Monotonic extensions on economic domains

The property of "monotonicity" is necessary, and in many contexts, sufficient, for a solution to be Nash implementable (Maskin 1977). In this paper, we follow Sen (1995) and evaluate the extent to which a solution may fail monotonicity by identifying the minimal way in which it has to be enlarged so as to satisfy the property. We establish a general result relating the "minimal monotonic extensions" of the intersection and the union of a family of solutions to the minimal monotonic extensions of the members of the family. We then calculate the minimal monotonic extensions of several solutions in a variety of contexts, such as classical exchange economies, with either individual endowments or a social endowment, economies with public goods, and one-commodity economies in which preferences are single-peaked. For some of the examples, very little is needed to recover monotonicity, but for others, the required enlargement is quite considerable, to the point that the distributional objective embodied in the solution has to be given up altogether.