Robust Predictions in Games with Incomplete Information

Transcription

1 Princeton University - Economic Theory Center Reserch Pper No Robust Predictions in Gmes with Incomplete Informtion Mrch 31, 2013 Dirk Bergemnn Stephen Morris

2 Cowles Foundtion for Reserch in Economics t Yle University Cowles Foundtion Discussion Pper No. 1821RRR ROBUST PREDICTIONS IN GAMES WITH INCOMPLETE INFORMATION Dirk Bergemnn nd Stephen Morris September 2011 Revised Mrch 2013 An uthor index to the working ppers in the Cowles Foundtion Discussion Pper Series is locted t: This pper cn be downloded without chrge from the Socil Science Reserch Network Electronic Pper Collection: Electronic copy vilble t:

3 Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn y Stephen Morris z Mrch 31, 2013 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. set of outcomes tht cn rise in equilibrium for some informtion structure is equl to the set of Byes correlted equilibri. 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 new optiml informtion policies vi the Byes correlted equilibri. We lso 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, Liner Best Responses, Qudrtic Pyo s. We cknowledge nncil support through NSF Grnt SES The present pper formed the bsis of the Fisher- Schultz Lecture given by Stephen Morris t the 2012 Europen Meeting of the Econometric Society in Mlg. We would like to thnk the Co-Editor, Mtt Jckson, nd two nonymous referees for their detiled comments. We bene tted from converstions with Steve Berry, Vincent Crwford, Mtthew Gentzkow, Phil Hile, Emir Kmenic, Mrc Henry, Arthur Lewbel, Lrry Smuelson, Elie Tmer, Tkshi Ui 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 The 1 Electronic copy vilble t:

4 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 bsic gme. To nlyze behvior in this setting, we lso hve to specify wht gents believe bout the pyo sttes, bout wht others believe, nd so on. Cll this the informtion structure. A stndrd incomplete informtion gme consists of combintion of bsic gme nd n informtion structure. Insted of sking wht hppens in xed incomplete informtion gme, in this pper we will chrcterize wht my hppen in given bsic gme for ny informtion structure. 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 bsic gme nd for some informtion structure. There re number of resons why this exercise is both trctble nd interesting. The informtion structure 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 informtion structure (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 bsic gme which re independent of - nd in tht sense robust to - the speci ction of the informtion structure. Conversely, if we re ble to identify mpping from bsic gmes to outcomes which does not depend on the informtion structure, then we cn lso study the inverse of the mp, seeing which bsic gme prmeters re consistent with n observed outcome. This mpping gives us frmework for prtilly identifying the bsic gme without ssumptions bout the informtion structure. Thus we cn crry out robust identi ction of the prmeters of the bsic gme. Chrcterizing the set of ll equilibri for ll informtion structures 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 informtion structure, 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 bsic gme equls the set of Byes Nsh equilibri tht could rise for ny informtion structure. The Byes correlted equilibrium chrcteriztion cn lso be used to nlyze the strtegic vlue of informtion. Economists re often interested in nlyzing which informtion structure is best for some 2 Electronic copy vilble t:

5 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 (2013), we pursue this reserch gend for generl gmes (with nite plyers, ctions nd sttes). In this pper, we exmine these issues in trctble bsic gme with continuum of plyers, symmetric qudrtic pyo functions nd normlly distributed uncertinty. Thus the best response is liner in the (expecttions of) the stte nd the verge ction in the popultion. The bsic gme is then one of "peer e ects", in which the pyo of ech gent depends on the verge ction tken by ll the gents, nd the pyo stte. Thus, the bsic gme cn ccommodte lrge number of environments rnging from the beuty contest, competitive mrkets nd networks, nd we relte these environments to the present nlysis in some detil in Section 2. We consider trctble informtion structure, consisting of noisy privte nd noisy public signl of the pyo relevnt stte. The combintion of trctble bsic gme nd informtion structure 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, lthough we sometimes note how results would extend without these ssumptions. 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 informtion structure in our bivrite clss of informtion structures. Integrting out the gents signls, we show tht ech informtion structure nd its (unique) 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 (2013), within the clss of symmetric norml distributions. Byes correlted equilibri re two dimensionl in this environment, i.e. we cn express them completely in terms of two correltion coe cients representing the correltion between: (i) the pyo stte nd the individul ction nd (ii) the individul ctions of ny pir of gents, nd thus simple two dimensionl clss of informtion structures is lrge enough to rech ll Byes correlted equilibri. Then to understnd wht is driving the structure of Byes correlted 3

