Monotonicity and revenue equivalence domains by monotonic transformations in differences

In a mechanism design setting with quasilinear preferences, a domain D of admissible valuations of an agent 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 revenue equivalence. Carbajal and Müller (2015) introduced the notions of monotonic transformations in differences and showed that if D admits these transformations then it is a revenue equivalence and monotonicity domain. Here, we show that various economic domains, with countable or uncountable allocation sets, admit monotonic transformations in differences. Our applications include public and private supply of divisible public goods, multi-unit auction-like environments with increasing valuations, and allocation problems with externalities. Single-peaked domains admit only a modified version of monotonic transformations in differences. We show that this property implies too that single-peaked domains are revenue and monotonicity domains.