Algorithmic Game Theory (Lecture 5: Revenue-Maximizing Auctions)

This lecture introduces revenue-maximizing auctions, contrasting them with surplus-maximizing auctions (like Vickrey). Surplus maximization is simpler, achievable even with unknown valuations. Revenue maximization requires a model of bidder valuations (e.g., Bayesian model), optimizing expected revenue. A key result is a formula expressing expected revenue in terms of allocation rules and "virtual valuations," enabling the design of optimal auctions, which in simple cases (e.g., IID uniform bidders) reduce to Vickrey auctions with reserve prices. This segment uses a simplified single-bidder, single-item auction to highlight the stark contrast between surplus and revenue maximization. It demonstrates how surplus maximization leads to a straightforward, ex-post optimal solution (R=0), while revenue maximization introduces complexity and ambiguity, setting the stage for the introduction of more sophisticated models. This segment contrasts ex-post revenue maximization (knowing the bidder's valuation) with the more realistic scenario where the valuation is unknown. It clearly illustrates the challenge of revenue maximization in the absence of complete information, emphasizing the need for a robust model to handle uncertainty. The speaker draws a compelling parallel between choosing a revenue-maximizing auction and selecting the best algorithm for a given task. This analogy effectively communicates the inherent difficulty of optimizing for revenue across various inputs, highlighting the need for a model that considers performance across different scenarios.This segment introduces the Bayesian model (or average-case analysis) as a framework for addressing the challenges of revenue maximization. It explains how positing a distribution over inputs allows for a principled approach to optimizing expected revenue, providing a crucial theoretical foundation for the subsequent analysis. This segment provides a concrete example of calculating expected revenue in a single-bidder auction with a posted price. It clearly defines expected revenue as the product of the posted price and the probability of a sale, illustrating how the Bayesian model allows for quantifiable analysis of auction performance. This segment extends the analysis to a two-bidder, single-item auction, introducing the concept of reserve prices in the Vickrey auction. It demonstrates how adding a reserve price can improve expected revenue by strategically balancing the potential for higher prices with the risk of no sale, showcasing the trade-offs involved in auction design. The speaker introduces the challenge of finding optimal auctions to maximize revenue, highlighting the complexity of a direct revenue formula. They then outline a plan to derive a second, more useful formula for expected revenue using Myerson's lemma, focusing on the derivation process and its importance for optimization. This segment revisits Myerson's payment formula, initially presented for piecewise constant allocation rules. The speaker explains the formula's application in sponsored search and its extension to a continuous form, setting the stage for a more general approach to revenue optimization. The speaker transitions from the discrete sum representation of Myerson's payment formula to a continuous integral form, justifying this change for ease of manipulation and generalization beyond piecewise constant allocation rules. The discussion touches upon the mathematical properties of monotone allocation rules and their suitability for integration.The speaker explains the difficulty of directly optimizing the revenue formula involving payments and proposes an alternative strategy: re-expressing the revenue in terms of allocation rules (x's), which are easier to optimize. This shift in focus lays the groundwork for the subsequent derivation.The speaker details the step-by-step plan for deriving the simplified revenue formula. The initial steps involve fixing bidder valuations, expanding the expected revenue using Myerson's payment formula, and setting up a double integral for further simplification.This segment focuses on the simplification of the double integral through reversing the order of integration and evaluating one of the resulting integrals. The speaker highlights the strategic choices made during the simplification process and the positive signs indicating progress towards a simpler formula. This segment details the process of defining an optimal allocation rule by maximizing virtual surplus for each valuation profile. It discusses the implications of this rule, including the possibility of no winner if all virtual values are negative, and connects this to the intuition of setting a reserve price to maximize revenue. This segment addresses the crucial issue of monotonicity in the allocation rule. It explains that the allocation rule's monotonicity is equivalent to the fee function being non-decreasing, a condition that determines whether the optimal auction is implementable. The segment clarifies the relationship between theoretical optimality and practical implementability. This segment introduces simplifying assumptions (single-item auction, IID valuations) to make the optimization problem more tractable. It then explains the concept of pointwise optimization: maximizing the virtual surplus for each possible valuation profile to achieve the highest overall expected revenue. This approach provides a clear, step-by-step method for designing optimal auctions. This segment explains how to interpret the proven formula for expected auction revenue, connecting it to the concept of virtual surplus. It highlights that maximizing expected revenue is equivalent to maximizing expected virtual surplus, providing a crucial framework for optimizing auction mechanisms. The explanation connects theoretical concepts to practical implications for auction design. The speaker applies integration by parts to further simplify the integral expression, demonstrating the steps and explaining how certain terms vanish, leading to a more manageable and interpretable formula for expected revenue. This segment introduces the concept of "regular" distributions, where the fee function is non-decreasing, ensuring the monotonicity of the allocation rule. It shows that for IID regular bidders, the optimal auction is equivalent to a Vickrey auction with a reserve price, providing a concrete and practical application of the theoretical framework. The segment concludes by stating that for IID bidders from a uniform distribution, the optimal auction is a Vickrey auction with a reserve price of 1/2.