Preserving coalitional rationality for non-balanced games

Abstract

International audienceIn cooperative games, the core is one of the most popular solution concept since it ensures coalitional rationality. For non-balanced games however, the core is empty, and other solution concepts have to be found. We propose the use of general solutions, that is, to distribute the total worth of the game among groups rather than among individuals. In particular, the k-additive core proposed by Grabisch and Miranda is a general solution preserving coalitional rationality which distributes among coalitions of size at most k, and is never empty for k ≥ 2. The extended core of Bejan and Gomez can also be viewed as a general solution, since it implies to give an amount to the grand coalition. The k-additive core being an unbounded set and therefore difficult to use in practice, we propose a subset of it called the minimal negotiation set. The idea is to select elements of the k-additive core mimimizing the total amount given to coalitions of size greater than 1. Thus the minimum negotiation set naturally reduces to the core for balanced games. We study this set, giving properties and axiomatizations, as well as its relation to the extended core of Bejan and Gomez. We give a method of computing the minimum bargaining set, and lastly indicate how to eventually get classical solutions from general ones

Autonomous coalitions

“…Theorem 9 (Gonzalez and Grabisch 2015). Let v be a game on N , and S be a coalition of N such that |S| ≥ 2.…”

“…Given x ∈ R n , and S ⊆ N , denote by x(S) the sum i∈S x i with the convention that x(∅) = 0. Gonzalez and Grabisch (2015) is a vector x ∈ R 2 N \∅ that assigns to coalition S ⊆ N the payoff x S .…”

“…One way to escape this situation, among the many others which have been studied in the literature, is that we allow for a more general situation of cooperation (the grand coalition may be not the only state of cooperation), or a more general notion of allocation. In the latter case, Grabisch and Miranda (2008) have proposed to share the worth of the grand coalition not only among individual players, but also among all coalitions, up to a certain size k. This is called the k-additive core, and this notion has been further refined in (Gonzalez and Grabisch 2015), leading to the negotiation set. In the former case, there exists natural generalizations of the core, once it is admitted that players may eventually form either disjoint coalitions, i.e., a partition of the grand coalition, or a collection of coalitions which has the property to be balanced: interpreting the balancing weights as amounts of time and assuming that each player has one unit of time to spend [see for example Peleg and Sudhölter (2003)], this unit can be split among different coalitions of this collection.…”

“…The aim of this paper is to study this question in depth. One of the main contribution of this paper is that the "strong" coalitions are the autonomous coalitions, introduced in (Gonzalez and Grabisch 2015). An autonomous coalition of a game is defined as a coalition such that any d-core element yields a payment for that coalition equal to its worth.…”

Optimal deterrence of cooperation

“…Even in the case where the core is empty the literature has investigated the possibility to enforce a stable cooperative agreement by introducing other core solution concepts: the strong and the weak ε-cores (Shapley and Shubik, 1966), the (weak) least core (Maschler et al, 1979;Young et al, 1982), the aspiration core (Albers, 1979;Cross, 1967;Bennett, 1983), the extended core (Bejan and Gómez, 2009), the negotiation set (Gonzalez and Grabisch, 2015b) and the d-multicoalitional core (Gonzalez and Grabisch, 2016). Some of these variants of the core are non-empty when applied to non-balanced TU-games and coincide with the core on the set of balanced TU-games.…”