrium predictions which re vlid for, nd in this sense robust to, ll possible privte informtion structures tht the gents my hve. We completely chrcterize the set of Byes correlted equilibri in clss of gmes with qudrtic pyo s nd normlly distributed uncertinty in terms of restrictions on the rst nd second moments of the equilibrium ction-stte distribution. We derive exct bounds on how prior knowledge bout the privte informtion re nes the set of equilibrium predictions. We consider informtion shring mong rms under demnd uncertinty nd nd newly optiml informtion policies vi the Byes correlted equilibri. Finlly, we reverse the perspective nd investigte the identi ction problem under concerns for robustness to privte informtion. The presence of privte informtion leds to set rther thn point identi ction of the structurl prmeters of the gme. Jel Clssifiction: C72, C73, D43, D83. Keywords: Incomplete Informtion, Correlted Equilibrium, Robustness to Privte Informtion, Moments Restrictions, Identi ction, Informtions Bounds. We cknowledge nncil support through NSF Grnt SES 0851200. We bene tted from comments of Steve Berry, Vincent Crwford, Mtthew Gentzkow, Phil Hile, Emir Kmenic, Mrc Henry, Arthur Lewbel, Lrry Smuelson, nd Elie Tmer, nd reserch ssistnce from Brin Bis nd Aron Tobis. We would like to thnk seminr udiences t Boston College, the Collegio Crlo Alberto, Ecole Polytechnique, Europen University Institute, HEC, Microsoft Reserch, Northwestern University, the Pris School of Economics, Stnford University nd the University of Colordo for stimulting converstions; nd we thnk Dvid McAdms for his discussion t the 2011 North Americn Winter Meetings of the Econometric Society. y Deprtment of Economics, Yle University, New Hven, CT 06520, U.S.A., dirk.bergemnn@yle.edu. z Deprtment of Economics, Princeton University, Princeton, NJ 08544, U.S.A. smorris@princeton.edu 1
1 Introduction In gmes of incomplete informtion, the privte informtion of ech gent typiclly induces posterior belief bout the pyo sttes, nd posterior belief bout the beliefs of the other gents. The posterior belief bout the pyo stte represents knowledge bout the pyo environment, wheres the posterior belief bout the beliefs of the other gents represents knowledge bout the belief environment. In turn, the privte informtion of the gent, the type in the lnguge of Byesin gmes, in uences the optiml strtegy of the gent, nd ultimtely the equilibrium distribution over ctions nd sttes. The objective of this pper is to obtin equilibrium predictions for given pyo environment which re independent of - nd in tht sense robust to - the speci ction of the belief environment. We de ne the pyo environment s the complete description of the gents preferences nd the common prior over the pyo sttes. The fundmentl uncertinty bout the set of fesible pyo s is thus completely described by the common prior over the pyo sttes, which we lso refer to s the fundmentl sttes. We de ne the belief environment s the complete description of the common prior type spce over nd bove the informtion contined in the common prior distribution of the pyo sttes. The belief environment then describes potentilly rich type spce which is subject only to the constrint tht the mrginl distribution over the fundmentl sttes coincides with the common prior. A pir of pyo environment nd belief environment form stndrd Byesin gme. Yet importntly, for given pyo environment, there re mny belief environments, nd ech distinct belief environment my led to distinct equilibrium distribution over outcomes, nmely ctions nd fundmentls. The objective of the pper is to describe the equilibrium implictions of the pyo environment for ll possible belief environments reltive to the given pyo environment. Consequently, we refer to the (prtil) chrcteriztion of the equilibrium outcomes tht re independent of the belief environment s robust predictions. We exmine these issues in trctble clss of gmes with continuum of plyers, symmetric pyo functions, nd liner best response functions. A possible route towrds comprehensive description of the equilibrium implictions stemming from the pyo environment lone, would be n exhustive nlysis of the Byes Nsh equilibri of ll possible belief environments ssocited with given pyo environment. Here we shll not pursue this direct pproch. Insted we shll use relted equilibrium notion, nmely the notion of Byes correlted equilibrium to obtin comprehensive chrcteriztion. We begin with n epistemic result tht estblishes the equivlence between the clss of Byes Nsh equilibrium distributions for ll possible belief environments nd the clss of Byes correlted equilibrium distributions. This result is nturl extension of seminl result by Aumnn (1987). In gmes with complete informtion bout the pyo environment, he estblishes the equivlence between the set of Byes Nsh equilibri nd the set of correlted equilibri. We present the epistemic result for the clss of 2
gmes with continuum of gent nd symmetric pyo functions, nd show tht the insights of Aumnn (1987) generlizes nturlly to this clss of gmes with incomplete informtion. Subsequently we use the epistemic result to provide complete chrcteriztion of the Byes correlted equilibri in the clss of gmes with qudrtic pyo s. With qudrtic gmes, the best response function of ech gent is liner function nd in consequence the conditionl expecttions of the gents re linked through liner conditions which in turn permits n explicit construction of the equilibrium sets. The clss of qudrtic gmes hs fetured prominently in mny recent contributions to gmes of incomplete informtion, for exmple the nlysis of rtionl expecttions in competitive mrkets by Guesnerie (1992), the nlysis of the beuty contest by Morris nd Shin (2002) nd the equilibrium use of informtion by Angeletos nd Pvn (2007). We o er chrcteriztion of the equilibrium outcomes in terms of the moments of the equilibrium distributions. In the clss of qudrtic gmes, we show tht the expected men is constnt cross ll equilibri nd provide shrp inequlities on the vrince-covrince of the joint outcome stte distributions. If the underlying uncertinty bout the pyo stte nd the equilibrium distribution itself re normlly distributed then the chrcteriztion of the equilibrium is completely given by the rst nd second moments. If the distribution of uncertinty or the equilibrium distribution itself is not normlly distributed, then the chrcteriztion