Auction theory is a field of applied economics that uses the tools of game theory to analyze and design auctions.
It studies how bidders behave in different auction formats and how the rules of an auction can influence outcomes such as efficiency (the item going to the person who values it the most) and revenue for the seller.
Key Concepts in Auction Theory
- Valuation: This refers to the value a bidder places on an item. There are two main types:
- Private Value: Each bidder’s valuation of the item is independent of the other bidders’ valuations. An example is a collector bidding on a painting for their personal collection.
- Common Value: The true value of the item is the same for all bidders, but each bidder has different, private information (or “signals”) about that value. An example is an oil company bidding on a drilling lease, where the actual value of the lease depends on how much oil is in the ground.
- Winner’s Curse: This is a phenomenon that occurs in common-value auctions. The winner of the auction is often the bidder with the most optimistic, and likely overestimated, view of the item’s value. In an equilibrium, rational bidders will “shade” their bids—bidding less than their true estimate—to account for the risk of overpaying.
- Game Theory and Equilibrium: Auction theory is fundamentally a branch of game theory. It models auctions as strategic games where bidders choose their actions (bids) to maximize their own utility.
- Dominant Strategy: A strategy that is the best for a player regardless of what the other players do.
- Nash Equilibrium: A set of strategies, one for each player, where no player can improve their outcome by unilaterally changing their strategy, given the strategies of the other players. In auctions with incomplete information (where bidders don’t know the other bidders’ valuations), a more complex concept, the Bayesian-Nash equilibrium, is often used.
The Four Basic Auction Types
Most of auction theory revolves around four fundamental auction types, which can be classified based on whether bids are open or sealed, and whether prices are ascending or descending.
- English Auction (Open Ascending-bid):
- Rules: The price starts low and is raised by successive bids until only one bidder remains.
- Strategy: Bidders have an incentive to continue bidding until the price reaches their true valuation. This auction format is generally considered to be efficient, as the item is allocated to the bidder with the highest valuation.
- Dutch Auction (Open Descending-bid):
- Rules: The price starts high and is gradually lowered until a bidder accepts the current price. That bidder wins and pays the accepted price.
- Strategy: This format is fast-paced. Bidders must decide when to jump in, balancing the risk of waiting for a lower price and losing the item to another bidder. Bidders have an incentive to “shade” their bids below their true valuation to increase their profit margin.
- First-Price Sealed-Bid Auction:
- Rules: Bidders submit a single, sealed bid simultaneously, without knowing the bids of others. The highest bidder wins and pays the amount they bid.
- Strategy: Bidders must strategically “shade” their bids below their true valuation. If they bid their true value, they would gain no surplus if they won. The optimal bid depends on their beliefs about the other bidders’ valuations.
- Second-Price Sealed-Bid Auction (Vickrey Auction):
- Rules: Bidders submit a single, sealed bid. The highest bidder wins, but pays a price equal to the second-highest bid.
- Strategy: This auction has a unique and powerful property: it is a dominant strategy for every bidder to bid their true valuation. This is because their bid only determines whether they win, not the price they pay. Bidding higher than their value risks overpaying, and bidding lower risks losing to a bidder with a lower valuation.
The Revenue Equivalence Theorem
A foundational result in auction theory, the Revenue Equivalence Theorem states that under certain assumptions (including risk-neutral bidders, independent private values, and symmetric bidders), all four of the basic auction types yield the same expected revenue for the seller.
However, these assumptions often do not hold in the real world. For example, when bidders are risk-averse, they may bid more aggressively in a first-price sealed-bid auction to increase their chance of winning, which can lead to higher expected revenue for the seller. This is a key reason why auction design is so important.
Applications
Auction theory has moved beyond just the sale of art and antiques. It is used to design complex auctions for a variety of goods and services, including:
- Government assets: The Federal Communications Commission (FCC) has used auction theory to design and conduct multi-billion dollar auctions for radio frequency spectrum licenses.
- Online advertising: The systems used by Google and other tech companies to sell ad space are often based on a form of second-price auction.
- Electricity markets and financial instruments: Auctions are used to allocate resources and set prices in various markets.