Cowles Foundtion for Reserch in Economics t Yle University Cowles Foundtion Discussion Pper No. 1821RR ROBUST PREDICTIONS IN GAMES WITH INCOMPLETE INFORMATION Dirk Bergemnn nd Stephen Morris September 2011 Revised December 2011 Revised October 2012 An uthor index to the working ppers in the Cowles Foundtion Discussion Pper Series is locted t: http://cowles.econ.yle.edu/p/u/index.htm This pper cn be downloded without chrge from the Socil Science Reserch Network Electronic Pper Collection: http://ssrn.com/bstrct=2163606 Electronic copy vilble t: http://ssrn.com/bstrct=2163606
Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn y Stephen Morris z September 26, 2012 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. The present pper formed the bsis of the Fisher- Schultz Lecture given by Stephen Morris t the Europen Econometric Society Meeting 2012. We bene tted from comments of Steve Berry, Vincent Crwford, Mtthew Gentzkow, Phil Hile, Emir Kmenic, Mrc Henry, Arthur Lewbel, Lrry Smuelson, Elie Tmer, nd Xvier Vives, s well s 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 Electronic copy vilble t: http://ssrn.com/bstrct=2163606
1 Introduction Suppose tht some economic gents ech hve set of fesible ctions tht they cn tke nd their pyo s depend on the ctions tht they ll tke nd pyo stte with known distribution. Cll this scenrio the pyo environment. To nlyze behvior in this setting, we lso hve to specify wht gents believe bout the pyo sttes, wht others believe, nd so on. Cll this the belief environment. A stndrd incomplete informtion gme consists of combintion of pyo environment nd belief environment. Insted of sking wht hppens in xed incomplete informtion gme, in this pper we will chrcterize wht my hppen in given pyo environment for ny given belief environment. In prticulr, we identify which outcomes, i.e., probbility distributions over ction pro les nd pyo sttes, could rise in Byes Nsh equilibrium for xed pyo environment nd for some belief environment. There re number of resons why this exercise is both trctble nd interesting. The belief environment will generlly be very hrd to observe, s it is in the gents minds nd does not necessrily hve n observble counterprt. We know tht outcomes re very sensitive to the belief environment (Rubinstein (1989), Kjii nd Morris (1997) nd Weinstein nd Yildiz (2007)). If we cn chrcterize equilibrium outcomes independent of the informtion structure, we cn identify robust predictions for given pyo environment which re independent of - nd in tht sense robust to - the speci ction of the belief environment. Conversely, if we re ble to identify mpping from pyo environments to outcomes which does not depend on the belief environment, then we cn lso study the inverse of the mp, seeing which pyo environments re consistent with n observed outcome. This mpping gives us frmework for prtilly identifying the pyo environment without ssumptions bout the belief environment. Thus we cn crry out robust identi ction of the pyo environment. Chrcterizing the set of ll equilibri for ll belief environments sounds dunting, but it turns out tht it is often esier to chrcterize wht hppens for ll or mny informtion structures t once thn it is for xed informtion structure. Suppose tht insted of explicitly modelling the belief environment, we use the clssicl gme theoretic metphor of meditor who mkes privte ction recommendtions to the gents. In prticulr, suppose tht meditor ws ble to mke privte, perhps correlted, ction recommendtions to the gents s function of the pyo stte. If the gents hve n incentive to follow the meditor s recommendtion, we sy tht the resulting joint distribution of pyo s nd ctions is Byes correlted equilibrium. We cn show tht the set of Byes correlted equilibri for given pyo environment correspond one to one with the set of Byes Nsh equilibri tht could rise for ny belief environment. The Byes correlted equilibrium chrcteriztion cn lso be used to nlyze the strtegic vlue of 2 Electronic copy vilble t: http://ssrn.com/bstrct=2163606
informtion. Economists re often interested in nlyzing which informtion structure is best for some welfre mesure or some subset of gents in given setting. In nlyzing such problems, it is usul to focus on low dimensionl prmeterized set of informtion structures becuse working with ll informtion structures seems intrctble. Our results suggest n lterntive pproch: one cn nd the Byes correlted equilibrium tht mximizes some objective, nd then reverse engineer the informtion structure tht genertes tht distribution s Byes Nsh equilibrium. In Bergemnn nd Morris (2011), we pursue this reserch gend for generl gmes. In this pper, we exmine these issues in trctble pyo environment with continuum of plyers, symmetric pyo functions, liner best response functions nd normlly distributed uncertinty. We consider trctble belief environment, consisting of noisy privte nd noisy public signl of the pyo relevnt stte. This combintion of trctble pyo environment nd belief environment is widely studied, see Morris nd Shin (2002) nd Angeletos nd Pvn (2007) mong mny others. The nlysis in this pper provides powerful illustrtion of the logic nd usefulness of the more generl pproch, s well s providing new results bout n importnt economic environment tht is widely used in economic pplictions. Symmetry nd normlity ssumptions re mintined throughout the nlysis. Byes correlted equilibri in this environment re symmetric norml distributions of