of rst nd second moments remins vlid, but of course it is not complete chrcteriztion in the sense tht the determintion of the higher moments is incomplete. The reltionship between the Byes Nsh equilibrium nd the Byes correlted equilibrium is shown to led to new insights into the reltionship between informtion structure nd the nture of the Byes Nsh equilibrium. The compct representtion of the Byes correlted equilibri llows us to ssess the privte nd socil welfre cross the entire set of possible informtion structures nd ssocited Byes Nsh equilibri. We illustrte this in the context of informtion shring mong rms. A striking result by Clrke (1983) ws the nding tht rms, when fcing uncertinty bout common prmeter of demnd, will never nd it optiml to shre informtion. The present nlysis of the Byes correlted equilibrium llows us to modify this insight - implicitly by llowing for richer informtion structures thn previously considered - nd we nd tht the Byes correlted equilibrium tht mximizes the privte welfre of the rms is not necessrily obtined with either zero or full informtion disclosure. The initil equivlence result between Byes correlted nd Byes Nsh equilibrium relied on very wek ssumptions bout the belief environment of the gents. In prticulr, we llowed for the possibility tht the gents my hve no dditionl informtion beyond the common prior bout the pyo stte. Yet, in some circumstnces the gents my be commonly known to hve some given prior informtion, or bckground informtion. Consequently, we then nlyze how lower bound on either the public or the 3
privte informtion of the gents, cn be used to further re ne the robust predictions nd impose dditionl moment restrictions on the equilibrium distribution. The pyo environment is speci ed by the (ex-post) observble outcomes, the ctions nd the pyo stte. By contrst, the elements of the belief environment, the beliefs of the gents, the beliefs over the beliefs of the gents, etc. re rrely directly observed or inferred from the reveled choices of the gents. The bsence of the observbility (vi reveled preference) of the belief environment then constitutes seprte reson to be skepticl towrds n nlysis which relies on very speci c nd detiled ssumptions bout the belief environment. Finlly, we therefore reverse the perspective of our nlysis nd consider the issue of identi ction rther thn prediction. Nmely, we re sking whether the observble dt, ctions nd pyo sttes, cn identify the structurl prmeters of the pyo functions, nd thus of the gme, without overly nrrow ssumptions on the belief environment. The question of identi ction is to sk whether the observble dt imposes restrictions on the unobservble structurl prmeters of the gme given the equilibrium hypothesis. Similrly to the problem of robust equilibrium prediction, the question of robust identi ction then is which restrictions re common to ll possible belief environments given speci c pyo environment. We nd tht we cn robustly identify the sign of some interction prmeters, but hve to leve the sign nd size of other prmeters, in prticulr whether the gents re plying gme of strtegic substitutes or complements, unidenti ed. The identi ction results here, in prticulr the contrst between Byes Nsh equilibrium nd Byes correlted equilibrium, re relted to, but distinct from the results presented in Ardills-Lopez nd Tmer (2008). In their nlysis of n entry gme with incomplete informtion, they document the loss in identi ction power tht rises with more permissive solution concept, i.e. level k-rtionlizbility. As we compre Byes Nsh nd Byes correlted equilibrium, we show tht the lck of identi ction is not necessrily due to the lck of common prior, s ssocited with rtionlizbility, but rther the richness of the possible privte informtion structures (but ll with common prior). In recent yers, the concern for robust equilibrium nlysis in gmes of incomplete informtion hs been rticulted in mny wys. In mechnism design, where the rules of the gmes cn be chosen to hve fvorble robustness properties, number of positive results hve been obtined. Dsgupt nd Mskin (2000), Bergemnn nd Välimäki (2002), Bergemnn nd Morris (2005), nd Perry nd Reny (2002) mong others, show tht the e cient socil lloction cn be implemented in n ex-post equilibrium nd hence in Byes Nsh equilibrium for ll type spces, with or without common prior. 1 But in given rther thn designed gmes, such strong robustness results seem out of rech for most clsses of gmes. In 1 Jehiel nd Moldovnu (2001) nd Jehiel, Moldovnu, Meyer-Ter-Vehn, nd Zme (2006) demonstrte the limits of these results by considering multi-dimensionl pyo types. 4
prticulr, mny Byesin gmes simply do not hve ex post or dominnt strtegy equilibri. In the bsence of such globl robustness results, nturl rst step is then to investigte the robustness of the Byes Nsh equilibrium to smll perturbtion of the informtion structure. For exmple, Kjii nd Morris (1997) consider Nsh equilibrium of complete informtion gme nd sy tht the Nsh equilibrium is robust to incomplete informtion if every incomplete informtion gme with pyo s lmost lwys given by the complete informtion gme hs n equilibrium which genertes behvior close to the Nsh equilibrium. In this pper, we tke di erent pproch nd use the dichotomy between the pyo environment nd the belief environment to nlyze the equilibrium behvior in given pyo environment while llowing for ny rbitrry, but common prior, type spce, s long s it is consistent with the given common prior of the pyo type spce. Chwe (2006) discusses the role of sttisticl informtion in single-gent nd multi-gent decision problems. In series of relted settings, he rgues tht the correltion between the reveled choice of n gent, referred to s incentive comptibility, nd rndom vrible, not controlled by the gent, llows n nlyst to infer the nture of the pyo interction between the gent s choice nd the rndom vrible. In the current contribution, we trce out the Byes Nsh equilibri ssocited with ll possible informtion structures. A relted literture seeks to identify the best informtion structure consistent with the given common prior over pyo types. For exmple, Bergemnn nd Pesendorfer (2007) chrcterizes the revenue-mximizing informtion structure in n uction with mny bidders. Similrly, in clss of sender-receiver gmes, Kmenic nd Gentzkow (2011) derive the sender-optiml informtion structure. The reminder of the pper is orgnized s follows. Section 