Maximizing Revenue with Limited Correlation: The Cost of Ex-Post Incentive Compatibility

In a landmark paper in the mechanism design literature, Cremer and McLean (1985) (CM for short) show that when a bidders valuation is correlated with an external signal, a monopolistic seller is able to extract the full social surplus as revenue. In the original paper and subsequent literature, the focus has been on ex-post incentive compatible (or IC) mechanisms, mechanisms where truth telling is an ex-post Nash equilibrium. This has been viewed as an innocuous assumption due to its sufficiency for full revenue extraction in the CM world. However, more realistic models are unlikely to satisfy the assumptions of CM. In this paper, we explore the implications of Bayesian versus ex-post IC in a correlated valuation setting. We generalize the full surplus extraction result to settings of limited correlation, and we give necessary and sufficient conditions for full surplus extraction that strictly relax the original conditions given in CM. These more general conditions characterize the situations under which requiring ex-post IC leads to a decrease in expected revenue relative to Bayesian IC. We also demonstrate that the expected revenue from the optimal ex-post IC mechanism guarantees at most a (|Θ| + 1)/4 approximation to that of a Bayesian IC mechanism, where |Θ| is the number of bidder types. Finally, by using techniques from automated mechanism design, we are able to show that, for randomly generated distributions, the average expected revenue achieved by Bayesian IC mechanisms is significantly larger than that for ex-post IC mechanisms.