6 equilibri, we nlyze the comprtive sttics of the Byes Nsh equilibrium with respect to the bivrite informtion structure. An increse in the precision of the public signl leds to substntil increse in the correltion of ction cross gents nd only to modest increse in the correltion between individul ction nd stte of the world. By contrst, n increse in the precision of the privte signl increses the correltion between ction nd stte, but t the sme time increses the dispersion cross gents. Hence, for ll but high levels of the precision, it ctully decreses the correltion in the ctions of the gents. We cn identify robust predictions in terms of restrictions on the rst nd second moments of the joint distribution over ctions nd the 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 permit 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 (i.e., the popultion ction) 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. We show how our pproch cn be used to nlyze the strtegic vlue of informtion by considering informtion shring mong rms. Clrke (1983) showed the striking result 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 nlysis contrsts two extremes: either nothing is known bout the informtion structure, 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 informtion structure. 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. The comprtive sttic results with respect to the informtion structure described bove, now provide hint t the emerging restrictions. For given correltion between ctions, n increse in the precision of the public signl renders impossible equilibri with either very low or very high correltion 4

7 between the individul ction nd the stte, wheres n increse in the precision of the privte signl renders equilibri with either very low correltion between the individul ction nd the stte impossible. We use our chrcteriztion of wht hppens in intermedite informtion structures to nlyze the robust identi ction question in depth. We re sking whether observble dt bout ctions nd sttes cn identify the structurl prmeters of the pyo functions without overly nrrow ssumptions on the informtion structure. 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 informtion structures given speci c bsic gme. With no restrictions on the informtion structure, 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. However, we lso identify conditions on the informtion structure under which we re ble to identify the sign of the interction prmeter. Given the peer e ect structure of the gme, the identi ction results lso extend the in uentil nlysis of Mnski (1993) to environments with incomplete rther thn complete informtion. The 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 informtion structures but less permissive in restricting ttention to common prior informtion structures. The reminder of the pper is orgnized s follows. Section 2 de nes the bsic gme, clss of qudrtic gmes with normlly distributed uncertinty, nd the informtion structure. We lso de ne the relevnt solution concepts, nmely Byes Nsh equilibrium nd Byes correlted equilibrium. Section 3 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. Section 4 then contrsts the nlysis of the Byes correlted equilibrium with the stndrd pproch to gmes of incomplete informtion nd nlyses the Byes Nsh equilibri under bivrite informtion structure. Here ech gent receives privte nd public signl bout the pyo stte. In Section 5 we consider the optiml shring of informtion mong rms. In Section 6 we nlyze how prior restrictions bout the informtion structure cn further restrict the equilibrium predictions. By rephrsing the choice of informtion policy s 5

8 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 Bsic Gme 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) where u (; A; ) is plyer s pyo if she chooses ction, the verge (or popultion) ction is A nd the stte is. A bsic gme is thus prmeterized by (u; ). The Byes correlted equilibrium depends on the bsic gme lone. 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 ssumption of symmetry cross plyers. 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 (A; ) 2 R. De nition 1 (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 ; (3) for ech 2 R nd 0 2 R; nd mrg = : (4) 6

9 The condition (3) 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, nlogous to the best response condition in the de nition of correlted equilibrium for complete informtion gmes in Aumnn (1987). The condition (4) 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 (2013). The de nition here is written for the prticulr gmes with continuum of plyer studied in this pper, mintining symmetry nd normlity, nd with plyers conditioning on their ctions only nd not on ny dditionl informtion Informtion Structure Strting with the bsic gme (u; ) described in the previous subsection, we cn dd description of the informtion structure, i.e., wht plyers know bout the stte nd others beliefs. The bsic gme nd the informtion structure together de ne gme of incomplete informtion. Now, 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 = + ". (5) The rndom vribles " i nd " re ssumed 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. 1 In Bergemnn nd Morris (2013) we stte the generl de nition of Byes correlted equilibrium for generl gmes (with nite plyers, ctions, nd sttes). The generl de nition llows the plyers to hve dditionl informtion bout the stte beyond the common prior, n extension we llow for in this pper strting in Section 6. 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 (2013), 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 seems contrived 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 informtion structures. 7

