Collusion and Distribution of Profits Under Differential Information*

eference Trees", Psychological Re­ Collusion and distribution of profits under (1988) "Contingent Weighting in differential information* r Review 95, 371 - 384. ~ttribute Decision Rules", in: L. ,s.), Human Decision Making, Bok­ Konstantinos Serfes 1 and Nicholas C. Yannelis2

'The Effect of Attribute Ranges on '1 Department of Economics, SUNY at Stony Brook, Stony Brook, NY :l.surements", lvIanagement Science 11794-4384, U.S.A 2 Department of Economics, University of Illinois at Urbana-Champaign, 330 (1982) "Costs and Payoffs in Per­ Commerce West Building, 1206 South Sixth Street,Champaign, IL 61820, U.S.A <:tin 91, 609- 622. (1986) Decision Analysis and Be­ ity Press: Cambridge. Abstract. We examine a Cournot game with differential private informa­ "Behavioral Influences on vVeight tion. 'Ne study collusion under different information rules, i.e., when firms \flaking", European Journal of O~ pool their private information, use their common knowledge information, or decide not to share their private information at all. We put the industry nong Models of Contextually In­ profits under the three different information schemes in a hierarchy. In addi­ f Experimental Psychology: Learn­tion, we look at the incentive compatibility problem and we show that only 78. collusion under common knowledge information is incentive compatible. al Reports of Internal Events: A Finally, we deal with the issue of how the industry profits are distributed f Bern", Psychological Review 87, among the firms, in a way that asymmetries are captured. vVe propose the Shapley va"lue as a proper way to distribute the industry profits among the ased Decision Strategies: Sequels firms. We also point out that the a-core associated with the Cournot game s in the Management Sciences 6, 'with differential information is non-empty. i1) "Man- Computer Cooperation JEL Classification Number: C71, C72, D82 , IRE Transactions of the Profes­ ctronics, HFE 2, 20-26. Keywords: Collusion, differential information, Shapley value, alpha-core

1 Introduction We study the collusion of firms with differential information. A game with differential information consists of a finite number of firms, where each fum is characterized by its strategy set, its payoff function, its private informa­ tion (which is a partition of an exogeneously given probability measure space) and a prior. When firms collude, they choose an output level that maximizes joint expected profits. The information firms can use in the col­ lusive agreement varies. Firms may pool their information, may use their private information, or they may choose to use their common knowledge information. Each type of information sharing yields different profits and most importantly creates different incentives to the individual firms for

"The paper benefited from disc1lssions with Jingallg Zhao.

ot"'" 456 misreporting their true information. outcome may not be coalitional i The main focus of our paper is to address the following questions: i) that coalitions of firms, rather th: How should firms share their private information in collusive agreement not to report truthfully the reali: such that industry profits are the highest? ii) How are the collusive prof­ a sensible rule for allocating prod its under different types of information sharing compared to profits from the firms is according to the Shar non-cooperative production? iii) How should colluding firms share their pri­ rule yields individually rational a vate information in a coalitional incentive compatible way? iv) How should informational asymmetries betwe industry profit.s be distributed among firms in a way which captures t.he each firm to the total profits. V contribution or the "worth" of each firm to total profits? different.ial information game is Other work on the subject [e.g., Donsimoni et al. (1986) and Crampton that illustrate and clarify our res and Palfrey (1990)] follow an approach similar to ours by assuming that The rest of the paper is organiz firms abide by the cartel agreement. The problem of explicit collusion in an notation and definitions. In Secti industry with heterogeneous firms and private information was first con­ contains existence results. In Sec sidered formally by Roberts (1983). He derives properties of the incentive and in Section 6 we rank the indm compatibility constraints associated with a revelation game. He found that rules. Section 7 addresses the in, without side payments, if firms are sufficiently similar, then monopoly col­ the issue of profit di.stribution is i lusion cannot be achieved, but if side payments are allowed such collusion is illustrative examples. possible with a dominant strategy mechanism essentially equivalent to the Vickrey (second-price) auction. Rotemberg and Saloner (1990) investigate a price leadership scheme in a differentiated products duopoly in which the 2 Notation and definiti firms are asymmetrically informed. Crampton and Palfrey (1990) st.udy the 2.1 Notation issue of cartel enforcement when the cost of each firm is private informa­ tion. An enforceable cartel is one which is feasible, incentive compatible and ffi.l denotes the l-fold Cartesian p individually rational. They show that if defection results in eitlier Cournot ffi.~ denotes the positive cone of II or Bertrand competition, the incentive problem in large cartels is severe ffi.~+ denotes the st.rictly positive enough to prevent the cartel from achieving the monopoly outcome. Laf­ 2A denotes the set of all non-emr font and Martimort (1997) study collusion of agents whose objectives are odenotes the empty set. not aligned with that of their organization under asymmetric information. \ denotes set theoretic subtractic Our model is different from the above ones. In particular, we have a gen­ eral model and we address the issue of collusion in a differential information 2.2 Definitions game for the first time. We show that collusion under the pooled informa­ If X and Y are sets, the gmph I tion yields the highest industry profits. However, this type of information dence), ¢; : X ---7 2Y is denoted b) sharing is not coalitional incentive compatible. 1 We present examples with two firms where one firm can distinguish between two states of nature and Gq, = {(X,y) the other cannot and the firm with the "superior" information finds it prof­ itable to misreport the true state of nature to the other firm. Only collusion Let (0, F, f..L) be a complete, £lni under the common knowledge information is coalitional incentive compati­ Banach space. T he set-valued fUl ble. It is important to emphasize here that we look at the coalitional incen­ sumble graph if Gq, Q9 f3( X), wher tive compatibility and not at the individual incentive compatibility as, for and c>9 denotes the product (I-alg( example, Crampton and Palfrey (1990). An individual i:ncentive compat.ible is said to be lower measurable or V of X, the set {w E n 1 A mllnsivc agreement. i~ coalit.ional incent.ive compat.ible wben t.here does not. exist a cO