2 de nes the relevnt solution concepts nd estblishes the epistemic result which reltes the set of Byes Nsh equilibri to the set of Byes correlted equilibri. Beginning with Section 3, we con ne our ttention to clss of qudrtic gmes with normlly distributed uncertinty bout the pyo stte. Section 4 reviews the stndrd pproch to gmes with incomplete informtion nd nlyses the Byes Nsh equilibri under bivrite belief environment in which ech gent receives privte nd public signl bout the pyo stte. Section 5 begins with the nlysis of the Byes correlted equilibrium. We give complete description of the equilibrium set in terms of moment restrictions on the joint equilibrium distribution. In Section 6 we nlyze how prior informtion bout the belief environment cn further restrict the equilibrium predictions. In Section 7 we consider the optiml shring of informtion mong rms. By rephrsing the choice of informtion policy s choice over informtion structures, we derive newly optiml informtion policies through the lens of Byes correlted equilibri. In Section 8, we turn from prediction to the issue of identi ction. Section 9 discusses some possible extensions nd o ers concluding remrks. The Appendix collects the proofs from the min body of the text. 5
2 Set-Up We rst de ne the solution concept of Byes correlted equilibrium. We then relte the notion of Byes correlted equilibrium to robust equilibrium predictions in clss of continuum plyer gmes with symmetric pyo. In the compnion pper, Bergemnn nd Morris (2011), we develop this solution concept nd its reltionship to robust predictions in cnonicl nite plyer nd nite ction gmes. In the compnion pper, we lso show how the results there cn be dpted nd re ned rst to symmetric pyo s nd then to the continuum of gents nd continuum of ctions nlyzed here. There is continuum of plyers nd n individul plyer is indexed by i 2 [0; 1]. Ech plyer chooses n ction 2 R. There will then be relized popultion ction distribution h 2 (R). There is pyo stte 2. All plyers hve the sme pyo function u : R (R)! R, where u (; h; ) is plyer s pyo if she chooses ction, the popultion ction distribution is h nd the stte is. There is prior distribution 2 (). A pyo environment is thus prmeterized by (u; ). We my lso refer to (u; ) s the "bsic gme" s 2 () only speci es the common prior distribution over the pyo stte 2 wheres it does not specify the privte informtion the gents my hve ccess to. We will be interested in probbility distributions 2 ( (R) ) with the interprettion tht is the joint distribution of the popultion ction distribution h nd the stte. For ny such, we write b for the induced probbility distribution on R (R) if (h; ) 2 (R) re drwn ccording to nd there is then conditionlly independent drw of 2 R ccording to h. For ech 2 R, we write b (j) for the probbility on (R) conditionl on (we will write s if it is uniquely de ned). De nition 1 (Byes Correlted Equilibrium ) A probbility distribution 2 ( (R) ) is Byes correlted equilibrium (BCE) of (u; ) if E b(j) u (; h; ) E b(j) u 0 ; h; (1) for ech 2 R nd 0 2 R; nd mrg = : (2) This de nition extends the notion of correlted equilibrium in Aumnn (1987) to n environment with uncertin pyo s, represented by the stte of the world. We will show tht Byes correlted equilibrium cptures ll behvior tht could rise if plyers observed dditionl privte informtion (in symmetric wy) nd plyed ccording to symmetric Byes Nsh equilibrium. To formlize this, we rst introduce the relevnt (symmetric) informtion structures for this continuum gent economy. Ech plyer will observe signl (or relize type) t 2 T. In ech stte of the 6
world 2, there will be relized distribution of signls g 2 (T ) drwn ccording to distribution k 2 ( (T )). Let :! ( (T )) give the distribution over signl distributions. Thus the belief environment, or lterntively n informtion structure, is prmeterized by (T; ). The pyo environment (u; ) nd the belief environment (T; ) together de ne gme of incomplete informtion ((u; ) ; (T; )). A symmetric strtegy in the gme is then de ned by : T! (R). The interprettion is tht (t) is the relized distribution of ctions mong those plyers observing signl t (i.e., we re "ssuming the lw of lrge numbers" on the continuum). A distribution of signls g 2 (T ) nd 2 induce probbility distribution g 2 (R). The prior 2 () nd signl distribution :! (T ) induce probbility distribution 2 ( (T ) ). As before, write [ for the probbility distribution on T (T ) if (g; ) 2 (T ) re drwn ccording to nd there is then conditionlly independent drw of t 2 T ccording to the relized g 2 (T ). For ech t 2 T, we write [ (jt) for the probbility on (T ) conditionl on t (we will write s if it is uniquely de ned). De nition 2 (Byes Nsh Equilibrium) A strtegy 2 is Byes Nsh equilibrium (BNE) of ((u; ) ; (T; )) if E [ (jt) u (; g ; ) E [(jt) u 0 ; g ; ; for ll t 2 T, in the support of ( jt) nd 0 2 R. Let be the probbility distribution on ( (R) ) induced if (g; ) 2 (T ) re drwn ccording to nd h 2 (R) is set equl to g. De nition 3 (Byes Nsh Equilibrium Distribution) A probbility distribution 2 ( (R) ) is BNE ction stte distribution of ((u; ) ; (T; )) if there exists BNE of ((u; ) ; (T; )) such tht =. We re now in position to relte the Byes correlted equilibri with the Byes Nsh equilibri. Proposition 1 A probbility distribution 2 ( (R) ) is Byes correlted equilibrium of (u; ) if nd only if it is BNE ction stte distribution ((u; ) ; (T; )) for some informtion structure (T; ). Aumnn (1987) estblishes the reltion between Nsh equilibri nd correlted equilibri in gmes with complete informtion. In the compnion pper, Bergemnn nd Morris (2011), we estblish the relevnt epistemic results for cnonicl gme theoretic environments in more detil. 7
