A General Model of Information Sharing in Oligopoly

Transcription

1 journal of economc theory 71, (1996) artcle no A General Model of Informaton Sharng n Olgopoly Mchael Rath* ECARE, Unverste Lbre de Bruxelles, 39, Avenue Frankln Roosevelt, 1050 Bruxelles, Belgum Receved February 3, 1994; revsed November 2, 1995 Under whch condtons do olgopolsts have an ncentve to share prvate nformaton about a stochastc demand or stochastc costs? We present a general model whch encompasses vrtually all models of the exstng lterature on nformaton sharng as specal cases. Wthn ths unfyng framework we show that n contrast to the apparent nconclusveness of prevous results some smple prncples determnng the ncentves to share nformaton can be obtaned. Exstng results are generalzed, some prevous nterpretatons are questoned, and new explanatons offered, leadng to a sngle general theory for a large class of models. Journal of Economc Lterature Classfcaton Numbers C72, C73, D82, L Academc Press, Inc. 1. INTRODUCTION Theoretcal research on nformaton sharng n olgopoly was poneered by Novshek and Sonnenschen [13], Clarke [2], and Vves [21]. Snce then, numerous contrbutons on ths topc have appeared. Whle the models analyzed vary along several dmensons, ther basc structure s the same. Accordng to the receved vew on the current state of ths feld, there s no general theory regardng the ncentves of frms to share prvate nformaton; rather, the results of the models depend delcately on the specfc assumptons. * I would partcularly lke to thank John Sutton, an anonymous referee, and an assocate edtor for valuable comments and suggestons. Prevous versons have beneftted from comments by and dscussons wth Frank Bckenbach, Rembert Brkfeld, Patrck Bolton, Clemens Esser, Gudo Frebel, Benny Moldovanu, Georg No ldeke, Urs Schwezer, Avner Shaked, Mark Spoerer, and semnar partcpants n Bonn, Louvan-la-Neuve, Jerusalem, Tel-Avv, and Chana (Crete). All remanng errors are my own. Fnally, fnancal support by the Deutsche Forschungsgemenschaft (Sonderforschungsberech 303, Unversty of Bonn), the Deutsche Akademsche Austauschdenst, and the Suntory-Toyota Internatonal Centre for Economcs and Related Dscplnes (at the London School of Economcs) s gratefully acknowledged Copyrght 1996 by Academc Press, Inc. All rghts of reproducton n any form reserved. 260

2 INFORMATION SHARING IN OLIGOPOLY 261 In ths paper we analyze a general model of nformaton sharng n olgopoly. The model s constructed so as to encompass vrtually all models n the current lterature as specal cases, resultng from approprate specfcaton of the parameters. The analyss of the model not only leads to results more general than prevous ones; more mportantly, a small number of forces drvng the ncentves to share nformaton n most types of models can be dentfed. It s argued that prevous nterpretatons of nformaton sharng models are not always consstent wth the formal analyses, and we suggest a new explanaton for the ncentves to reveal nformaton. The structure of our model s the same as that used n prevous works: () In an n-frm olgopoly wth dfferentated goods, frms face ether a stochastc ntercept of a lnear demand functon or a stochastc margnal cost, whch can be dfferent for each frm. The devaton of the vector of demand nterceptscosts from ts mean, the ``State of Nature,'' s unknown to the frms. () Instead, each frm receves a prvate sgnal wth nformaton about the true State of Nature. For example, frms mght receve nosy sgnals about the ntercept of a common demand functon, or they mght know ther own costs exactly, but not the costs of the rval frms. () Prvate nformaton can be exchanged, where we assume that frms commt themselves ether to reveal ther prvate nformaton to other frms or to keep t prvate before recevng any prvate nformaton. (v) In the last stage, the ``olgopoly game,'' frms noncooperatvely set prces or quanttes so as to maxmze expected profts condtonal on the avalable prvate and revealed nformaton. Followng the lterature, we use two dfferent approaches to analyze the revelaton behavor of frms: In the smpler case, we determne under whch condtons ndustry-wde contracts on nformaton sharng are proftable, by comparng the expected equlbrum profts wth and wthout nformaton sharng. Alternatvely, we assume that frms decde on ther revelaton behavor smultaneously and ndependently, thus allowng for asymmetrc revelaton decsons. How, then, do prcesquanttes and expected profts wth and wthout nformaton sharng depend on the characterstcs of the market, and how does ths affect the ncentves for frms to exchange nformaton n the frst place? I wll not revew n detal the varous contrbutons addressng these questons; a bref survey can be found n Vves [23]. It has been noted by Vves [23] and others that the results concernng the ncentves to share nformaton seem to depend senstvely on the specfc assumptons of the model: A change from Cournot to Bertrand, from substtutes to complements, from demand to cost uncertanty, or from a common value

3 262 MICHAEL RAITH to ``prvate values,'' referrng to an n-dmensonal State of Nature for n frms, may lead to completely dfferent outcomes. More dsturbngly, apparently smlar models often lead to contrastng results. Three ponts shall llustrate that the exstng lterature cannot satsfactorly explan the dversty of results, or worse, that only lttle seems to be known about the forces drvng each partcular result. () Accordng to the receved vew, there are two man effects of nformaton sharng from the vewpont of the frms (excludng colluson n the prcequantty settng stage). On one hand, each frm s better nformed about the prevalng market condtons, whch s presumably proftable. On the other hand, the homogenzaton of nformaton among frms leads to a change n the correlaton of the strateges. An ncrease n the correlaton n turn s proftable for Bertrand competton but not for Cournot competton. The overall proftablty s then determned by the sum of these two effects. Except for some specal cases, however, ths well-known reasonng s ether napplcable or flawed: Frst, f a frm s perfectly nformed about ts own cost, t s n general not true that t benefts from obtanng nformaton about ts rvals as well (Fred [3], Saka [16]). Second, the change n the correlaton of strateges s tself endogeneous and not easly predcted. We show that contrary to what s sometmes beleved, t s n general not true that nformaton sharng always leads to an ncrease n the correlaton, or that the correlaton ncreases wth a common value and decreases wth prvate values. Fnally, t s also shown that the relatonshp mentoned above, between a change n the correlaton of strateges and ts effect on expected profts, s not qute as generally vald as s usually assumed. () Vves [21], Gal-Or [4], and L [10] have shown that n a Cournot olgopoly wth homogeneous goods and demand uncertanty frms do not share nformaton n the equlbrum of the two-stage game descrbed above. In contrast, Fred [3], L [10], and Shapro [19] have shown that n a Cournot market wth uncertanty about prvate costs frms completely reveal nformaton n the equlbrum. Ths contrast has been attrbuted to the dfference between a common value, e.g., the ntercept of a common demand functon, and prvate values, e.g., dfferent margnal costs for the frms. But the results for prvate values n many models hold even f the correlaton of margnal costs approaches unty, although economcally, ths stuaton s equvalent to a model wth a common value. Therefore, ths nterpretaton s nconsstent wth the models as t suggests a dscontnuty of profts n the underlyng parameters whch one would not expect n ths class of models. () Vves [21] shows that n a duopoly wth dfferentated products and demand uncertanty, a change from substtutes to complements or

