Implementability under monotonic transformations in differences

In a social choice setting with quasilinear preferences and monetary transfers, a domain D of admissible valuations is called a monotonicity domain if every 2-cycle monotone allocation rule is truthfully implementable (in dominant strategies). D is called a revenue equivalence domain if every implementable allocation rule satisfies the revenue equivalence property. We introduce the notions of monotonic transformations in differences, which can be interpreted as extensions of Maskin's monotonic transformations to quasilinear environments, and show that if D admits these transformations then it is a monotonicity and revenue equivalence domain. Our proofs are elementary and do not rely on strenuous additional machinery. We illustrate monotonic transformations in differences for settings with finite and infinite allocation sets