a general model of information sharing in oligopoly · theoretical research on information sharing...

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Journal of Economic Theory � ET2191

journal of economic theory 71, 260�288 (1996)

A General Model of Information Sharing in Oligopoly

Michael Raith*

ECARE, Universite� Libre de Bruxelles, 39, Avenue Franklin Roosevelt,1050 Bruxelles, Belgium

Received February 3, 1994; revised November 2, 1995

Under which conditions do oligopolists have an incentive to share privateinformation about a stochastic demand or stochastic costs? We present a generalmodel which encompasses virtually all models of the existing literature on informationsharing as special cases. Within this unifying framework we show that in contrastto the apparent inconclusiveness of previous results some simple principles deter-mining the incentives to share information can be obtained. Existing results aregeneralized, some previous interpretations are questioned, and new explanationsoffered, leading to a single general theory for a large class of models. Journal ofEconomic Literature Classification Numbers C72, C73, D82, L13. � 1996 Academic

Press, Inc.

1. INTRODUCTION

Theoretical research on information sharing in oligopoly was pioneeredby Novshek and Sonnenschein [13], Clarke [2], and Vives [21]. Sincethen, numerous contributions on this topic have appeared. While themodels analyzed vary along several dimensions, their basic structure is thesame. According to the received view on the current state of this field, thereis no general theory regarding the incentives of firms to share private infor-mation; rather, the results of the models depend delicately on the specificassumptions.

article no. 0117

2600022-0531�96 �18.00Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

* I would particularly like to thank John Sutton, an anonymous referee, and an associateeditor for valuable comments and suggestions. Previous versions have benefitted fromcomments by and discussions with Frank Bickenbach, Rembert Birkfeld, Patrick Bolton,Clemens Esser, Guido Friebel, Benny Moldovanu, Georg No� ldeke, Urs Schweizer, AvnerShaked, Mark Spoerer, and seminar participants in Bonn, Louvain-la-Neuve, Jerusalem,Tel-Aviv, and Chania (Crete). All remaining errors are my own. Finally, financial support bythe Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 303, University of Bonn), theDeutsche Akademische Austauschdienst, and the Suntory-Toyota International Centre forEconomics and Related Disciplines (at the London School of Economics) is gratefullyacknowledged.


In this paper we analyze a general model of information sharing inoligopoly. The model is constructed so as to encompass virtually all modelsin the current literature as special cases, resulting from appropriatespecification of the parameters. The analysis of the model not only leads toresults more general than previous ones; more importantly, a small numberof forces driving the incentives to share information in most types ofmodels can be identified. It is argued that previous interpretations ofinformation sharing models are not always consistent with the formalanalyses, and we suggest a new explanation for the incentives to revealinformation.

The structure of our model is the same as that used in previous works:(i) In an n-firm oligopoly with differentiated goods, firms face either astochastic intercept of a linear demand function or a stochastic marginalcost, which can be different for each firm. The deviation of the vector ofdemand intercepts�costs from its mean, the ``State of Nature,'' is unknownto the firms. (ii) Instead, each firm receives a private signal with informa-tion about the true State of Nature. For example, firms might receive noisysignals about the intercept of a common demand function, or they mightknow their own costs exactly, but not the costs of the rival firms. (iii)Private information can be exchanged, where we assume that firms committhemselves either to reveal their private information to other firms or tokeep it private before receiving any private information. (vi) In the laststage, the ``oligopoly game,'' firms noncooperatively set prices or quantitiesso as to maximize expected profits conditional on the available private andrevealed information.

Following the literature, we use two different approaches to analyze therevelation behavior of firms: In the simpler case, we determine under whichconditions industry-wide contracts on information sharing are profitable,by comparing the expected equilibrium profits with and without informationsharing. Alternatively, we assume that firms decide on their revelationbehavior simultaneously and independently, thus allowing for asymmetricrevelation decisions.

How, then, do prices�quantities and expected profits with and withoutinformation sharing depend on the characteristics of the market, and howdoes this affect the incentives for firms to exchange information in the firstplace?

I will not review in detail the various contributions addressing thesequestions; a brief survey can be found in Vives [23]. It has been noted byVives [23] and others that the results concerning the incentives to shareinformation seem to depend sensitively on the specific assumptions ofthe model: A change from Cournot to Bertrand, from substitutes tocomplements, from demand to cost uncertainty, or from a common value

261INFORMATION SHARING IN OLIGOPOLY


to ``private values,'' referring to an n-dimensional State of Nature for nfirms, may lead to completely different outcomes. More disturbingly,apparently similar models often lead to contrasting results. Three pointsshall illustrate that the existing literature cannot satisfactorily explain thediversity of results, or worse, that only little seems to be known about theforces driving each particular result.

(i) According to the received view, there are two main effects ofinformation sharing from the viewpoint of the firms (excluding collusion inthe price�quantity setting stage). On one hand, each firm is better informedabout the prevailing market conditions, which is presumably profitable. Onthe other hand, the homogenization of information among firms leads to achange in the correlation of the strategies. An increase in the correlation inturn is profitable for Bertrand competition but not for Cournot competi-tion. The overall profitability is then determined by the sum of these twoeffects.

Except for some special cases, however, this well-known reasoningis either inapplicable or flawed: First, if a firm is perfectly informed aboutits own cost, it is in general not true that it benefits from obtaininginformation about its rivals as well (Fried [3], Sakai [16]). Second, thechange in the correlation of strategies is itself endogeneous and not easilypredicted. We show that contrary to what is sometimes believed, it is ingeneral not true that information sharing always leads to an increase in thecorrelation, or that the correlation increases with a common value anddecreases with private values. Finally, it is also shown that the relationshipmentioned above, between a change in the correlation of strategies and itseffect on expected profits, is not quite as generally valid as is usuallyassumed.

(ii) Vives [21], Gal-Or [4], and Li [10] have shown that in aCournot oligopoly with homogeneous goods and demand uncertainty firmsdo not share information in the equilibrium of the two-stage gamedescribed above. In contrast, Fried [3], Li [10], and Shapiro [19] haveshown that in a Cournot market with uncertainty about private costs firmscompletely reveal information in the equilibrium. This contrast has beenattributed to the difference between a common value, e.g., the intercept ofa common demand function, and private values, e.g., different marginalcosts for the firms. But the results for private values in many models holdeven if the correlation of marginal costs approaches unity, althougheconomically, this situation is equivalent to a model with a common value.Therefore, this interpretation is inconsistent with the models as it suggestsa discontinuity of profits in the underlying parameters which one wouldnot expect in this class of models.

(iii) Vives [21] shows that in a duopoly with differentiated productsand demand uncertainty, a change from substitutes to complements or

262 MICHAEL RAITH


from Cournot to Bertrand yields opposite results as to the incentives toshare information. This may be attributed to a change in the slope of thereaction curves. However, in the private-values, cost-uncertainty model ofGal-Or [5] there is only a difference between Cournot and Bertrand butnot between substitutes and complements. Finally, in Sakai's [17] modelfirms always share information, regardless of whether they set prices orquantities, or whether the goods are substitutes or complements. Hencefrom these results, very little can be concluded about the role of the typeof competition and the characteristics of goods.

The results of the analysis in this paper imply that these interpretativeproblems can all be resolved.

1. We argue (prior to the formal analysis) that the distinctive charac-teristic of most so-called private-value models is not the independence of(say) costs, but the fact that firms are perfectly informed about their owncost. We therefore introduce a new distinction between independent-valuemodels and what we label ``perfect-signal'' models.