4 INFORMATION SHARING IN OLIGOPOLY 263 from Cournot to Bertrand yelds opposte results as to the ncentves to share nformaton. Ths may be attrbuted to a change n the slope of the reacton curves. However, n the prvate-values, cost-uncertanty model of Gal-Or [5] there s only a dfference between Cournot and Bertrand but not between substtutes and complements. Fnally, n Saka's [17] model frms always share nformaton, regardless of whether they set prces or quanttes, or whether the goods are substtutes or complements. Hence from these results, very lttle can be concluded about the role of the type of competton and the characterstcs of goods. The results of the analyss n ths paper mply that these nterpretatve problems can all be resolved. 1. We argue (pror to the formal analyss) that the dstnctve characterstc of most so-called prvate-value models s not the ndependence of (say) costs, but the fact that frms are perfectly nformed about ther own cost. We therefore ntroduce a new dstncton between ndependent-value models and what we label ``perfect-sgnal'' models. 2. The correlaton of strateges ncreases wth nformaton sharng n the case of a common value; wth ndependent values or perfect sgnals, however, the drecton of change depends on the slope of the reacton curves. 3.(a) For Cournot markets, and for Bertrand markets wth demand uncertanty, there are some smple general results underlyng almost all prevous results: Wth perfect sgnals or uncorrelated demandscosts, or wth a common value and strategc complements, complete nformaton poolng s an equlbrum of the two-stage game (whch s effcent from the vewpont of the frms), regardless of all other parameters. Wth a common value and strategc substtutes, no poolng s the equlbrum soluton. Ths soluton s effcent n Cournot markets wth homogeneous goods and neffcent for a large degree of product dfferentaton. It s shown that the proftablty of nformaton sharng n most models wth cost uncertanty s drven merely by the assumpton that frms know ther own costs wth certanty, whch refutes prevous nterpretatons attrbutng these results to other factors. (b) For the remanng case, Bertrand markets wth cost uncertanty, the ncentves are rather ambguous. It turns out that results derved for duopoly models (Gal-Or [5]) may be reversed n the case of many frms. More mportantly, ths case provdes a counterexample to a common belef accordng to whch, e.g., an ncrease n the correlaton of strateges due to nformaton sharng s proftable f the reacton curves are upward-slopng. 4. For the cases mentoned under 3(a), we suggest a new explanaton for the ncentves to reveal prvate nformaton whch rests on two

5 264 MICHAEL RAITH prncples: () Lettng the rvals acqure a better knowledge of ther respectve proft functons leads to a hgher correlaton of strateges, the proftablty of whch s determned by the slope of the reacton curves. () Lettng the rvals acqure a better knowledge of one's own proft functon always ncreases the own expected profts by nducng a change n the correlaton of strateges n the proftable drecton. The ncentve to reveal nformaton s then determned by the sum of these two effects. 2. GENERAL MODEL In ths secton, a stochastc n-frm olgopoly model wth prvate nformaton s ntroduced at ts most general level. In later sectons, when we analyze partcular aspects of nformaton sharng, we wll have to mpose addtonal symmetry assumptons. We frst dscuss the man elements of the model: the State of Nature, prvate nformaton, nformaton sharng, and strateges and payoffs. Subsequently, explct game formulatons are gven. State of Nature. The State of Nature s denoted by the random varable {=({ 1,..., { n )$, where { s the devaton of ether the margnal cost or the ntercept of a lnear demand functon of frm from ts mean, dependng on the type of uncertanty under consderaton. 1 Note that for demand uncertanty, the ntercepts may be dfferent for each frm as well as for cost uncertanty. The varables { are normal wth zero mean, varance t s, and covarance t n #[0,t s ]. For I denotng the n-dmensonal unt 1,..., 1)$, and I the covarance matrx of { s therefore gven by t s I+t n I =: T. Prvate nformaton. The State-of-Nature varable { enters nto frm 's proft functon (see below) but s unknown to. Instead, before settng a prce or quantty, the frmcostlesslyreceves a nosy sgnal y about { as prvate nformaton: y :={ +'. The sgnal nose ' s normal wth zero mean, varance u, and covarance u n # [0, mn [u ]]. Thus the covarance matrx of '=(' 1,..., ' n ) s dag(u 11,..., u nn )+u n I :=U. Furthermore, we assume that { and ' are ndependent, whch mples Cov(y)=T+U=: P. The precson of frm 's sgnal s gven by u &1 : If u =0, frm s perfectly nformed about { (e.g., ts own cost); a postve u mples a nosy sgnal, and for u =, y does not convey any nformaton. We follow Gal-Or [4] n allowng that the sgnal errors ' be correlated. For example, publcly accessble predctons about busness cycles mght 1 () The prme denotes transposton. () For convenence, both the random varable and ts realzatons (hence partcular States of Nature) are denoted by {.

