Resumen de Characterization results for convex games by means of bargaining sets

Josep María Izquierdo Aznar, Carles Rafels Pallarola

• Convex games (Shapley, 1971) play a central role in the cooperative game theory.

  Most of the solutions de ned for cooperative games have a nice behavior whenever they are applied to a convex game; in particular its core is non-empty. The bargaining set is de ned by means of objections and counter-objections to payo distributions. Di erent de nitions of bargaining set can be found depending on how objections and counterobjections are made (see Davis and Maschler, 1963, and Mas-Colell, 1989). For a convex game, it holds that these bargaining sets coincide with the core of the game but this fact does not characterize its convexity. This work provides a two-fold characterization. Firstly, we state that a balanced game is convex if and only if its DM bargaining set is equal to its Weber set (Weber, 1988). Secondly, we consider a modi cation of the MC bargaining set and we prove that a game is convex if and only if its core coincides with its modi ed MC bargaining set.