2. The correlation of strategies increases with information sharing inthe case of a common value; with independent values or perfect signals,however, the direction of change depends on the slope of the reactioncurves.

3.(a) For Cournot markets, and for Bertrand markets with demanduncertainty, there are some simple general results underlying almost allprevious results: With perfect signals or uncorrelated demands�costs, orwith a common value and strategic complements, complete informationpooling is an equilibrium of the two-stage game (which is efficient from theviewpoint of the firms), regardless of all other parameters. With a commonvalue and strategic substitutes, no pooling is the equilibrium solution. Thissolution is efficient in Cournot markets with homogeneous goods andinefficient for a large degree of product differentiation. It is shown that theprofitability of information sharing in most models with cost uncertainty isdriven merely by the assumption that firms know their own costs withcertainty, which refutes previous interpretations attributing these results toother factors.

(b) For the remaining case, Bertrand markets with cost uncertainty,the incentives are rather ambiguous. It turns out that results derived forduopoly models (Gal-Or [5]) may be reversed in the case of many firms.More importantly, this case provides a counterexample to a common beliefaccording to which, e.g., an increase in the correlation of strategies due toinformation sharing is profitable if the reaction curves are upward-sloping.

4. For the cases mentioned under 3(a), we suggest a new explana-tion for the incentives to reveal private information which rests on two



principles: (i) Letting the rivals acquire a better knowledge of their respec-tive profit functions leads to a higher correlation of strategies, the profita-bility of which is determined by the slope of the reaction curves. (ii) Lettingthe rivals acquire a better knowledge of one's own profit function alwaysincreases the own expected profits by inducing a change in the correlationof strategies in the profitable direction. The incentive to reveal informationis then determined by the sum of these two effects.

2. GENERAL MODEL

In this section, a stochastic n-firm oligopoly model with private informa-tion is introduced at its most general level. In later sections, when weanalyze particular aspects of information sharing, we will have to imposeadditional symmetry assumptions.

We first discuss the main elements of the model: the State of Nature,private information, information sharing, and strategies and payoffs.Subsequently, explicit game formulations are given.

State of Nature. The State of Nature is denoted by the random variable{=({1 , ..., {n)$, where {i is the deviation of either the marginal cost or theintercept of a linear demand function of firm i from its mean, depending onthe type of uncertainty under consideration.1 Note that for demand uncer-tainty, the intercepts may be different for each firm as well as for costuncertainty. The variables {i are normal with zero mean, variance ts , andcovariance tn # [0, ts]. For I denoting the n-dimensional unit matrix,@=(1, 1, ..., 1)$, and I� =@@$&I, the covariance matrix of { is therefore givenby tsI+tn I� =: T.

Private information. The State-of-Nature variable {i enters into firm i 'sprofit function (see below) but is unknown to i. Instead, before setting aprice or quantity, the firm��costlessly��receives a noisy signal yi about {i

as private information: yi :={i+'i . The signal noise 'i is normal with zeromean, variance uii , and covariance un # [0, mini[uii]]. Thus the covariancematrix of '=('1 , ..., 'n) is diag(u11 , ..., unn)+unI� :=U. Furthermore, weassume that { and ' are independent, which implies Cov(y)=T+U=: P.

The precision of firm i 's signal is given by u&1ii : If uii=0, firm i is

perfectly informed about {i (e.g., its own cost); a positive uii implies a noisysignal, and for uii=�, yi does not convey any information.

We follow Gal-Or [4] in allowing that the signal errors 'i be correlated.For example, publicly accessible predictions about business cycles might

264 MICHAEL RAITH

1 (i) The prime denotes transposition. (ii) For convenience, both the random variable andits realizations (hence particular States of Nature) are denoted by {.


enter into all yi inducing a correlation which has nothing to do withthe true State of Nature. Hence the private signals may be correlated(i) due to a correlation of the components of the State of Nature and(ii) due to correlation of the signal errors.2, 3 We assume throughout thatthe correlation of the signal errors does not exceed the correlation of theState-of-Nature components. This is stated more precisely in

Assumption COR. tnuii�ts un \i.Let tn �ts=: \{ and un�- uiiujj=: \ij

' (for uii , ujj>0) denote the correlationcoefficients of { and ', respectively. Then COR implies \ij

'�\{ for all i andj. If the uii are all equal, the two statements are equivalent. AssumptionCOR is automatically satisfied for all models of the literature. In thegeneral model this assumption has to be made explicitly; its significancewill become clear in the next section.

Information revelation. Firms reveal their private information completely,partially, or not at all, to all other firms by means of a signal yi :=yi+!i ,where !i is normal with zero mean and variance ri . The !i are independentof each other and of { and ', hence for r=(r1 , ..., rn)$ and y=( y1 , ..., yn)$we have Cov(!)=diag(r) and Cov(y)=T+U+diag(r)=: Q. The varianceri of the noise added to the true signal yi expresses the revelation behaviorof firm i: for ri=0, yi is completely revealed to the other firms; for ri=�a noisy signal with infinite variance is revealed, which is equivalent toconcealing private information. For 0<ri<�, private information isrevealed partially: the signal yi is distorted by the noise !i , which reducesthe informativeness of yi according to the variance ri . Note that yi cannotbe strategically distorted, since !i and yi are independent and !i has zeromean. Hence apart from random noise, private information is (if at all)revealed truthfully, or equivalently, revealed information can be verified atno cost.4

Strategies and payoffs. Finally, we turn to the market structure of themodel. Demand and cost functions are not explicit elements of the model.Instead, we directly formulate the profit functions. Each firm i controls thevariable si , which is either the price of the good produced by i (Bertrand


2 For analytical reasons we require that the covariances between the signal errors are thesame, which is a limitation of the model if the signal precisions are asymmetric. Thus we mayeither study the implications of correlated signal errors, assuming equal precisions, or analyzethe effects of asymmetric precisions, assuming uncorrelated signal errors.

3 In Gal-Or's [4] model, however, the conditional correlation of the signal errors for agiven State of Nature is nonpositive, an assumption for which Gal-Or does not provide aneconomic rationale.

4 This concept of partial revelation is due to Gal-Or [4]. A different but qualitativelyequivalent approach is used by Vives [21] and Li [10].


markets) or the quantity supplied (Cournot). The payoff for firm i is givenby

?i=:i ({i)+ :j{i

(;n+#n{i&=si) sj+(;ii+#s{i&$si) si , (2.1)

where :i ({i) is any function of {i , and ;ii , ;n , #s , #n , $ and = are parameters.We assume that $>0 and = # (&$�(n&1), $].

The parametric profit function (2.1) suits a large range of standardoligopoly models, in particular, all types discussed in the information sharingliterature (see Table I further below). This includes Cournot models witha linear demand system and linear or quadratic costs and Bertrand modelswith a linear demand system and linear costs, both for n firms producingheterogeneous goods. On the other hand, this also implies that there are noclear-cut economic interpretations of the parameters.

For all models with demand uncertainty (Cournot or Bertrand), #s

equals 1, and for Cournot models with cost uncertainty, #s equals &1. Inall these cases, #n equals zero. Hence (i) for Cournot competition, #s

indicates the source of uncertainty, and (ii) only in the case of a Bertrandmarket with cost uncertainty, #n will take a nonzero value, the importanceof which will be seen in later sections.

The linear-quadratic specification (2.1) arises from an underlying lineardemand system with a coefficient matrix of the form D=$I+=I� in both theCournot and the Bertrand case. Such a demand system can be derived asthe first-order condition of a representative consumer's maximization of anappropriately defined utility function (cf. Vives [21], Sakai�Yamato [18]),which in turn requires that the matrix D (or D&1, respectively) be positivedefinite, leading to the restriction on = and $ stated above.

