A lower bound on seller revenue in single buyer monopoly auctions

We consider a monopoly seller who optimally auctions a single object to a single potential buyer, with a known distribution of valuations. We show that a tight lower bound on the seller's expected revenue is $1/e$ times the geometric expectation of the buyer's valuation, and that this bound is uniquely achieved for the equal revenue distribution. We show also that when the valuation's expectation and geometric expectation are close, then the seller's expected revenue is close to the expected valuation.Comment: 5 pages. To appear in Operations Research Letter

Selling to a No-Regret Buyer

We consider the problem of a single seller repeatedly selling a single item to a single buyer (specifically, the buyer has a value drawn fresh from known distribution $D$ in every round). Prior work assumes that the buyer is fully rational and will perfectly reason about how their bids today affect the seller's decisions tomorrow. In this work we initiate a different direction: the buyer simply runs a no-regret learning algorithm over possible bids. We provide a fairly complete characterization of optimal auctions for the seller in this domain. Specifically: - If the buyer bids according to EXP3 (or any "mean-based" learning algorithm), then the seller can extract expected revenue arbitrarily close to the expected welfare. This auction is independent of the buyer's valuation $D$, but somewhat unnatural as it is sometimes in the buyer's interest to overbid. - There exists a learning algorithm $\mathcal{A}$ such that if the buyer bids according to $\mathcal{A}$ then the optimal strategy for the seller is simply to post the Myerson reserve for $D$ every round. - If the buyer bids according to EXP3 (or any "mean-based" learning algorithm), but the seller is restricted to "natural" auction formats where overbidding is dominated (e.g. Generalized First-Price or Generalized Second-Price), then the optimal strategy for the seller is a pay-your-bid format with decreasing reserves over time. Moreover, the seller's optimal achievable revenue is characterized by a linear program, and can be unboundedly better than the best truthful auction yet simultaneously unboundedly worse than the expected welfare

Liquidity constraints and credit subsidies in auctions

I consider an auction with participants that differ in valuation and access to liquid assets. Assuming credit is costly (e.g. due to moral hazard considerations) different auction rules establish different ways of screening valuation-liquidity pairs. The paper shows that standard auction forms result in different allocation rules. When the seller can deny access to capital markets or offer credit subsidies, she gains an additional tool to screen agents. The paper derives conditions under which the seller increases profits by way of subsidizing loans. In particular, in a second price auction, the seller always benefits from offering small subsidies. The result extends to a non-auction setting to show that a monopolist may use credit subsidies as a price discrimination device

Mechanism Design with Limited Information: The Case of Nonlinear Pricing

We analyze the canonical nonlinear pricing model with limited information. A seller offers a menu with a finite number of choices to a continuum of buyers with a continuum of possible valuations. By revealing an underlying connection to quantization theory, we derive the optimal finite menu for the socially efficient and the revenue-maximizing mechanism. In both cases, we provide an estimate of the loss resulting from the usage of a finite n-class menu. We show that the losses converge to zero at a rate proportional to 1/n^2 as n becomes large.Mechanism design, Limited information, Nonlinear pricing, Quantization, Lloyd-max optimality

LIQUIDITY CONSTRAINTS AND CREDIT SUBSIDIES IN AUCTIONS

I consider an auction with participants that differ in valuation and access to liquid assets. Assuming credit is costly (e.g. due to moral hazard considerations) different auction rules establish different ways of screening valuation-liquidity pairs. The paper shows that standard auction forms result in different allocation rules. When the seller can deny access to capital markets or offer credit subsidies, she gains an additional tool to screen agents. The paper derives conditions under which the seller increases profits by way of subsidizing loans. In particular, in a second price auction, the seller always benefits from offering small subsidies. The result extends to a non-auction setting to show that a monopolist may use credit subsidies as a price discrimination device.