CONVEXITY of PRODUCTION, COMMON POOL and OLIGOPOLY GAMES

A cooperative game with transferable utility (TU) is a pair 〈N, v〉, where N is a nonempty, finite set and v : 2N → R is a characteristic function, defined on the power set of N , satisfying v(∅) := 0. An element of N (notation: i ∈ N) and a nonempty subset S of N (notation: S ⊆ N or S ∈ 2N with S 6= ∅) is called a player and coalition respectively, and the associated real number v(S) is called the worth of coalition S. Concerning the modelling part of game theory, it is customary to investigate whether or not any appropriate class of cooperative TU games satisfies one or another appealing property. Without going into details, we state that the so-called convexity property of the characteristic function v is a widely-spread concept through the game theory literature. Any cooperative TU game 〈N, v〉 is said to be a convex game (cf. Shapley, [6], 1971) if it holds v(S) + v(T ) ≤ v(S ∪ T ) + v(S ∩ T ) for all S, T ⊆ N or equivalently, for all i ∈ N , j ∈ N , i 6= j, and all S ⊆ N\{i, j}, it holds