Consistent solutions for cooperative games with externalities

A recursive core for partition function form games

“…Note that the result is always an outcome, but it might be inferior to the status quo. The r-theory (Huang and Sjöström, 2003) converts a normal form game (over a strategy space and utility function u: → IR N ) to a CF using the same concept for the response as for the original game. They define the worth W (S|T , P N \T ) of coalition S given that players in N\T have formed P N \T , which will then help to determine the set of strategies C(S|S, P N \S ) that can possibly be played after the departure of P N \T , using the solution, for example the core:…”

“…Slightly obscured by the complex notation used in normal form games, Huang and Sjöström (2003) have also used consistency, although in their model residual games are similar to, but not identical with an original game 1 and therefore the consistency argument is less natural. The r-core is only defined if all residual cores are non-empty -a demanding condition for large games, not even satisfied by all CFF games and hence the r-core is not a generalisation of the core.…”

“…This model generalises the α-core (Aumann and Peleg, 1960), the optimistic approach by Shenoy (1979), the γ -core (Chander and Tulkens, 1997) or the status quo or the δ-approach (see also Hart and Kurz, 1983), each of which can arise in particular games. The r-core (Huang and Sjöström, 2003) also allows arbitrary reactions, but works via a characteristic function that is often undefined.…”

The Nash bargaining solution in general n-person cooperative games

“…Huang and Sjöström [14] and Kóczy [17] define the notion of a recursive core (abbreviated by 'r-core') by a theory of a solution's consistency that is similar to that of the Nash core. Roughly, the recursive core of a cooperative game with externality 4 is a variant of the core defined by the assumption that when a coalition forms, its members predict an outcome in a core of the "reduced game" composed of other players.…”

“…First, the works of Huang and Sjöström [14] and Kóczy [17] are motivated by a cooperative game approach with the aim of refining the core solution in a consistent manner. In contrast, our approach is a noncooperative one.…”