In our compnion pper, Bergemnn nd Morris (2011), the notion of Byes correlted equilibrium is de ned in cnonicl gme theory environment - with nite number of ctions, gents nd sttes - where the plyers hve dditionl informtion from n informtion structure (T; ), nd thus Byes correlted equilibrium is joint distribution over ction, sttes nd types, i.e. distribution 2 (A T ). In the lnguge of the more generl notion o ered there, the Byes correlted equilibrium de ned here is the Byes correlted informtion with the null informtion structure, i.e. the cse in which the gents re not ssumed priori to hve ccess to speci c informtion structure. Here, we choose this miniml notion of Byes correlted equilibrium to obtin robust predictions for n observer who only knows the pyo environment but hs null informtion bout the belief environment of the gme. But, just s in the compnion pper, Bergemnn nd Morris (2011), we cn nlyze the impct of privte informtion on the size of the Byes correlted equilibrium set. In fct in Section 6, we nlyze how prior knowledge of the belief environment cn re ne the set of equilibrium predictions. We mintin our restriction to normlly distributed uncertinty, now normlly distributed types, to obtin explicit descriptions of the resulting restriction on the equilibrium set. By contrst, in Bergemnn nd Morris (2011), we llow for generl informtion structures nd derive mny plyer generliztion of the ordering of Blckwell (1953) s necessry nd su cient condition to order the set of Byes correlted equilibrium. However, within this generl environment, we do not obtin n explicit nd compct description of the equilibrium set in terms of the rst nd second moments of the equilibrium distributions, s we do in the present nlysis. The generl notion of Byes correlted informtion lso fcilittes the discussion of the reltionships between the notion of Byes correlted equilibrium, nd relted, but distinct notions of correlted equilibrium in gmes of incomplete informtion, most notbly in the work of Forges (1993), which is titled nd identi es ve legitimte de nitions of correlted equilibrium in gmes with incomplete informtion. We refer to the reder to the compnion pper, Bergemnn nd Morris (2011) for detiled discussion nd comprison. 3 Liner Best Response nd Norml Uncertinty For the reminder of this pper, we shll consider pyo environment with liner best responses nd normlly distributed uncertinty. Thus we ssume tht plyer i sets his ction equl to liner function of his expecttions of the verge ction of others A nd pyo relevnt stte. Thus we hve i = re i (A) + se i () + u, (3) where r; s; u 2 R re the prmeters of the best response function nd re ssumed to be identicl cross plyers. The verge ction of the ll plyers but i is represented by A. In the cse of nite number I of 8
plyers, it is therefore the sum: A, 1 I 1 nd in the cse of continuum of gents, j 2 [0; 1], it is n integrl: A, Z 1 0 X j, (4) j6=i j dj. (5) With the liner best response, the equilibrium behvior with nite, but lrge number of plyers converges to the equilibrium behvior with continuum of plyers. The model with continuum of plyers hs the dvntge tht we do not need to keep trck of the reltive weight of the individul plyer i, nmely 1=I, nd the weight of ll the other plyers, nmely (I 1) =I. In consequence, the expression of the equilibrium strtegies re frequently more compct with continuum of plyers. In the subsequent nlysis, we will focus on the gme with continuum of plyers, but report on the necessry djustments with nite plyer environment. The prmeter r represents the strtegic interction mong the plyers, nd we therefore refer to it s the interction prmeter. If r < 0, then we hve gme of strtegic substitutes, if r > 0, then we hve gme of strtegic complementrities. The cse of r = 0 represents the cse of single person decision problem where ech plyer i simply responds the stte of the world, but is not concerned bout his interction with the other plyers. The prmeter s represents the informtionl response of plyer i, nd it cn be either negtive or positive. We shll ssume tht the stte of the world mtters for the decision of gent i, nd hence s 6= 0. We shll ssume tht the interction prmeter r is bounded bove, or r 2 ( 1; 1) ; (6) which is necessry nd su cient condition for the complete informtion gme to hve n interior Nsh equilibrium. In fct, with the restriction (6), the Nsh equilibrium in the gme with complete informtion is unique nd given by: i () = u 1 r + s, for ll i nd. (7) 1 r Moreover, under complete informtion bout the stte of the world, even the correlted equilibrium is unique; Neymn (1997) gives n elegnt rgument. The pyo stte, or the stte of the world, is ssumed to be distributed normlly with N ; 2. (8) The present environment of liner best response nd normlly distributed uncertinty encompsses wide clss of interesting economic environments. The following three pplictions re prominent exmples nd we shll return to them throughout the pper to illustrte some of the results. 9
Exmple 1 (Beuty Contest) In Morris nd Shin (2002), continuum of gents, i 2 [0; 1], hve to choose n ction under incomplete informtion bout the stte of the world. Ech gent i hs pyo function given by: u i ( i ; A; ) = (1 r) ( i ) 2 r ( i A) 2. The weight r re ects concern for the verge ction A tken in the popultion. Morris nd Shin (2002) nlyze the Byes Nsh equilibrium in which ech gent i hs ccess to privte (idiosyncrtic) signl nd public (common) signl of the world. In terms of our nottion, the beuty contest model set s = 1 nd u = 0 with 0 r < 1. Exmple 2 (Competitive nd Strtegic Mrkets) Guesnerie (1992) presents n nlysis of the stbility of the competitive equilibrium by considering continuum of producers with qudrtic cost of production nd liner inverse demnd function. If there is uncertinty bout the demnd intercept, we cn write the demnd curve s p (A) = s + ra + u with r < 0 while the cost of rm i is c ( i ) = 1 2 2 i. Individul rm pro ts re now given by i p (A) c ( i ) = (ra + s + u) i 1 2 2 i. In n lterntive interprettion, we cn hve common cost shock, so the demnd curve is p (A) = ra + u with r < 0 while the cost of rm i is c ( i ) = of lrge, but nite, Cournot mrkets, s shown by Vives (1988), (2011). s i + 1 2 2 i. Such n economy cn be derived s the limit Exmple 3 (Qudrtic Economies nd the Socil Vlue of Informtion) Angeletos nd Pvn (2009) consider generl clss of qudrtic economies (gmes) with continuum of gents nd privte informtion bout common stte 2 R. There the pyo of gent i is given by symmetric qudrtic utility function u i ( i ; A; ), which depends on the individul ction i, the verge ction A nd the pyo stte 2 R: u i ( i ; A; ), 1 2 0 B @ i A 1 C A 0 0 B @ 1 0 1 U U A U i U A U AA U C B A A @ A U U A U C A ; (9) where the mtrix U = fu kl g represents the pyo structure of the gme. In the erlier working pper version, Bergemnn nd Morris (2011b), we lso represented the pyo structure of the gme by the mtrix U. Angeletos nd Pvn (2009) ssume tht the pyo s re concve in the own ction: U < 0; nd tht the interction of the individul ction nd the verge ction (the indirect e ect ) is bounded by the own ction (the direct e ect ): U A =U < 1, U + U A < 0. (10) r 10