outcome may not be coalitional incentive compatible which in turn means address the following questions: i) that coalitions of firms, rather than individual firms, may have an incentive information in collusive agreement not to report truthfully the realized state of nature. We also propose that est? ii) How are the collusive prof­ a sensible rule for allocating production and distributing the profits among 1 sharing compared to profits from the firms is according to the Shapley value of each firm. The Shapley value lOuld colluding firms share their pri­ rule yields individually rational and Pareto optimal outcomes, captures the ve compatible ....vay? iv) How should informational asymmetries between the firms as well as the contribution of firms in a way which captures the each firm to the total profits. We also point out that the (X-core of the n to total profits? differential information game is non-empty. We provide several examples simoni et al. (1986) and Crampton that illustrate and clarify our results. 1 similar to ours by assuming that The rest of the paper is organized as follows. In Section 2, we provide the e problem of explicit collusion in an notation and definitions. In Section 3, we present the model and Section 4 private information was first con­ contains existence results. In Section 5, we outline three information rules derives properties of the incentive and in Section 6 we rank the industry profits under the different information h a revelation game. He found that rules. Section 7 addresses the incentive compatibility issue. In Section 8, ciently similar, then monopoly col­ the issue of profit distribution is addressed. Finally, Section 9 contains two fments are allowed such collusion is illustrative examples. ,anism essentially equivalent to the ,erg and Saloner (1990) investigate lted products duopoly in which the 2 Notation and definitions 1pton and Palfrey (1990) study the 2.1 Notation st of each firm is private informa­ 3 feasible, incentive compatible and ]Rl denotes the I-fold Cartesian product of the set of real numbers. defection results ill either Cournot ]R~ denotes the positive cone of ]RI. problem in large cartels is severe ]R~+ denotes the strictly positive elements. !Villg the monopoly outcome. Laf­ 2A denotes the set of all non-empty subsets of the set A. ion of agents whose objectives are (/) denotes the empty set. :)11 under asymmetric information. \ denotes set theoretic subtraction. ones. In particular, we have a gen­ lusion in a differential information 2.2 Definitions Illusion under the pooled informa­ If X and Y are sets, t.he graph of the set-valued function (or correspon­ However, this type of information dence), ¢ : X ---) 2Y is denoted by ~tible.l We present examples with between two states of nature and G,t> = {(x,y) E X x Y: y E ¢(x) }. uperior" information finds it prof­ e to the other firm. Only collusion Let (D, Y, fL) be a complete, finite measure space, and X be a separable x n is coalitional incentive compati­ Banach space. The set-valued function ¢ : D ---) 2 is said to have a mea­ .t we look at the coalitional incen­ surable graph if Gd,> ® (3(X), where (3(X) denotes the Borel a-algebra on X x ual incentive compatibility as, for and 150 denotes the product a-algebra. The set-valued function ¢ : D -t 2 "u individual incentive compatible is said to be lower measurable or just measurable if for every open subset V of X, the set {w ED: ¢(w) n V -::j: eI} rcompatible whell t.hen! dOCR not. cxist. x k, at,c of nat.ure and benefit. it,s m embers. is an element of F. It is well known that if ¢ : D -t 2 has a measurable graph, then ¢ is lower measurable. Furthermore, if ¢(- ) is closed valued and lower measurable then ¢ : D -t 2x has a measurable graph. A theorem 458 of Aumann tells us that if (n, F , fJ.) is a complete finite measure space, ii) 7ri : Q(w) --t lR is the random x X is a separable metric space and ¢ : n --t 2 is a non-empty valued Ql(W) x ... X Qn(w)); correspondence having a measurable graph, then ¢(- ) admits a measurable iii) Fi is a sub a-algebra of :T, v selection, i.e., there exists a measurable function f : n --t X such that few) E ¢(w),fJ. - a. e. firm i; Let (n, F, fJ.) be a finite measure space and X be a Banach space. Fol­ iv) fJ. is a probability measure 0 lowing Diestel-Uhl (1977), the function f : n --t X is called simple if there exist Xl, X2, .. . , Xn in X and 0'1 , 0'2, ... , O'n in F such that L~l XiXo:, where Let LQ, denote the set of all B Xc<, (w) = 1 if w E ct'i and Xc< ,(w) = 0 if w ~ O'i. A function f ; n --t X lections from the production set ( is said to be f.L-measurable if there exists a sequence of simple function fn : n --t X such that timn-->CX) Ilfn(w) - f(w)11 = 0 for almost all wEn. A fJ.-meas'umble function f : n --t X is said to be Bochner integrable if there exists a sequence of simple functions {jn : n = 1,2, ... } such that Let LQ = LQ l X ... x LQ,,' Giver firm i is an element qi E LQ,' The ex-ante expected profit fun as4 In this case we define for each E E F the integral to be Ili(qi, q-i) = 1 wE! r f(w)dfJ.(w) = lim r fn(w)dfJ.(w) . .JE n-->CX) } E A Cournot-Nash equilibrium fe element q* E LQ such that for all It can be shown [see Diestel-Uhl (1977), Theorem 2, p.45] that if f : n --t X is a fJ.- measurable function then f is Bochner integrable if and only if IIi(q*) = , In IIf(w)lldfJ.(W) < 00. For 1 ::; p < 00, we denote by Lp(fJ., X) the space of equivalence classes \Ve can now state the assumpt of X-valued Bochner integrable functions X : n --t X normed by Cournot-Nash equilibrium. (A.l)

It is a standard result that normed by the functional II · lip above, Lp(fJ. , X) 2 If pew) : Q(w) --> R is the inverse Ii becomes a Banach space [see Diestel-Uhl (1977), p.50]. function of firm i, then 7r;(q(w )) = p( , the payoff fnncUo n 7r to depend a/so on remain valid. 3 The Cournot game with differential information 3The entire analysis 1V0uid go t.hroL used the interim one. That is, the cand l We assume that there are n firms, {i = 1, ... , n}, that produce an output i ITi (" ') : LQi x Q-i(W) --> R is define q = {ql, ... ,qn}' The subscript -i will be used to denote all firms other ITi (qi, q-i) = 1 7r; than firm i. Let (n, F, fJ.) be a complete probability measure space. We w'EEi(w) interpret n as the states of nature of the world and assume that it is large where enough to include all events that we consider to be interesting. As usual F denotes the u-algebra of events and fJ. is a common probability measure. ki (w'IE;(w)) = { 0 Let Y be a separable Banach space denoting the production space. r;; is the prior of Io\gent i, (where k; is a R.ad. Definition 3.1: A Cournot game with differential information is a set 1 and Ei(W) denot.es the event in firm , C = {(Qi, 7fi,Fi,fJ.): i = 1, .. . ,n}, where nature). Y 4 For sim plicity we assume t.hat the i) Qi : n --t 2 is the random production set of fum i ; mentioned above all t.he results of the r 459

2 a complete finite measure space, ii) ITi : Q(w) -t IR is the random profitfu,nction of firm i, (where Q(w) = n -) 2x is a non-empty valued Ql (w) X ... X Qn(w)); ~ ph, then ¢(- ) admits a measurable iii) Fi is a sub O"-algebra of F, which denotes the private 'information of e function f : n --+ X such that firm ij ce and X be a Banach space. Fol­ iv) f-L is a probability measure on n denoting the common prior. f : n -) X is called simple if there Let LQi denote the set of all Bochner integrable and Fi-measurable se­ ~ in F such that 2:7=1 XiXQ , where if W tt O:i · A function f : n --+ X lections from the production set Qi of firm i, i.e., sts a sequence of simple function f(w)1 1= 0 for almost all wEn. A LQ. = {qi E Ll (f-L, Y) : qi : n --+ Y is Fi - measurable and d to be Bochner -integrable if there qi(W) E Qi(W) and f-L - a.e.}. ,, : n = 1, 2, ... } such that Let LQ = LQl X ... x LQn' Given a Cournot game, a production plan for fum i is an element qi E LQi . W)lldf-L(w) = O. 3 The ex-ante expected profit fu,nction of firm i, TIi : LQ --+ IR. is defined as4 Ie integral to be TIi(qi, q-i) = 1 IT i(qi (W),q-i(W ))df-L(W). wEn kfn(W)df-L(W) . A Cournot-Nash equilibrium for C = {(Qi,ITi,Fi,f-L): i = 1, ... ,n} is an element q" E LQ such that for all i, rheorem 2, p.45] that if f : n --+ X Bochner integrable if and only if TIi (q·) = max TIi(q:'i, Yi). YiELQi r) the space of equivalence classes We can now state the asstunptions needed to prove the existence of a IS x : n -) X normed by Cournot-Nash equilibritun.