From (2.1), �2?i��si �sj=&=. Hence for =>0 (e.g., Cournot withsubstitute goods or Bertrand with complements), we have a game ofstrategic substitutes, i.e., downward-sloping reaction curves, and strategiccomplements for =<0.

Game structures. We can now formulate the explicit game(s) that willbe analyzed. The model consists of the following stages:

(i) Firms decide on their revelation behavior by setting ri . We willconsider two variants: (a) firms enter into a contract specifying that infor-mation shall be revealed completely or not at all, i.e., ri=0 \i or ri=� \i;(b) firms set the ri 's simultaneously, where we exclude partial revelationbut allow asymmetric behavior, i.e., ri # [0, �] \i.

(ii) The State of Nature { is determined randomly. The playersknow the distribution of { but not its realization.

266 MICHAEL RAITH


(iii) Each firm i receives a private signal yi . The distribution of y iscommon knowledge.

(iv) yi is revealed completely, partially, or not at all, to all otherfirms by means of yi . The revelation behavior is given by ri , and r is knownto all firms.

(v) Firms play the oligopoly game, i.e., each firm i sets the price�quantity si conditional on the information zi :=( yi , y$)$ available to firm i.

Information structures. In Section 4, we will focus on some special casesof information structures to which the literature has restricted its attention.The first is the case of a Common Value, where according to the usualmodelling the State of Nature is a scalar entering into all firms' profits.Equivalently (since we are concerned with statistical decisions), we canassume that the n (identically distributed) components of the State ofNature are perfectly correlated, since then all {i are equal with probabilityone. We refer to this case as

Assumption CV (Common Value). tn=ts=: t. All other cases, in whichthe State of Nature is a nondegenerate n-vector, have been referred to asprivate-value models. However, here we will distinguish two different kindsof those models. The first is the case where the components of the State ofNature are uncorrelated:

Assumption IV (Independent Values). tn=un=0, where setting un tozero (uncorrelated signal errors) follows from COR. In fact, the work ofGal-Or [5] is the only one in which assumption IV is made. In most of theother private-value models, any correlation between the State-of-Naturecomponents is allowed for. But it is additionally assumed that firms receivesignals without noise, i.e., acquire perfect knowledge about their ``own'' {i .We refer to this case as

Assumption PS (Perfect Signals). uii=un=0 \i. Hence in this case, 'degenerates to a zero distribution. Our separation of models classified asprivate-value models in the literature into two categories has two reasons:First, it seems more appropriate to refer to a ``common value'' and ``privatevalues'' as limit cases of the correlation of the {i lying between 0 and 1rather than speak of a common value in the case of perfect correlation andof private values for any other case, including both independence and acorrelation arbitrarily close to 1. Second and more important, only bytaking the impact of signal noise into account, the apparent inconsistencypointed out in the Introduction between the results in common-valuemodels and the results in certain ``private-value'' models can beexplained: The existence or nonexistence of signal noise is the only



TABLE I

Previous Models as Special Cases of the General Model

Model n :i ({i ) = cH #n ;ii ;n T, U r

Clarke [2] n 0 $ 1 0 Equal 0 CV, \'=0 r # [0, � @]Fried [3] 2 0 $ 1 0 Diff. 0 PS ri # [0, �]Vives [21] 2 0 Any 1 0 Equal 0 CV, \'=0 ri # [0, �]Gal-Or [4] (1) n 0 $ 1 0 Equal 0 CV, un=&tn ri # [0, �]Gal-Or [4] (2) 2 0 $ 1 0 Equal 0 CV, \'�0 ri # [0, �]Li [10] (1) n 0 $ 1 0 Equal 0 CV, \'=0 ri # [0, �]Li [10] (2) n 0 $ &1 0 Equal 0 PS ri # [0, �]Gal-Or [5] (1) 2 0 Any &1 0 Equal 0 IV ri # [0, �]Gal-Or [5] (2) 2 &;ii {i Any $ = Equal 0 IV ri # [0, �]Shapiro [19] n 0 $ &1 0 Equal 0 PS r # [0, � @]Sakai [17] 2 0 Any 1 0 Diff. 0 PS ri # [0, �]Kirby [9] n 0 Any 1 0 Equal 0 CV, \'=0 ri # [0, �]Sakai�Yamato [18] n 0 Any &1 0 Equal 0 PS r # [0, � @]

remaining difference between these types of models. The role of signal noisehas not received any attention in previous work.

Almost all models of the literature are special cases of the modeldeveloped here, resulting by appropriately specifying the parameters.5 Thesespecifications are shown in Table I. Note in particular that all models belongto one of the three classes, CV, IV, and PS, introduced above.

3. NASH EQUILIBRIUM OF THE OLIGOPOLY GAME

In this section, we derive the Bayesian Nash equilibrium of the oligopolygame. At this last stage, the revelation behavior r=(r1 , ..., rn)$ is known toall firms, and each firm i has information zi=( yi , y$)$. The Bayesian Nashequilibrium s* of this subgame is characterized by

si*(zi)=arg maxs i # Rn

E{, ' &i[?i (si , s*&i | zi)] (i=1, ..., n),

268 MICHAEL RAITH

5 (i) Li [10] and Shapiro [19] have generalized the normality assumption by allowing forany distribution (e.g., one with compact support) for which all conditional expectations areaffine functions of the given information variables. (ii) Kirby [9] has studied informationsharing agreements where nonrevealing firms are excluded from the pooled information. (iii)Hviid [6] analyzes information sharing between duopolists that are risk-averse. (iv) Shapiro[19] considers (in our notation) {i 's with different variances and the same correlation; Sakai's[17] perfect-signal duopoly model allows for arbitrary matrices D and T. (v) The model ofNovshek�Sonnenschein [13] does not fit into our framework except for the uninteresting caseof a common value and perfect signals (cf. the discussion in Clarke [2]). These are the onlyexceptions.


leading to the reaction functions

si=1

2$ _;ii+#sE({i | z i)&= :j{i

E(sj | zi)& (i=1, ..., n), (3.1)

where expectations are formed over all random variables unknown at thisstage, i.e., the State of Nature and the signal errors '&i of the rival firms.Following the usual procedure, we derive the equilibrium strategies in twosteps: First, we establish existence and uniqueness of an equilibrium withstrategies si that are affine functions of zi . In the second step, thecoefficients of these functions are computed.

Proposition 3.1. There exists a unique Nash equilibrium of theoligopoly game for given information vectors zi (i=1, ..., n). The equilibriumstrategies si (zi) are affine in zi ; i.e., for all i, there exist ai , bi # R and ci # Rn

such that si=ai+biyi+c$i y.

For the proofs of all results, see the Appendix.Having established linearity of the equilibrium strategies, we now

compute the coefficients ai , bi , ci . To evaluate the first-order conditions(3.1), we first compute the conditional expectations E({i | zi) and E( yj | zi).

Let pii :=ts+uii \i and pn :=tn+un denote the variances and covariancesof the signals yi , respectively. Furthermore, define mi :=( pii& pn+ri)

&1

and m :=(m1 , ..., mn)$. Finally, let ei denote the i th unit vector.

Proposition 3.2. For given zi , the conditional expectations for {i and yj are

E({i | zi)=gi yi+g$i y and E(sj | zi)=hij yi+h� $ij y ( j{i),

where

gi= t�p, gi=(tnpii&tspn)(m&miei) p&1,

hij= pnrjmj p&1, h� ij=( pjj& pn)mjej+hij ( pii& pn)(m&miei),

and

ti=ts+ pn(ts&tn) :j{i

mj , pi= pii+ pn( pii& pn) :j{i

mj .

