A decomposition for the space of games with externalities

Abstract

The main goal of this paper is to present a di¤erent perspective than the more 'traditional' approaches to study solutions for games with externalities. We provide a direct sum decomposition for the vector space of these games and use the basic representation theory of the symmetric group to study linear symmetric solutions. In our analysis we identify all irreducible subspaces that are relevant to the study of linear symmetric solutions and we then use such decomposition to derive some applications involving characterizations of classes of solutions.

Solutions for network games and symmetric group representations

“…The calculation of value may involve both costs and benefits, and is a richer object than a characteristic function of a cooperative game 5 , as it allows the value that accrues to depend on the network structure and not only on the coalition of players involved.…”

“…al. [4] for games in characteristic function form, and Sánchez-Pérez [5] for games in partition function form. By contrast, for a survey of the ways in which the representation theory of the symmetric group is used in voting theory, see Crisman and Orrison [2].…”

“…For instance, a buyer's expected utility from trade may depend on how many sellers that buyer is negotiating with, on how many other buyers they are connected to, etc. Similarly, a network where players have very few acquaintances with whom they share information will result in different employment patterns from the one where players have many such acquaintances 5. Formally, a cooperative game in characteristic function form is defined as a function v:2 N   with the property that v(AE) = 0.…”