the stte nd the ctions in the continuum popultion with the "obedience" property tht plyer with no informtion beyond the ction tht he is to ply would not hve n incentive to choose di erent ction. We compre Byes correlted equilibri with Byes Nsh equilibri for every belief environment in our bivrite clss of informtion structures. Integrting out the gents signls, we show tht ech informtion structure nd its Byes Nsh equilibrium gives rise to Byes correlted equilibrium. Conversely, ech Byes correlted equilibrium corresponds to the unique Byes Nsh equilibrium for some informtion structure in the bivrite clss. This result illustrtes the more generl equivlence in Bergemnn nd Morris (2011), within the clss of symmetric norml distributions. Becuse Byes correlted equilibri re two dimensionl in this environment, simple two dimensionl clss of informtion structures is lrge enough to rech ll Byes correlted equilibri. We cn identify robust predictions in terms of restrictions on the rst nd second moments of the joint distribution over ctions nd stte. 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. 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 men of the individul ctions (nd of the popultion ction) is constnt cross ll equilibri nd provide shrp inequlities on the vrince-covrince of the joint outcome stte 3
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. We show how our pproch cn be used to nlyze the strtegic vlue of informtion by considering 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. Our benchmrk model contrsts two extremes: either nothing is known bout the belief environment, or it is perfectly known. For both robust prediction nd robust identi ction results, it is nturl to consider intermedite cses where there is prtil informtion bout the belief environment. In prticulr, we nlyze how lower bound on either the public or the privte informtion of the gents, cn be used to further re ne the robust predictions nd impose dditionl moment restrictions on the equilibrium distribution. We use our chrcteriztion of wht hppens in intermedite belief environments to nlyze the robust identi ction question in depth. 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. With no restrictions on the belief 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. 1 However, we lso 1 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). 4
identify conditions on the belief environment under which we re ble to identify the sign of the interction prmeter. This present work exmines how the nlysis of xed gmes cn be mde robust to informtionl ssumptions. This work prllels work in robust mechnism design, where gmes re designed so tht equilibrium outcomes re robust to informtionl ssumptions (our own work in this re beginning with Bergemnn nd Morris (2005) is collected in Bergemnn nd Morris (2012)). While the endogeneity of the gme design mkes the issues in the robust mechnism design literture quite di erent, in both litertures informtionl robustness cn be studied with richer, more globl, perturbtions of the informtionl environment nd more locl ones. This pper is very permissive in llowing for rich clss of belief environments but less permissive in restricting ttention to common prior belief environments. The reminder of the pper is orgnized s follows. Section 2 de nes the pyo environment, clss of qudrtic gmes with normlly distributed uncertinty, nd the belief environment. We lso de ne the relevnt solution concepts, nmely Byes Nsh equilibrium nd Byes correlted equilibrium. Section 3 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 4 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 5 we consider the optiml shring of informtion mong rms. In Section 6 we nlyze how prior restrictions bout the belief environment cn further restrict the equilibrium predictions. 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 7, we turn from prediction to the issue of identi ction. Section 8 discusses some possible extensions nd o ers concluding remrks. The Appendix collects the proofs from the min body of the text. 2 Set-Up 2.1 Pyo Environment There is continuum of plyers nd n individul plyer is indexed by i 2 [0; 1]. Ech plyer chooses n ction i 2 R. The verge ction of ll plyers is represented by A 2 R nd is the integrl: A, Z 1 0 j dj. (1) There is pyo relevnt stte 2 with prior distribution 2 (). All plyers hve the sme pyo function u : R R! R; (2) 5
where u (; A; ) is plyer s pyo if she chooses ction, the verge (or popultion) ction is A nd the stte is. A pyo environment is thus prmeterized by (u; ). We will restrict ttention to 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 A nd the pyo relevnt stte : i = re i [A] + se i [] + k, (3) where r; s; k 2 R re the prmeters of the best response function nd re ssumed to be identicl cross plyers. 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 sensitivity of plyer i, the responsiveness to the stte, 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) : (4) Under this ssumption, 2 there is unique Nsh equilibrium of the gme with complete informtion given by: i () = k 1 r + s, for ll i nd. (5) 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. (6) The present environment of liner best response nd normlly distributed uncertinty encompsses wide clss of interesting economic environments. The following four pplictions re prominent exmples nd we shll return to them throughout the pper to illustrte some of the results. 2 If r > 1, the Nsh equilibrium is unstble nd, if ctions sets were bounded, there would be multiple Nsh equilibri. 6