Setting ri=� for all i implies m=0 and gi=h� ij=0 for all i. Thus no useis made of the revealed signals y, which is equivalent to a situation withoutinformation sharing.6


6 For finite variances of ri or uii we sometimes form limits to apply expressions of the kindof Proposition 3.2. For example, ``for ri=�, ri mi=1'' is meant in the sense thatlimr i � � ri mi=1.


The expression for gi makes the significance of assumption COR clear:since tn pii&tspn=tnuii&tsun , COR implies that the components of gi arenonnegative, which in turn ensures that a correlation of yi and yj isattributed to a correlation of {i and {j rather than to a correlation of thesignal errors.

Substituting E(sj | zi)=aj+bjE( yj | zi)+c$j y and the expressions fromProposition 3.2 in (3.1) yields

si (zi)=1

2$ _\;ii&= :j{i

aj++\#s gi&= :j{i

bj hij+ yi

+\#s g$i&= :j{i

bjh� $ij+c$j + y& . (3.2)

On the other hand, si=ai+bi yi+c$i y. Identification of these coefficientswith the corresponding terms in (3.2) leads to the main result of thissection:

Proposition 3.3. In the Bayesian Nash equilibrium of the oligopolygame each firm i (i=1, ..., n) has the strategy si (zi)=ai+biyi+c$i y, where

ai=1

d� \;ii&=

d�:n

j=1

;ii+ , bi=#s

vi \ ti&=pn:n

j=1 (rimi ti �vi)1&=pn :n

j=1 (rimi�vi)+ ,

ci=2$

d� _( pii& pn) bi&=

d�:n

j=1

( pjj& pn) bj&#s(ts&tn)

d� & m

&_( pii& pn) bi&#s(ts&tn)

d� & miei ,

and

d� =2$&=, d� =2 $+(n&1) =, vi=2 $p&=pnrimi .

The equilibrium strategies of the models of other works result ascorollaries of Proposition 3.3: this applies for Clarke [2], Fried [3], Vives[21] (Propositions 2, 2a), Gal-Or [4, Theorems 1 and 2; 5, Lemmas 1 and2], Shapiro [l9], Li [10, first model, Proposition 1], Kirby [9], Sakai[17], and Sakai�Yamato [18].

An inspection of the expressions of Propositions 3.2 and 3.3 shows thatalthough firm i does not use yi for the expectations about {i or yj , yi , herstrategy si does depend on yi since it enters into E(sj | zi).

The strategies for the situation without information sharing follow as alimit case from Proposition 3.3 by setting ri=� for all i, which impliesci=0 for all i.

270 MICHAEL RAITH


Using Proposition 3.3 we can derive the expected profits for the equi-librium of the oligopoly game, where expectations are formed for unknownzi (i.e., before firms receive private information) but known revelationbehavior r, for simplicity denoted E(?(s)).

Proposition 3.4. In the equilibrium given by Proposition 3.3, the expectedprofit for firm i is

E(?i (s))=E(:i ({i))+$a2i +;n :

j{i

aj+$ Var(si)+#n :j{i

(tnbj+c$j ti). (3.3)

For Cournot markets and for Bertrand markets with demand uncertainty,#n=0 and hence the last term in (3.3) vanishes. For most of the followingsections we restrict the analysis to these cases. Only Section 4.6 is devotedto the remaining case, Bertrand markets with cost uncertainty.

4. THE INCENTIVES TO SHARE INFORMATION

Several authors have noted that the incentives to share private informa-tion are largely determined by the change in the correlation of strategiesinduced by the pooling of information. However, how this correlation isactually affected in different settings has never been treated analytically.Sections 4.1 and 4.2 address this question. We then study the twoapproaches to the determination of revelation behavior introduced inSection 2: First, we analyze the incentives to completely pool information,compared with no pooling. Alternatively, we derive the equilibrium of thetwo-stage game where firms first independently decide on their revelationbehavior. A discussion in Section 4.5 draws the threads together. Finally,we turn to the case excluded for most of this paper, Bertrand markets withcost uncertainty.

For the rest of the paper we assume that ;ii=;s for all i. For mostapplications, this means that the firms have the same expected demandintercepts and marginal costs. Moreover, except for Section 4.6 we hence-forth assume that #n=0.

4.1. No-Sharing Case

As noted above, complete concealing of private information correspondsto ri=� \i, and from Proposition 3.3 we obtain (because of mi=0 andrimi=1)

bi=#sts

2 $pii&=pn � j{i vi �vjand ci=0 \i (4.1)



Without information sharing, therefore,

Var(si)= pii b2i and Cov(si , sj)=bi bj E( yiyj)= pnbibj ( j{i), (4.2)

hence for the correlation of these strategies, \ijs , we have \ij

s = pn�- pii pjj ,or if pii := ps\i: \s= pn�ps . Thus the correlation of si and sj equals thecorrelation of the private signals yi and yj . Without sharing their privatesignals, players are not able to discriminate between the underlying Stateof Nature and the signal errors; therefore the correlation of the strategiesdoes not depend on how the parameters of { and ' enter into pii and pn .

Next, we investigate how strategies and profits are influenced by theprecision and correlation of the signals. From Proposition 3.4, changes inthe information structure affect profits only inasmuch they affect Var(si).We frequently use the notation atb to denote sign(a)=sign(b).

Proposition 4.1.7 Without information sharing,

(a)�bi

�piit&#s , (b)

�E(?i)�pii

<0, (c)�bi

�pjjt#s =, (d )

�E(?i)�pjj

t=.

Both the absolute value of bi (the sign of which is determined by #s) andi 's expected profit increase with the precision of yi (parts a, b), whereasthey are decreasing (increasing) in the precision of another firm's signal forstrategic substitutes (complements) (parts c, d).

For the rest of the paper, we assume that the private signals have equalprecisions; i.e., pii= ps \i. Then (4.1) implies bi=#s ts[ ps[2$+(n&1)_=\y]]&1=: b, where \y= pn�ps is the correlation of the signals. FromE(?i (s))t psb2 we immediately obtain (without proof)

Proposition 4.2. In the completely symmetric model without informationsharing,

(a) �E(?i)��ps |\ y const.<0, i.e., for a given correlation of signals, auniform increase in the precision of the signals increases expected profits;

(b) �E(?i)��\y | ps const. t&=, i.e, for a given precision of signals, anincrease in the correlation leads to higher expected profits for strategiccomplements and to lower expected profits for strategic substitutes.

In contrast to the case of Proposition 4.1(b), no relative informationadvantages of players are involved in result (a). Hence the precision of the

272 MICHAEL RAITH

7 The shorthand notation E(?i) refers to the expected profits for equilibrium strategies. Part(a) implies Lemma 1a in Vives [21], and (b) implies Lemma 3a. From (b), Proposition 1 inFried [3] follows. Parts (c) and (d) imply parts of Lemmas 1b and 3b in Vives [21].


private signal matters absolutely as well as in relation to the signals of therival firms.

Some intuition on the well-known result (b) can be gained by consideringa Cournot market with demand uncertainty (cf. Vives [21]): For a positivesignal yi , a higher correlation of signals implies a higher probability thatthe rival firms have received a high signal as well and supply a largerquantity. Since the reaction curves are downward-sloping, this induces areduction of the own quantity si . As a result, i reacts less sensitively to yi ,which reduces the expected profit.

