Estimating Approximate Incentive Compatibility

In practice, most mechanisms for selling, buying, matching, voting, and so on are not incentive compatible. We present techniques for estimating how far a mechanism is from incentive compatible. Given samples from the agents' type distribution, we show how to estimate the extent to which an agent can improve his utility by misreporting his type. We do so by first measuring the maximum utility an agent can gain by misreporting his type on average over the samples, assuming his true and reported types are from a finite subset---which our technique constructs---of the type space. The challenge is that by measuring utility gains over a finite subset of the type space, we might miss pairs of types t and t' where an agent with type t can greatly improve his utility by reporting the type t'. Our technique discretizes the type space by constructing a learning-theoretic cover in a higher-dimensional space. The key technical contribution is proving that the maximum utility gain over this finite subset nearly matches the maximum utility gain overall, despite the volatility of the utility functions we study. We apply our tools to the single-item and combinatorial first-price auctions, generalized second-price auction, discriminatory auction, uniform-price auction, and second-price auction with spiteful bidders. To our knowledge, these are the first guarantees for estimating approximate incentive compatibility from the mechanism designer's perspective.