A cooperative game of information trading: The core, the nucleolus and the kernel

A certain trade of the information about a technological innovation between the initial owner of the information and n identical producers is studied by means of a cooperative game theoretic approach. The information trading situation is modelled as a cooperative (n + 1)-person game with side payments. The symmetrical strong e-cores (including the core), the nucleolus and the kernel of the cooperative game model are determined. Interpretations of these game theoretic solutions and their implications for the information trading problem are given. 1 An Economic Model of Information Trading We consider an industry consisting of a fixed number of identical producers of which all produce the same homogeneous output with the same technology. The profit level of each producer is identical and expressed in terms of monetary units. In addition to the producers there is another agent who possesses the information about a technological innovation. The new technology may increase the monetary profit level of the producers who acquire and utilize it, but may decrease the profit level of the producers who don't purchase it and continue to use the old technology. Due to an external diseconomy, the profit level of every producer decreases as the number of producers who purchase the new technology grows. The initial owner of the information about the technological innovation has no means of production. As such, we regard the initial owner of the information as the seller of the technological innovation and the producers as the potential buyers of the new technology. To be exact, the initial owner attempts to sell the information about the new technology to all or some of the producers. We aim to describe the profit that can be realized by the initial owner of the information if a trade of the information has been carried out. Concerning the pro1 Theo Driessen, Department of Applied Mathematics, University of Twente, Enschede, The Netherlands. 2 Shigeo Muto, Faculty of Economics, Tohoku University, Kawauchi, Sendai, Japan. 3 Mikio Nakayama, Department of Economics, Hosei University, Tokyo, Japan. 03409422/92/1/55 72 $2.50 9 1992 Physica-Verlag, Heidelberg 56 T. Driessen et al. ducers, the relevant comparison is between the profit level with the use of the new technology and the profit level obtained before the technological innovation. Our analysis of the above information trading situation will be based on a cooperative game model in which the players consist of the seller and the potential buyers of the new technology. In other words, we define a so-called cooperative game in characteristic function form in such a way that the corresponding characteristic function reflects the cooperative behaviour between the seller and the potential buyers. In the context of the relevant game theoretic approach, it is permitted that the players communicate with one another in order to make binding agreements regarding the coalition formation. For instance, the producers can make a binding agreement not to purchase the new technology if they want to do so. Further, a trade of the information can only be carried out on the understanding that resales of the new technology are completely prohibited. The exact cooperative game model of the information trading situation will be treated in Section 2. Following the cooperative setting of Section 2, the profit shares to the seller and the producers can be prescribed by basic solution concepts. The general notions of three main solution concepts the strong e-cores (including the core), the nucleolus and the kernel are discussed in Section 3. Our attention is mainly directed to the core and the nucleolus solution concepts. In the Sections 4, 5 and 6 respectively, we determine the strong e-cores, the nucleolus (as well as the core) and the kernel of the cooperative game model of the information trading situation. Section 7 deals with the monopolistic structure of the information trading situation and the implications for the solution concepts of the corresponding cooperative game model. Concluding remarks are given in Section 8. The most important results can be summarized as follows. The symmetrical strong e-core elements can be described in terms of two realvalued functions f and g of the real number e (cf. Lemma 4.2 and Theorem 4.4). The sign of the critical number e* satisfying f ( e* )= g(e*) determines whether the core is empty or not (cf. Theorem 5.2). According to the nucleolus concept, the profit share to any producer equals either the amount f(e*) or the bound E* ( n 1) induced by the individual rationality principle (cf. Theorem 5.1). As a consequence, the core is nonempty if and only if the potential