1 ,II Pdf-L(w))P. (A.I) le functional II·li pabove, Lp(f-L , X) 2IJ p(w) : Q(w) --+ R is the inverse dem and Junction lind C i : Qi(W) --+ R is the cost 1 (1977), p.50]. Junction oj jirmi, then "i(q(W)) = p(q(W ))qi(W) - Ci(qi(W)). We could have allowed the payoff Junct ion IT to depend also on. the state oj nature"'). The TP-snits oj the pape'" remain uahd. ifferential informat ion 3The entire analysi~ would go through if ill~t. e[\d of the ex-allte profit function we used the interim one. That is, the cond'itional ( interim) c :Lp~cted profit Junction of firm 1, . . . ,n} , that produce an output i fIi C, ,) : Lqi x Q-i(W) --+ R is defined as be used to denote all firms other fIi (qi,<1-;) = r 'Tr;(W',qi,<1-i (W' ))ki(W'IE i (W))dtl (w' ), te probability measure space. We J W'EEi(w) ~ world and assume that it is large where insider to be interesting. As usual 0 ifw' rt- Ei(W) is a common probability measure. k;(w'IEi(W)) = ( ') J. 9; w n if w' E E,(w) )ting the production space. I wEEi (w'l ql ndW ~ w is the prior ofagellt i, (where k; i~ a Radon-Nikodym uerivative ~\lch that f k;(w)dtl(w ) = L is a set diffeTential information 1 and Ei(W) deuot.es the event in firm ·i's partition which contaius the realized state of nat1ll'e). 4 For siUlplicit.y we assume thilt the profit fUllctioll does not depenu 011 n. As we hon set of firm i; mentioned above all the results of the paper remain VAlid . 460

Y Qi : n ----+ 2 , is a non-empty, convex, weakly compact-valued and Since I1(q) is a concave functi( integrably bounded correspondence having an Fi-measurable graph, i.e., valued. By virtue of Berge's Max GQ; E Fi (>(I B(Y). u.s.c. Finally, an appeal to Wei! is also a non-empty valued corrE (A.2) Now since each 'Pi is non-emp i) For each i, 7ri(-,' ) : Q(w) ----+ JR, is weakly continuous. it follows that likewise is F : satisfies aU the conditions of t. ii) The function 7ri is concave in the i-th coordinat.e for all i. Consequently, there exists some

iii) 7ri is integrably bounded. 5 Collusion under diffE 4 Existence of a Cournot-Nash equilibrium I t is a well known result that Pareto optimal. In other words, We can now st.ate the first existence result. '/lie assume that t.here exists a tracted by the firms. If t.he fin finite or countable partition Ai, (i = 1, ... , n) of D, and the a-algebra Fi is t.hen a Pareto optimal outcome generated by Ai. examined when firms have sym] Theorem 4.1: Let C = {(Qi, 7ri, Fi, /1) : i = 1, ... ,n} be a Cournot game ence of differential information satisfying (A.1) - (A.2). Then, there exists a Cournot-Nash equilibrium. to collude, depending on how t.hE

LQ Before we proceed, let's define tb Proof: For each i, define the correspondence 'Pi : LQ_; ----+ 2 ; by will consider in the sequel. Definition 5.1: A Pooled in/or their information, i.e., the info Also define the correspondence F : LQ ----+ 2LQ by 1, .. . ,n, where V denotes the]01 Definition 5.2: A Private info their own private information, i.( As in Yannelis and Rustichini (1991), we will show that the correspondence Definition 5.3: A Common kn( F satisfies all the hypotheses of the Fan-Glicksberg Fixed Point Theorem. It can then be easily checked that a fixed point of the correspondence F firms use only the information is by construction a Cournot-Nash equilibrium for C. We will complete the l\i=1Fi, j = 1, ... , n, where 1\ del proof in t.hree steps. Let L~i denote the set of all n. 1. LQ is non-empty, convex, weakly compact and metrizable. selections from the production SE The non-emptiness of LQ follows from the Aumann measurable selection theorem. Also, since each Qi is non-empty, convex and weakly compact, it {qi E L1 (IL, Y) : qi follows from Diestel's Theorem that each LQi is a weakly compact subset E Qi(W), /1 - a.e.}. of L 1(/1 ,Y ). Obviously, each LQ ; is convex. Furthermore, since each LQ; is a weakly compact subset of a separable Banach space L1 (/1, Y), it is also Also let L~ = L~, X ... x L~ ". metrizable [for more details see Yarmelis and Rustichini (1991), Theorem Let LQ; denote t.he set of all Bo 5.1j. selections from the production SE II. The function IT; is weakly continuous for each i. {qi E L 1(/1 ,Y) : qi Since, by assumption, 7ri is concave, weakly continuous and 7ri is inte­ grably bounded, the result follows by an application of Theorem 2.8 in E Qi(w) ,/1- a.e.}. Balder and Yannelis (1993). LQi III. Each correspondence 'Pi : LQ _; ----+ 2 , is non-empty, convex valued ;, That. is t.h e smallest. CT-algchra CO ll i and weakly u.s.c. fiTl);)!. is t.he largest. CT-algcbrfl COllt.fl 461

::mvex, weakly compact-valued and Since II(q) is a concave function of qi on LQi' it follows that