10 We cn now describe the stndrd pproch to nlyze gmes of incomplete informtion by mens of xed informtion structure (or type spce) nd the ssocited Byes Nsh equilibri. A symmetric pure strtegy in the gme is then de ned by s : R 2! R. De nition 2 (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 Liner Qudrtic Pyo s We restrict ttention to bsic gme 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, (6) 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) : (7) 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. (8) 1 r Moreover, under complete informtion bout the stte of the world, even the correlted equilibrium is unique; Neymn (1997) gives n elegnt rgument. 2 If r > 1, the Nsh equilibrium is unstble nd, if ctions sets were bounded, there would be multiple Nsh equilibri. 8

11 The pyo stte, or the stte of the world, is ssumed to be distributed normlly with N ; 2. (9) The present environment of liner best response nd normlly distributed uncertinty encompsses wide clss of interesting economic environments. The following pplictions re prominent exmples nd we shll return to them to illustrte some of the results. 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 informtion structure 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 inverse demnd curve s p (A) = s + ra + k; (10) with r < 0; while the cost of rm i of output i is c ( i ) = i. Individul rm pro ts re now given by i p (A) c ( i ) = (ra + s + k) i 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 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; ), i A 1 C A U U A U i U A U AA U C B A A U U A U 9 C A ; (11)

12 where the mtrix U = fu kl g represents the pyo structure of the gme. In the erlier working pper version, Bergemnn nd Morris (2013b), 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. (12) 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 (12) is equivlent to the present restriction (7). 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 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 re importnt in generl for welfre nlysis (the focus of Angeletos nd Pvn (2009)). 5 In the one welfre nlysis in this pper, these terms re nywy set equl to zero, so we set them equl to zero throughout the pper without loss of generlity for our results. 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 X j. (13) With the liner best response (6), 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. 10

13 Exmple 5 (Network Gmes) Network gmes often lso re nlyzed s noncoopertive gmes where ech plyer decides how much ction to exert s function of the weighted verge of the behvior of the other plyers (see Jckson nd Zenou (2013) for survey). Thus in Bllester, Clvo-Armengol, nd Zenou (2006), ech plyer chooses n e ort x i to mximizes his biliner pyo : u i (x 1 ; :::; x I ) = i x i iix 2 i + X ij x i x j ; j6=i nd so the best response of gent i is given by liner function: i + ii x i + X j6=i ij x j = 0, x i = i P j6=i ijx j ii. Thus if we considered the nite plyer version of our results nd llowed for symmetry, we would tie in with tht literture. Our results would then hve nlogues where the strtegic interction prmeter r ws replced with mtrix of strtegic interction prmeters. With the exception of few contributions, such s Gleotti, Goyl, Jckson, Veg-Redondo, nd Yriv (2010), this literture hs minly focused on gmes of complete informtion. By contrst, recent nd ongoing work by de Mrti nd Zenou (2011) llows the mrginl return or cost of e ort, i, to be common to ll gents, but only prtilly known by the gents. Thus, they consider model with common vlues nd privte informtion much like the present model. Their nlysis emphsizes, just s we observe in Proposition 1, tht the equilibrium behvior under either complete or incomplete informtion, is to lrge extent determined by the chrcteristics of the interction mtrix. 3 Byes Correlted Equilibrium We begin the nlysis with the chrcteriztion of the Byes correlted equilibri. We restrict ttention to symmetric nd normlly distributed correlted equilibri nd discuss the extent to which these restrictions 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 stte of the world, 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 the stte of the world nd n rbitrry pir of individul ctions, i nd j. The rst pproch puts more emphsis on the distributionl properties of 11