nucleolus profit share f(e*) to any producer exceeds his monetary profit level obtained before the technological innovation. Further, the nonemptiness of the core implies that the new technology is actually acquired by at least one producer or even all producers (cf. Theorem 5.3). Finally, the kernel coincides with the nucleolus (cf. Theorem 6.2) and the monopolistic structure of the information trading situation yields an empty core (cf. Theorem 7.2). A Cooperative Game of Information Trading: The Core, the Nucleolus and the Kernel 57 2 A Cooperative Game Model of Information Trading First of all, we specify four assumptions on the underlying economic model of the information trading situation as presented in Section 1. Recall that the initial owner seeks to sell the information about the technological innovation to all or some of the n producers (n_> 2). (A.1) Given any nontrivial number of actual buyers of the new technology, an actual buyer attains at least as many profits as a nonbuyer. (A.2 3) The external diseconomy applies to both the actual buyers and the nonbuyers. That is, the profit level of any type of a producer decreases as the number of actual buyers grows. (A.4) The profit level of a unique buyer is more than the positive profit level obtained before the technological innovation. If t producers have purchased the new technology, then E(t) and E* (t) respectively denote the monetary profit level of any producer who has purchased the new technology and not. The collection {E(t),E* (t)l t ~ {0,1, . . . , n}} is called a profit structure of the information trading situation if it satisfies the above four assumptions, i.e., (A.1) E(t)>_E*(t) for all t~{1,2 . . . . . n l } , (A.2) E(t)>_E(t+l) for all t~{1,2 . . . . . n l } , (A.3) E*(t)>_E*(t+l) for all t~{O, 1 . . . . . n l } , (A.4) E ( 1 ) > E * ( 0 ) > 0 , E ( 0 ) = E * ( n ) : = 0 . For the sake of mathematical convenience, we put E(0) = E* ( n ) : = 0. Next we model the information trading situation as a cooperative (n + 1)-person game where its player set N O consists of the seller 0 and the potential buyers 1,2 . . . . . n of the new technology. That is, N O = [0} u N where N: = {1,2 . . . . . n} represents the set of all n producers. Any nonempty subset S of the player set N O (notation: S C NO) is called a coalition. A coalition which includes the seller is always denoted by S O in such a way that S O = {0} u S with S C N. The number of potential buyers in a coalition S O and S is denoted by s. The cornerstone of the cooperative game model is the so-called characteristic function v: 2N~ which assigns to every coalition its maximal joint profit. To be exact, the worth v(S ~ of any coalition S O represents the largest possible monetary profit what the producers in S O can achieve by the cooperative behaviour between themselves and the initial owner of the information about the technological innovation. Notice that the coalition S O must determine the profit independently of the members of the complementary coalition N O S O because 58 T. Driessen et al. resales of the new technology are completely prohibited. In point of fact, the new technology is acquired and utilized by a suitable number of producers in S O so as to maximize the joint profit of production. Moreover, if the initial owner of the information is not a member of a coalition S, then the producers in S can not acquire the new technology by cooperation within S. Consequently, the worth v(S) of any coalition S represents the joint profit of production under the worst conceivable circumstance that all producers outside S do purchase the new technology. Definition 2.1: The cooperative game (NO; v) of the information trading situation is given by v(S ~ : = max (tE(t) + ( s t)E* ( t ) l t e {0, 1 . . . . . s}) for all S O C N O , v ( S ) : = s E * ( n s ) for all S C N . Obviously, the above information trading game (NO; v) satisfies v (N ~ _> n E* (0) = v (N) , v ({0}) = 0 , v({i})=E*(n-1) , v({O,i})=E(1) for all i e N . 3 Game Theoretic Notions The profit shares to the seller and the n producers will be described by means of a specific (n + D-dimensional payoff vector x = (Xo, Xl . . . . . xn) e [R n+l where xi represents the payoff (the profit share) to player i. Following the cooperative game model of Section 2 and the solution part of cooperative game theory, it is customary to require that any payoff vector meets the efficiency and individual rationality principles. In other words, any payoff vector for the information trading game (NO; v) should belong to the imputation set I(v) which is defined to be I (v) : = [(Xo,Xl . . . . . xn) e Nn+l xj= v(N ~ and j ~ N ~ xi>-v({i}) for all i e N ~ 1 A Cooperative Game of Information Trading: The Core, the Nucleolus and the Kernel 59