The best response in the qudrtic economy (with complete informtion) is given by: i = U AA + U U : The qudrtic term of the own cost, U simply normlizes the terms of the strtegic nd informtionl externlity, U A nd U. In terms of the present nottion we hve r = U A U ; s = U U. Their restriction (10) is equivlent to the present restriction (6). The entries in the pyo mtrix U which do not refer to the individul ction, i.e. the entries in the lower submtrix of U, nmely 2 3 4 U AA U A U A U re not relevnt for the determintion of either the Byes Nsh or the Byes correlted equilibrium. These entries my be relevnt for welfre nlysis (s in Angeletos nd Pvn (2009)), but for the welfre nlysis in this pper they re not nd cn be uniformly set to zero. 5 4 Byes Nsh Equilibrium We rst report s benchmrk stndrd pproch to nlyzing this clss of gmes of incomplete informtion. Strting with the pyo environment described in the previous section, we dd description of the belief environment, i.e., wht plyers know bout the stte nd others beliefs. Speci clly, we ssume tht ech plyer observes two-dimensionl signl. In the rst dimension, the signl is privtely observed nd idiosyncrtic to the gent, wheres in the second dimension, the signl is publicly observed nd common to ll the gents. In either dimension, the signl is normlly distributed nd centered round the true stte of the world. In this clss of normlly distributed signls, speci c type spce is determined by the vrince of the noise long ech dimension of the signl. For given vrinces, nd hence for given type spce, we then nlyze the Byes Nsh equilibrium/ of the bsic gme. We shll then proceed to nlyze the bsic gme with the notion of Byes correlted equilibrium nd estblish which predictions re robust cross ll of the privte informtion environments, independent of the speci c bivrite nd norml type spce to be considered now. Accordingly, we consider the following bivrite norml informtion structure. Ech gent i is observing privte nd public noisy signl of the true stte of the world. The privte signl x i, observed only by gent i, is de ned by: x i = + " i ; (11) 11
nd the public signl, common nd commonly observed by ll the gents is de ned by: y = + ". (12) The rndom vribles " i nd " re normlly distributed with zero men nd vrince given by 2 x nd 2 y, respectively; moreover " i nd " re independently distributed, with respect to ech other nd the stte. This model of bivrite normlly distributed signls ppers frequently in gmes of incomplete informtion, see Morris nd Shin (2002) nd Angeletos nd Pvn (2007) mong mny others. It is t times convenient to express the vrince of the rndom vribles in terms of the precision: x, 2 x ; y, 2 y ;, 2 nd, 2 + 2 x + 2 y ; we refer to the vector ( x ; y ) s the informtion structure of the gme. A specil cse of the noisy environment is the environment with zero noise. In this environment, the complete informtion environment, ech gent observes the stte of the world without noise. We begin the equilibrium nlysis with the complete informtion environment. The best response: i = ra + s + u, re ects the, possibly con icting, objectives tht gent i fces. Ech gent hs to solve prediction-like problem in which he wishes to mtch his ction, with the stte nd the verge ction A simultneously. The interction prmeters, s nd r, determine the weight tht ech component, nd A, receives in the delibertion of the gent. If there is zero strtegic interction, or r = 0, then ech gent fces pure prediction problem. Now, we observed erlier, see (7), tht the resulting Nsh equilibrium strtegy is given by: (), u 1 r + s. (13) 1 r We refer to the terms in equilibrium strtegy (13), u= (1 r) nd s= (1 r), s the equilibrium intercept nd the equilibrium slope, respectively. Next, we nlyze the gme with incomplete informtion, where ech gent receives bivrite noisy signl (x i ; y). In prticulr, we shll compre how responsive the strtegy of ech gent is to the underlying stte of the world reltive to the responsiveness in the gme with complete informtion. In the gme with incomplete informtion, gent i receives pir of signls, x i nd y, generted by the informtion structure (11) nd (12). The prediction problem now becomes more di cult for the gent. First, he does not observe the stte, but rther he receives some noisy signls, x i nd y, of. Second, since he does not observe the other gents signls either, he cn only form n expecttion bout their ctions, but gin hs to rely 12
on the signls x i nd y to form the conditionl expecttion. The best response function of gent i then requires tht ction is justi ed by the conditionl expecttion, given x i nd y: i = re [A jx i ; y ] + se [ jx i ; y ] + u. In this liner qudrtic environment with norml distributions, we conjecture tht the equilibrium strtegy is given by function liner in the signls x i nd y: (x i ; y) = 0 + x x i + y y. The equilibrium is then identi ed by the liner coe cients 0 ; x ; y ; which we expect to depend on the interction terms (r; s; u) nd the informtion structure ( x ; y ). Proposition 2 (Liner Byes Nsh Equilibrium) The unique Byes Nsh equilibrium is liner equilibrium: (x; y) = 0 + xx + yy, with the coe cients given by: 0 = u 1 r + s 1 r ; x x = s ; y = s r x r x 1 r y r x : (14) The derivtion of the liner equilibrium strtegy lredy ppered in mny contexts, e.g., in Morris nd Shin (2002) for the beuty contest model, nd for the present generl environment, in Angeletos nd Pvn (2007). With the normliztion of the verge ction given by (4) nd (5), the bove equilibrium strtegy is independent of the number of plyers, nd in prticulr independent of the nite or continuum version of the environment. The Byes Nsh equilibrium shres the uniqueness property with the Nsh equilibrium, its complete informtion counterprt. We observe tht the liner coe cients x nd y disply the following reltionship: y x = y x 1 1 r. (15) Thus, if there is no strtegic interction, or r = 0, then the signls x i nd y receive weights proportionl to the precision of the signls. The fct tht x i is privte signl nd y is public signl does not mtter in the bsence of strtegic interction, ll tht mtters is the bility of the signl to predict the stte of the world. By contrst, if there is strtegic interction, r 6= 0, then the reltive weights lso re ect the informtiveness of the signl with respect to the verge ction. Thus if the gme displys strtegic complements, r > 0, then the public signl y receives lrger weight. The commonlity of the public signl cross gents mens tht their decision is responding to the public signl t the sme rte, nd hence in equilibrium the public signl is more informtive bout the verge ction thn the privte signl. By contrst, if the gme displys strtegic substitutbility, r < 0, then ech gent would like to move wy from the verge, nd hence plces smller weight on the public signl y, even though it still contins informtion bout the underlying stte of the world. 13
