Nash implementation with little communication

The paper considers the communication complexity (measured in bits or real numbers) of Nash implementation of social choice rules. A key distinction is whether we restrict to the traditional one‐stage mechanisms or allow multistage mechanisms. For one‐stage mechanisms, the paper shows that for a large and important subclass of monotonic choice rules—called intersection monotonic —the total message space size needed for one‐stage Nash implementation is essentially the same as that needed for “verification” (with honest agents who are privately informed about their preferences). According to Segal (2007), the latter is the size of the space of minimally informative budget equilibria verifying the choice rule. However, multistage mechanisms allow a drastic reduction in communication complexity. Namely, for an important subclass of intersection‐monotonic choice rules (which includes rules based on coalitional blocking such as exact or approximate Pareto efficiency, stability, and envy‐free allocations), we propose a two‐stage Nash implementation mechanism in which at most 5 alternatives plus 4 N  log 2   N bits are announced in any play. Such two‐stage mechanisms bring about an exponential reduction in the communication complexity of Nash implementation for discrete communication measured in bits or a reduction from infinite‐ to low‐dimensional continuous communication.