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 ; 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 for the belief environment nlyzed in this pper. nottion, the beuty contest model set s = 1 r nd k = 0 with 0 r < 1. In terms of our 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 + k 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 + k) i 1 2 2 i. In n lterntive interprettion, we cn hve common cost shock, so the demnd curve is p (A) = ra + k 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 ; A; ), which depends on the individul ction i, the verge ction A nd the pyo stte 2 R: u ( 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 ; (7) 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. (8) 7
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 = Their restriction (8) is equivlent to the present restriction (4). The entries in the pyo mtrix U which U U. 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 highlighted in Angeletos nd Pvn (2009)), but for the welfre nlysis discussed in this pper they re not nd cn be uniformly set to zero. Exmple 4 (Qudrtic Economies with Finite Number of Agents) In the cse of nite number I of plyers, the verge ction of ll plyers but i is represented by the sum: A, 1 I 1 5 X j. (9) With the liner best response (3), 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 j6=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. 2.2 Belief Environment Strting with the pyo environment (u; ) described in the previous subsection, we dd description of the belief environment, i.e., wht plyers know bout the stte nd others beliefs. We rst introduce the relevnt belief (or signl) environment for this continuum gent economy. Ech plyer is ssumed to observe two-dimensionl signl. In the rst dimension, the signl x i is privtely observed nd idiosyncrtic to the plyer i, wheres in the second dimension, the signl y is publicly observed nd common to ll the plyers: x i = + " i ; y = + ". (10) 8
The respective rndom vribles " i nd " re ssume to be 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. The type of ech plyer is therefore given by the pir of signls: (x; y i ). 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, 2 x nd 2 y. 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. 2.3 Solution Concepts We rst describe the stndrd pproch to nlyze gmes of incomplete informtion by mens of xed belief environment (or type spce) nd the ssocited Byes Nsh equilibri. We then introduce the notion of Byes correlted equilibrium which will be de ned s function of the pyo environment lone. 2.3.1 Byes Nsh Equilibrium The pyo environment nd the belief environment together de ne gme of incomplete informtion. A symmetric pure strtegy in the gme is then de ned by s : R 2! R. De nition 1 (Byes Nsh Equilibrium) A (symmetric) pure strtegy s is Byes Nsh equilibrium (BNE) if E [u (s (x i ; y) ; A; ) jx i ; y ] E u 0 ; A; jx i ; y ; for ll x i ; y 2 R nd 0 2 R. 2.3.2 Byes Correlted Equilibrium The Byes correlted equilibrium depends on the pyo environment lone, nd does not refer to the belief environment. A Byes correlted equilibrium is de ned to be joint distribution over sttes nd plyers ctions which hs the property tht plyer who knows only wht ction he is supposed to ply hs no incentive to choose di erent ction. In ddition, in this pper, we mintin the ssumptions of normlity nd symmetry. Ech plyer chooses n ction 2 R nd there will then be relized verge or popultion ction A. There is pyo relevnt stte 2. We re interested in probbility distributions 2 (R R ) with the interprettion tht is the joint distribution of the individul, the verge ction nd the stte. For ny such, we write (j) for the conditionl probbility distribution on R if (; A; ) 2 R R. 9
De nition 2 (Byes Correlted Equilibrium ) A probbility distribution 2 (R R ) is symmetric Byes correlted equilibrium (BCE) if E (j) [u (; A; ) j] E (j) u 0 ; A; j ; (11) for ech 2 R nd 0 2 R; nd mrg = : (12) The condition (11) sttes tht whenever plyer is sked to choose, he cnnot pro tbly devite by choosing ny di erent ction 0. This is the obedience condition, by direct nlogy with the best response condition in the correlted equilibrium for gmes of complete informtion in Aumnn (1987). The condition (12) sttes tht the mrginl of the Byes correlted equilibrium distribution over the pyo stte spce hs to be consistent with the common prior distribution. This de nition is specil cse of concept introduced in Bergemnn nd Morris (2011). The de nition here is written for the prticulr continuum plyer gmes studied in this pper, mintining symmetry nd normlity, nd with plyers conditioning on their ctions only nd not on ny dditionl informtion. The generl de nition of Byes correlted equilibrium in Bergemnn nd Morris (2011) is de ned for generl gmes nd llows for plyers to hve some informtion bout the stte - giving rise to dditionl incentive constrints - but llowing for rbitrry dditionl correltion in their ctions. In Section 6 of this pper, we nlogously nlyze how prior knowledge of the belief environment cn re ne the set of equilibrium predictions, mintining our restriction to symmetric norml distributions. There is signi cnt literture on lterntive de nitions of correlted equilibrium in incomplete informtion environments, with Forges (1993) providing clssic txonomy. As we discuss in Bergemnn nd Morris (2011), our de nition of Byes correlted equilibrium is weker thn the wekest de nition in the literture nd Forges (1993), intuitively becuse we llow the meditor to know the pyo stte which no individul plyer knows. While this ssumption my seem extreme when de ning solution concepts de novo, we will see how it precisely delivers the solution concept tht cptures the entire set of possible equilibrium outcomes for ll possible belief environments. 3 Byes Nsh Equilibrium We rst report, s benchmrk, the stndrd pproch to nlyzing this clss of gmes of incomplete informtion. We 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. 10