5 Collusion under different information rules ash equilibrium It is a well known result that a Cournot-Nash equilibrium may not be ult. We assume that there exists a Pareto optimal. In other words, there is a surplus that has not been ex­ .. ,n) of n, and the a-algebra Fi is tracted by the firms. If the firms collude and play a cooperative game, then a Pareto optimal outcome will be reached. This problem has been examined when firms have symmetric information. However, in the pres­ : i = 1, ... , n} be a Cournot game ence of differential information there may be different ways for the firms sts a Cournot-Nash equilibrium. to collude, depending on how they want to share their private information. lence 'Pi : LQ _i ---'l 2L Q ; by Before we proceed, let's define the three different information rules that we wiH consider in the sequel. = max IIi (q:'i, Yi)}. YiEL Qi Definition 5.1: A Pooled information r'ule is the one where firms share their information, i.e., the information they use is, Fj = Vi~lFi' j ) 2L q by 1, ... ,n, where V denotes the join.5 i(q:'i)' Definition 5.2: A Private information rule is the one where firms use their own private information, i.e., Fi, i = 1, ... , n. ~7 ill show that the correspondence :;licksberg Fixed Point Theorem. Definition 5.3: A Common knowledge information rule is the one where d point of the correspondence F firms use only the information that is common to them, i.e. , Fj = 6 'rium for C. vVe will complete the I\i= 1Fi, j = 1, ... , n, ."here 1\ denotes the meet. Let L~; denote the set of all Bochner integrable and Vi=lFi- measurable ~pact and metrizable. selections from the production set Q;; of firm i, i.e., lIe Aumann measurable selection " convex and weakly compact, it L~, {qi E Ll (p, Y) : qi : n ---t Y is Vf=l Fi - measurable and LQi is a weakly compact subset qi(W) E Qi(W), p- a.e.} . . Furthermore, since each L Q .; is :anach space Ll (p, Y), it is also Also let LP LP x··· x LP . Q = Q1 Q" md Rustichini (1991), Theorem Let L'Q , denote the set of all Bochner integrable and l\i=lFi - measurable selections from the production set Qi of firm i, i.e. , . faT each i. akly continuous and 7fi is inte­ L'Q , {qi E Ll (p, Y) : qi : n ---'I Y is I\?=l Fi - measurable and application of Theorem 2.8 in qi(W) E Qi(w),p-a.e.}.

'2;! i_s non-empty, convex valued "That is the smallest. u-algebra cont.aining a ll of the sub u-algcbras F i , i = 1, ... , n. °That is t.he largest u-alg(!]'l'a contained in all of the sub u-algebras Fi, i = 1, ... , n. 462

Also let LQ = LQl X ... x LQ" . Proof: Obvious. 0 A collusion equilibrium under the pooled information rule for C However, as the following pro {(Qi, 'If;, F , p.) : 1, ... , n} is an element q* E L~ such that i industry profits derived under t.. n rived from the Cournot-Na.c;h ga using the corrunon information, , information, which means lower A collusion eqv,ilibrium under the private information rule for C mization alone, gives them highE tell which effect outweighs the 0 {(Qi, 'lfi, Fi, p.) : i = 1, .. , , n} is an element q* E LQ such that

n Proposition 6.3: The indusi11 the common knowledge informc Cournot-Nash game are not corr

A collusion equilibrium under the common knowledge information rule Proof: Consider a Cournot gal nature, i.e., = {a, b, c} and onE for C = {(Qi, 1fi, Fi, p.) : i = 1, ... ,n} is an element q* E LQ such that n with the same probability. Each n following partition of the state s

FI = {a,b,c}

Next we present the second existence result of this paper. The inverse demand function is, function which is measurable wit] Theorem 5.1: Under assumptions (A.l)-(A.2), a collusion equilibrium exists for all the information rules. is: For firm 1, C1 (w, ql (w)) = .4q

Proof: Notice that the objective function is weakly continuous and L~, LQ and LQ are non-empty and weakly compact. Therefore the maximum is attained and the argmax is the set of all equilibrium points. 0

6 Comparison of profits under the three information Notice that firm 1 has trivial inf< rules mation. The following productio: In this section, we will put the industry profits under the three different information rules in a hierarchy. It is known that under symmetric informa­ tion the industry profits when firms collude are greater than or equal t.o the industry profits derived from the Cournot-Nash game. But what happens under differential information? Let lIP(q*) , fI(q*) and lIC(q*) be the value functions under the three information rules, as defined in the previous section. Proposition 6.1: lIP(q*) 2 fI(q*) 2 lIC(q*). Observe that the production pIa firm's private information. The e::l Proof: First observe that /\i=lFi ~ Fi, i = 1, ... ,n, ~ Vi=1 Fi. This implies Nash game are 3.296. that LQ ~ LQ ~ L~. Since the objective functions are the same, the desired ow assume that finns collude result follows . 0 rule. Since the information that Let lIN (q*) denote the industry profits derived from the Nash game. information, the production plan Proposition 6.2: lIP(q*) 2 fI(q*) 2 fIN (q*).

/ " Proof: Obvious. 0 the pooled information rule for C = aent q* E L~ such that However, as the foHowing proposition indicates, we cannot compare the n industry profits derived under the common knowledge rule with those de­ ~f* 2: fIi(q). rived from the Cournot-Nash game. The reason is that when firms collude q i:1 using the cornmon information, essentially they throwaway some valuable information, which means lower profit. On the other hand, the joint maxi­ ~e private information rule for C == mization alone, gives them higher profits. In tills general setting we cannot lement q* E LQ such that tell which effect outweighs the other. n Proposition 6.3: The industry profits derived from the collusion under ~ 2: fIi(q). the common knowledge infor-mation rule and the ones derived from the ~q ;=1 Cournot-Nash game are not comparable. common knowledge information rule Proof: Consider a Cournot game wit.h two firms {I, 2}, three states of is an element q* E L'Q such that nature, i.e., D = {a, b, c} and one homogeneous output q. Each state occurs n with the same probability. Each firm's private information is given by the ~ 2: fIi(q) . following partition of the state space, Qi=l F1 = {a, b, c}, F2 = {{a}, {b} , {c} }. :e result of this paper. The inverse demand fWlction is, p(w ) = 5 - 1.5(q1(W) + q2(W)). The cost (A.l)-(A.2), a collUSion equilibrium function which is measurable with respect to each firm's private information is: For firm 1, C1(W ,q1(W)) = .4qf for all wED and for firm 2, on is weakly continuous and L~ , LQ Impact. Therefore the maximum is q~ ifw = a 11 equilibrium points. 0 C2(W, Q2(W)) = .4Q~ if w = b { .8q~ if w = c. fer the three information Notice that firm 1 has trivial information, while firm 2 has complete infor­ mation. The following production plan is a Cournot-Nash equilibrium, y profits under the three different wn that under symmetric informa­ q1(W) = 1.00275, for all wED; de are greater than or equal to the Jt-Nash game. But what happens .699 if w = a

value functions under the three Q2(W) = .9~9 ~f w = b { DUS section. .709 Ifw=c. '.]*). Observe that the production plan is also measurable with respect to each = 1, ... , n, ~ Vi~1:G. This implies firm 's private information. The expected industry profits from the Cournot­ lUctions are the same, the desired Nash game are 3.296. Now assume that firms collude using the common knowledge information rule. Since the information that is common to both of them is the trivial derived from the Nash game. information, the production plan must be constant in all states. This is, 7*) . Q1(W) = .9194,Q2(w) = .502, for all wED.

