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<l.lition of firms that can miHreport. t.he t.rue state of nat.ure and benefit. it,s members, is an element of F. It is well knc For a precise definit.ion see definition 7,1. graph, then ¢; is lower measurable lower measurable then ¢ : n -1 457 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) }.