Now, if we compre the equilibrium strtegies under complete nd incomplete informtion, (13) nd (14), we nd tht in the incomplete informtion environment, ech gent still responds to the stte of the world, but his response to is noisy s both x i nd y re noisy reliztions of, but centered round : x i = + " i nd y = + ". Now, given tht the best response, nd hence the equilibrium strtegy, of ech gent is liner in the expecttion of, the vrition in the ction is explined by the vrition in the true stte, or more generlly in the expecttion of the true stte. Proposition 3 (Attenution) The men ction in equilibrium is: E [] = 0 + x + y = u + s 1 r ; nd the sum of the weights, x + y, is: x + y = s 1 r 1 s r x 1 r. Thus, the men of the individul ction, E [], is independent of the informtion structure ( x ; y ). In ddition, we nd tht the liner coe cients of the equilibrium strtegy under incomplete informtion re (wekly) less responsive to the true stte thn under complete informtion. In prticulr, the sum of the weights is strictly incresing in the precision of the noisy signls x i nd y. The equilibrium response to the stte of the world is diluted by the noisy signls, tht is the response is ttenuted. The residul is lwys picked up by the intercept of the equilibrium response. Now, if we sk how the joint distribution of the Byes Nsh equilibrium vries with the informtion structure, then Proposition 3 estblished tht it is su cient to consider the higher moments of the equilibrium distribution. But given the normlity of the equilibrium distribution, it follows tht it is su cient to consider the second moments, tht is the vrince-covrince mtrix. The vrince-covrince mtrix of the equilibrium joint distribution over individul ctions i ; j, nd stte is given by: i ; j ; = 2 6 4 2 2 2 2 2 3 7 5 : (16) We denote the correltion coe cient between ction i nd j shorthnd by rther thn. With continuum of gents, we cn describe the equilibrium distribution, fter replcing the individul ction j by the verge ction A, through the triple ( i ; A; ). The covrince between the individul, but symmetriclly distributed, ctions i nd j, given by 2 hs to be equl to the vrince of the verge 14
ction, or 2 A = 2. 2 Similrly, the covrince between the individul ction nd the verge ction hs to be equl to the covrince of ny two, symmetric, individul ction pro les, or A A = 2. Likewise, the covrince between the individul (but symmetric) ction i nd the stte hs to equl to the covrince between the verge ction nd the stte, or or = A A. With the chrcteriztion of the unique Byes Nsh equilibrium in Proposition 2, we cn express the vrince-covrince mtrix of the equilibrium joint distribution over ( i ; A; ) in terms of the equilibrium coe cients ( x ; y ) nd the vrinces of the underlying rndom vribles (; " i ; "): i ;A; = 2 6 4 2 x 2 x + 2 y 2 y + 2 ( x + y ) 2 2 y 2 y + 2 ( x + y ) 2 2 ( x + y ) 2 y 2 y + 2 ( x + y ) 2 2 y 2 y + 2 ( x + y ) 2 2 ( x + y ) 2 ( x + y ) 2 ( x + y ) 2 3 7 5. (17) Conversely, given the structure of the vrince-covrince mtrix, we cn express the equilibrium coe cients x nd y in terms of the vrince nd covrince terms tht they generte: x = y; y = y q 2. (18) Thus, we ttribute to the privte signl x, through the weight x, the residul correltion between nd, where the residul is obtined by removing the correltion between nd which is due to the public signl. In turn, the weight ttributed to the public signl is proportionl to the di erence between the correltion cross ctions nd cross ction nd signl. We recll tht the ctions of ny two gents re correlted s they respond to the sme underlying fundmentl stte. Thus, even if their privte signls re independent conditionl on the true stte of the world, their ctions re correlted due to the correltion with the hidden rndom vrible. Now, if these conditionlly independent signls were the only sources of informtion, nd the correltion between ction nd the hidden stte where, then ll the correltion of the gents ction would hve to come through the correltion with the hidden stte, nd in consequence the correltion cross ctions rises indirectly, in two wy pssge through the hidden stte, or =. In consequence, ny correltion beyond this indirect pth, or 2 is generted by mens of common signl, the public signl y. Since the correltion coe cient of the ctions hs to be nonnegtive, the bove representtion suggest tht s long s the correltion coe cient ( ; ) stisfy: 0 1, nd 2 0; (19) 2 With nite number of gents nd the de nition of the verge ction given by: A = (1= (I 1)) P j6=i j, the vrince of A is given by 2 1 A = nd hence the vrince-covrince mtrix in the continuum version is only n + I 2 I 1 I 1 2 pproximtion, but not exct. We present the exct restrictions in Corollry 1 in the next section. 15
we cn nd informtion structures ( x ; y ) such the coe cients resulting from (18) re indeed the equilibrium coe cients of the ssocited Byes Nsh equilibrium strtegy. Proposition 4 (Informtion nd Correltion) For every ( ; ) such tht 0 1, nd 2 0; there exists unique informtion structure ( x ; y ) such tht the ssocited Byes Nsh equilibrium displys the correltion coe cients ( ; ): nd x = y = (1 ) 2 (1 ) + (1 r) 2 2 ; 2 2 2 (1 r) 2 (1 ) + (1 r) 2 2 : 2 In the two-dimensionl spce of the correltion coe cients ; 2, the set of possible Byes Nsh equilibri is described by the re below the 45 degree line. We illustrte how prticulr Byes Nsh equilibrium with its correltion structure ( ; ) is generted by prticulr informtion structure ( x ; y ). In Figure 1, ech level curve describes the correltion structure of the Byes Nsh equilibrium for prticulr precision x of the privte signl. A higher precision x genertes higher level curve. The upwrd sloping movement represents n increse in informtiveness of the public signl, i.e. n increse in the precision y. An increse in the precision of the public signl therefore leds to n increse in the correltion