." 464

The expected industry profits are 3.5537. and such that members of S an Thus, in this example the profits from collusion with common knowledge has actually occurred. Formally, information are higher than the profits from the Cournot-Nash game. Next compatible for C if it is not true t we present an example where the profits from the Cournot - Nash game are with!> a E nirtSEi(b), such that9 ~ higher than the profits from collusion with common knowledge information. all i E S, that is, each firm in co Consider now a Cournot game with two firms {1,2} that produce a ho­ that state b occurred rather thaJ mogeneous output q. Uncertainty is generated by the marginal cost func­ unable to distinguish between st, tions Ci : -+ lR+ , i = 1,2. Firm 1 has trivial information and its cost n It turns out that a COlITllot-l' function is: C cIqr, where CI 5 for all wEn. Firm 2 has complete I = = Also a collusion equilibrium unde information and its cost function is: C = c2q~. \Ve assume that C2 is dis­ 2 is coalitional incentive compatibl tributed uniformly on [O,10J. The abov information about the Cournot the pooled information rule and game is common knowledge. Moreover, the inverse demand function is not be coalitional incentive comr: p(w) = 50 - 1.5(ql(W) + q2(W)). Since firm 1 has trivial information, its production plan will be constant across all states, while firm's 2 produc­ Proposition 7.1: A Cournot-N tion plan will be contingent on each realization of the random variable C2. i = 1, ... , n} i8 incentive com.pat~ Therefore, the Cournot-Nash equilibrium is Proof: Since we are dealing with 22 .5122 ate to reduce the coalition S to tl ql(W) = 3.317, for aU wEn and q2(W) = ( ). 1.5 + C2 w -i denotes all the firms but i. S equilibrium and there exist a, b, 1 The expected industry profits from the Cournot-Nash game are 174.747. Now assume that firms collude using the common knowledge information rule. This now implies that production must be constant across all states. The production plan is F irst, since q:"i is F_i-measurablt for all wE Ei(a) n E_;(a) and t I qdw) = q2(W) = 3.125, for all wEn.

The expected industry profits now are 156.25. 0 Now consider the following prodt 7 Coalitional incentive compatibility qi(w) = q One of the basic questions that one may ask is whether the different equi­ qi(W) ~ { librium notions we defined previously are coalitional incentive compatible. qi(w) That is, whether a coalition of firms has an incentive to misreport the I t follows that true state of nature and benefit its members. This is an important ques­ tion especially for the collusion equilibrium. If a collusion equilibrium is not coalitional incentive compatible, then it is not sustainable. We define rigorously betow the notion of coalitional incentive compatibility which is related to the one in Krasa-Yannelis (1994). This contradicts the fact that q* Definition 7.1: An output function q E LQ is said to be coalitional Proposition 7.2: A collusion e incentive compatible if and only if the following does not hold: There exist rule for C = {(Q~, 7l'i, Fi, j.t) : i = a coalition of firms7 ScI and two states a, b that. members of 1\ S cannot compatible. distinguish (i.e., a and b are in the same partition for the firms in 1\ S)

" E i(b) , is t.he event. in firm s' inform. 7 I i~ the set of all firms. I) qS and qI\S are vectors of out.put; 465

and such that members of 5 are better off by announcing b whenever a collusion with common knowledge has actually occurred. Formally, q E LQ is said to be coalitional incentive om the Cournot-Nash game. Next compatible for C if it is not true that there exist coalition 5, and states a, b from the Cournot - Nash game are with8 a E ni~s Ei(b), such that9 7ri(qS(b), ql\S(b)) > 7ri(qS(a), ql\S(b)), for common knowledge information. a,ll i E 5, that is, each firm in coalition 5 is strictly better off announcing 1'0 firms {l,2} that produce a ho­ that state b occurred rather than the true state a and firms not in 5 are era ted by the marginal cost func­ unable to distinguish between state a and b. s trivial information and its cost It turns out that a Cournot-Nash equilibrium is incentive compatibie. : all wEn. Firm 2 has complet.e Also a collusion equilibrium under the common knowledge information rule = c2 q~. We assume that C2 is dis­ is coalitional incentive compatible. However, a collusion equilibrium under 2 information about the Cournot. the pooled information rute and under the private information rule may the inverse demand function is not be coalitional incentive compatible. irm 1 has trivial informat.ion, its all states, while firm's 2 produc­ P roposition 7.1: A Cournot-Nash equilibrium for C = {(Qil 7ri, Fi, (1) : ization of the random variable C2. i = :1, ... , n} is incentive compatible. 1 IS Proof: Since we are dealing with a non-cooperative concept it is appropri­ 22.5122 ate to reduce the coalition 5 to the singleton coalition, i.e., 5 = {i}. Then, Id q2(W) = 1.5 + C2(W) -i denotes all the firms but i. Suppose that q* E LQ is a Cournot-Nash equilibrium and there exist a, b, where a E E-i (b), such that ~ournot-Nash game are 174.747. le common knowledge information 7ri(q~(b), q:'i(b)) > 7ri(qi(a), q:'i(b)). oust be constant across all states. First, since q:'.i is F_i-measurable, it is implied that q:'i(a) = q:'Jb). Thus, for aU W E Ei(a) n E_i(a) and t E Ei(b) n E_i(a), , for all wEn. 7ri(q~(t) , q:'i(t)) > 7ri (q:; (w), q:'i(w)) , )6.25. 0 Now consider the following production plan for firm i, atibility qi(W) = { qi(w) = qi(t) if wE E;(a) n E_i(a) ask is whether the different equi­ qi (w) otherwise. ~ coalitional incentive compatible. as an incentive to misreport the Ibers. This is an important ques­ It follows that lum. If a collusion equilibrium is n it is not sustainable. We define J7ri(qi(w) ,q:'i(w))d(1 > J7ri(qi (w),q:'i(w))d(1 . 1 incentive compatibility which is )4). This contradicts the fact that q* is a Nash equilibrium. 0 1 E LQ is said to be coalitional Proposition 7.2: A COll1lsion equilibrium under- the pr-ivate information lowing does not hold: There exist rule fOT C = {(Qi, 7ri, Fi, f.l) : i = 1, ... , n} may not be coalitional incentive a, b that members of 1\5 cannot compatible. ) partition for the firms in I \ 5)

"Ei(b), iH the event. il,l finus' information part.ition that Lontains the realized st.ate b. 9 qS illld ql\ S are V0.ct.ors of out.puts for firms in cOA.lition S A.lld 1 \ S respect.ively. 466

Proof: Consider two firms {1,2} that produce a homogeneous output. 8 Distribution of profl The state space is n = {a, b, c} where each state occurs with probabil­ So far, we showed that when firm ity ~ and the private information of each firm is: ;::1 = {{a, b}, {c}} and information rules), industry pre ;::2 = {{a,c}, {b}}. We denote by ql(W),q2(W) the production in state W by the Cournot-Nash game. Me of firm 1 and firm 2 respectively. The inverse demand that firms face is collude under the common kno\', p = (5 - 1.5(ql(W) + Q2(W))), The marginal cost function of each firm, remains to be answered is how which is measurable with respect to each firm's private information is among the firms in a way that the total profits. if W = a,b

if W = c 8.1 The private value prod We propose that each firm shoulc .25 if W = a, c to the total profits. One way to to its Shapley value. Then a pI C2(W) = {. 1 if w = b. each firm will get its Shapley val production pla.n. A collusion equilibrium under the private information rule is As in the definition of the st, first derive a transferable profit! Q;(a) = Q~(b) = .916,Q~(c) = .416, weighted by a factor Ai, (i = 1, .. In the value allocation itself no si q;(a) = q;(c) = .916, q;(b) = .416, L--J side payments is then defined as and the profit of each firm in each state is Definition 8.1.1: A game witi finite set of agents (firms) 1 = { function V defined on 21 such coalition and V(S) is the "worH 7r2(a) = 1.833, 7r2(b) = .832, 7r2(C) = 2.52.

