Continuous analogue of bilateral inequalities implying cap-monotone rules

Establish whether bilateral value inequalities alone imply cap-monotone allocation rules for continuous type distributions, including a continuous counterpart of the finite minimax and smallest-mass construction with measurable selections and valid limiting arguments.

Background

The matching-market reduction requires local allocation rules whose trade probabilities are monotone in the seller cap. For finite supports, the paper derives such rules from bilateral value inequalities using a minimax argument and a smallest-mass construction.

For continuous distributions, the authors instead construct the MHR rules directly. They leave unresolved whether bilateral inequalities by themselves suffice to generate cap-monotone rules in the continuous setting, together with the associated measurability and limiting technicalities.