The result also explains parts (c) and (d) of Proposition 4.1: anexogenous increase in the precision of another player's signal (leaving thecovariance unaffected) does not necessarily per se, i.e., because of an infor-mation advantage of the other firm, lead to a change of the expected profit,but rather through the increased correlation of strategies.8 Hence theprofitability depends on the sign of =.

4.2. Complete Pooling: Correlation of Strategies

Setting ri=0 \i we obtain the case of complete information sharing: allyi are revealed without noise; all players have the same information. Withrimi=0 and m&1

i = ps& pn , Proposition 3.3 implies

b=#sts+(n&1) pn(ts&tn)�( ps& pn)

2$[ ps(n&1) pn], ci=\b&

#s

d�

ts&tn

ps& pn+\2$

d�@&ei+ .

(4.3)

Henceforth, we usually focus on the cases CV, IV, and PS introducedabove. For the case of a common value (CV), where ts=tn=t, the firms'strategies are identical and affine in the sample mean of the signals:si=a+(2 $b�d� ) @$y. In the case of perfect signals (PS), where ps=ts andpn=tn , all parameters of random variables cancel out in (4.3), as alluncertainty has vanished (cf. Shapiro [19]).

Using (4.3) we can derive the variance and covariance of the equilibriumstrategies, the sign of the correlation, and subsequently the direction ofchange with respect to the oligopoly without information pooling:

Proposition 4.3. For CV, the correlation of equilibrium strategiesalways increases if information is completely pooled. For IV and PS, thecorrelation decreases for strategic substitutes and increases for strategiccomplements.


8 Vives [21] distinguishes the correlation effect and an information advantage of the rivalfirm, both affecting expected profits negatively. However, it does not follow from his analysisthat there exists a negative information advantage effect if the correlation of the signals is heldconstant.


(In particular, with independent values and strategic substitutes, informationsharing leads to a negative correlation of previously uncorrelatedstrategies.)

Proposition 4.3 thus shows that the conjecture that the correlationincreases with a common value and decreases with private values (cf. Li[10], Gal-Or [5]) is correct for Cournot oligopolies with substitute goods,but not in general.

It is important to notice that Proposition 4.3 does not require that#n=0; i.e., it is valid for both Cournot and Bertrand, for demand and costuncertainty.

4.3. Incentives to Share I: Contractual Approach

The firms' incentives to enter into industry-wide contracts on informa-tion sharing are determined by the difference between expected profits withand without information sharing. Using 2E(?i)t2 Var(si) the results ofSection 4.1, and (A.10), we have

E(?CPi )&E(?NP

i )=$#2s { 1

d� 2 _(ts+(n&1) pnt~ )t�

p� n

+(n&1) t~ \4$2+(n&1) =2

d� 2(ts&tn)&ts+&&

pst2s

v2 = , (4.4)

where t� :=ts+(n&1) tn , p� k := ps+(k&1)pn and t~ =(ts&tn)�( ps& pn).The sign of this difference does not depend on the sign of #s . Hence at leastfor Cournot models, the source of uncertainty��demand or cost��affectsthe signs of the strategies but is irrelevant for expected profits.

Instead of treating the IV and PS cases separately, we can derive moregeneral results by taking an important similarity between these two casesinto account: In both cases, firms do not acquire any new informationabout their {i 's by the pooling of information. With perfect signals, firm ialready knows {i , whereas with uncorrelated signals, it cannot infer any-thing about {i from the other firms' signals. In the model, this is reflectedin the fact that in both cases, gi=0. Evaluation of (4.4) leads to

Proposition 4.4. If gi=0, hence in particular for IV and PS, completepooling is always profitable. For CV, pooling is profitable if and only if

4$($&=) ps&(n&1) =2( ps+npn)>0.

The expression on the l.h.s. is positive if + :==�$ is less than 2�(n+1) andnegative if + is greater than 2(- n&1)�(n&1)<1, and otherwise depends onthe magnitudes of ps and pn .

274 MICHAEL RAITH


As corollaries follow the corresponding results of Clarke [2], Fried [3,Proposition 2], Li [10, Proposition 2], Shapiro [19, Theorem 1], Sakai[17, Theorem 1], Kirby [9, Proposition 2], and Vives [21, Proposition 5].

In a common-value Cournot oligopoly with sufficiently homogeneousgoods (= close to $), complete sharing is unprofitable. In contrast, for smallpositive =��corresponding to a large degree of product differentiation, or,for quadratic costs, to quickly increasing marginal costs (cf. Kirby[9])��information sharing is profitable, as well as for negative = (strategiccomplements).

The most important consequence of Proposition 4.4 is that with perfectsignals, complete pooling is always profitable, regardless of any otherparameters of the model. This result is in sharp contrast with the inter-pretations of Fried [3], Shapiro [19], Li [10], Sakai [17] andSakai�Yamato [18], who have attributed the profitability of informationsharing to the ``private-value'' character of their models or the uncertaintyabout costs as opposed to demand uncertainty. Rather, the result iscompletely determined by the assumption that firms have perfectknowledge of their own costs, or in general, of their {i .

The proposition suggests that the unprofitability of informationexchange in a homogeneous Cournot market with uncertainty about acommon value is a rather exceptional case. Hence in general Clarke's [2]argument that observing an agreement on information sharing may betaken as a prima facie evidence for collusion does not apply.

4.4. Incentives to Share II: Noncooperative Approach

We now analyze the two-stage game in which firms simultaneouslydecide on their revelation behavior before playing the oligopoly game.While the ``noncooperativeness'' of this model structure obviously onlyrelates to the revelation decisions, leaving their commitment characterunaffected, studying the two-stage game can nevertheless yield importantinsights about the stability of information sharing arrangements. Inparticular, we will analyze under which circumstances firms have adominant revelation strategy in the sense that they commit to a certainrevelation behavior (e.g., always to reveal the own signal) regardless ofhow the other firms decide, in anticipation of the equilibrium of theoligopoly game resulting from the first-stage decisions.

In this subsection, therefore, we allow for asymmetric revelationbehavior. However, we exclude partial revelation, i.e., each firm has todecide whether to reveal completely or not at all.9


9 In analyzing the two-stage game, Vives [21] and Gal-Or [4, 5] allow for partialrevelation, whereas this is excluded by Li [10], Fried [3], and Sakai [17].


Without loss of generality we assume that the first k players(k # [0, ..., n]) reveal, whereas the last n&k players conceal theirinformation. A nonrevealing firm (for given k) has an incentive to reveal ifE(?i (s R, k+1

i , sk+1&i ))&E(?i (sN, k

i , sk&i))>0, where sR, k+1

i denotes thestrategy of a Revealing firm (increasing the number to k+1) and sN, k

i thestrategy of a Nonrevealing firm (where the number of revealing firmsremains k). If this inequality is valid for all k, i has a dominant strategy toreveal (in the sense explained above), and vice versa if the inequality isnever fulfilled (cf. Li [10]).

Setting ri = 0 for i # [1, ..., k] and ri = � for i # [k + 1, ..., n] forgiven k, we can derive the equilibrium strategies for revealing and con-cealing firms from Proposition 3.3, calculate their variances, and com-pute the expected profits. This leads to one of the main results of thissection:

Proposition 4.5. For CV and gi=0 (including IV and PS), there alwaysexists a dominant revelation strategy (in the above sense). Informationrevelation is the dominant strategy if gi=0, and for CV and strategic com-plements. For CV and strategic substitutes, nonrevelation is the dominantstrategy.

There are many corresponding results in the literature: Proposition 3 in Li[10] and Proposition 3 in Fried [3] follow as corollaries; and similarresults are provided by Vives [21], Gal-Or [4, 5], Li [10], and Sakai[17].