The ex-ante expected profit for the industry is III + II2 = 1. 725 + 1.725 = The Shapley value of the garr 3.45. to each firm i a "payoff", Shi, g Suppose that state b occurs. Firm 2's profit is .832. However, if firm 2 reports that state a occurred and produce as if a had actually occurred, its Shi(V) = L (ISI-1 profit is 1.146. Since 1.146 > .832 the collusion equilibrium with differential S C I s ::> ( i ) information is not incentive compatible. 0 Remark : It follows from the above proposition that a coHusion equilibrium The Shapley value has the prol under the pooled information rule may not be incentive compatible as well. Shapley value is Pareto efficient. The reason is that although information is now symmet.ric, still firms cannot Shi ;::: V({i}),Vi. distinguish between the states that could not distinguish before the pooling We now define for each Coun took place. Hence, the above proposition is applicable here as well. and for each set of weights, {Ai side payments (1 ,VI) (we also IE Proposition 7.3: A collusion equilibrium under the common knowledge game) as follows: information rule is incentive compatible.

Proof: It is rather obvious, since now there do not exist states a and b LO See Emmolls <\lId Scafuri (1985, P such that one firm can distinguish between the two and the others cannot. It The Shaplcy valllc nwasure is t.he: o call make 1.0 all I.he coalit.ions of whid 467

produce a homogeneous output. 8 Distribution of profits each state occurs with probabil­ So far, we showed that when firms collude (under the pooled and the private :h firm is: Fl = {{a, b}, {c}} and information rules), industry profits are higher than the profits obtained ,Q2(W) the production in state w by the Cournot-N"ash game. Moreover, profits may be higher when fums .nverse demand that fums face is collude under the common knowledge information rule. The question that 'ginal cost function of each firm, remains to be answered is how are these extra profits being distributed 1 firm's private information is among the firms in a way that captures the contribution of each firm to the total profits. 'w =a,b

W=C 8.1 The private value production plan We propose that each firm should be rewarded according to its contribution f w = a, C to the total profits. One way to do this is to reward each firm according to its Shapley value. Then a production plan will be determined, so as fw = b. each fum win get its Shapley value. This production plan will be the value ate information rule is production plan. As in the definition of the standard value allocation concept, we must , Q~(c) = .416, first derive a transferable profit game (TP) in which each fum's profits are weighted by a factor Ai , (i = 1, . . . , n), which allows for profits comparisons. In the value allocation itself no side payments are necessary.lO A game with ,q;(b) = .416, side payments is then defined as follows: is Definition 8.1.1: A game with side payments r = (I, V) consists of a .52,1fl(C) = .832, finite set of agents (fums) 1= {1 , ... ,n} and a superadditive, real valued function V defined on 21 such that V(0) = O. Each ScI is called a coalition and V(S) is the "worth" of the coalition S. 332 ,1f2(C) = 2.52. -' ,try is III + II2 = 1.725 + 1.725 = The Shapley value of the game r (Shapley 1953) is a rule that assigns to each firm i a "payoff" , S hi, given by the formula, 11 profit is .832. However, if firm 2 ~ as if a had actually occmred, its Shi(V) = L (lSI - 1),!~!;I-ISI)! [V(S) - V(S \ {i} )]. usion equilibrium with differential SC I o s ::> (i)

ition that a collusion equilibrium The Shapley value has the property that 2:iE1 Shi (V) = V(I), i.e., the It be incentive compatible as well. Shapley value is Pareto efficient. Moreover, it is individually rational, i.e., now symmetric, still firms cannot Shi ;:::: V(fi}),Vi. [lot distinguish before the pooling We now define for each Cournot game with differential information, C, is applicable here as well. and for each set of weights, {Ai : i = 1, . .. ,n}, the associated game with side payments (I,Vn (we also refer to this as a "transferable profits" (TP) .Lm under the common knowledge game) as follows:

!here do not exist states a and b IOSec Emmolls and Scafuri (1985, p.60) for further discussion. m the two and the others cannot. 11 Thc Shapley valuc measure is the sum of the expcct.ed marginal contribut.iol1s a firl1l call make t.o all the coalitiolls of which it is a mcmber. 468

For each coalition ScI, let knowledge production plan is COl it is of great importance if we kn the surplus (if there is any) in a r is rewarded. We now define for each Courn We are now ready to define the private value production plan. and for each set of weights, {Ai : side payments (I, Vn (we also ref, Definition 8.1.2: An output plan q E LQ is said to be a private value game) as follows: production plan of the Cournot game wit.h differential information C if For each coalition ScI, let there exist Ai ~ 0 (i = 1, ... , n, which are not all equal to zero), with ' V{(S) = max LAi q iES

where S hi (VI) is the Shapley value of firm i derived from the game (I, VI), subject to defined in (8.1). i) for each i, qi is J\ r~l Fj-meas The above definition says that the expected profits of each firm multiplied The common knowledge vahle by its weight Ai must be equal to its Shapley value derived from the (TP) definition (8.1.2), except that we game (I, VI). by VC. An immediate consequence of Definition 8.1.2 is that the private value Thus, in contrast to the private production plan is individually rational (profits for firm i are greater than production plan within a coalitio of equal from the ones derived from the Cournot-Nash game). This follows information. Notice that the corr immediately from the fact that the game (I, VI) is superadditive for all coalitional incentive compatible. weights. In addition, it is Pareto efficient. 12 tence theorem. In fact, if, for eXI We are now ready to state the first existence result of this section. and the other has "full" informa implies that the trivial informati< Theorem 8.1.1: LetC = {(Qi, 7'fi,Fi,IJ·) : i = 1, .. . ,n} be a Cournot game additivity condition of the functi as defined in Section 3, satisfying assumptions {A.l}-{A.2}. exist coalitions S , T with S nT = Then, a private value production plan exists in C. the proof of proposition 6.3, we i Proof: This result can be proved along the lines of Krasa and Yannelis the Cournot-Nash equilibrium y (1996), Theorem 1. 0 common knowledge information, tion. This causes problems with t Remark: One can easily show that a pooled information (where the in­ production plan. Therefore, we ( formation that is being used is the pooled information) value production of a common knowledge value pr plan1J exists as well.