of ction cross gents s well s in the correltion between individul ction nd stte of the world. For low levels of precision in the privte nd the public signl, n increse in the precision of the public signl rst leds to n increse in the correltion of ctions, nd then only lter into n incresed correltion with the stte of the world. In Figure 2, we remin in the unit squre of the correltion coe cients ; 2. But this time, ech level curve is identi ed by the precision y of the public signl. As the precision of the privte signl increses, the level curve bends upwrd nd rst bckwrd, nd eventully forwrd. At low levels of the precision of the privte signl, n increse in the precision of the privte signl increses the dispersion cross gents nd hence decreses the correltion cross gents. But s it gives ech individul more informtion bout the true stte of the world, n increse in precision lwys leds to n increse in the correltion with the true stte of the world, this is the upwrd movement. As the precision improves, eventully the noise becomes su ciently smll so tht the underlying common vlue generted by domintes the noise, nd then serves to both increse the correltion with the stte nd cross ctions. But in contrst to the privte informtion, where the equilibrium sets moves mostly northwrds, i.e. where the improvement occurs mostly in the direction of n increse in the correltion between the stte nd the individul gent, the public informtion leds the equilibrium sets to move mostly estwrds, i.e. most of the chnge leds to 16
2 1.0 0.8 45 o 0.6 0.4.5 x 1 0.2.01.2.1 0.2 0.4 0.6 0.8 1.0 Figure 1: Byes Nsh equilibrium of beuty contest, r = 1=4, with vrying degree of precision x of privte signl. n increse in the correltion cross ctions. In fct for given correltion between the individul ctions, represented by, n increse in the precision of the public signl leds to the elimintion of Byes Nsh equilibri with very low nd with very high correltion between the stte of the world nd the individul ction. 5 Byes Correlted Equilibrium We now chrcterize the set of Byes correlted equilibri. We restrict ttention to symmetric nd normlly distributed correlted equilibri nd discuss the extent to which these re without loss of generlity t the end of this Section. We begin the nlysis with continuum of gents nd subsequently describe how the equilibrium restrictions re modi ed in nite plyer environment. We cn chrcterize the Byes correlted equilibri in two distinct, yet relted, wys. With continuum of gents, we cn chrcterize the equilibri in terms of the relized verge ction A nd the devition of the individul ction i from the verge ction, i A. Under the continuum hypothesis, the distribution round the relized verge ction A represents the exct distribution of ctions by the gents, conditionl on the relized verge ction A. Alterntively we cn chrcterize the equilibri in terms of n rbitrry pir of individul ctions, i nd j, nd the stte of the world. The rst pproch puts more emphsis on the distributionl properties of the correlted equilibrium, nd is convenient when we go beyond symmetric nd normlly distributed equilibri, wheres the second pproch is closer to the description of the Byes Nsh equilibrium in terms of the individul ction. 17
2 1.0 0.8 0.6 0.4.5 y 1 0.2.001.2.1.01 0.2 0.4 0.6 0.8 1.0 Figure 2: Byes Nsh equilibrium of beuty contest, r = 1=4, with vrying degree of precision x of public signl. 5.1 Equilibrium Moment Restrictions We consider the clss of symmetric nd normlly distributed Byes correlted equilibri. With the hypothesis of normlly distributed Byes correlted equilibrium, the ggregte distribution of the stte of the world nd the verge ction A is described by: 0 1 00 1 0 @ A A N @@ A 2 A ; @ A A A A 2 A 11 AA : In the continuum economy, we cn describe the individul ction s centered round the verge ction A with some dispersion 2, so tht = A+, for some N 0; 2. In consequence, the joint equilibrium distribution of (; A; ) is given by: 0 1 00 B @ A C A N BB @@ A 1 0 11 2 A A A A C A ; B @ A A 2 A 2 CC A AA : (20) A A 2 A 2 A + 2 The nlysis of the Byes correlted equilibrium proceeds by deriving restrictions on the joint equilibrium distribution (20). In other words, we seeks to identify the restrictions on the moments of the equilibrium distribution. Given tht we presently restrict ttention to multivrite norml distribution, it is su cient to derive restrictions in terms of the rst nd second moments of the equilibrium distribution (20). The equilibrium restrictions rise from two sources: (i) the best response conditions of the individul gents: i = re [A j i ] + se [ j i ] + u, for ll i nd i 2 R, (21) nd (ii) the consistency condition, see De nition 1, where the lter condition, nmely tht the mrginl distribution over is equl to the common prior over, is stis ed by construction of the joint equilibrium 18
distribution (20). The best response condition (21) of the Byes correlted equilibrium llows the gent to form his expecttion over the verge ction A nd the stte of the world by conditioning on the informtion tht is contined in his recommended equilibrium ction i. As the best response condition (21) uses the expecttion of the individul gent, it is convenient to introduce the following chnge of vrible for the equilibrium rndom vrible. symmetric equilibrium, we hve: = A nd 2 = 2 A + 2. By hypothesis of the The covrince between the individul ction nd the verge ction is given by A A = 2 A ;nd is identicl, by construction, to the covrince between the individul ctions: 2 = 2 A. (22) We cn therefore express the correltion coe cient between individul ctions,, s: = 2 A 2 A +, (23) 2 nd the correltion coe cient between individul ction nd the stte s: = A A. (24) In consequence, we cn rewrite the joint equilibrium distribution of (; A; ) in terms of the moments of the stte of the world nd the individul ction s: 0 1 00 1 0 11 2 B @ A C A N BB @@ C A ; B @ 2 2 CC AA : (25) 2 2 With the joint equilibrium distribution described by (25), we now use the best response property (21), to completely chrcterize the moments of the equilibrium distribution. imposing the obedience condition (1) in the generl setting of Section 2. Note tht this corresponds to As the best response property (21) hs to hold for ll i in the support of the correlted equilibrium, it follows tht the bove condition hs to hold in expecttion over ll i, or by the lw of totl expecttion: E [ i ] = u + se [E [ j i ]] + re [E [A j i ]]. (26) But by symmetry, it follows tht the expected ction of ech gent is equl to expected verge ction A, nd hence we cn use (26) to solve for the men of the individul ction nd the verge ction: E [ i ] = E [A] = u + se [] 1 r 19 = u + s 1 r. (27)