First of all, we observe that in all cases considered there aredominant revelation strategies. Furthermore, the result for PS complementsProposition 4.4: the results obtained by Fried [3], Li [10], and Sakai[17] are not due to cost uncertainty or ``private values'' but are determinedby the mere assumption of perfect signals.

Comparing Propositions 4.4 and 4.5, we see that for most cases, theequilibrium of the two-stage game is efficient from the point of view of thefirms. Only for CV with strategic substitutes and small = (large degree ofproduct differentiation) a Prisoner's Dilemma situation arises: completesharing is profitable but does not occur in the two-stage game (cf. Vives[21]).

This, in turn, suggests that studying exclusionary disclosure rules (i.e.,where only revealing firms have access to information revealed by others;such rules have been considered by Kirby [9] and Shapiro [19]) mightnot yield very interesting new insights, since ``quid-pro-quo-agreements''(Kirby [9]) only become interesting in Prisoner's Dilemma situationswhere firms insist on the ``quo''. For exclusionary agreements among all nfirms, of course, the results of Section 4.3 apply.

276 MICHAEL RAITH


4.5. Discussion of the Results

Excluding Bertrand markets with cost uncertainty from the analysis, wehave shown: For CV and strategic complements, and for IV and PS in anycase, complete information pooling is an efficient equilibrium of thetwo-stage game, regardless of all other parameters. For CV and strategicsubstitutes, no pooling is the equilibrium solution, which is efficient(inefficient) for a small (large) degee of product differentiation. Except forGal-Or's [5] Bertrand model with cost uncertainty, these statementssummarize all results of the literature on the incentives to share informationin symmetric models.

To explain the results of Sections 4.3 and 4.4, we start with the well-known common-value case (cf. Vives [21]). The pooling of informationhas two effects: First, each firm has better information about the prevailingmarket conditions; second, strategies are perfectly correlated. The firsteffect increases expected profits, whereas the profitability of the secondeffect depends on the slope of the reaction curves (cf. Proposition 4.2). Forstrategic complements, then, information sharing is unambiguouslyprofitable. For strategic substitutes (say, a Cournot market with substitutegoods), the correlation effect is negative. It outweighs the precision effect inthe case of fairly homogeneous goods. With more differentiated goods, incontrast, the precision effect dominates (Proposition 4.4) since there is lessintense competition, implying that the adverse effect of a higher correlationof strategies is smaller.10

In the noncooperative model, the decision to reveal only depends on the cor-relation effect, since the knowledge of the State of Nature is not influenced bythe own revelation behavior (cf. Proposition 3.2). This explains the differencebetween Propositions 4.4 and 4.5, which gives rise to a Prisoner's Dilemma.

While the distinction of a precision and a correlation effect is very usefulin explaining the common-value case, it is of little use for the understandingof the IV and PS cases, as pointed out in the Introduction: Recall that {i

denotes the component of the State of Nature which enters into firm i 'sprofit function. First, information sharing cannot improve firm i 's informa-tion on {i (cf. 4.3), and it is not clear why it would benefit from improvedinformation about the other {j as such. In fact, Fried [3] and Sakai [16]provide examples in which firms prefer never to receive any signals aboutthe rival's profit function. Second, while the change in the correlation ofstrategies is clearly important, it is endogenous and not easily predicted. Inparticular, our results show that in the IV and PS cases information sharingchanges the correlation of strategies exactly in the direction that isprofitable for the firms! This, of course, undermines the explanatory powerof the correlation effect.


10 For an alternative interpretation in the model with quadratic costs, see Kirby [9].


Therefore, we proceed to explain our results in terms of two ratherdifferent, and more general, effects in the change of strategies due to infor-mation sharing: ``direct adjustments,'' due to an improved knowledge of theown {i , and ``strategic adjustments,'' due to an improved knowledge of therival firms' information and hence their actions.11

These effects can be readily identified analytically: While the directadjustment is directly related to the magnitude of the parameters of gi , thestrategic adjustments are determined by the parameters of h� ij (cf. (3.1) andProposition 3.2).

The significance of our new distinction is immediately clear: In bothcases IV and PS, since gi=0, there are only strategic adjustments. FromProposition 4.5 we may thus conclude that for IV and PS, unilateralrevelation of information to the other firms is profitable because and aslong as this induces only strategic adjustments by the rival firms.

Our distinction also sheds new light on the common-value case: Whilestrategic adjustments always alter the correlation of strategies in the direc-tion profitable for the firms (Propositions 4.2b, 4.3, and 4.4), directadjustments always lead to a higher correlation. Thus with strategiccomplements, both adjustments are profitable, whereas with strategicsubstitutes, the negative effect of highly correlated strategies may prevail.For CV, in particular, the components of gi have their maximal value (cf.Proposition 3.3), implying maximal direct adjustments.

Turning to intermediate cases between IV and CV, in markets withstrategic substitutes firms face a trade-off: a firm has an incentive to revealits private information as long as this does not significantly improve otherfirms' knowledge of their {j , which would induce direct adjustments bythese firms and thereby lead to more intense competition (cf. Fried [3],Proposition 4]).

4.6. Bertrand Markets with Cost Uncertainty

We briefly turn to Bertrand markets with cost uncertainty. Consider thesimplest example with the demand function qi=a&$pi&= �j{i pj and arandom marginal cost ci , for simplicity with zero mean. In terms of theprofit function (2.1), #n==, which is the coefficient for the productci �j{i pj . Such terms vanish in each of the other cases we have beenconsidering but play an important role in this case. Therefore, Bertrandcompetition with cost uncertainty is structurally different from the otherthree cases.

With #n {0, the last term in (3.3) does not vanish. As a consequence, theanalysis becomes considerably more complicated, and the results are much

278 MICHAEL RAITH

11 This terminology is borrowed from Fried [3], who uses the terms ``direct adjustments''and ``counteradjustments'' in the same way but in a slightly different context.


more ambiguous, than in the other cases. Therefore, I will only summarizethe results without presenting the formal analysis in full detail. As to themethod of analysis, it suffices to analyze the last term in (3.3) underdifferent informational settings, proceeding exactly as in the previoussections, and then combine the results with the corresponding resultsderived in 4.1�4.4.

In contrast to the simple results in the previous sections, the profitabilityof industry-wide contracts on information sharing in general depends onthe magnitudes of $, =, and n. Similarly, the profitability of unilateral infor-mation revelation depends on the these parameters. Moreover, in generalthere do not even exist dominant revelation strategies.

Only in the case of independent values does a dominant revelationstrategy exist. This strategy depends on the difference between expectedprofits for unilateral revelation vs concealing, which has the same sign as&4(2$&=)&(n&1) =(4$&3=). If =>0 or n=2, to conceal information isa dominant revelation strategy. This was shown by Gal-Or [5] for theduopoly case. However, for negative = and n � �, revealing becomes adominant strategy. Thus, even when a dominant revelation strategy exists,whether this strategy involves revelation or not depends on the specificparameters. Results obtained for duopolies do not extend to larger markets.

To see why this case is so different, contrast the profit functions for aBertrand duopoly for demand uncertainty, ?i= pi (a&{i&$pi&=pj ), andfor cost uncertainty, ?i=( pi&{i)(a&$pi&=pj ). For substitutes, = isnegative. In both cases, a positive {i will affect the profit negatively.