14 the correlted equilibrium, nd is convenient when we go beyond symmetric nd normlly distributed equilibri, wheres the second pproch is closer to the (subsequent) description of the Byes Nsh equilibrium in terms of the speci ction of the individul ctions. 3.1 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: A A A 2 A A A A A 2 A 11 AA : In the continuum economy, we cn describe the individul ctions s centered round the verge ction A with some dispersion 2, so tht = A +, for some N 0; 2. If the joint distribution (; A; ) is multivrite norml distribution, then the distribution of the individul ction hs to hve the bove liner form, in prticulr, the dispersion 2 cnnot depend on the reliztion of A. In consequence, the joint equilibrium distribution of (; A; ) is given by: 0 A 1 C A N 00 A A A A A C A ; A A 2 A 2 CC A AA : (14) A A 2 A 2 A + 2 The nlysis of the Byes correlted equilibrium proceeds by deriving restrictions on the joint equilibrium distribution (14). Given tht we re restricting ttention to multivrite norml distribution, it is su cient to derive restrictions in terms of the rst nd second moments of the equilibrium distribution (14). 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, (15) nd (ii) the consistency condition of De nition 1, nmely tht the mrginl distribution over is equl to the common prior over, is stis ed by construction of the joint equilibrium distribution (14). The best response condition (15) 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 (15) uses the expecttion of the individul gent, it is convenient to introduce the following chnge of vrible for the equilibrium rndom vribles. By hypothesis of the 12

15 symmetric equilibrium, we hve: = A nd 2 = 2 A + 2. 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. (16) We cn therefore express the correltion coe cient between individul ctions,, s: = 2 A 2 A +, (17) 2 nd the correltion coe cient between individul ction nd the stte s: = A A. (18) 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: A C A N C A ; 2 2 CC AA : (19) 2 2 With the joint equilibrium distribution described by (19), we now use the best response property (15), to completely chrcterize the moments of the equilibrium distribution. As the best response property (15) 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 ]]. (20) By symmetry, the expected ction of ech gent is equl to expected verge ction A, nd hence we cn use (20) 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. (21) 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 chrcteriztion of the second moments of the equilibrium distribution gin uses the best response property of the individul ction, see (15). 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 13

16 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. (22) d i d i Given the multivrite norml distribution (19), 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 ; (23) 1 r E [Aj i ] = k + s 1 r (1 ) + i : (24) The optimlity of the best response property cn then be expressed, using (23) nd (24) s 1 = s + r. (25) 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. (26) The remining restrictions on the correltion coe cients nd re coming in the form of inequlities from the chnge of vribles in (16)-(18), where 2 = 2 A 2 A 2 = 2 A. (27) 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 (27) 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. Proposition 1 (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 ; (28) 14

17 2. the stndrd devition of the individul ction is: = 3. the correltion coe cients nd stisfy the inequlities: s 1 r ; nd (29) 2 nd s 0. (30) Thus the robust predictions of the liner best response model re: (i) the men of the individul ction is pinned down by the prmeters of the model, see (28) nd (ii) there is one dimensionl restriction on the remining free endogenous vribles ( ; ; ), see (29) the correltion coe cients re less stringent. Notbly, the robust predictions bout 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. formlly in Proposition 12 tht ny vlue of the interction prmeter r 2 ( In prticulr, we will rgue 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 1. The restriction on the correltion coe cients, nmely 2, emerged directly from the bove chnge of vrible, see (16)-(18). 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 (19) 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. (31) 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 3.3) nd to prior restrictions on the privte informtion of the 15

18 Figure 1: Set of Byes correlted equilibrium in terms of correltion coe cients nd j j 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. These two procedures nturlly estblish the sme equilibrium restrictions. If the ction nd stte re independent ( = 0), the vrince of the individul ction 2 hs to be equl to zero by (26), 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 []. The condition on the vrince of the individul ction, given by (26), ctully follows the sme logic s the condition on the men of the individul ction, given by (21), in the following sense. For the men, we used the lw of totl expecttion to rrive t the equlity restriction. Similrly, we could obtin the bove restriction (26) by using the lw of totl vrince nd covrince. More precisely, we could require, using the equlity (15), 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 (26). Here we chose to directly use the liner form of the conditionl 16

19 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. We conclude by brie y describing 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 (given our normliztion) 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 (28) nd (29), re un ected by the number, in prticulr the niteness, of the plyers. The only modi ction rises with the chnge of vrible, see (16)-(18), 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 (28), (29), nd the correltion coe cients nd stisfy the inequlities: 1 I 1, I 1 ; s 0. (32) It is immedite to verify tht the restrictions of the correltion structure in (32) converge towrds the one in (30) s I! 1. We observe tht the restrictions in (32) 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. 3.2 Voltility nd Dispersion Proposition 1 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 2 (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: 17

