Owen Value

The Owen value (Ω) is a value for cooperative games in characteristic function form, based on a bargaining model that takes account of a priori unions in coalition formation. This value, proposed by Guillermo Owen, is generally understood as a modification of the Shapley value of a game for cases in which certain sets of players may be more likely to act together than others. In the same [Page 467]work, Owen proposed an extension of his value to union structure hierarchies, that is, to subgroups of a priori unions.

Some preliminary definitions are given below, together with a presentation of the Owen value by means of an example. This is followed by a mathematical formalization of the value. The extension of the Owen value to union structure hierarchies will be introduced using another example. (Familiarity with concepts relating to the Shapley value is assumed.)

Let N={1,2,…, n} be a set of players. A cooperative game on N is said to be expressed in characteristic function form if a function v (the characteristic function) is given that assigns a value v(S) to every coalition SN where v(Ø) = 0. The game characterized by v is usually called game v. An a priori union structure is a partition T={T1 T2,…, Tm} of the set N where coalitions Tj T are unions of players who have agreed to collaborate. Of course, these unions are pairwise disjoint, and their union is N.

The Owen value, like the Shapley value, assigns each player his or her expected marginal contribution to coalitions with respect to a random order of entering players. But while the Shapley value assumes that all orders are equiprobable, the Owen value restricts the set of orders according to the coalition structure. In other words, taking all possible permutations of the Shapley value into account, the Owen value deals with the subset that keeps the members of each Tj T together.