It thus follows tht the men of the individul ction nd the men of the verge ction is uniquely determined by the men vlue of the stte of the world nd prmeters (r; s; u) cross ll correlted equilibri. The complete description of the set of correlted equilibri then rests on the description of the second moments of the multivrite distribution. The chrcteriztion of the second moments of the equilibrium distribution gin uses the best response property of the individul ction, see (21). But, now we use the property of the conditionl expecttion, rther thn the iterted expecttion to derive restrictions on the covrites. The recommended ction i hs to constitute best response in the entire support of the equilibrium distribution. Hence the best response hs to hold for ll i 2 R, nd thus the conditionl expecttion of the stte E [ j i ] nd of the verge ction, E [A j i ], hve to chnge with i t exctly the rte required to mintin the best response property: 1 = s de [ j i ] + r de [A j i ] ; for ll i 2 R. d i d i Given the multivrite norml distribution (25), the conditionl expecttions E [ j i ] nd E [A j i ] re liner in i nd given by nd E [j i ] = 1 s 1 r + i u ; (28) 1 r E [Aj i ] = u + s 1 r (1 ) + i : (29) The optimlity of the best response property cn then be expressed, using (28) nd (29) s 1 = s + r. It follows tht we cn express either one of the three elements in the description of the second moments, ( ; ; ) in terms of the other two nd the primitives of the gme s described by (r; s). In fct, it is convenient to solve for the stndrd devition of the individul ctions, or = s 1 r. (30) The remining restrictions on the correltion coe cients nd re coming in the form of inequlities from the chnge of vribles in (22)-(24), where 2 = 2 A 2 A 2 = 2 A. (31) Finlly, the stndrd devition hs to be positive, or 0. Now, it follows from the ssumption of moderte interction, r < 1, nd the nonnegtivity restriction of implied by (31) tht 1 r > 0, nd thus to gurntee tht 0, it hs to be tht s 0. Thus the sign of the correltion coe cient hs to equl the sign of the interction term s. We summrize these results. 20
1.0 0.8 0.6 0.4 0.2. 0.2 0.4 0.6 0.8 1.0 Figure 3: Set of Byes correlted equilibrium in terms of correltion coe cients nd j j Proposition 5 (First nd Second Moments of BCE) A multivrite norml distribution of ( i ; A; ) is symmetric Byes correlted equilibrium if nd only if 1. the men of the individul ction is: E [ i ] = u 1 r + s 1 r ; (32) 2. the stndrd devition of the individul ction is: = 3. the correltion coe cients nd stisfy the inequlities: s 1 r ; nd (33) 2 nd s 0. (34) The chrcteriztion of the rst nd second moments suggests tht the men nd the vrince 2 of the fundmentl vrible re the driving force of the moments of the equilibrium ctions. The liner form of the best response function trnsltes into liner reltionship in the rst nd second moment of the stte of the world nd the equilibrium ction. In the cse of the stndrd devition, the liner reltionship is ected by the correltion coe cients nd which ssign weights to the interction prmeter r nd s, respectively. The set of ll correlted equilibri is grphiclly represented in Figure 3. The restriction on the correltion coe cients, nmely 2, emerged directly from the bove chnge of vrible, see (22)-(24). Alterntively, but equivlently, we could hve disregrded the restrictions implied by the chnge of vribles, nd simply insisted tht the mtrix of second moments of (25) is indeed 21
legitimte vrince-covrince mtrix, i.e. tht is nonnegtive de nite mtrix. A necessry nd su cient condition for the nonnegtivity of the mtrix is tht the determinnt of the vrince-covrince mtrix is nonnegtive, or, 6 4 s4 (1 ) 2 (1 r) 4 0 ) 2. Lter, we extend the nlysis from the pure common vlue environment nlyzed here, to n interdependent vlue environment (in Section 5.5) nd to prior privte informtion (in Section 6). In these extensions, it will be convenient to extrct the equilibrium restrictions in form of the correltion inequlities, directly from the restriction of the nonnegtive de nite mtrix, rther thn trce them through the relevnt chnge of vrible. In ny cse, these two procedures estblish the sme equilibrium restrictions. We observe tht t = 0, the only correlted equilibrium is given by = 1, in other words, there is discontinuity in the equilibrium set t = 0. In the symmetric equilibrium, if = 0, then this mens tht the ction of ech gent is completely insensitive to the reliztion of the true stte. But this mens, tht the gents do not respond to ny informtion bout the stte of the world beyond the expected vlue of the stte, E []. Thus, ech gent cts s if he were in complete informtion world where the true stte of the world is the expected vlue of the stte. But, we know from the erlier discussion, tht in this environment, there is unique correlted equilibrium where the gents ll choose the sme ction nd hence = 1. At this point, it is pproprite to describe how the nlysis of the Byes correlted equilibrium would be modi ed by the presence of nite number I of gents. We remrked in Section 3 tht the best response function of the gent i is constnt in the number of plyers. As the best response is independent of the number of plyers, it follows tht the equilibrium equlity restrictions, nmely (32) nd (33), re un ected by the number, in prticulr the niteness, of the plyers. The only modi ction rises with the chnge of vrible, see (22)-(24), which relied on the continuum of gents. By contrst, the inequlity restrictions with nite number of plyers cn be recovered directly from the fct tht vrince-covrince mtrix 1 ;:::; I ; of the equilibrium rndom vribles ( 1 ; :::; I ; ) hs to be nonnegtive de nite mtrix. Corollry 1 (First nd Second Moments of BCE with Finitely Mny Plyers) A multivrite norml distribution of ( 1 ; :::; I ; ) is symmetric Byes correlted equilibrium if nd only if it stis es (32), (33), nd the correltion coe cients nd stisfy the inequlities: 1 I 1, 2 1 2 I 1 ; s 0. (35) It is immedite to verify tht the restrictions of the correltion structure in (35) converge towrds the one in (34) s I! 1. We observe tht the restrictions in (35) re more permissive with smller number 22