6 INFORMATION SHARING IN OLIGOPOLY 265 enter nto all y nducng a correlaton whch has nothng to do wth the true State of Nature. Hence the prvate sgnals may be correlated () due to a correlaton of the components of the State of Nature and () due to correlaton of the sgnal errors. 2, 3 We assume throughout that the correlaton of the sgnal errors does not exceed the correlaton of the State-of-Nature components. Ths s stated more precsely n Assumpton COR. t n u t s u n \. Let t n t s =: \ { and u n - u u jj =: \ j ' (for u, u jj >0) denote the correlaton coeffcents of { and ', respectvely. Then COR mples \ j \ ' { for all and j. If the u are all equal, the two statements are equvalent. Assumpton COR s automatcally satsfed for all models of the lterature. In the general model ths assumpton has to be made explctly; ts sgnfcance wll become clear n the next secton. Informaton revelaton. Frms reveal ther prvate nformaton completely, partally, or not at all, to all other frms by means of a sgnal y^ :=y +!, where! s normal wth zero mean and varance r. The! are ndependent of each other and of { and ', hence for r=(r 1,..., r n )$ and y^ =(y^ 1,..., y^ n )$ we have Cov(!)=dag(r) and Cov(y^ )=T+U+dag(r)=: Q. The varance r of the nose added to the true sgnal y expresses the revelaton behavor of frm : for r =0, y s completely revealed to the other frms; for r = a nosy sgnal wth nfnte varance s revealed, whch s equvalent to concealng prvate nformaton. For 0<r <, prvate nformaton s revealed partally: the sgnal y s dstorted by the nose!, whch reduces the nformatveness of y^ accordng to the varance r. Note that y cannot be strategcally dstorted, snce! and y are ndependent and! has zero mean. Hence apart from random nose, prvate nformaton s (f at all) revealed truthfully, or equvalently, revealed nformaton can be verfed at no cost. 4 Strateges and payoffs. Fnally, we turn to the market structure of the model. Demand and cost functons are not explct elements of the model. Instead, we drectly formulate the proft functons. Each frm controls the varable s, whch s ether the prce of the good produced by (Bertrand 2 For analytcal reasons we requre that the covarances between the sgnal errors are the same, whch s a lmtaton of the model f the sgnal precsons are asymmetrc. Thus we may ether study the mplcatons of correlated sgnal errors, assumng equal precsons, or analyze the effects of asymmetrc precsons, assumng uncorrelated sgnal errors. 3 In Gal-Or's [4] model, however, the condtonal correlaton of the sgnal errors for a gven State of Nature s nonpostve, an assumpton for whch Gal-Or does not provde an economc ratonale. 4 Ths concept of partal revelaton s due to Gal-Or [4]. A dfferent but qualtatvely equvalent approach s used by Vves [21] and L [10].

7 266 MICHAEL RAITH markets) or the quantty suppled (Cournot). The payoff for frm s gven by? =: ({ )+ : (; n +# n { &=s ) s j +(; +# s { &$s ) s, (2.1) j{ where : ({ ) s any functon of {, and ;, ; n, # s, # n, $ and = are parameters. We assume that $>0 and = #(&$(n&1), $]. The parametrc proft functon (2.1) suts a large range of standard olgopoly models, n partcular, all types dscussed n the nformaton sharng lterature (see Table I further below). Ths ncludes Cournot models wth a lnear demand system and lnear or quadratc costs and Bertrand models wth a lnear demand system and lnear costs, both for n frms producng heterogeneous goods. On the other hand, ths also mples that there are no clear-cut economc nterpretatons of the parameters. For all models wth demand uncertanty (Cournot or Bertrand), # s equals 1, and for Cournot models wth cost uncertanty, # s equals &1. In all these cases, # n equals zero. Hence () for Cournot competton, # s ndcates the source of uncertanty, and () only n the case of a Bertrand market wth cost uncertanty, # n wll take a nonzero value, the mportance of whch wll be seen n later sectons. The lnear-quadratc specfcaton (2.1) arses from an underlyng lnear demand system wth a coeffcent matrx of the form D=$I+=I n both the Cournot and the Bertrand case. Such a demand system can be derved as the frst-order condton of a representatve consumer's maxmzaton of an approprately defned utlty functon (cf. Vves [21], SakaYamato [18]), whch n turn requres that the matrx D (or D &1, respectvely) be postve defnte, leadng to the restrcton on = and $ stated above. From (2.1), 2? s s j =&=. Hence for =>0 (e.g., Cournot wth substtute goods or Bertrand wth complements), we have a game of strategc substtutes,.e., downward-slopng reacton curves, and strategc complements for =<0. Game structures. We can now formulate the explct game(s) that wll be analyzed. The model conssts of the followng stages: () Frms decde on ther revelaton behavor by settng r. We wll consder two varants: (a) frms enter nto a contract specfyng that nformaton shall be revealed completely or not at all,.e., r =0 \ or r = \; (b) frms set the r 's smultaneously, where we exclude partal revelaton but allow asymmetrc behavor,.e., r # [0, ] \. () The State of Nature { s determned randomly. The players know the dstrbuton of { but not ts realzaton.

8 INFORMATION SHARING IN OLIGOPOLY 267 () Each frm receves a prvate sgnal y. The dstrbuton of y s common knowledge. (v) y s revealed completely, partally, or not at all, to all other frms by means of y^. The revelaton behavor s gven by r, and r s known to all frms. (v) Frms play the olgopoly game,.e., each frm sets the prce quantty s condtonal on the nformaton z :=( y, y^ $)$ avalable to frm. Informaton structures. In Secton 4, we wll focus on some specal cases of nformaton structures to whch the lterature has restrcted ts attenton. The frst s the case of a Common Value, where accordng to the usual modellng the State of Nature s a scalar enterng nto all frms' profts. Equvalently (snce we are concerned wth statstcal decsons), we can assume that the n (dentcally dstrbuted) components of the State of Nature are perfectly correlated, snce then all { are equal wth probablty one. We refer to ths case as Assumpton CV (Common Value). t n =t s =: t. All other cases, n whch the State of Nature s a nondegenerate n-vector, have been referred to as prvate-value models. However, here we wll dstngush two dfferent knds of those models. The frst s the case where the components of the State of Nature are uncorrelated: Assumpton IV (Independent Values). t n =u n =0, where settng u n to zero (uncorrelated sgnal errors) follows from COR. In fact, the work of Gal-Or [5] s the only one n whch assumpton IV s made. In most of the other prvate-value models, any correlaton between the State-of-Nature components s allowed for. But t s addtonally assumed that frms receve sgnals wthout nose,.e., acqure perfect knowledge about ther ``own'' {. We refer to ths case as Assumpton PS (Perfect Sgnals). u =u n =0 \. Hence n ths case, ' degenerates to a zero dstrbuton. Our separaton of models classfed as prvate-value models n the lterature nto two categores has two reasons: Frst, t seems more approprate to refer to a ``common value'' and ``prvate values'' as lmt cases of the correlaton of the { lyng between 0 and 1 rather than speak of a common value n the case of perfect correlaton and of prvate values for any other case, ncludng both ndependence and a correlaton arbtrarly close to 1. Second and more mportant, only by takng the mpact of sgnal nose nto account, the apparent nconsstency ponted out n the Introducton between the results n common-value models and the results n certan ``prvate-value'' models can be explaned: The exstence or nonexstence of sgnal nose s the only

