Monotonicity and implementability: extended abstract

Consider a model with finite number of goods, and with buyers with private values and quasi-linear utility functions. A domain of valuation functions for a buyer is a monotonicity domain if every finite-valued monotone randomized allocation rule defined on it is implementable, in the sense that there exists a randomized truth-telling direct mechanism that implements this allocation rule. We prove that a domain of valuations of dimension at least 2 is a monotonicity domain if and only if its closure is convex.