Classical Cooperative Theory I: Core-Like Concepts

Pure bargaining games discussed in the previous two lectures are a special case of n-person cooperative games. In the general setup coalitions other than the grand coalition matter as well. The primitive is the coalitional form (or, "coalitional function", or "characteristic form"). The primitive can represent many different things, e.g., a simple voting game where we associate to a winning coalition the worth 1 and to a losing coalition the worth 0, or an economic market that generates a cooperative game. Von Neumann and Morgenstern (1944) suggested that one should look at what a coalition can \ guarantee (a kind of a constant-sum game between a coalition and its complement); however, that might not always be appropriate. Shapley and Shubik introduced the notion of a C;-game (see Shubik (1982)): it is a game where there is no doubt on how to define the worth of a coalition. This happens, for example, in exchange economies where a coalition can reallocate its own resources, independent of what the complement does. We assume we are given a coalitional function. Let N denote the set of players; a subset ScN is called a coalition; V(S) is the set of feasible outcomes for S. How is an outcome defined? Assuming that some underlying utility functions for the players are specified, one can represent outcomes by the players' utilities. V:1ethus use a payoffvectora=(ai)ieS in 9{S to rep-resent an outcome, where al is player ilth utility of the outcome. So V(S)c9\S. Usually there are some assumptions made on the set V(S); e.g., comprehensive, closed, convex, etc. There are two special classes of games: 1) Pure bargaining games (FE): In these games only the grand coalition matters. Here V(S)={ x E9\S such that xiso for all j eS} for all2 S=t:-N.