9 268 MICHAEL RAITH TABLE I Prevous Models as Specal Cases of the General Model Model n : ({ ) = c H # n ; ; n T, U r Clarke [2] n 0 $ 1 0 Equal 0 CV, \ ' =0 r # Fred [3] 2 0 $ 1 0 Dff. 0 PS r # [0, ] Vves [21] 2 0 Any 1 0 Equal 0 CV, \ ' =0 r #[0,] Gal-Or [4] (1) n 0 $ 1 0 Equal 0 CV, u n =&t n r #[0,] Gal-Or [4] (2) 2 0 $ 1 0 Equal 0 CV, \ ' 0 r #[0,] L [10] (1) n 0 $ 1 0 Equal 0 CV, \ ' =0 r #[0,] L [10] (2) n 0 $ &1 0 Equal 0 PS r #[0,] Gal-Or [5] (1) 2 0 Any &1 0 Equal 0 IV r #[0,] Gal-Or [5] (2) 2 &; { Any $ = Equal 0 IV r #[0,] Shapro [19] n 0 $ &1 0 Equal 0 PS r # Saka [17] 2 0 Any 1 0 Dff. 0 PS r # [0, ] Krby [9] n 0 Any 1 0 Equal 0 CV, \ ' =0 r # [0, ] SakaYamato [18] n 0 Any &1 0 Equal 0 PS r # remanng dfference between these types of models. The role of sgnal nose has not receved any attenton n prevous work. Almost all models of the lterature are specal cases of the model developed here, resultng by approprately specfyng the parameters. 5 These specfcatons are shown n Table I. Note n partcular that all models belong to one of the three classes, CV, IV, and PS, ntroduced above. 3. NASH EQUILIBRIUM OF THE OLIGOPOLY GAME In ths secton, we derve the Bayesan Nash equlbrum of the olgopoly game. At ths last stage, the revelaton behavor r=(r 1,..., r n )$ s known to all frms, and each frm has nformaton z =(y, y^ $)$. The Bayesan Nash equlbrum s* of ths subgame s characterzed by s*(z )=arg max E {, ' & [? (s, s* & z )] (=1,..., n), s # R n 5 () L [10] and Shapro [19] have generalzed the normalty assumpton by allowng for any dstrbuton (e.g., one wth compact support) for whch all condtonal expectatons are affne functons of the gven nformaton varables. () Krby [9] has studed nformaton sharng agreements where nonrevealng frms are excluded from the pooled nformaton. () Hvd [6] analyzes nformaton sharng between duopolsts that are rsk-averse. (v) Shapro [19] consders (n our notaton) { 's wth dfferent varances and the same correlaton; Saka's [17] perfect-sgnal duopoly model allows for arbtrary matrces D and T. (v) The model of NovshekSonnenschen [13] does not ft nto our framework except for the unnterestng case of a common value and perfect sgnals (cf. the dscusson n Clarke [2]). These are the only exceptons.

10 INFORMATION SHARING IN OLIGOPOLY 269 leadng to the reacton functons s = 2$_ 1 ; +# s E({ z )&= : E(s j z ) & (=1,..., n), (3.1) j{ where expectatons are formed over all random varables unknown at ths stage,.e., the State of Nature and the sgnal errors ' & of the rval frms. Followng the usual procedure, we derve the equlbrum strateges n two steps: Frst, we establsh exstence and unqueness of an equlbrum wth strateges s that are affne functons of z. In the second step, the coeffcents of these functons are computed. Proposton 3.1. There exsts a unque Nash equlbrum of the olgopoly game for gven nformaton vectors z (=1,..., n). The equlbrum strateges s (z ) are affne n z ;.e., for all, there exst a, b # R and c # R n such that s =a +b y +c$ y^. For the proofs of all results, see the Appendx. Havng establshed lnearty of the equlbrum strateges, we now compute the coeffcents a, b, c. To evaluate the frst-order condtons (3.1), we frst compute the condtonal expectatons E({ z ) and E( y j z ). Let p :=t s +u \ and p n :=t n +u n denote the varances and covarances of the sgnals y, respectvely. Furthermore, defne m :=( p & p n +r ) &1 and m := (m 1,..., m n )$. Fnally, let e denote the th unt vector. Proposton 3.2. For gven z, the condtonal expectatons for { and y j are where and E({ z )=g y +g^ $ y^ and E(s j z )=h j y +h $ j y^ ( j { ), g =t^ p^, g^ =(t n p &t s p n )(m&m e ) p^ &1, &1 h j = p n r j m j p^, h j=(p jj & p n )m j e j +h j ( p & p n )(m&m e ), t^ =t s + p n (t s &t n ) : m j, p^ = p + p n ( p & p n ) : m j. j{ Settng r = for all mples m=0 and g^ = h j=0 for all. Thus no use s made of the revealed sgnals y^, whch s equvalent to a stuaton wthout nformaton sharng. 6 6 For fnte varances of r or u we sometmes form lmts to apply expressons of the knd of Proposton 3.2. For example, ``for r =, r m =1'' s meant n the sense that lm r r m =1. j{

