Racionalitat i monotonia en jocs cooperatius: possibilitats i impossibilitats

Abstract

[en] A single-valued solution for cooperative games suggest how to allocate what all agents can get if they cooperate. In this work, we analyze which of the most important solutions satisfy certain rationality and/or monotonicity properties. In particular we study the behavior of well-known solutions such as the Shapley value, the prenucleolus, and the per-capita prenucleolus regarding the core selection property and monotonicity properties such as coalitional monotonicity. Core selection imposes that whenever it is possible, the recommendation made by a solution should not give incentives to individual agents or any coalition to break cooperation. It is for this reason that we consider it as a rationality property. On the other hand, coalitional monotonicity requires that as long as one coalition becomes stronger (and the value of the rest of coalitions does not vary), no member of the coalition is strictly worse off. We show how the imposition of these two properties will make it impossible to find a solution that satisfies both. Next, we relax the monotonicity property to aggregate monotonicity. This property demands that if the coalition of all players becomes stronger (while the value of the rest of coalitions remains the same), no agent is strictly worse off. In this case, not only the perapita prenucleolus satisfies both properties, but also a whole set of solutions does, of which we study its geometry. Finally, we define a new monotonicity property, weak coalitional monotonicity, and we leave the door open to a future study of whether or not it is compatible with core selection.