How do you teach or explain the revenue equivalence theorem in auctions to students or non-experts?

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Have you ever wondered how different types of auctions affect the seller's expected revenue? If so, you might be interested in learning about the revenue equivalence theorem, a fundamental result in game theory that applies to auctions and other mechanisms. In this article, we will explain what the theorem says, why it is important, and how you can teach or understand it with some simple examples and intuition.

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Daniel Zaldana

??LinkedIn Top Voice in Artificial Intelligence | Algorithms | Thought Leadership

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MEE SUNTHORN

R at MOFRD Partner Ecosystem | Business Development | Emerging Technology | tiny.ee/View-And-Download-Now | Ready to…
Dario á.

Senior Game Producer

1 What is the revenue equivalence theorem?

The revenue equivalence theorem (RET) states that, under certain conditions, any auction format that awards the item to the bidder with the highest valuation and induces truthful bidding will yield the same expected revenue for the seller. This means that whether you use a first-price, second-price, or any other type of auction, as long as the bidders have independent and private values and bid according to their true valuations, you can expect to earn the same amount on average.

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MEE SUNTHORN

R at MOFRD Partner Ecosystem | Business Development | Emerging Technology | tiny.ee/View-And-Download-Now | Ready to Make an Impact in the Industry | t.ly/ZK0mO
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It can be difficult to explain to non-experts. One way to approach it is to start with the basics of auction theory and build up to the theorem gradually. First, explain the basic concepts of auction design, such as the winner's curse and the Vickrey auction. Next, introduce the idea of revenue equivalence by explaining the equivalence between the English auction and the Vickrey auction. Finally, tie it all together by explaining the revenue equivalence theorem and its implications for auction design.

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Dario á.

Senior Game Producer
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Think of it like a game where each player, or bidder, decides how much an auctioned item is worth to them. They all secretly bid this amount. Whether the auction is 'highest bid wins and pays' or 'highest bid wins but pays second-highest bid', the seller's revenue ends up being similar. This happens only if everyone bids their true values honestly. It’s like different game rules leading to the same score. RET in game theory illustrates how the auction type matters less than how players strategize based on their item valuations.

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2 Why is the revenue equivalence theorem important?

The RET is important because it shows that the seller's revenue does not depend on the auction format, but on the distribution of the bidders' valuations and the reservation price of the seller. This implies that the seller cannot increase their revenue by changing the rules of the auction, unless they also change the information or incentives of the bidders. The RET also provides a benchmark for comparing different auction formats and evaluating their efficiency and fairness.

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Dario á.

Senior Game Producer
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In the gaming industry, understanding RET is crucial, particularly for auctions related to game assets or special editions. This theorem enlightens us on how different auction types can yield similar revenues. This insight shifts our focus from just revenue to considering what auction format resonates best with our players. Also provides a deeper understanding of bidder behavior, key in an industry driven by player engagement and perceived value. Applying RET principles, we can design auctions that are fair, transparent, and align with gamer preferences. Such knowledge is valuable not only theoretically but also in practical scenarios like planning special releases or in-game auctions, balancing financial objectives with player satisfaction

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3 How to prove the revenue equivalence theorem?

The RET can be proved mathematically using a technique called the envelope theorem, which relates the expected payoff of a bidder to their bid function. The idea is to compare the expected payoff of a bidder in any auction format to their expected payoff in a second-price auction, which is known to induce truthful bidding. By applying the envelope theorem, we can show that any bid function that maximizes the expected payoff in any auction format must be equal to the true valuation of the bidder, and therefore generate the same expected revenue for the seller as a second-price auction.

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4 How to illustrate the revenue equivalence theorem with examples?

One way to illustrate the RET with examples is to use numerical simulations or experiments to show how different auction formats yield the same expected revenue for the seller. For instance, you can generate random values for two bidders from a uniform distribution between 0 and 100, and then calculate their bids and the seller's revenue in a first-price, second-price, and all-pay auction. By repeating this process many times, you can observe that the average revenue of the seller is the same across all formats.

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5 How to explain the revenue equivalence theorem with intuition?

Another way to explain the RET with intuition is to use the concept of rent extraction, which measures how much of the bidders' surplus (the difference between their valuation and their payment) is captured by the seller. The RET implies that the rent extraction is the same in any auction format that induces truthful bidding, and therefore the seller's revenue is the same. To see why, consider a first-price auction and a second-price auction. In a first-price auction, the winner pays their bid, which is lower than their valuation, so they keep some surplus. In a second-price auction, the winner pays the second-highest bid, which is higher than their own bid, but lower than their valuation, so they also keep some surplus. The RET says that these two surpluses are equal on average, and therefore the seller's revenue is equal on average.

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6 How to apply the revenue equivalence theorem to real-world auctions?

The RET can be applied to real-world auctions to understand how different factors affect the seller's revenue and the bidders' behavior. For example, the RET can help explain why some online platforms use second-price auctions with a reserve price (a minimum acceptable bid) to sell advertising space. The reserve price allows the seller to increase their revenue by excluding low-value bidders and extracting more rent from high-value bidders. The second-price auction encourages truthful bidding and reduces the risk of overbidding or underbidding. The RET can also help evaluate the impact of introducing fees, taxes, subsidies, or other interventions on the efficiency and equity of auctions.

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Daniel Zaldana

??LinkedIn Top Voice in Artificial Intelligence | Algorithms | Thought Leadership
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Imagine you’re buying a rare item—say, a vintage comic book. There are different ways sellers might auction it: one might ask for sealed bids, another might let everyone shout out their bids. The Revenue Equivalence Theorem (RET) suggests that if certain conditions are met (like bidders having the same information and bidding rationally), the seller should expect to make roughly the same amount of money no matter which auction format they choose. Imagine an auction where a fee is imposed on the winning bid. RET can help predict how this fee might change bidders' strategies—perhaps they bid lower to account for the extra cost. RET also helps assess whether the seller's revenue will stay the same after accounting for the fees or taxes.

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How do you teach or explain the revenue equivalence theorem in auctions to students or non-experts?

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1 What is the revenue equivalence theorem?

2 Why is the revenue equivalence theorem important?

3 How to prove the revenue equivalence theorem?

4 How to illustrate the revenue equivalence theorem with examples?

5 How to explain the revenue equivalence theorem with intuition?

6 How to apply the revenue equivalence theorem to real-world auctions?

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