We consider the bivrite norml informtion structure given by privte nd public signl for ech gent i: x i = + " i ; y = + ". (13) The respective rndom vribles " i nd " re ssume to be normlly distributed with zero men nd vrince given by 2 x nd 2 y. 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 ; nd 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 + k, 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 (5), tht the resulting Nsh equilibrium strtegy is given by: (), k 1 r + s. (14) 1 r We refer to the terms in equilibrium strtegy (14), k= (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 (13). 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 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 ] + k. 11
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; k) nd the informtion structure ( x ; y ). Proposition 1 (Liner Byes Nsh Equilibrium) The unique Byes Nsh equilibrium is liner equilibrium: (x; y) = 0 + xx + yy, with the coe cients given by: 0 = k 1 r + s 1 r ; x x = s ; y = s r x r x 1 r y r x : (15) 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 (1) nd (9), 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. (16) 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. Now, if we compre the equilibrium strtegies under complete nd incomplete informtion, (14) nd (15), 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 12
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 2 (Attenution) The men of the individul ction in equilibrium is: E [] = 0 + x + y = k + s 1 r ; (17) 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. Moreover, with continuum of gents, by the lw of lrge numbers, the relized verge ction A lwys stis es the equlity (17) for every reliztion of the stte, or: A = 0 + x + y = k + s 1 r ; 8, nd hence the relized verge ction is lso independent of the informtion structure. Now, if we sk how the joint distribution of the Byes Nsh equilibrium vries with the informtion structure, then Proposition 2 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 : (18) 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 13
ction, or 2 A = 2. 3 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 1, 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. (19) 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. (20) 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; (21) 3 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. 14
we cn nd informtion structures ( x ; y ) such the coe cients resulting from (20) re indeed the equilibrium coe cients of the ssocited Byes Nsh equilibrium strtegy. Proposition 3 (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 ech individul receives more informtion bout the stte, n increse in precision lwys leds to n increse in the correltion with the 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 the ctions of the other gents. 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 15
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. the chnge leds to 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. 4 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. With continuum of gents, the distribution round the relized verge ction A cn be tken to represent 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. 16
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. 4.1 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 : (22) A A 2 A 2 A + 2 The nlysis of the Byes correlted equilibrium proceeds by deriving restrictions on the joint equilibrium distribution (22). In other words, we seek 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 (22). The equilibrium restrictions rise from two sources: (i) the best response conditions of the individul gents: i = re [A j i ] + se [ j i ] + k, for ll i nd i 2 R, (23) nd (ii) the consistency condition of De nition 2, nmely tht the mrginl distribution over is equl to the common prior over, is stis ed by construction of the joint equilibrium distribution (22). The best 17
response condition (23) 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 (23) uses the expecttion of the individul gent, it is convenient to introduce the following chnge of vrible for the equilibrium rndom vribles. 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. (24) We cn therefore express the correltion coe cient between individul ctions,, s: = 2 A 2 A +, (25) 2 nd the correltion coe cient between individul ction nd the stte s: = A A. (26) 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 : (27) 2 2 With the joint equilibrium distribution described by (27), we now use the best response property (23), to completely chrcterize the moments of the equilibrium distribution. As the best response property (23) 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 ] = k + se [E [ j i ]] + re [E [A j i ]]. (28) By symmetry, the expected ction of ech gent is equl to expected verge ction A, nd hence we cn use (28) to solve for the men of the individul ction nd the verge ction: E [ i ] = E [A] = k + se [] 1 r = k + s 1 r. (29) 18
