On the Computational Complexity of the Bargaining Set and the Kernel in Compact Coalitional Games

This paper deals with the complexity of cooperative solution concepts, notably, the bargaining set and the kernel, for coalitional games in compact form. Deng and Papadimitriou have left open a number of issues regarding those concepts which this paper provides a thorough answer to. Open issues (and correspondent answers we provide) are as follows. Given a graph game G and an imputation x: (a) it was conjectured that checking for x to belong to the bargaining set of G is \Pi^P_2-complete--the conjecture will be shown to hold; (b) it was conjectured that checking for x to belong to the kernel of G is NP-hard--the conjecture will be shown to hold, in particular, by proving the tighter bound that it is in fact \Delta^P_2-complete; (c) it was asked for the complexity of checking x to belong to either the bargaining set or the kernel of G (membership, as the hardness are immediately implied by (a) and (b) above) for games G in general compact form, where the game is given implicitly by an algorithm for computing v(S)--we formally define such general compact games and prove that those complexities are in \Pi^P_2 and \Delta^P_2, resp., provided that the worth-computing algorithm runs in non-deterministic polynomial time.