Resumen de Monotonicity of Social Optima with respect to Participation Constraints

Lars P. Osterdal, Jens L. Hougaard, Hans Keiding

• In this paper we consider the problem of selecting a social optimum satisfying a participation constraint for any coalition, formulated as a problem of selecting from the core. In the theory of allocation rules, a fundamental result is that of Young (1985) stating that there is no rule for selecting an element of the core of all transferable utility (TU) games which satisfies coalitional monotonicity (in the sense that when the worth of a coalition increases, and the worth of all other coalitions remain unchanged, then the payoff given to any member of the coalition according to the rule should not decrease).

  The result shows that there is a certain trade-off between coalitional stability (that is, selecting from the core) and monotonicity. Monotonicity is an important property in the context of cost sharing rules where its failure means that some agents may lack incentives for reducing cost (cf. e.g. Shubik, 1962;

  Young, 1994). On the other hand, the assumption of coalitional stability is also a desirable one, and therefore there is a need for investigating the extent to which the two conficting goals may be reconciled. That this is indeed possible is witnessed by several contributions to the literature dealing with convex TU games: The Shapley value is indeed coalitionally monotonic (Shapley, 1971;

  Sprumont, 1990; Rosenthal, 1990), whereas the nucleolus is not (Hokari, 2000).

  In the present paper, we consider a class of core selection rules where the payoff chosen is found by maximizing a social welfare function on the core of the game. When this social welfare function is given, the solution for any given game will depend only on the core of the game and not on other structural characteristics of the game. This contracts with the situation for, say the nucleolus, which may yield different results for two games with identical cores, a feature which may make it difficult to justify a solution from a social planner�s point of view.

  While we concentrate on a particular class of core selection rules, we put no restrictions on the class of TU games considered (except of course that the core should be nonempty). Since our basic aim is to investigate the trade-offs between coalitional stability and monotonicity, and keeping in mind the generel impossibility result, we shall be satisfied with monotonicity with respect to the worth of any particular coalition (in the following called S-monotonicity).

  It turns out that there are such rules, and moreover, all such rules can be characterized in terms of the social desirability of giving payoff to the members of the coalition S. As it turns out, this social desirability has a simple expression in terms of the marginal social welfare of each individual�s payoff.

  While there are S-monotonic core selections on the domain of all TU games with nonempty core, this turns out to be a feature which is closely connected with the transferability of utility. Indeed, if NTU games are considered, then the previously derived results (characterizing S-monotonic core selection rules) may be exploited to yield an impossibility result: Even for any fixed coalition S there cannot be core selection rules that are monotonic. This shows that the non-monotonicity in coalitional worth is a feature which is inherent in any solution satisfying the coalitional stability conditions, that is in any core selection.

  The paper is structured as follows: In Section 2, we give the necessary definitions (in the context of TU games); the basic characterization result is given as Theorem 1 in Section 3; it exploits the characterization of games with nonempty core by balancedness and consequently needs some details concerning suitable balanced or non-balanced families of coalitions, which are developed as we proceed. In the following section, we establish the relationship between S-monotonicity and the partial derivatives of the social welfare function. Finally, in Section 5 we extend the investigation to NTU games and give the impossibility result in this context.