An Axiomatization of the Core of Market Games

As shown by Peleg, the core of market games is characterized by nonemptiness, individual rationality, superadditivity, the weak reduced game property, the converse reduced game property, and weak symmetry. It was not known whether weak symmetry was logically independent. With the help of a certain transitive 4-person TU game it is shown that weak symmetry is redundant in this result. Hence the core on market games is axiomatized by the remaining five properties, if the universe of players contains at least four members. In this note we solve an open problem of the theory of cooperative games which arises in a natural way in the context of the characterization of the core of market games due to Peleg ([1],[2]). Theorem 2 of [2] shows that the core of market games is characterized by nonemptiness (NE), individual rationality (IR), superadditivity (SUPA), the weak reduced game property (WRGP), the converse reduced game property (CRGP), and weak symmetry (WS). (Precise definitions of properties are recalled below.) The assertion without the assumption of WS was formulated in [1], but it turned out that WS, which is a weak variant of anonymity, was needed in addition in the proof of the uniqueness part. The problem whether WS is logically independent was mentioned in [2]. With the help of a “cyclic” 4-person game (its symmetry group is generated by a cyclic permutation) we show that the core of market games is characterized by NE, IR, SUPA, WRGP, and CRGP (see Theorem 3). Thus, WS is redundant. We adopt the notation of [1]. Let U be a set of players satisfying |U | ≥ 4, let us say M := {1, 2, 3, 4} is contained in U . A (cooperative TU) game is a pair (N, v) such that ∅ 6 = N ⊆ U is finite and v : 2N → IR, v(∅) = 0. For any game (N, v) let X∗(N, v) = {x ∈ IR | x(N) ≤ v(N)} and X(N, v) = {x ∈ IR | x(N) = v(N)} denote the set of feasible and Pareto optimal feasible payoffs (preimputations), respectively. The core of (N, v) is given by C(N, v) = {x ∈ X∗(N, v) | x(S) ≥ v(S) ∀S ⊆ N}. ∗The first author was partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). †Also at the Institute of Mathematical Economics, University of Bielefeld. Email: [email protected] ‡Email: [email protected] A game is balanced, if its core is nonempty, and totally balanced, if every subgame is balanced. A solution σ on a set Γ of games associates with each game (N, v) ∈ Γ a subset of X∗(N, v). Let θ denote the set of all totally balanced games. Let σ be a solution on a set Γ of games. σ satisfies nonemptiness (NE), if σ(N, v) 6= ∅ for every (N, v) ∈ Γ. σ is covariant under strategic equivalence (COV), if for (N, v), (N,w) ∈ Γ with w = αv + β∗ for some α > 0, β ∈ IRN the equation σ(N,w) = ασ(N, v) + β holds. Here, (N, β∗) is the inessential (additive) game given by β∗(S) = ∑ i∈S βi. In this case the games v and w are called strategically equivalent. σ is individually rational (IR), if xi ≥ v({i}) holds for every (N, v) ∈ Γ, for every x ∈ σ(N, v), and for every i ∈ N . σ is superadditive (SUPA), if σ(N, v) + σ(N,w) ⊆ σ(N, v + w), when (N, v), (N,w), (N, v + w) ∈ Γ. Let (N, v) be a game, x ∈ X∗(N, v), and ∅ 6= S ⊆ N . The reduced game (S, vx,S) with respect to S and x is defined by vx,S(∅) = 0, vx,S(S) = v(N)− x(N \ S) and vx,S(T ) = maxQ⊆N\S v(T ∪Q)− x(Q) for every T 6= ∅, S. σ satisfies the weak reduced game property (WRGP), if (S, vx,S) ∈ Γ and xS ∈ σ(S, vx,S) for every (N, v) ∈ Γ, for every S ⊆ N with 1 ≤ |S| ≤ 2, and every x ∈ σ(N, v). Here xS denotes the the restriction of x to S. σ satisfies the converse reduced game property (CRGP), if for every (N, v) ∈ Γ with |N | ≥ 2 the following condition is satisfied for every x ∈ X(N, v): If, for every S ⊆ N with |S| = 2, (S, vx,S) ∈ Γ and xS ∈ σ(S, vx,S), then x ∈ σ(N, v). Remark 1 It is well-known (see, e.g., [1]) that the core satisfies IR, SUPA, CRGP, and COV on every set Γ of games. Moreover, it satisfies NE and WRGP on θ. The 4-person game (M,u), defined by u(S) =  0 , if S ∈ {M, {1, 2}, {2, 3}, {3, 4}, {4, 1}, ∅} , −1 , if |S| = 3 , −4 , otherwise , will be used in the proof of Theorem 3. Note that the symmetry group of (M,u) is generated by the cyclic permutation, which maps 1 to 2, 2 to 3, 3 to 4, and 4 to 1; thus the game is transitive. (A game is called transitive, if its symmetry group is transitive.) Remark 2 (1) The core of (M,u) is the line segment with the extreme points (−1, 1,−1, 1) and (1,−1, 1,−1), i.e., C(M,u) = {(γ,−γ, γ,−γ) ∈ IR | −1 ≤ γ ≤ 1}. Indeed, every member xγ := (γ,−γ, γ,−γ) ∈ IRM with −1 ≤ γ ≤ 1 of this line segment belongs to the core. On the other hand, every core element assigns zero to M and, thus, to the members of the partitions {{1, 2}, {3, 4}} and {{2, 3}, {4, 1}}. Therefore the core is contained in the line {xγ | γ ∈ IR}. The facts xγ({1, 2, 3}) < −1 ∀γ < −1 and xγ({2, 3, 4}) < −1 ∀γ > 1 show that the core has the claimed shape. (2) Let x = xγ ∈ C(M,u). Then the reduced coalitional function w := ux,{1,2} is given by w({1}) = maxQ⊆{3,4} u({1} ∪Q)− x(Q) = u({4, 1})− x4 = γ, w({2}) = maxQ⊆{3,4} u({2} ∪Q)− x(Q) = u({2, 3})− x3 = −γ, w({1, 2}) = u(M)− x({3, 4}) = 0, and w(∅) = 0,