11 270 MICHAEL RAITH The expresson for g^ makes the sgnfcance of assumpton COR clear: snce t n p &t s p n =t n u &t s u n, COR mples that the components of g^ are nonnegatve, whch n turn ensures that a correlaton of y and y j s attrbuted to a correlaton of { and { j rather than to a correlaton of the sgnal errors. Substtutng E(s j z )=a j +b j E(y j z )+c$ j y^ and the expressons from Proposton 3.2 n (3.1) yelds s (z )= 2$_\ 1 ; &= : a j{ j+ + \ # s g &= : b j h j{ j+ y + \# sg^ $ & = : j { b j h $ j +c$ j+ y^ &. (3.2) On the other hand, s =a +b y +c$ y^. Identfcaton of these coeffcents wth the correspondng terms n (3.2) leads to the man result of ths secton: Proposton 3.3. In the Bayesan Nash equlbrum of the olgopoly game each frm (=1,..., n) has the strategy s (z )=a +b y +c$ y^, where and a = 1 d \ ; & = d c = 2$ d n : j=1 _ ( p & p n ) b & = d ; +, b = # s v \ n : j=1 & _( p & p n ) b & # s(t s &t n ) d & m e, t^ : n &=p (r j=1 m t^ v ) n 1&=p n : n (r j=1 m v )+ (p jj & p n ) b j & # s(t s &t n ) d & m d =2$&=, d =2 $+(n&1) =, v =2 $p^ &=p n r m. The equlbrum strateges of the models of other works result as corollares of Proposton 3.3: ths apples for Clarke [2], Fred [3], Vves [21] (Propostons 2, 2a), Gal-Or [4, Theorems 1 and 2; 5, Lemmas 1 and 2], Shapro [l9], L [10, frst model, Proposton 1], Krby [9], Saka [17], and SakaYamato [18]. An nspecton of the expressons of Propostons 3.2 and 3.3 shows that although frm does not use y^ for the expectatons about { or y j, y^, her strategy s does depend on y^ snce t enters nto E(s j z ). The strateges for the stuaton wthout nformaton sharng follow as a lmt case from Proposton 3.3 by settng r = for all, whch mples c =0 for all.,

12 INFORMATION SHARING IN OLIGOPOLY 271 Usng Proposton 3.3 we can derve the expected profts for the equlbrum of the olgopoly game, where expectatons are formed for unknown z (.e., before frms receve prvate nformaton) but known revelaton behavor r, for smplcty denoted E(?(s)). Proposton 3.4. proft for frm s In the equlbrum gven by Proposton 3.3, the expected E(? (s))=e(: ({ ))+$a 2 +; n : a j +$ Var(s )+# n : (t n b j +c$ j t ). (3.3) j{ For Cournot markets and for Bertrand markets wth demand uncertanty, # n =0 and hence the last term n (3.3) vanshes. For most of the followng sectons we restrct the analyss to these cases. Only Secton 4.6 s devoted to the remanng case, Bertrand markets wth cost uncertanty. j{ 4. THE INCENTIVES TO SHARE INFORMATION Several authors have noted that the ncentves to share prvate nformaton are largely determned by the change n the correlaton of strateges nduced by the poolng of nformaton. However, how ths correlaton s actually affected n dfferent settngs has never been treated analytcally. Sectons 4.1 and 4.2 address ths queston. We then study the two approaches to the determnaton of revelaton behavor ntroduced n Secton 2: Frst, we analyze the ncentves to completely pool nformaton, compared wth no poolng. Alternatvely, we derve the equlbrum of the two-stage game where frms frst ndependently decde on ther revelaton behavor. A dscusson n Secton 4.5 draws the threads together. Fnally, we turn to the case excluded for most of ths paper, Bertrand markets wth cost uncertanty. For the rest of the paper we assume that ; =; s for all. For most applcatons, ths means that the frms have the same expected demand ntercepts and margnal costs. Moreover, except for Secton 4.6 we henceforth assume that # n = No-Sharng Case As noted above, complete concealng of prvate nformaton corresponds to r = \, and from Proposton 3.3 we obtan (because of m =0 and r m =1) # b = s t s and c =0 \ (4.1) 2 $p &=p n j{ v v j

13 272 MICHAEL RAITH Wthout nformaton sharng, therefore, Var(s )=p b 2 and Cov(s, s j )=b b j E(y y j )=p n b b j ( j{), (4.2) hence for the correlaton of these strateges, \ j s, we have \ j s = p n - p p jj, or f p := p s \: \ s = p n p s. Thus the correlaton of s and s j equals the correlaton of the prvate sgnals y and y j. Wthout sharng ther prvate sgnals, players are not able to dscrmnate between the underlyng State of Nature and the sgnal errors; therefore the correlaton of the strateges does not depend on how the parameters of { and ' enter nto p and p n. Next, we nvestgate how strateges and profts are nfluenced by the precson and correlaton of the sgnals. From Proposton 3.4, changes n the nformaton structure affect profts only nasmuch they affect Var(s ). We frequently use the notaton atb to denote sgn(a)=sgn(b). Proposton Wthout nformaton sharng, (a) b t&# s, (b) E(? ) <0, (c) b t# s =, (d) E(? ) t=. p p p jj Both the absolute value of b (the sgn of whch s determned by # s ) and 's expected proft ncrease wth the precson of y (parts a, b), whereas they are decreasng (ncreasng) n the precson of another frm's sgnal for strategc substtutes (complements) (parts c, d). For the rest of the paper, we assume that the prvate sgnals have equal precsons;.e., p = p s \. Then (4.1) mples b =# s t s [p s [2$+(n&1) _=\ y ]] &1 =: b, where \ y = p n p s s the correlaton of the sgnals. From E(? (s))tp s b 2 we mmedately obtan (wthout proof) Proposton 4.2. sharng, p jj In the completely symmetrc model wthout nformaton (a) E(? )p s \y const.<0,.e., for a gven correlaton of sgnals, a unform ncrease n the precson of the sgnals ncreases expected profts; (b) E(? )\ y ps const. t&=,.e, for a gven precson of sgnals, an ncrease n the correlaton leads to hgher expected profts for strategc complements and to lower expected profts for strategc substtutes. In contrast to the case of Proposton 4.1(b), no relatve nformaton advantages of players are nvolved n result (a). Hence the precson of the 7 The shorthand notaton E(? ) refers to the expected profts for equlbrum strateges. Part (a) mples Lemma 1a n Vves [21], and (b) mples Lemma 3a. From (b), Proposton 1 n Fred [3] follows. Parts (c) and (d) mply parts of Lemmas 1b and 3b n Vves [21].

