Values for cooperative games with incomplete information: An eloquent example

Myerson (1984b)’s extension of the λ-transfer value to cooperative games with incomplete information focus among other things on the strength of the incentive constraints at the solution for determining the power of coalitions. We construct an intuitive three-player game where the player whose only contribution is to partly release the two other players from the incentive constraints they face when they cooperate, receives a zero payoff, according to Myerson’s solution. On the contrary, the random order arrival procedure attributes a strictly positive payoff to him. Our example is thus an analog of the banker game of Owen (1972) that was designed for evaluating the λ-transfer value under complete information. Asymmetric information now takes up the role that was formerly attributed to the lack of transferability of utilities. JEL Classification: C71. Aspirant FNRS; CORE, Université catholique de Louvain, Belgium; e-mail: [email protected]. Special thanks to Professors Jean-François Mertens and Enrico Minelli; errors are mine. This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the author. Introduction Shapley (1969) designed the fictitious transfer procedure in order to extend the value he introduced in 1953, to the class of NTU games. The pertinence of the so-defined λ-transfer value, was tested afterwards in many different ways. In particular, the banker game of Owen (1972) appeared to be a very constructive example. Two players generate some surplus by cooperating (e.g. via the provision of a public good, or via mutually beneficial exchanges), but are limited in their ability to share this surplus. The third player (the “banker”) can only release them from this restriction, but creates no real additional surplus. In this context, should the banker receive a strictly positive payoff or not? According to the λ-transfer value, the answer is no. Other fairness criteria, such as the random order arrival procedure (cf. Maschler and Owen (1989)), imply a positive answer to the question. Myerson (1984b) generalized the fictitious transfer procedure in order to extend Shapley (1953)’s value, to the class of cooperative games with incomplete information. The so-defined M-solution focus among other things on the strength of the incentive constraints at the solution for determining the power of coalitions. Starting from the bargaining problem studied in section 10 of Myerson (1984a), we add a third player whose only contribution is to partly release the two original players from the incentive constraints they face when they cooperate. Some may consider this contribution important enough for giving a strictly positive payoff to player 3 in any fair solution. The random order arrival procedure appears to abide to this principle, while Myerson (1984b)’s solution does not. The Example Let us consider the following cooperative game with incomplete information: (using concepts and notations introduced in section 2 of Myerson (1984b)) N = {1, 2, 3}, T1 = {L, H}, T2 = T3 = {∗}, Prob(L) = 0.2, D{1} = {d1}, D{2} = {d2}, D{3} = {d3}, D{1,3} = {[d1, d3]}, D{2,3} = {[d2, d3]}, D{1,2} = {[d1, d2], d12, d21}, D{1,2,3} = {[d1, d2, d3], [d12, d3], [d21, d3], d213, d312}, and [d1, d2, d3] [d12, d3] [d21, d3] d213 d312 u1(., L) 0 30 −60 0 0 u1(., H) 0 90 0 0 0 u2(., L) 0 0 90 30 0 u2(., H) 0 0 90 90 0 u3(., L) 0 0 0 0 30