Now fix pi as a random variable and consider how the expected profitsin both cases depend on the rival's strategy pj . For demand uncertainty, theterm &=E( pipj) enters into the expected profits. Hence, we obtain the well-known result that expected profits are increasing in the correlation of thefirms' strategies. With cost uncertainty, in contrast, we get &=E( pipj)+=E({i pj). With information sharing, the second term counterbalances thefirst, since (with #s=$>0) pi and {i are positively correlated. Consideringthe effect of this second term, therefore, it is not surprising that theprofitability of information sharing depends on the parameters of thespecific model.

As noted above, Proposition 4.3 also covers Bertrand markets with costuncertainty. The important conclusion is that while information sharingleads to the unambiguous change in the correlation of strategies stated inProposition 4.3 (depending on the information structure), only for thethree cases considered in Sections 4.4 and 4.5 is it true that an increasein the correlation of the firms' strategies is profitable for strategiccomplements, contrary to what is usually believed. In contrast toGal-Or's [5] interpretation, therefore, the nonprofitability of informationsharing in the Bertrand duopoly with cost uncertainty does not arise



because of a decrease in the correlation of strategies, but rather despite anincrease.

What is remarkable, then, is not the ambiguity observed here but thesimplicity of the results of the previous sections. This simplicity hinges ona simple relationship between expected profits and the variances andcovariances of the equilibrium strategies which does not exist in the caseconsidered here.

5. CONCLUDING REMARKS

Previous work has fostered the impression that the incentives to shareinformation delicately depend on the details of the model. In contrast, wehave shown��building on our more general results��that the results for themajority of the specific models can be summarized in a very simple way.

Our analysis suggests that some generalizing interpretations of thoseresults found in other works are invalid: (i) As we have argued in 4.5, theassertion that one major determinant encouraging firms to exchange infor-mation is an improvement of the information about market conditions isvalid only as far as information about own demand or cost is concerned,but then does not apply to models in which firms cannot improve thisinformation, viz. in independent-value or perfect-signal models (which, infact, comprise at least one half of those used in the literature). (ii) Theassertion that the other major determinant of the profitability of informa-tion sharing is the induced change in the correlation of strategies is unhelp-ful, since this change in the correlation is itself endogenous and not easilypredicted without explicit formal analysis. In particular, we have shownthat for independent-value or perfect-signal models, the correlation alwayschanges in the direction which is profitable for the firms, although thisdirection depends on the details of the profit function.

Our new alternative interpretation, which applies to all informationstructures and to all market types considered except for Bertrand marketswith cost uncertainty, rests on two separate effects which determine theincentives to reveal information: (1) Letting the rivals acquire a betterknowledge of their respective profit functions leads to a higher correlationof strategies, the profitability of which is determined by the slope of thereaction curves. (2) Letting the rivals acquire a better knowledge of one'sown profit function is always profitable.

An analysis of the welfare effects of information sharing, not included inthis paper, leads to less clear-cut results than with the incentives for firmsto share information. In many cases, the direction of change of consumersurplus and total welfare depends on the magnitudes of the parameters of

280 MICHAEL RAITH


the model. The intuitive conjecture, often found in the nonformal literature,that without collusion information sharing is socially beneficial can by andlarge be supported, as far as overall welfare is concerned. However, it failsto take into account the impact of a change in the correlation of strategieson profits and the effect on consumers' surplus. In general, producers andconsumers have conflicting interests, making a weighting of these interestsnecessary (cf. Shapiro [19]).

The understanding of the role of information in oligopoly could befurther improved by studying asymmetric information structures in moredetail. Situations where firms have differently precise private informationhave been analyzed by Clarke [2], Fried [3], and Sakai [17].

While in this paper the incentives for firms to reveal private informationhave been emphasized, two other related issues are the incentives to acquire(costly) information about the own profit functions and finally to receiveinformation about other firms. The first has been pursued by Li et al. [11],Vives [22], and recently by Hwang [7], the second by Fried [3], Sakai[16], and Jin [8].

Despite the generality of the structure used, the present model might beconsidered restrictive in some respects. It rests on linear-quadratic profitfunctions, normally distributed random variables, and various symmetryassumptions. Commitment to a revelation strategy and truthtelling areimposed by assumption. In this respect the paper is no more general thanprevious work. This is certainly a justified criticism which calls for thedevelopment of still more general frameworks which allow for an assessmentof the robustness of our results.

But while recent research has moved on to generalize certain elements ofthe earlier models, or��probably more fruitfully��to develop strategicallymore sophisticated models which build on the methodological criticismsraised against the standard models (Zvi [24] makes a step in this direc-tion), all the old questions which gave rise to this sort of research havebeen left unanswered, as forcefully argued by Vives [23]. This paper is acontribution to fill this gap; i.e., though adhering to a restrictiveframework, it does lead to a general theory for a large class of modelswhich has been the focus of research for ten years, thus providing abenchmark for departures from this framework.

APPENDIX: PROOFS

Most proofs involve very tedious but straightforward algebra which hasbeen omitted here. The details can be found in Raith [15].

Proof of Proposition 3.1. Equations (3.1) can be obtained as the first-order conditions of an appropriately defined team-decision problem in the



sense of Radner [14]. Similarly, all higher order conditions for the Nashequilibrium and the solution of the team decision problem are identical.Therefore, we immediately obtain the result by applying Theorem 5 ofRadner [14], where in particular the assumption of normal distributionsand the positive definiteness of D ensure that all assumptions of thattheorem are satisfied.12 K

Proof of Proposition 3.2.

1. The proofs of Section 3 make use of the following result: formatrices A ( p_p, nonsingular), U, V (q_p), and S (q_q) we have

(A+U$SV)&1=A&1&A&1U$S+SVA&1U$S)&1 SVA&1 (A.1)

(see Madansky [12], p. 9]). Now let a, b # R, a # Rn, and a� =(1�a1 , ..., 1�an)$. Then from (A.1) it follows that

[diag(a)+b@@$]&1=diag(a� )&b

1+b@$a�a� a� $ (A.2)

and

[aI+b@@$]&1=a&1I&(a(a+bn))&1 b@@$. (A.3)

2. Since {, ', and ! are independent, we have Var({)=Cov({, y)=Cov({, y)=T, Var(y)=Cov(y, y)=P, and Var(y)=Q. Writing T and P asrows of column vectors, T=(t1 , ..., tn) and P=(p1 , ..., pn), we obtain

E({i | ( yi , y$)$)=(ts , t$i) V( yi , y$)$

and E( yi | ( yi , y$)$)=( pn , pj$) V( yi , y$)$,

where

V=\pii

pi

pi$Q+

&1

=\ C1

&C1Q&1pi

&C1pi$Q&1

Q&1+C1Q&1pipi$Q&1+and C1=( pii&p$iQ

&1pi)&1 (cf. Theil [20], p. 17�18]). From this we

obtain

E({i | ( yi , y$)$)= gi yi+g$i y and E( yi | ( yi , y$)$)=hij yi+h� $ij y,

282 MICHAEL RAITH

12 See Basar�Ho [1] for a similar application of Radner's theorem to a duopoly model andVives [22] for the application to both oligopolies and competitive markets.


where

gi=ts&t$i Q

&1pi

pii&p$i Q&1pi

, gi=Q&1(ti&gipi),

hij=pn&p$j Q

&1pi

pii&p$i Q&1pi

, h� ij=Q&1(pj&hijpi).

With m as defined in the main text it follows thatQ=diag(m&1

1 , ..., m&1n )+ pn @@$, and hence from (A.2), Q&1=diag(m)&

(1+ pn �nj=1 mj )

&1 pn mm$. Writing ti=tn @+(ts&tn)ei etc., straight-for-ward calculations then lead to the expressions stated in the proposition. K


1. From (3.2), 2 $ai=;ii&= � j{i aj or (2$&=) ai=;ii&= �nj=1 aj \i,

from which the expression for :i follows in an obvious way.