14 INFORMATION SHARING IN OLIGOPOLY 273 prvate sgnal matters absolutely as well as n relaton to the sgnals of the rval frms. Some ntuton on the well-known result (b) can be ganed by consderng a Cournot market wth demand uncertanty (cf. Vves [21]): For a postve sgnal y, a hgher correlaton of sgnals mples a hgher probablty that the rval frms have receved a hgh sgnal as well and supply a larger quantty. Snce the reacton curves are downward-slopng, ths nduces a reducton of the own quantty s. As a result, reacts less senstvely to y, whch reduces the expected proft. The result also explans parts (c) and (d) of Proposton 4.1: an exogenous ncrease n the precson of another player's sgnal (leavng the covarance unaffected) does not necessarly per se,.e., because of an nformaton advantage of the other frm, lead to a change of the expected proft, but rather through the ncreased correlaton of strateges. 8 Hence the proftablty depends on the sgn of = Complete Poolng: Correlaton of Strateges Settng r =0 \ we obtan the case of complete nformaton sharng: all y are revealed wthout nose; all players have the same nformaton. Wth r m =0 and m &1 = p s & p n, Proposton 3.3 mples t s +(n&1) p n (t s &t n )(p s &p n ) s b=# s, c = 2$[p s (n&1) p n ] \b&# d t s &t n p s &p n+\ d +. (4.3) Henceforth, we usually focus on the cases CV, IV, and PS ntroduced above. For the case of a common value (CV), where t s =t n =t, the frms' strateges are dentcal and affne n the sample mean of the sgnals: s =a+(2 $bd In the case of perfect sgnals (PS), where p s =t s and p n =t n, all parameters of random varables cancel out n (4.3), as all uncertanty has vanshed (cf. Shapro [19]). Usng (4.3) we can derve the varance and covarance of the equlbrum strateges, the sgn of the correlaton, and subsequently the drecton of change wth respect to the olgopoly wthout nformaton poolng: Proposton 4.3. For CV, the correlaton of equlbrum strateges always ncreases f nformaton s completely pooled. For IV and PS, the correlaton decreases for strategc substtutes and ncreases for strategc complements. 8 Vves [21] dstngushes the correlaton effect and an nformaton advantage of the rval frm, both affectng expected profts negatvely. However, t does not follow from hs analyss that there exsts a negatve nformaton advantage effect f the correlaton of the sgnals s held constant.

15 274 MICHAEL RAITH (In partcular, wth ndependent values and strategc substtutes, nformaton sharng leads to a negatve correlaton of prevously uncorrelated strateges.) Proposton 4.3 thus shows that the conjecture that the correlaton ncreases wth a common value and decreases wth prvate values (cf. L [10], Gal-Or [5]) s correct for Cournot olgopoles wth substtute goods, but not n general. It s mportant to notce that Proposton 4.3 does not requre that # n =0;.e., t s vald for both Cournot and Bertrand, for demand and cost uncertanty Incentves to Share I: Contractual Approach The frms' ncentves to enter nto ndustry-wde contracts on nformaton sharng are determned by the dfference between expected profts wth and wthout nformaton sharng. Usng 2E(? )t2 Var(s ) the results of Secton 4.1, and (A.10), we have E(? CP )&E(? NP )=$# 2 s { 1 d 2_ (t s+(n&1) p n t~ ) t n p +(n&1) t~ \ 4$2 +(n&1) = 2 d 2 (t s &t n )&t s+& &p st 2 s 2 v^ =, (4.4) where t :=t s +(n&1) t n, p k := p s +(k&1)p n and t~ =(t s &t n )(p s &p n ). The sgn of ths dfference does not depend on the sgn of # s. Hence at least for Cournot models, the source of uncertantydemand or costaffects the sgns of the strateges but s rrelevant for expected profts. Instead of treatng the IV and PS cases separately, we can derve more general results by takng an mportant smlarty between these two cases nto account: In both cases, frms do not acqure any new nformaton about ther { 's by the poolng of nformaton. Wth perfect sgnals, frm already knows {, whereas wth uncorrelated sgnals, t cannot nfer anythng about { from the other frms' sgnals. In the model, ths s reflected n the fact that n both cases, g^ = 0. Evaluaton of (4.4) leads to Proposton 4.4. If g^ =0, hence n partcular for IV and PS, complete poolng s always proftable. For CV, poolng s proftable f and only f 4$($&=) p s &(n&1) = 2 (p s +np n )>0. The expresson on the l.h.s. s postve f + :==$ s less than 2(n+1) and negatve f + s greater than 2(- n&1)(n&1)<1, and otherwse depends on the magntudes of p s and p n.

16 INFORMATION SHARING IN OLIGOPOLY 275 As corollares follow the correspondng results of Clarke [2], Fred [3, Proposton 2], L [10, Proposton 2], Shapro [19, Theorem 1], Saka [17, Theorem 1], Krby [9, Proposton 2], and Vves [21, Proposton 5]. In a common-value Cournot olgopoly wth suffcently homogeneous goods (= close to $), complete sharng s unproftable. In contrast, for small postve =correspondng to a large degree of product dfferentaton, or, for quadratc costs, to quckly ncreasng margnal costs (cf. Krby [9])nformaton sharng s proftable, as well as for negatve = (strategc complements). The most mportant consequence of Proposton 4.4 s that wth perfect sgnals, complete poolng s always proftable, regardless of any other parameters of the model. Ths result s n sharp contrast wth the nterpretatons of Fred [3], Shapro [19], L [10], Saka [17] and SakaYamato [18], who have attrbuted the proftablty of nformaton sharng to the ``prvate-value'' character of ther models or the uncertanty about costs as opposed to demand uncertanty. Rather, the result s completely determned by the assumpton that frms have perfect knowledge of ther own costs, or n general, of ther {. The proposton suggests that the unproftablty of nformaton exchange n a homogeneous Cournot market wth uncertanty about a common value s a rather exceptonal case. Hence n general Clarke's [2] argument that observng an agreement on nformaton sharng may be taken as a prma face evdence for colluson does not apply Incentves to Share II: Noncooperatve Approach We now analyze the two-stage game n whch frms smultaneously decde on ther revelaton behavor before playng the olgopoly game. Whle the ``noncooperatveness'' of ths model structure obvously only relates to the revelaton decsons, leavng ther commtment character unaffected, studyng the two-stage game can nevertheless yeld mportant nsghts about the stablty of nformaton sharng arrangements. In partcular, we wll analyze under whch crcumstances frms have a domnant revelaton strategy n the sense that they commt to a certan revelaton behavor (e.g., always to reveal the own sgnal) regardless of how the other frms decde, n antcpaton of the equlbrum of the olgopoly game resultng from the frst-stage decsons. In ths subsecton, therefore, we allow for asymmetrc revelaton behavor. However, we exclude partal revelaton,.e., each frm has to decde whether to reveal completely or not at all. 9 9 In analyzng the two-stage game, Vves [21] and Gal-Or [4, 5] allow for partal revelaton, whereas ths s excluded by L [10], Fred [3], and Saka [17].

