Consider agents who undertake costly effort to produce stochastic outputs observable by a principal. The principal can award a prize deterministically to the agent with the highest output, or to all of them with probabilities that are proportional to their outputs. We show that, if there is...

We consider abstract exchange mechanisms wherein individuals submit "diversified" offers in m commodities, which are then redistributed to them. Our first result is that if the mechanism satisfies certain natural conditions embodying "fairness" and "convenience" then it admits unique prices, in...