Optimizing time and convenience in group scheduling

We consider the situation in which a group of agents are trying to find an agreeable time for a meeting. Each agent's availability is governed by a known prior probability distribution, but the realization of it is private information. The goal (of the convener) is to find an agreeable time slot while optimizing in two dimensions: Expected time to reach agreement and expected inconvenience caused during the process. We take the most common web-based tool in use today -- Doodle -- as the starting point. We generalize Doodle to a class of B-Doodle mechanisms (\"Batched Doodle\") and study the trade-off between the Time-Inconvenience dimensions. In a B-Doodle mechanism, the convener iteratively asks agents about their availability for one or more time slots, until an agreement can be reached. Among (exponentially) many B-Doodle mechanisms, the main objective is to find a B-Doodle mechanism that minimizes the aggregate cost of Time and Inconvenience (such as a linear combination of the two). We provide an efficient algorithm that achieves this for a broad class of cost functions, and show in simulations that the result substantially outperforms Doodle in many realistic settings.