8.2 The common knowledge value production plan 9 Examples We now introduce another notion of a value production plan for the Below we give examples with tW( Cournot game with differential information. The difference stems from the collude using the common knowlE measurability restriction on the type of production plans. It is an ana­ of profits is determined by a co log of the coarse core of Yannelis (1991). We call it a common knowledge These examples illustrate how tl determined and also show that fu va~ue production plan, since the information that is being used is the com­ mon knowledge information. As we saw in the previous section a common ing the other characteristics of . higher Shapley value and higher

12For lllore

knowledge production plan is coalitional incentive compatible. Therefore, it is of great importance if we know that there is also a way to distribute the surplus (if there is any) in a manner that the contribution of each firm ))d{l(w) = max: L AiIIi(q). (8.1) qELQ iES is rewarded. We now define for each Cournot game with differential information, C, l value production plan. and for each set of weights, {Ai : i = 1, ... ,n}, the associated game with side payments (I, V,X) (we also refer to this as a "transferrable profits" (TP) LQ is said to be a p'rivate value game) as follows: ith differential information, C, if For each coalition ScI, let 2 not all equal to zero), with

Vn,Vi, V{(S) = max: L Ai J7ri(qS(W), l\S(w))d{-L(w) (8.2) q iES

11 i derived from the game (I ,V1), subject to i) for each i, qi is I\i= 1Fy-measurable. :ted profits of each firm multiplied The common knowledge value pTOduction plan can now be defined as in pley value derived from the (TP) definition (8. 1.2), except that we replace (8.1) by (8.2) and also replace VP by VC. )n 8.1.2 is that the private value Thus, in contrast to the private value production plan, we now require the profits for firm i are greater than production plan within a coalition to be based on the common knowledge ;ournot-Nash game). This follows information. Notice that the common knowledge value production plan is e (I, Vn is superadditive for all coalitional incentive compatible. However, we cannot prove a general exis­ 12 tence theorem. In fact, if, for example, one firm has "trivial" information stence result of this section. and the other has "full" information, the common knowledge information implies that the trivial information must be used and therefore the super­ : i = 1",. ,n} be a Cournot game additivity condition of the function V{(· ) may be violated, i.e. , there can ,tions (A.l)- (A.2). exist coalitions S,T with SnT = I/) and V{(S) + V{(T) > V{(SUT).14 In e.Tists in C. the proof of proposition 6.3, we present an example with two firms where the lines of Krasa and Yannelis the Cournot-Nash equilibrium yields higher profits than collusion under common knowledge information, which destroys the superadditivity condi­ tion. This causes problems with the existence of a common knowledge value poled information (where the in­ production plan. Therefore, we cannot prove a general existence theorem !d information) value production of a common knowledge value production plan.

:production plan 9 Examples value production plan for the Below ,ve give examples with two firms, with differential information, that m. The difference stems from the collude using the common knowledge information rule and the distribution production plans. It is an ana­ of profits is determined by a common knowledge value production plan. We call it a common knowledge These examples illustrate how the common knowledge production plan is on that is being used is the com­ determined and also show that firms with superior information, while keep­ 1 the previous section a common ing the other cbaracteristics of the firms (i.e., marginal cost) fixed, have higher Shapley value and higher share of the industry profits. The profits

16),1',169. Klt examine it. thuroughly. 14See Kmsit alld Yanllelis (1996), p.177, for lllore details, 470

from collusion are higher than the Cournot-Nash profits and the Shapley (5 -(ql+ value of each firm captures its contribution to the tota1 industry profits. It Hence, a value production plan. is important to note here that in these examples the value allocation is in conclude, the firm with the superic the core and therefore the cartel can be viewed as stable, in the sense that production plan, by being assignE no coalition of firms can deviate from the cartel agreement and become higher profits (III = 3.25521 and 1 strictly better off. In the next example, the asymn Example 9.1 side. Consider two firms {I, 2} that produce a homogeneous product. The Example 9.2 state space is n = {a, b} where each state occurs with probability ~ and Consider two firms {I, 2} that the private information of each firm is: Fl = {{a} , {b} } and F2 = {{a, b} }. state space is n = {a, b} where e We denote by ql (w), q2 (w) the production in state w of firm 1 and firm 2 the private information of each fir respectively. The inverse demand that firms face is: p = (5 - (3(W)(ql (w) + We denote by ql(W),q2(W) the pH q2(W))). The marginal cost is zero for both firms. The slope (3 takes on the respectively. The inverse demand following values: q2(W))) , The marginal cost of ead .8 if w = a to each firm 's private information (3(w) = { 1.2 if w = b A Cournot-Nash equili bri urn is o,(w) ~ {

q~(a) = 2.2916, q;(b) = 1.25, q;(a) = q;(b) = 1.666. c,(w) ~ ! Notice that the production is measurable with respect to each firm's private information. The ex-ante expected profit for the industry is TIl + I:h = A Cournot-Nash equilibrium is 3.03819 + 2.77778 = 5.81597. Now suppose that the two firms collude under the common knowledge information rule. The information they use now is the trivial information and the optimum total production is 2.5. The expected industry profits are: q;(a) = .955~ 6.25. The problem that arises is how this surplus will be distributed among The ex-ante expected profit for the two firms. Or put it in different words, what is the production that will 1.19185 = 2.37704. be assigned to each firm? Without taking the information superiority of the Now suppose that the two fim first firm into account, both firms are identical. Hence, one solution would information rule. The informatiOl be just to split the profits. However, this is not a "fair solution" since firm and the optimum total production 1 contributes more to the coalition than firm 2 does. The value production are 2.66667. plan allocation we discussed above provides a more sensible outcome. The Shapley value of the two fi The Shapley value of the two firms (with )1} = A2 = 1) is 1 . Sh l = 2[2.66667 - 1 Sh l = ~[6.25 - 2.77778] + ~[3.03819] = 3.25521 2 2 1 . 1 1 Sh2 = 2[2.66667 - 1.1 Sh2 = 2[6.25 - 3.03819] + 2[2.77778] = 2.99479. Then, a va1ue production wiiil be a solution to the following problem: Then, a value production will be (5 - 1.5(ql ­ 471

lrnot-Nash profits and the Shapley (5 - (ql + q2))q2 = 2.99479. ;ion to the total industry profits. It Hence, a value production plan is ql = 1.30208 and q2 = 1.19792. To examples the value allocation is in conclude, the firm with the superior information gets rewarded in the value viewoo as stable, in the sense that production plan, by being assigned a higher level of production and thus the cartel agreement and become higher profits (Ill = 3.25521 and II2 = 2.99479). In the next example, the asymmetry of information comes from the cost side. uce a homogeneous product.. The Example 9.2 ate occurs wit.h probability ! and Consider two fu'ms {1,2} that produce a homogeneous product. The /:"1 = {{a},{b}} and .1'2 = {{a,b}}. state space is = where each state OCcurS with probability ~ and on in state w of firm 1 and firm 2 n {a, b} the private information of each firm is: .1'2 {a}, {b} } and .1'] = { {a, b} }. rms face is: p = (5 - {3(w)( q] (w) + = { ,th firms. The slope {3 takes on the We denote by Ql(W),q2(W) the production in state w of firm 1 and firm 2, respectively. The inverse demand that firms face is p = (5 - 1.5(q](w) + q2(W))). The marginal cost of each firm, which is measurable with respect if w = a to each firm's private information, is ifw = b .8 if w = a C2 (w) = { 1. 2 if w = b (b) = 1.25, = l.666. : ifw =a with respect to each firm's private "(W) ~{ ifw = b. t for the industT',lj is II] + II2 = A Cournot-Nash equilibrium is 1e under the common knowledge q;(a) = q; (b) = .888889, [se now is t.he trivial information The expected industry profits are: Q2(a) = .955556, q2(b) = .822222. 3nrplus will be distributed among , what is the production that will The ex-ante e:xpected profit for the industry is III + II2 = 1.18519 + the information superiority of the 1.19185 = 2.37704. nticaI. Hence, one solution would Now suppose that the two firms collude under the common knowledge is not a "fair solution" since firm information rule. The information they use now is the trivial information lnn 2 does. The value production and the optimum total production is 1.33333. The expected industry profits ies a more sensible outcome. are 2.66667. th A] = A2 = 1) is The Shapley value of the two firms (with A] = A2 = 1) is 1 1 [3.03819] = 3.25521 Sh] = 2[2.66667 - l.19185] + 2[1 .18519] = 1.33,