17 276 MICHAEL RAITH Wthout loss of generalty we assume that the frst k players (k # [0,..., n]) reveal, whereas the last n&k players conceal ther nformaton. A nonrevealng frm (for gven k) has an ncentve to reveal f R, k+1 E(? (s, s k+1 & ))&E(? (s N, k, s k k+1 &))>0, where sr, denotes the strategy of a Revealng frm (ncreasng the number to k+1) and s N, k the strategy of a Nonrevealng frm (where the number of revealng frms remans k). If ths nequalty s vald for all k, has a domnant strategy to reveal (n the sense explaned above), and vce versa f the nequalty s never fulflled (cf. L [10]). Settng r = 0 for # [1,..., k] and r = for # [k+ 1,..., n] for gven k, we can derve the equlbrum strateges for revealng and concealng frms from Proposton 3.3, calculate ther varances, and compute the expected profts. Ths leads to one of the man results of ths secton: Proposton 4.5. For CV and g^ =0 (ncludng IV and PS), there always exsts a domnant revelaton strategy (n the above sense). Informaton revelaton s the domnant strategy f g^ =0, and for CV and strategc complements. For CV and strategc substtutes, nonrevelaton s the domnant strategy. There are many correspondng results n the lterature: Proposton 3 n L [10] and Proposton 3 n Fred [3] follow as corollares; and smlar results are provded by Vves [21], Gal-Or [4, 5], L [10], and Saka [17]. Frst of all, we observe that n all cases consdered there are domnant revelaton strateges. Furthermore, the result for PS complements Proposton 4.4: the results obtaned by Fred [3], L [10], and Saka [17] are not due to cost uncertanty or ``prvate values'' but are determned by the mere assumpton of perfect sgnals. Comparng Propostons 4.4 and 4.5, we see that for most cases, the equlbrum of the two-stage game s effcent from the pont of vew of the frms. Only for CV wth strategc substtutes and small = (large degree of product dfferentaton) a Prsoner's Dlemma stuaton arses: complete sharng s proftable but does not occur n the two-stage game (cf. Vves [21]). Ths, n turn, suggests that studyng exclusonary dsclosure rules (.e., where only revealng frms have access to nformaton revealed by others; such rules have been consdered by Krby [9] and Shapro [19]) mght not yeld very nterestng new nsghts, snce ``qud-pro-quo-agreements'' (Krby [9]) only become nterestng n Prsoner's Dlemma stuatons where frms nsst on the ``quo''. For exclusonary agreements among all n frms, of course, the results of Secton 4.3 apply.

18 INFORMATION SHARING IN OLIGOPOLY Dscusson of the Results Excludng Bertrand markets wth cost uncertanty from the analyss, we have shown: For CV and strategc complements, and for IV and PS n any case, complete nformaton poolng s an effcent equlbrum of the two-stage game, regardless of all other parameters. For CV and strategc substtutes, no poolng s the equlbrum soluton, whch s effcent (neffcent) for a small (large) degee of product dfferentaton. Except for Gal-Or's [5] Bertrand model wth cost uncertanty, these statements summarze all results of the lterature on the ncentves to share nformaton n symmetrc models. To explan the results of Sectons 4.3 and 4.4, we start wth the wellknown common-value case (cf. Vves [21]). The poolng of nformaton has two effects: Frst, each frm has better nformaton about the prevalng market condtons; second, strateges are perfectly correlated. The frst effect ncreases expected profts, whereas the proftablty of the second effect depends on the slope of the reacton curves (cf. Proposton 4.2). For strategc complements, then, nformaton sharng s unambguously proftable. For strategc substtutes (say, a Cournot market wth substtute goods), the correlaton effect s negatve. It outweghs the precson effect n the case of farly homogeneous goods. Wth more dfferentated goods, n contrast, the precson effect domnates (Proposton 4.4) snce there s less ntense competton, mplyng that the adverse effect of a hgher correlaton of strateges s smaller. 10 In the noncooperatve model, the decson to reveal only depends on the correlaton effect, snce the knowledge of the State of Nature s not nfluenced by the own revelaton behavor (cf. Proposton 3.2). Ths explans the dfference between Propostons 4.4 and 4.5, whch gves rse to a Prsoner's Dlemma. Whle the dstncton of a precson and a correlaton effect s very useful n explanng the common-value case, t s of lttle use for the understandng of the IV and PS cases, as ponted out n the Introducton: Recall that { denotes the component of the State of Nature whch enters nto frm 's proft functon. Frst, nformaton sharng cannot mprove frm 's nformaton on { (cf. 4.3), and t s not clear why t would beneft from mproved nformaton about the other { j as such. In fact, Fred [3] and Saka [16] provde examples n whch frms prefer never to receve any sgnals about the rval's proft functon. Second, whle the change n the correlaton of strateges s clearly mportant, t s endogenous and not easly predcted. In partcular, our results show that n the IV and PS cases nformaton sharng changes the correlaton of strateges exactly n the drecton that s proftable for the frms! Ths, of course, undermnes the explanatory power of the correlaton effect. 10 For an alternatve nterpretaton n the model wth quadratc costs, see Krby [9].

