Core Solutions in Vector-Valued Games

Cost Sharing under Uncertainty: An Algorithmic Approach to Cooperative Interval-Valued Games

“…See also Branzei, Tijs, and Alparslan-Gök (2008) for a survey. A cooperative interval-valued game is defined by a pair (N, c I ) where N is an index set of players as before, and c I : 2 N → I(I R) is a characteristic (cost) function which assigns to every coalition S ∈ 2 N a closed interval c I (S) = [c I (S);c I (S)] (with c I (∅) = [0; 0]) where I(I R) is the set of all closed intervals in I R. It is easy to see that classical cooperative games are a special case of cooperative interval-valued games where c I (S) =c I (S) for all coalitions S ⊆ N. It should also be remarked that vector-valued games (see Fernandez, Hinojosa, and Puerto 2002) are somewhat different to interval-valued games. To define a core variant that applies to interval- The interval I is called weakly better than interval J (I J or J I), iff I ≤ J andĪ ≤J.…”

Partially ordered cooperative games: extended core and Shapley value

“…On the other hand, it applies to the important class of vector valued games (Fernández et al 2002a) (i.e. those games whose values are taken on R m ).…”

“…Our analysis is completely different instead of imposing conditions on the argument of the characteristic function we extend the nature of the payoffs. Particular instances of this model have been already considered in (Fernández et al 2002a;Granot 1977;Nishizaki and Sakawa 2001;Suijs 2000;Suijs et al 1998Suijs et al , 1999Timmer 2001).…”

“…It is worth noting that the well-known scalar, vector-valued (Fernández et al 2002a) and stochastic cooperative games (Fernández et al 2002b;Suijs et al 1999) are particular cases of this formulation, just considering ℵ = R with the ≥ order over the reals, R n with the component-wise order and L 1 (R) with the stochastic dominance order, respectively. The cases above are examples that can be found in the literature of game theory although one can think of many other interesting structures where this approach can be applied.…”

A multi‐objective model for bank ATM networks

“…An extensive analysis for the class of vector-valued transferable utility (TU) games is found in Hinojosa [6], and Fernández, Hinojosa and Puerto [4] where the classical individual and collective rationality principles are extended using two dif-ferent orderings in the payoff space. The first ordering reflects a very compromising attitude in negotiation with coalitions admitting payoffs that are not worse in all the components than any payoffs that they can ensure by themselves.…”