20 2. If the gme displys strtegic substitutes, r < 0, then: (i) is decresing in nd incresing in j j; (ii) the mximl is obtined t = j j 2 = min 1 r ; 1. (33) In prticulr, we nd tht s the correltion in the ctions cross individuls increses, the vrince in the ction is mpli ed in the cse of strtegic complements, but ttenuted in the cse of strtegic substitutes. An interesting impliction of the ttenution of the individul vrince is tht the mximl vrince of the individul ction my not be ttined under miniml or mximl correltion of the individul ctions but rther t n intermedite level of correltion. In prticulr, if the interction e ect r is lrge, nmely jrj > 1, then the mximl vrince is obtined with n interior solution. Of course, in the cse of strtegic complements, the positive feed-bck e ect implies tht the mximl vrince is obtined when the ctions re mximlly correlted. We hve described the Byes correlted equilibrium in terms of the triple (; A; ). representtion cn be given in terms of (; A; di erence, An equivlent A) : the stte, the verge ction A, the idiosyncrtic A. In gmes with continuum of gents, we cn interpret the conditionl distribution of the gents ction round the men A s the exct distribution of the ctions in the popultion. The idiosyncrtic di erence A describes the dispersion round the verge ction, nd the vrince of the verge ction A cn be interpreted s the voltility of the gme. The dispersion, A, mesures how much the individul ction cn devite from the verge ction, yet be justi ed consistently with the conditionl expecttion of ech gent in equilibrium. The lnguge for voltility nd dispersion in the context of this environment ws erlier suggested by Angeletos nd Pvn (2007). The dispersion is described by the vrince of A, which is given by (1 ) 2 wheres the ggregte voltility is given by 2 A = 2. Proposition 3 (Voltility nd Dispersion) 1. The voltility is incresing in j j, nd incresing in if nd only if r 1= ; 2. The dispersion is incresing in j j nd reches n interior mximum t: The dispersion, = 2 = 1 2 r. A, mesures how much the individul ction cn devite from the verge ction. The mximl level of dispersion occurs when the correltion with respect to the stte is lrgest. But it reches its mximum t n interior level of the correltion cross the individul ctions s we might expect. We note tht reltive to the vrince of the individul ction, see Proposition 2, the voltility, is incresing in the correltion coe cient for lrger rnge of strtegic interction prmeters, including moderte strtegic substitutes. 18

21 3.3 Interdependent Vlue Environment So fr, we hve restricted our nlysis to the common vlue environment in which the stte of the world is the sme for every gent. However, the nlysis of the Byes correlted equilibrium set esily extends to model with interdependent, but not necessrily common, vlues. Here we describe suitble generliztion of the common vlue environment to n interdependent vlue environment: the pyo type of gent i is now given by i = + i, where is the common vlue component nd i is the privte vlue component. The distribution of the common component is given, s before, by N ; 2, nd the distribution of the privte component i is given by i N 0; 2. It follows tht by incresing 2 t the expense of 2, we cn move from model of pure common vlues to model of pure privte vlues, nd in between we re in cnonicl model of interdependent vlues. The nlysis of the Byes correlted equilibrium cn proceed s in Section 3.1. The erlier representtion of the Byes correlted equilibrium in terms of the vrince-covrince mtrix of the individul ction, the ggregte ction A nd the common vlue simply hs to be ugmented by distinguishing between the common vlue component nd the privte vlue component : The new correltion coe cient ;A;; = : represents the correltion between the individul ction nd the individul vlue, the privte component. The set of the Byes correlted equilibri re ected by the introduction of the privte component in systemtic mnner. The equilibrium conditions, in terms of the best response, re given by: = re [A j] + se [ + j] + k. (34) As the privte component hs zero men, it is centered round the common vlue, the privte component does not chnge the men ction in equilibrium. However, the ddition of the privte vlue component does ect the vrince nd covrince of the Byes correlted equilibri. In fct, the best response condition (34), restricts the vrince of the individul ction to: = s ( + ) ; 1 r so tht the stndrd devition of the individul ction is now composed of the weighted sum of the common nd privte vlue sources of pyo uncertinty. Finlly, the dditionl restrictions tht rise from the requirement tht the mtrix ;A;; is indeed vrince-covrince mtrix, i.e. tht it is positive 19

