Mechanism Design with Bandit Feedback

We study a multi-round welfare-maximising mechanism design problem in instances where agents do not know their values. On each round, a mechanism assigns an allocation each to a set of agents and charges them a price; then the agents provide (stochastic) feedback to the mechanism for the allocation they received. This is motivated by applications in cloud markets and online advertising where an agent may know her value for an allocation only after experiencing it. Therefore, the mechanism needs to explore different allocations for each agent, while simultaneously attempting to find the socially optimal set of allocations. Our focus is on truthful and individually rational mechanisms which imitate the classical VCG mechanism in the long run. To that end, we define three notions of regret for the welfare, the individual utilities of each agent and that of the mechanism. We show that these three terms are interdependent via an $\Omega(T^{\frac{2}{3}})$ lower bound for the maximum of these three terms after $T$ rounds of allocations, and describe a family of anytime algorithms which achieve this rate. Our framework provides flexibility to control the pricing scheme so as to trade-off between the agent and seller regrets, and additionally to control the degree of truthfulness and individual rationality.