Final Activity Report Summary - D2CG (Introducing Democracy to Coalitional Games)
While for non-cooperative games the Nash equilibrium is widely accepted as the solution in the sense that if there is a solution, it must be the Nash equilibrium, for cooperative games, where players can form credible commitments and thus actions by groups of players, coalitions must also be taken into account and the debate is far from settled. All scientists agree that any sensible solution must be like the core, defined as the set of imputations where no coalition can earn by deviating; nevertheless for some games the core is empty and thus makes no prediction whatsoever. When coalition formation creates externalities, that is, effects beyond the coalitions formed, even the definition of the core is subject to debate.
During our research we had three groups of results. For games with externalities, such as international environmental agreements, we introduced the recursive core. In that case, where the core was seen as an equilibrium concept, the deviation of a coalition or coalitions of players resulted in a residual game and the deviating players assumed that the residual game settled to the same equilibrium. Further, we showed that this solution concept could be supported by order-independent equilibria of a non-cooperative coalition formation process.
For games with coalition structures we introduced the minimal dominant set: a nonempty solution that coincided with a nonempty core. Thus, it was the closest proper solution one could get to the core: in case the core 'worked', the two coincided, while the minimal dominant set always provided a prediction of the outcome of the game. The results were shown using a very restricted, democratic form of dominance, where deviating players could not influence other coalitions' payoffs.
The minimal dominant set was motivated by a short-sighted, but dynamic interpretation of cooperative games. We also showed that if the grand coalition formed, that is, if all players agreed to cooperate and the debate was only about sharing the value of the grand coalition, a mediator could lead the players to core imputation in a bounded number of steps. Seemingly a technical detail, this result showed that negotiations make sense: if an agreement were possible, negotiations could end before a predefined deadline.
During our research we had three groups of results. For games with externalities, such as international environmental agreements, we introduced the recursive core. In that case, where the core was seen as an equilibrium concept, the deviation of a coalition or coalitions of players resulted in a residual game and the deviating players assumed that the residual game settled to the same equilibrium. Further, we showed that this solution concept could be supported by order-independent equilibria of a non-cooperative coalition formation process.
For games with coalition structures we introduced the minimal dominant set: a nonempty solution that coincided with a nonempty core. Thus, it was the closest proper solution one could get to the core: in case the core 'worked', the two coincided, while the minimal dominant set always provided a prediction of the outcome of the game. The results were shown using a very restricted, democratic form of dominance, where deviating players could not influence other coalitions' payoffs.
The minimal dominant set was motivated by a short-sighted, but dynamic interpretation of cooperative games. We also showed that if the grand coalition formed, that is, if all players agreed to cooperate and the debate was only about sharing the value of the grand coalition, a mediator could lead the players to core imputation in a bounded number of steps. Seemingly a technical detail, this result showed that negotiations make sense: if an agreement were possible, negotiations could end before a predefined deadline.