22 de nite mtrix, simply pper integrted in the originl conditions: 2 0; (35) In other words, to the extent tht the individul ction is correlted with the privte component, it imposes bound on how much the individul ctions cn be correlted, or 1 2. Thus to the extent tht the individul gent s ction is correlted with the privte component, it limits the extent to which the individul ction cn be relted with the public component, s by construction, the privte nd the public component re independently distributed. In Section 6, we consider the impct of prior restrictions of the informtion structures on the shpe of the equilibrium set. There, nturl restriction in the context of interdependent vlues is tht ech gent knows his own pyo stte, i = + i, but does not know the composition of his own pyo stte in terms of the public component nd the privte component i. In Bergemnn, Heumnn, nd Morris (2013) we investigte this interdependent vlue environment, which encompsses both the pure privte nd the pure common vlue model, in more detil. 3.4 Beyond Norml Distributions nd Symmetry Beyond Norml Distributions The chrcteriztion of the men nd vrince/covrince of the equilibrium distribution ws obtined under the ssumption tht the distributions of the fundmentl vrible nd resulting joint distribution ws multivrite norml distribution. Now, even if the distribution of the stte of the world is normlly distributed, the joint equilibrium distribution does not necessrily hve to be norml distribution itself. If the equilibrium distribution is not multivrite norml distribution nymore, then the rst nd second moments lone do not completely chrcterize the equilibrium distribution nymore. In other words, the rst nd second moment only impose restrictions on the higher moments, but do not completely identify the higher moments nymore. We observe however tht the restrictions regrding the rst nd second moment remin to hold. In prticulr, the result regrding the men of the ction is independent of the distribution of the equilibrium or even the normlity of the fundmentl vrible. With respect to the restrictions on the second moments, the restrictions still hold, but outside of the clss of multivrite norml distribution, the inequlities my not necessrily be chieved s equlities for some equilibrium distributions. The equilibrium chrcteriztion of the rst nd second moments could lterntively be obtined by using the lw of totl expecttion, nd its second moment equivlents, the lw of totl vrince nd covrince. These lws, insofr s they relte mrginl probbilities to conditionl probbilities, nturlly ppered in the equilibrium chrcteriztion of the best response function which introduce the conditionl expecttion over the stte nd the verge ction, nd hence the conditionl probbilities. For higher-order 20

23 moments, n elegnt generliztion of this reltionship exists, see Brillinger (1969), sometimes referred to s lw of totl cumulnce, nd s such would deliver further restrictions on higher-order moments if we were to consider equilibrium distributions beyond the norml distribution. Beyond Symmetry The chrcteriztion of the men nd vrince of the equilibrium distribution pertined to the symmetric equilibrium distribution. But ctully, the chrcteriztion remins entirely vlid for ll equilibrium distributions if we focus on the verge ction rther thn the individul ction. In ddition, the result bout the men of the individul ction remins true for ll equilibrium distributions, nd not only the symmetric equilibrium distribution. This lter result suggests tht the symmetric equilibri only o er richer set of possible second moments distributions cross gents. Interestingly, in the nite gent environment, the symmetry in the second moments does not led to joint distributions over ggregtes outcomes nd stte which cnnot be obtin lredy with symmetric equilibrium distributions. Essentilly, the symmetry vnishes in the ggregte outcome s the ggregte outcome verges over the individul best responses, ll of which re required to be blnced by the sme, symmetric, interction condition. 4 Byes Nsh Equilibrium We now contrst the nlysis of the Byes correlted equilibrium with the conventionl solution concept for gmes with incomplete informtion, the Byes Nsh equilibrium. Here we need to ugment the description of the bsic gme with speci ction of n informtion structure. We consider bivrite norml informtion structure given by privte signl x i nd public signl y for ech gent i: x i = + " i ; y = + ". (36) The respective rndom vribles " i nd " re ssumed 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, 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, the complete informtion environment, in which ech gent observes the stte of the world without noise. We begin the equilibrium nlysis with the complete informtion environment where the best response: i = ra + s + k, (37) 21