Grand Mechanism and Population Uncertainty

This paper studies an informed mechanism designer problem in which the principal's private information is a number of agents. The principal designs both a protocol structure with respect to how to reveal the information and a collection of sub-mechanisms. The former is a mapping from numbers of agents to probability distributions over information revelation rules, and the latter's different sub-mechanisms may each contain unique allocation and payment functions depending on the principal's private information as well as a rule. Since choosing a protocol structure and such a collection are interwoven, we establish the existence of the grand mechanism using an expected payoff linearity. Then, we show that there is a single threshold for the optimal grand mechanism if a sub-mechanism cannot depend on the principal's private information. Interestingly, the main result shows that if a sub-mechanism can also depend on his private information, the optimal grand mechanism is characterized by double thresholds such that the principal does not announce the number of agents if it is in the middle range. We further extend the protocol structure to include rich sets of rules.