  We study a multi-round welfare-maximising mechanism design problem in
instances where agents do not know their values. On each round, a mechanism
first assigns an allocation each to a set of agents and charges them a price;
at the end of the round, the agents provide (stochastic) feedback to the
mechanism for the allocation they received. This setting 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 so that it can
learn their values, 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 first 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 an
algorithm which essentially achieves this rate. Our framework also provides
flexibility to control the pricing scheme so as to trade-off between the agent
and seller regrets. Next, we define asymptotic variants for the truthfulness
and individual rationality requirements and provide asymptotic rates to
quantify the degree to which both properties are satisfied by the proposed
algorithm.
