Optimal Mechanisms with Finite Agent Types

In some mechanism design problems, finite-type spaces may be natural ones to consider, yet the current literature is dominated by analyses of continuous-type spaces. This probably derives from the intellectual dominance of the early work in this area. Here we present an analysis of the finite-state case that unifies and generalizes current understanding of these problems. We analyze general quasi-linear utility functions among asymmetric agents with an arbitrary number of finite types, in the context of incentive-compatible direct revelation games. A key part of the analysis is the relationship between expected benefit functions that feature alternative forms of supermodularity that translate into relaxations or restrictions of the original problem. The required features can be attained with a range of assumptions on the model primitives, each of which can support the results. This unified approach can suggest a range of alternative assumption combinations, often more general than their counterparts in the continuous-space literature, and each of which can reduce the problem to a more tractable form. Also, mechanism design problems with finite-type spaces can require conscious attention to how the principal handles ties, which are probability-zero events, and hence innocuous in continuous spaces.