The Procedural Value

We propose a single-valued solution that extends both the consistent Shapley value of Maschler and Owen (1989) and Raiffa’s discrete bargaining solution to a large class of NTU games. Though not axiomatized, the solution is motivated via the Nash program. In this respect, we follow an approach that is similar to the one initiated by Hart and Mas-Collel (1996). 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 A game with non-transferable utility (NTU) specifies, for each coalition, the set of utility profiles that its members can guarantee themselves if they cooperate. Various values have been defined in this context, the main ones being defined by Harsanyi (1963), Shapley (1969) and Maschler and Owen (1992). Two subclasses of NTU games were especially studied: the two-player case and the games with transferable utility (TU). Up to now, the litterature was mainly concerned in defining solutions that extend both the Nash (1950) bargaining solution and the Shapley (1953) value. Axiomatic systems were designed in order to better understand and to discriminate among the various solutions (cf. Aumann (1985), Hart (1985) and de Clippel (2002)). Recently, the Nash program was also successfully applied by Hart and Mas-Collel (1996) in order to support the consistent Shapley value of Maschler and Owen (1992). In what follows, we define a single-valued solution (named the procedural value) that extends both the Shapley TU-value (even the consistent Shapley value of Maschler and Owen (1989)) and Raiffa’s discrete bargaining solution (cf. Luce and Raiffa (1957), pages 136-137) to a large class of NTU games. Though not axiomatized, the solution is motivated via the Nash program. The bargaining procedures we consider are closely related to those defined in Hart and Mas-Collel (1996). $1. Notations Let n be a positive integer and let N := {1, . . . , n} be the set of players. P (N) denotes the set of coalitions, that is the set of nonempty subsets of N . P ∗(N) denotes the set of coalitions of cardinality greater or equal to two. Elements of RN that represent a distribution of payoffs are called allocations. Let S be a coalition. For each i ∈ N , S + i (resp. S − i) stands for S ∪ {i} (resp. S \ {i}). RS is seen as a subset of RN : RS := {x ∈ RN |(∀i ∈ N \ S) : xi = 0}. Inequalities between vectors in RS are of three kinds: ≥S , >S , >>S (usual definitions). R S + (resp. R S ++) denotes the set of vectors in R S with nonnegative (resp. positive) components in S. When X is a subset of RN , projS(X) denotes the projection of X on RS . When x ∈ RN , xS denotes the projection of x on RS . When X is a subset of RS , intS(X) (resp. ∂SX) denotes the topological interior (resp. frontier) of X in RS . $2. Games A (cooperative) game V is a function that associates, to each coalition S, a nonempty closed comprehensive convex strict subset V (S) of RS such that: 1. (∀S ∈ P (N))(∀x ∈ ∂SV (S)) : {λ ∈ R+ \ {0}|λ.x = maxy∈V (S) λ.y} ⊆