19 278 MICHAEL RAITH Therefore, we proceed to explan our results n terms of two rather dfferent, and more general, effects n the change of strateges due to nformaton sharng: ``drect adjustments,'' due to an mproved knowledge of the own {, and ``strategc adjustments,'' due to an mproved knowledge of the rval frms' nformaton and hence ther actons. 11 These effects can be readly dentfed analytcally: Whle the drect adjustment s drectly related to the magntude of the parameters of g^, the strategc adjustments are determned by the parameters of h j (cf. (3.1) and Proposton 3.2). The sgnfcance of our new dstncton s mmedately clear: In both cases IV and PS, snce g^ =0, there are only strategc adjustments. From Proposton 4.5 we may thus conclude that for IV and PS, unlateral revelaton of nformaton to the other frms s proftable because and as long as ths nduces only strategc adjustments by the rval frms. Our dstncton also sheds new lght on the common-value case: Whle strategc adjustments always alter the correlaton of strateges n the drecton proftable for the frms (Propostons 4.2b, 4.3, and 4.4), drect adjustments always lead to a hgher correlaton. Thus wth strategc complements, both adjustments are proftable, whereas wth strategc substtutes, the negatve effect of hghly correlated strateges may preval. For CV, n partcular, the components of g^ have ther maxmal value (cf. Proposton 3.3), mplyng maxmal drect adjustments. Turnng to ntermedate cases between IV and CV, n markets wth strategc substtutes frms face a trade-off: a frm has an ncentve to reveal ts prvate nformaton as long as ths does not sgnfcantly mprove other frms' knowledge of ther { j, whch would nduce drect adjustments by these frms and thereby lead to more ntense competton (cf. Fred [3], Proposton 4]) Bertrand Markets wth Cost Uncertanty We brefly turn to Bertrand markets wth cost uncertanty. Consder the smplest example wth the demand functon q =a&$p &= j{ p j and a random margnal cost c, for smplcty wth zero mean. In terms of the proft functon (2.1), # n ==, whch s the coeffcent for the product c j{ p j. Such terms vansh n each of the other cases we have been consderng but play an mportant role n ths case. Therefore, Bertrand competton wth cost uncertanty s structurally dfferent from the other three cases. Wth # n {0, the last term n (3.3) does not vansh. As a consequence, the analyss becomes consderably more complcated, and the results are much 11 Ths termnology s borrowed from Fred [3], who uses the terms ``drect adjustments'' and ``counteradjustments'' n the same way but n a slghtly dfferent context.

20 INFORMATION SHARING IN OLIGOPOLY 279 more ambguous, than n the other cases. Therefore, I wll only summarze the results wthout presentng the formal analyss n full detal. As to the method of analyss, t suffces to analyze the last term n (3.3) under dfferent nformatonal settngs, proceedng exactly as n the prevous sectons, and then combne the results wth the correspondng results derved n In contrast to the smple results n the prevous sectons, the proftablty of ndustry-wde contracts on nformaton sharng n general depends on the magntudes of $, =, and n. Smlarly, the proftablty of unlateral nformaton revelaton depends on the these parameters. Moreover, n general there do not even exst domnant revelaton strateges. Only n the case of ndependent values does a domnant revelaton strategy exst. Ths strategy depends on the dfference between expected profts for unlateral revelaton vs concealng, whch has the same sgn as &4(2$&=)&(n&1) =(4$&3=). If =>0 or n=2, to conceal nformaton s a domnant revelaton strategy. Ths was shown by Gal-Or [5] for the duopoly case. However, for negatve = and n, revealng becomes a domnant strategy. Thus, even when a domnant revelaton strategy exsts, whether ths strategy nvolves revelaton or not depends on the specfc parameters. Results obtaned for duopoles do not extend to larger markets. To see why ths case s so dfferent, contrast the proft functons for a Bertrand duopoly for demand uncertanty,? = p (a&{ &$p &=p j ), and for cost uncertanty,? =(p &{ )(a&$p &=p j ). For substtutes, = s negatve. In both cases, a postve { wll affect the proft negatvely. Now fx p as a random varable and consder how the expected profts n both cases depend on the rval's strategy p j. For demand uncertanty, the term &=E( p p j ) enters nto the expected profts. Hence, we obtan the wellknown result that expected profts are ncreasng n the correlaton of the frms' strateges. Wth cost uncertanty, n contrast, we get &=E( p p j )+ =E({ p j ). Wth nformaton sharng, the second term counterbalances the frst, snce (wth # s =$>0) p and { are postvely correlated. Consderng the effect of ths second term, therefore, t s not surprsng that the proftablty of nformaton sharng depends on the parameters of the specfc model. As noted above, Proposton 4.3 also covers Bertrand markets wth cost uncertanty. The mportant concluson s that whle nformaton sharng leads to the unambguous change n the correlaton of strateges stated n Proposton 4.3 (dependng on the nformaton structure), only for the three cases consdered n Sectons 4.4 and 4.5 s t true that an ncrease n the correlaton of the frms' strateges s proftable for strategc complements, contrary to what s usually beleved. In contrast to Gal-Or's [5] nterpretaton, therefore, the nonproftablty of nformaton sharng n the Bertrand duopoly wth cost uncertanty does not arse

21 280 MICHAEL RAITH because of a decrease n the correlaton of strateges, but rather despte an ncrease. What s remarkable, then, s not the ambguty observed here but the smplcty of the results of the prevous sectons. Ths smplcty hnges on a smple relatonshp between expected profts and the varances and covarances of the equlbrum strateges whch does not exst n the case consdered here. 5. CONCLUDING REMARKS Prevous work has fostered the mpresson that the ncentves to share nformaton delcately depend on the detals of the model. In contrast, we have shownbuldng on our more general resultsthat the results for the majorty of the specfc models can be summarzed n a very smple way. Our analyss suggests that some generalzng nterpretatons of those results found n other works are nvald: () As we have argued n 4.5, the asserton that one major determnant encouragng frms to exchange nformaton s an mprovement of the nformaton about market condtons s vald only as far as nformaton about own demand or cost s concerned, but then does not apply to models n whch frms cannot mprove ths nformaton, vz. n ndependent-value or perfect-sgnal models (whch, n fact, comprse at least one half of those used n the lterature). () The asserton that the other major determnant of the proftablty of nformaton sharng s the nduced change n the correlaton of strateges s unhelpful, snce ths change n the correlaton s tself endogenous and not easly predcted wthout explct formal analyss. In partcular, we have shown that for ndependent-value or perfect-sgnal models, the correlaton always changes n the drecton whch s proftable for the frms, although ths drecton depends on the detals of the proft functon. Our new alternatve nterpretaton, whch apples to all nformaton structures and to all market types consdered except for Bertrand markets wth cost uncertanty, rests on two separate effects whch determne the ncentves to reveal nformaton: (1) Lettng the rvals acqure a better knowledge of ther respectve proft functons leads to a hgher correlaton of strateges, the proftablty of whch s determned by the slope of the reacton curves. (2) Lettng the rvals acqure a better knowledge of one's own proft functon s always proftable. An analyss of the welfare effects of nformaton sharng, not ncluded n ths paper, leads to less clear-cut results than wth the ncentves for frms to share nformaton. In many cases, the drecton of change of consumer surplus and total welfare depends on the magntudes of the parameters of