Nonempty Core-Type Solutions Over Balanced Coalitions in Tu-Games

Abstract

In this paper we introduce two related core-type solutions for games with transferable utility (TU-games) the B-core and the M-core. The elements of the solutions are pairs (x, B), where x, as usual, is a vector representing a distribution of utility and B is a balanced family of coalitions, in the case of the B-core, and a minimal balanced one, in the case of the M-core, describing a plausible organization of the players to achieve the vector x. Both solutions extend the notion of classical core but, unlike it, they are always nonempty for any TU-game. For the M-core, which also exhibits a certain kind of "minimality" property, we provide a nice axiomatic characterization in terms of the four axioms nonemptiness (NE), individual rationality (IR), superadditivity (SUPA) and a weak reduced game property (WRGP) (with appropriate modifications to adapt them to the new framework) used to characterize the classical core. However, an additional axiom, the axiom of equal opportunity is required. It roughly states that if (x, B) belongs to the M-core then, any other admissible element of the form (x, B ) should belong to the solution too.

Multicoalitional solutions

“…Remark 2. As we can see from the following example, unlike the d-core or the B-core (Cesco, 2012) it is not necessary to require that each player is active during exactly one unit of time. Moreover, (x, (α i ) i∈N ) in the d-multicoalitional core does not necessarily imply that (1) = α 1 (12) = α 1 (13) = 1, α 2 (12) = α 3 (13) = 1, and α i (S) = 0 otherwise.…”

“…In his article about the d-core, Cross (1967) described informally his solution as a set of stable coalitions with their associated payoffs. The B-core and the M-core proposed by Cesco (2012) are closely related to or constitute a continuation of the paper of Cross, by considering the set of coalitions which lead to a payoff into the d-core. Under this view, a cooperative TU-game solution should be not only composed of the payoffs given to each player but should also comprise the time alloted to each coalition which permits to achieve these payoffs.…”

Autonomous coalitions

“…Literature focuses essentially on the set of payoffs without considering the properties of the set of corresponding coalitions. Cesco (2012) gives and axiomatizes a new kind of solution which associates to each game not only an amount of the d-core but also the coalitions which permit to reach this amount, that is, the coalitions S such that there exists a maximising system λ * ∈ B(N ) which satisfies λ(S) > 0. Bejan and Gómez (2012a) implements the d-core in strong Nash equilibria, corresponding strategies consist of giving an amount of the d-core and an arrangement of the players in time given by a maximising system which permits to reach this amount.…”

“…It is important to note that the partition, or the balanced collection, which represents the final state of cooperation, is chosen so as to maximise the total worth obtained by cooperation, and for this reason, we call them maximising collections. In the case of a partition, the total worth is simply the sum of the worth of the coalitions in the partition, while for a balanced collection, one need to take into account the time allocated to each coalition: considering a utility linear in time, one simply compute the weighted sum of the worths, where the weights corresponds to the allotted times: see Cesco (2012) for an axiomatization of such kind of solution, and Bejan and Gómez (2012a), who implements the d-core in strong Nash equilibria.…”

Axiomatizing core extensions on NTU games

“…However, the power of the core concept is limited by the fact that the non-emptiness of the core may be assured only in certain ideal environments where the grand coalition formation is reasonable. A natural nonempty extension of the core is the aspiration core introduced by Cross (1967) [see also Albers (1979), Bennett (1983), Bejan and Gómez (2012) and Cesco (2012)]. The idea behind the aspiration core is to search those outcomes generated by non-trivial families of coalitions called balanced families that no coalition can improve.…”