[2.77778] = 2.99479. Sh2 = ~[2.66667 - l.18519] + ~[l.19185] = 1.33667. 'n to the following problem: Then, a value production will be a solution to the following problem:

=3.25521 (5 - 1.5(ql + q2))q] - ql = 1.33, 472

(5 - 1.5(ql + q2))q2 - q2 = 1.33667. Krasa, S., and N.C. Yannelis (19' with Differential Information," Hence, a value production plan is ql = .665 and q2 = .668333. To conclude, Krasa, S., and N.C. Yannelis (19 as in the above example, the firm with the superior information gets re­ Allocat.ion for an Economy w warded in the value production plan, by being assigned a higher level of Mathematical Economics, 25, : production and thus higher profits. Laffont, J.J., and D. Martimort formation," Econometrica, 65 , 10 Concluding remarks Roberts, K. (1983): "Self-Agree Stanford University. R emark 1: Alternatively, one could have used t.he notion of the a-core Rotemberg, J.J., and G. Saloner which is defined as follows: We say that q E LQ is an a-core of the game C Journal of Industrial Economi. if Shapley, L.A. (1953): "A Value A.W. Thcker, eds., Contributio: it is not true that there exist ScI and (Yi)iE8 E I1iE8LQ; such that ton University Press, Princetol for any zI\8 E I1 i~8 LQ i ,I1i(y8.zJ \ 8) > I1i(q) for all i E S. Yannelis, N.C., (1991) : "The Co It follows that under our assumptions in Section 3 the a-core is non­ mation," Economic Theor'Y, 1, empty [see Yannelis (1991)]. A collusive agreement that is an element of Yannelis, N.C. and A. Rust.ichi the a-core is individually rational, Pareto optimal and coalitional stable. Cooperative Random and Baye Although t.hese are clearly desirable properties, we do not have a straight­ Spaces, and Economics, C.D. j forward way of selecting an element. from the core that would capture the emburg, eds., Springer-Verlag. "worth" of each firm. To this end, the Shapley value provides a relatively Zhao, J., (1998): "A Necessary a: easy way of figuring out the contribut.ion of each fum to the total profits in Oligopoly Games," Matheme and how to distribute them among the firms. Remark 2: In the two firms case, the Shapley value is in the core and there­ fore in this case the duopoly with differential information can be viewed as stable. This is not the case for more than two firms unless t.he corre­ sponding TU game is convex. Zhao (1998) provides necessary and sufficient conditions for the deterministic TU game to be convex. In a subsequent. pa­ per we intend t.o examine the conditions which guarantee the convexity of the side-payments game defined in (8.1) or (8.2).

References Balder, E.J ., and .C. Yannelis (1993): "On the Continuity of the Expect.ed Utilit.y," Economic Theory," 3, 625-643. Crampton, P.C., and T.R. Palfrey (1990): "Cartel Enforcement with Un­ certainty About Costs," International Economic Review, 31, ]7-47. Diestel, J ., and J. Uhl (1977): "Vector' Measures, Mathematical Surveys," American Mathematical Society, 15. P rovidence. Dondimoni, M.P, .S. Economides, and H.M. Polemarchakis (1986): "Sta­ ble Cartels," International Economic Review, 27, 317-327. Emmons, D. and A.J. Scafuri (1985): "Value Allocations-An Exposdion, " in C.D. Aliprantis et al., eds., Advances in Equilibrium Theory. Springer, Berlin, Heidelberg, New York. 473

q2 = l.33667. Krasa, S., and N.C. Yannelis (1994): "The Value Allocation of an Economy with Differential Information," Econometrica, 62, 881-900. 35 and q2 = .668333. To conclude, Krasa, S., and N.C. Yannelis (1996): "Existence and Properties of a Value the superior information gets re­ Allocation for an Economy with Differential Information," Journal of o being assigned a higher level of Mathematical Economics, 25, 165-179. Laffont, J.J., and D. Martimort (1997): "Collusion under Asymmetric In­ formation," Econometrica, 65, 875-911. Roberts, K. (1983): "Self-Agreed Cartel Rules," working paper, INISSS, Stanford University. 1e used the notion of the (}: - core Rotemberg, J.J., and G. Saloner (1990): "Collusive Price Leadership," The r E LQ is an (}:-core of the game C JO'umal of Industrial Economics, XXXIX, 93-11l. Shapley, L.A. (1953): "A Value of n-person Gan1es," in H.W. Kuhn and A.\V. Thcker, eds., Contributions to the Theory of Games, VoL II. Prince­ and (Yi)iES E IliEsLQi such t hat ton University Press, Princeton, NJ, 307-317. ') > Ili(q) for all i E S. Yannelis, N.C., (1991): ;'The Core of an Economy with Differential Infor­ in Section 3 the (}:-core is non­ mation," Economic Theory, 1, 183-198. agreement that is an element of Yannelis, N.C. and A. Rustichini (1991): "Equilibrium Points of Non­ o optimal and coalitional stable. Cooperative Random and Bayesian Games," in Positive Operators, Riesz erties, we do not have a straight- Spaces, and Economics, C.D. Aliprantis, K.C. Border and W.A.J. Lux­ the core that would capture t he emburg, eds., Springer-Verlag. wpley value provides a relatively Zhao, J., (1998): "A Necessary and Sufficient Condition for the Convexity in Oligopoly Games," Mathematical Social Sciences, forthcoming. l of each firm to the total profits [·ms. pley value is in the core and there­ ,ntial information can be viewed than two firms unless the corre­ provides necessary and sufficient to be convex. In a subsequent pa­ which guarantee the convexity of )r (8.2).

)n the Continuity of the Expected

I: "Cartel Enforcement with Un­ Sconomic Review, 31. 17-47. reaSUTe8, lvlathematical Surveys," ovidence. l.M. Polemarchakis (1986): "Sta­ eview, 27, 317-327. 'alue Allocations-An Exposition, " in Equilibrium Theory. Springer,