Summaries: Assessing the Robustness of Cremer-McLean Mechanism Design and Maximizing Revenue with Limited Correlation: The Cost of Ex-Post Incentive Compatibility

and Introduction A result by Cremer and McLean in 1985 shows that ”if buyers’ valuations are sufficiently correlated, there exists a mechanism that allows the seller to extract the full surplus from efficient allocation as revenue”. This paper uses automated mechanism design to examine the sensitivity of the Cremer-McLean result when its main technical assumption is relaxed. The Cremer-McLean result assumes that each buyer valuation corresponds to a unique conditional distribution over external signal(s). It also assumes that these conditional distributions are linearly independent. This paper explores the relaxation of these assumptions by allowing multiple buyer types to have the same distribution over the signal(s)(but different valuations). In this paper, the authors partition the set of buyer types into subsets, where types within the same subset have identical conditional distributions. Ultimately, this paper finds that as the size of these subsets increase, the optimal revenue converges to that of a second-price auction with reserve. Full Surplus Extraction with Correlation (Review of Cremer and McLean (1985)) In this paper, the setting is as follows: A single buyer, a single external signal, and a single indivisible good that the seller values at 0. This simple case can be extended to the case of more than one buyer. The risk-neutral buyer has a valuation for the object, drawn from the set V = {v1, v2, ..., v|V |} ⊂ [0, v], where vi < vi+1 for all i. The buyer’s type(valuation along with possible additional information) is denoted θ ∈ Θ = {1, 2, ..., |Θ|}, and v(θ) denotes the valuation that is associated with type θ. In addition to the buyer, there is an external signal ω ∈ Ω = {1, 2, ..., |Ω|}, where |Ω| ≥ |V |. The joint distribution over θ and ω is given by π(θ, ω). We have the probability q(θ̂, ω) that the buyer is allocated the item, and the payment of the buyer m(θ̂, ω), where θ̂ is the reported outcome, and ω is the observed signal. Definition 1. (Buyer’s Utility). Given a true type θ ∈ Θ and reported type θ̂ ∈ Ω, the buyer’s expected utility under mechanism (q,m) is: