11.2 Revenue equivalence theorem

Auction theory explores how different auction formats affect bidder behavior and outcomes. The revenue equivalence theorem is a key result, showing that under certain conditions, various auction types yield the same expected revenue for the seller.

This theorem assumes symmetric bidders with independent private values and risk neutrality. It applies to auctions with the same allocation rule and payment based only on the winning bid, like first-price and second-price sealed-bid auctions.

Assumptions and Bidder Characteristics

Symmetric Bidders and Independent Private Values

Top images from around the web for Symmetric Bidders and Independent Private Values

• 

  Enforced symmetry: the necessity of symmetric waxing and waning [PeerJ] View original

  Is this image relevant?

• 

  Bayesian Probability Illustration Diagram | TikZ example View original

  Is this image relevant?

• 

  Enforced symmetry: the necessity of symmetric waxing and waning [PeerJ] View original

  Is this image relevant?

• 

  Bayesian Probability Illustration Diagram | TikZ example View original

  Is this image relevant?

1 of 2

Top images from around the web for Symmetric Bidders and Independent Private Values

• 

  Enforced symmetry: the necessity of symmetric waxing and waning [PeerJ] View original

  Is this image relevant?

• 

  Bayesian Probability Illustration Diagram | TikZ example View original

  Is this image relevant?

• 

  Enforced symmetry: the necessity of symmetric waxing and waning [PeerJ] View original

  Is this image relevant?

• 

  Bayesian Probability Illustration Diagram | TikZ example View original

  Is this image relevant?

1 of 2

• Bidders are symmetric have the same value distribution and preferences
• Each bidder's valuation is drawn independently from the same probability distribution
• Bidders' valuations are private information not known to other bidders or the auctioneer
• Valuations are independent across bidders not influenced by others' valuations (no common value component)

Risk Neutrality

• Bidders are risk-neutral aim to maximize their expected payoff
• Do not have preferences over risk or uncertainty
• Willing to participate in auctions as long as their expected payoff is non-negative
• Simplifying assumption allows for tractable analysis of bidding behavior and auction outcomes

Auction Design Elements

Allocation Rule

• Determines which bidder wins the auction and receives the item being sold
• Common allocation rules include:
  • First-price auction: highest bidder wins and pays their bid
  • Second-price auction: highest bidder wins but pays the second-highest bid
  • All-pay auction: highest bidder wins, but all bidders pay their bids regardless of winning
• Allocation rule influences bidders' incentives and strategies

Payment Rule

• Specifies how much the winning bidder must pay for the item
• Payment rules are linked to the allocation rule and can vary across auction formats
• Examples include paying the winning bid (first-price), paying the second-highest bid (second-price), or paying a predetermined price (posted-price)
• Payment rule affects bidders' willingness to pay and their bidding strategies

Bidding Strategies

• Bidders choose their bids to maximize their expected payoff given the auction format and their valuation
• Optimal bidding strategies depend on the allocation and payment rules, as well as assumptions about other bidders
• In a first-price auction, bidders shade their bids below their true valuation to maximize expected profit
• In a second-price auction, bidding one's true valuation is a weakly dominant strategy (truthful bidding)

Key Results

Revenue Equivalence Theorem

• Under certain assumptions (symmetric bidders, independent private values, risk neutrality), different auction formats yield the same expected revenue for the auctioneer
• Applies to auctions with the same allocation rule and payment rule that depends only on the winning bid
• Examples include first-price and second-price sealed-bid auctions, which are revenue equivalent
• Theorem allows for comparison and ranking of auction formats based on their expected revenue

Expected Revenue

• The average revenue the auctioneer expects to receive from the auction
• Calculated as the expected value of the payment made by the winning bidder
• Depends on the auction format, bidders' value distributions, and their equilibrium bidding strategies
• Revenue equivalence theorem implies that expected revenue is the same for auctions satisfying the assumptions
• Auctioneer's goal is often to design an auction that maximizes expected revenue subject to other constraints (efficiency, fairness)