10.2 Revenue equivalence theorem

The Revenue Equivalence Theorem is a cornerstone of auction theory. It states that under specific conditions, different auction formats yield the same expected revenue for sellers. This powerful idea simplifies auction design and provides a foundation for comparing various auction mechanisms.

Understanding this theorem is crucial for grasping the broader concepts in Auction Theory and Mechanism Design. It helps explain why certain auction formats are chosen in different situations and provides insights into optimal auction design strategies.

Revenue Equivalence Theorem

Assumptions and Implications

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• The revenue equivalence theorem assumes bidders are risk-neutral, have independent private values, and the auction is a standard auction mechanism
• Under these assumptions, any auction format that allocates the item to the highest bidder and where the lowest bidder expects zero surplus yields the same expected revenue for the seller
• Applies to sealed-bid auctions (first-price, second-price), open ascending-bid auctions (English auction), open descending-bid auctions (Dutch auction), and all-pay auctions
• Suggests the choice of auction format should not matter for the seller's expected revenue, as long as the assumptions hold
• Has important implications for auction design, suggesting factors other than the auction format, such as the reserve price or the number of bidders, may be more important for maximizing the seller's revenue

Mathematical Formulation

• Let $n$ be the number of bidders, $v_i$ be bidder $i$'s private value, and $F(v)$ be the cumulative distribution function of the private values
• Under the assumptions of the theorem, the expected payment of bidder $i$ with value $v_i$ is given by: $E[p_i(v_i)] = v_i \cdot P(v_i) - \int_{0}^{v_i} P(x) dx$

where $P(v_i)$ is the probability that bidder $i$ wins the auction with value $v_i$

• The expected revenue of the seller is the sum of the expected payments of all bidders: $E[R] = \sum_{i=1}^{n} E[p_i(v_i)]$

• The revenue equivalence theorem states that any two auction formats that result in the same allocation rule and the same expected payment for the lowest-value bidder will yield the same expected revenue for the seller

Applying Revenue Equivalence

Comparing Auction Formats

• Under the assumptions of the theorem, the expected revenue from a first-price sealed-bid auction equals the expected revenue from a second-price sealed-bid auction (Vickrey auction)
  • In a first-price sealed-bid auction, bidders submit bids simultaneously, and the highest bidder wins and pays their bid
  • In a second-price sealed-bid auction, the highest bidder wins but pays the second-highest bid
• The expected revenue from an open ascending-bid auction (English auction) equals the expected revenue from a second-price sealed-bid auction
• The expected revenue from an open descending-bid auction (Dutch auction) equals the expected revenue from a first-price sealed-bid auction
• The theorem can compare the expected revenues of more complex auction formats, such as all-pay auctions or auctions with reserve prices, as long as the assumptions hold

Optimal Auction Design

• The revenue equivalence theorem provides a foundation for the design of optimal auction mechanisms
• Myerson's optimal auction design theory builds upon the revenue equivalence theorem to characterize the optimal auction mechanism under more general conditions
• The optimal auction design maximizes the seller's expected revenue subject to incentive compatibility and individual rationality constraints
• Key insights from the optimal auction design theory include:
  • The optimal auction may involve setting a reserve price to exclude low-value bidders
  • The optimal auction may discriminate among bidders based on their value distributions
  • The optimal auction may involve bundling multiple items or using more complex pricing rules

Limitations of Revenue Equivalence

Relaxing Assumptions

• The revenue equivalence theorem relies on strict assumptions (risk-neutral bidders, independent private values, standard auction mechanism), and may not hold if these are violated
• If bidders are risk-averse, the expected revenue from a first-price sealed-bid auction may be higher than from a second-price sealed-bid auction
  • Risk-averse bidders tend to bid more aggressively in first-price auctions to increase their chances of winning
• If bidders have affiliated values (i.e., positively correlated valuations), the expected revenue from an open ascending-bid auction may be higher than from a second-price sealed-bid auction
  • Affiliated values lead to more aggressive bidding in open ascending-bid auctions as bidders update their valuations based on others' bids
• Relaxing the assumptions has led to the development of more general theories, such as the linkage principle and the optimal auction design theory

Extensions and Generalizations

• The theorem can be extended to include auctions with reserve prices, showing the optimal reserve price is the same across different auction formats that satisfy the assumptions
• It can also be extended to multi-unit auctions, where multiple identical items are sold simultaneously, under certain conditions
  • Example: Vickrey-Clarke-Groves (VCG) mechanism for multi-unit auctions
• The revenue equivalence theorem has been generalized to settings with interdependent values, where bidders' valuations depend on others' private information
• Researchers have also explored the implications of the theorem in settings with budget constraints, externalities, and costly participation

Relevance of Revenue Equivalence

Practical Applications

• The revenue equivalence theorem provides a useful benchmark for comparing the expected revenues of different auction formats, but its practical relevance may be limited by the strictness of its assumptions
• In real-world auctions, bidders may be risk-averse, have affiliated values, or face budget constraints, leading to deviations from the theorem's predictions
• The theorem does not account for factors such as participation costs, collusion among bidders, or externalities, which can affect the choice of auction format in practice
• Despite its limitations, the theorem has been influential in the development of auction theory and has provided valuable insights into the design of optimal auction mechanisms
• It has been applied to the design of auctions for a wide range of goods and services (spectrum licenses, oil and gas leases, government procurement contracts)

Empirical Evidence and Future Research

• Empirical studies have tested the predictions of the revenue equivalence theorem in various settings, with mixed results, highlighting the need for further research on factors influencing the performance of different auction formats
• Some studies have found support for the theorem's predictions in specific contexts (e.g., U.S. Forest Service timber auctions), while others have found deviations from revenue equivalence (e.g., online advertising auctions)
• Future research could focus on developing more robust auction models that account for real-world complexities and on empirically testing the implications of these models in different settings
• Advances in auction theory and empirical methods, combined with the increasing availability of auction data, offer opportunities for further refining our understanding of the revenue equivalence theorem and its applications