2. From (3.2), 2$bi=#sgi&= �j{i bj hij \i, which can be written as

b1 g1\2$

=h21

b

=hn1

=h12

2$

} } }

} } }

. . .} } }

=h1n

b

b

2$ + \ b +=#s \ b + . (A.4)

bn gn

Define %=( t1 , ..., tn) and m~ $=(r1m1 , ..., rnmn). Since gi= ti � pi andhij= pnrjmj� pi , it follows that

[diag(v)+=pn @m~ $]b=#s% or b=#s[diag(v)+=pn @m~ $]&1 %

Using (A.1) we obtain

[diag(v)+=pn @m~ $]&1=\I&=pn

1+=pnm~ $v�v� m~ + diag(v� ),

where v� :=(v&11 , ..., v&1

n ). The i th row of I&=pnv� m~ $�(1+=pnm~ $v� ) ise$i&=pnv� im~ $�(1+=pnm$v� ), which leads to the expression stated in theproposition.

3. From (3.2),

2$ci+= :j{i

cj=#s gi&= :j{i

bj h� ij \i. (A.5)



Now define C :=(c1 , ..., cn) and

Z :=\#s g1&= :j{1

bjh� 1j , ..., #s gn&= :j{n

bjh� nj+ .

Then the system (A.5) can be written as C(d� I+=I� )=Z. Using (A.3) weobtain

ci=Z1

d� \ei&=

d�@+=

1

d� _#s gi&= :j{i

bj h� ij&=

d�:n

j=1 \#s gj&= :k{j

bk h� jk+& .

(A.6)

Substituting the expressions for gi and h� ij given in Proposition 3.2 even-tually yields the result given in the proposition. K

Proof of Proposition 3.4. According to (2.1), firm i 's profit is

?i=:i ({i)+(;n+#n {i) :j{i

sj+_;ii+#s{i&= :j{i

sj&$si& si . (A.7)

Since i knows zi when she determines si , E(?i (s)) equals

E(:i ({i))+E _(;n+#n{i) :j{i

sj&+Ey i , y

__E \;ii+#s{i&= :j{i

sj&$si } zi+ si& . (A.8)

By (3.1), the last term in (A.8) reduces to $E(s2i ), since si is an equilibrium

strategy, and E(s2i )=E2(si)+Var(si)=a2

i +Var(si). For the second term in(A.8) we get

E _(;n+#n{i) :j{i

sj&= :j{i

(;naj+#nbj tn+#n c$j ti). K

Proof of Proposition 4.1. First, the restriction =>&(n&1)&1 $ impliesthat 1+=pn �n

j=1 (1�vj)>0 and hence bi t#s . From (4.1), we have

�bi

�pii=&

2 $bi

vi \1&=pn�vi

1+=pn :nj=1 (1�vj)+ .

284 MICHAEL RAITH


The two previous results then imply part (a). Part (b) follows from�E(?(s))��pii t�( piib2

i )��pii , �( piibi)��pii t&#s , pii bi tbi t#s , and part(a). Parts (c) and (d) can then be derived in a straightforward way. K

Proof of Proposition 4.3. 1. From E( y2i )=ps , E[(c$i y)2]=

E(c$i yy$ci)=c$iQci and E(byi y$ci)=bp$i ci we obtain

Var(sCPi )=E[(byi+c$i y)2]= ps b2+c$iQci+2bp$i ci . (A.9)

Using (4.3) and Q=( ps& pn) I+ pn @@$ and (4.3), (A.9) leads to

Var(sCPi )=

c2H

d� 2 _tst�

p� n+(n&1) t~ \pnt�

p� n&ts+4$2+(n&1) =2

d� 2(ts&tn)+& ,

(A.10)

where t� :=ts+(n&1) tn , p� k := ps+(k&1) pn and t~ =(ts&tn)�( ps& pn).Proceeding in the same way, we obtain a similar expression for thecovariance:

Cov(sCPi , sCP

j )=c2

H

d� 2 _tst�

p� n+(n&1) t~ \pnt�

p� n&tn+&

4$2+(n&1) =2

d� 2(ts&tn) t~ & , (A.11)

2. Let \s=Cov(si , sj)�Var(si) denote the correlation of equilibriumstrategies, where \CP

s and \NPs refer to the complete-pooling and no-pooling

cases, respectively.

CV: Trivial since for complete sharing the strategies are identical.

IV: The result follows immediately from Cov(sCPi , sCP

j )t= and\NP

s =0.

PS: With \CPs &\NP

s tps Cov(si , sj)& pn Var(si), which is using(A.10) and (A.11) can be shown to be t&=. K


1. CV: From (4.4) we obtain

E(?CPi )&E(?NP

i )t4$($&=)&(n&1) =2&n(n&1) =2 pn

ps.

Since pn�ps # [0, 1], the last expression is necessarily positive if4$($&=)&(n&1) =2>n(n&1) =2 and negative if 4$($&=)&(n&1) =2<0.Calculating the critical values of = for these inequalities then leads to theresult stated in the proposition.



2. gi=0: Since tnps=ts pn , we obtain from (4.4)

E(?CPi )&E(?NP

i )=(n&1) $=2 c2H

d� 2d� 2v2(ts&tn) p� nt~ [v(d� +d� )+d� d� ps]>0. K

Proof of Proposition 4.5. 1. We first calculate for a given k thecoefficients b and ci for revealing and nonrevealing firms in the generalcase. Using R as an index for a revealing firm i (i=1, ..., k) and N as anindex for a nonrevealing firm i (i=k+1, ..., n), we get from Proposition 3.3

bN=cHts+kpnt~

2$( ps+kpn)+(n&k&1) =pn, (A.12)

bR=cH

2$p� k _2$( ps+kpn)&=pn

v~ k(ts+kpnt~ )& pnt~ & (A.13)

and

ci=2$

d� \bi&=

d�:b

j=1

bj&cH

d�t~ + @k&\bi&

cH

d�t~ + eR

i , (A.14)

where v~ k :=2$( ps+kpn)+(n&k&1) =pn , @k is a vector with ones in thefirst k components and zeros in the last n&k, and eR

i =ei if i�k, andeR

i =0 otherwise.

2. CV: Equations (A.12) to (A.14) imply

sRi =a+

cH ts

d� p� k@$kyk and sN

i =a+cH ts

v~yi+

cHts

d� p� k

2$p� k&k=pn

v~@$kyk ,

where yk is the vector of signals y with zeros in the last (n&k) components.Noting that E[(@$ky)2]=@$kP@k=kp� k and @$kE(yk yi)=kpn , we can calculatethe variances and then obtain

2E(?) :=E(?i (s R, k+1i , sk+1

&i ))&E(?i (sN, ki , sk

&i))

=$[Var(sR, k+1i )&Var(sN, k

i )]t&=.

3. gi=g=0: From (A.12) to (A.14), we get

sRi =a+

cH

d�t~ yi&cH

=

d� d�p� n

p� k t~ @$kyk

286 MICHAEL RAITH


and

sNi =a+cH

p� k+1t~v~ k

yi&cH=

d� d�p� n

p� k t~ \1&d� pn

v~ k + @$kyk .

Noting that @$E( yiyk)= p� k and @$E( yiyk)=kpn , we can calculate thevariances and obtain

2E(?)t2d� d� ( p� k+1)2 [(n&1)( ps& pn)+knpn]+ p� n(d� p� k+1&v~ k)2,

where both terms are positive. K

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288 MICHAEL RAITH

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