It 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 the prmeters (r; s; k) 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 (23). 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 (27), 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 k ; (30) 1 r E [Aj i ] = k + s 1 r (1 ) + i : (31) The optimlity of the best response property cn then be expressed, using (30) nd (31) 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. (32) The remining restrictions on the correltion coe cients nd re coming in the form of inequlities from the chnge of vribles in (24)-(26), where 2 = 2 A 2 A 2 = 2 A. (33) 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 (33) 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. 19
Proposition 4 (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 ] = k 1 r + s 1 r ; (34) 2. the stndrd devition of the individul ction is: = 3. the correltion coe cients nd stisfy the inequlities: s 1 r ; nd (35) 2 nd s 0. (36) Thus the robust predictions of the liner model re tht the men of the individul ction is pinned down by the prmeters of the model nd there is one dimensionl restriction on the remining free endogenous vribles ( ; ; ). Notbly, there re (lmost) no robust predictions bout the correltion coe cients. The sign of is pinned down by the sign of s, nd there is sttisticl requirement tht 2, but beyond these restrictions, ny correltion coe cients re consistent with ny vlues of the prmeters of the model nd, in prticulr, with ny vlue of the interction prmeter r. In Section 7, we nlyze the issue of robust identi ction in the model. In prticulr, we will rgue formlly in Proposition 12 tht ny vlue of the interction prmeter r 2 ( 1; 1) is consistent with ny given observed rst nd second moments of the stte ( ; ) nd the endogenous vribles ( ; ; ; ). 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 (24)-(26). 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 (27) is indeed legitimte vrince-covrince mtrix, i.e. tht it is nonnegtive de nite mtrix. A necessry 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. (37) 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 In ddition, due to the specil structure of the present mtrix, nmely 2 A = 2, the bove inequlity is lso su cient condition for the nonnegtive de niteness of the mtrix. Lter, we extend the nlysis from the pure common vlue environment nlyzed here, to n interdependent vlue environment (in Section 4.4) nd to prior restrictions on the privte informtion of the gents (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 vrince of the individul ction 2 hs to be equl to zero by (32), nd hence if the individul ctions do not disply ny correltion with the pyo stte ; then the individul ction nd hence the verge ctions must be constnt. 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, E []. At this point, it is useful 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 2 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 (34) nd (35), re un ected by the number, in prticulr the niteness, of the plyers. The only modi ction rises with the chnge 21
of vrible, see (24)-(26), 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 (34), (35), nd the correltion coe cients nd stisfy the inequlities: 1 I 1, 2 1 2 I 1 ; s 0. (38) It is immedite to verify tht the restrictions of the correltion structure in (38) converge towrds the one in (36) s I! 1. We observe tht the restrictions in (38) re more permissive with smller number of gents, nd in prticulr llow for moderte negtive correltion cross individul ctions with nite number of gents. By contrst, with in nitely mny gents, it is sttisticl impossibility tht ll ctions re mutully negtively correlted. The condition on the vrince of the individul ction, given by (32), ctully follows the sme logic s the condition on the men of the individul ction, given by (29). To wit, for the men, we used the lw of totl expecttion to rrive t the equlity restriction. Similrly, we could obtin the bove restriction (32) by using the lw of totl vrince nd covrince. More precisely, we could require, using the equlity (23), tht the vrince of the individul ction mtches the sum of the vrinces of the conditionl expecttions. Then, by using the lw of totl vrince nd covrince, we could represent the vrince of the conditionl expecttion in terms of the vrince of the originl rndom vribles, nd obtin the exct sme condition (32). Here we chose to directly use the liner form of the conditionl expecttion given by the multivrite norml distribution. We explin towrds the end of the section tht the lter method, which restricts the moments vi conditioning, remins vlid beyond the multivrite norml distributions. 4.2 Voltility nd Dispersion Proposition 4 documents tht the reltionship between the correltion coe cients nd depends only on the sign of the informtion externlity s, but not on the strength of the prmeters r nd s. We cn therefore focus our ttention on the vrince of the individul ction nd how it vries with the strength of the interction s mesured by the correltion coe cients ( ; ). Proposition 5 (Vrince of Individul Action) 1. If the gme displys strtegic complements, r > 0; then: (i) is incresing in nd j j; (ii) the mximl is obtined t = j j = 1: 22