Dutch Auction¶

Definition¶

In a Dutch auction the price starts high and falls continuously (for example on a clock). The first bidder to stop the clock wins the object and pays the current price; the auction then ends. A strategy specifies, for each valuation $\theta_i$, the price at which the bidder will stop the clock if nobody has stopped it before.

The Dutch auction is strategically equivalent to the first-price auction: the two games have the same strategy sets (a planned stopping price is a bid) and the same mapping from strategy profiles to outcomes (the highest planned price wins and is paid).

Interpretation¶

The equivalence is easiest to see by writing the Dutch auction as a normal-form game. A strategy in an extensive-form game must say what to do at every point where the bidder could act. But the only information a bidder ever receives before the auction ends is that no one has stopped yet; once someone stops, the auction is over. So a complete plan reduces to one number: the price at which to stop. Bidder $i$ planning to stop at $b_i$ wins if and only if $b_i$ is the highest planned stopping price, and then pays $b_i$. That is exactly the first-price sealed-bid rule.

In the figure the clock falls from $1$. Bidder 1, with $\theta_1=0.8$$0.8$, plans to stop at $b_1=\theta_1/2=0.4$$0.4$, the two-bidder equilibrium of the Examples; bidder 2 plans to stop at $b_2=\theta_2/2$${\theta }_2/2$, drawn here below $0.4$. The clock reaches the higher planned price first, so bidder 1 wins and pays $0.4$, and the lower plan is never reached. Read the two dashed lines as sealed bids and the outcome is the same: the highest bid wins and its bidder pays it.

As a result the Dutch auction has the same equilibria as the first-price auction. With values i.i.d. $\operatorname{Unif}(0,1)$, the symmetric equilibrium stops the clock at $\frac{n-1}{n}\theta$. The bidder faces the same trade-off as in a sealed first-price bid: waiting lowers the price paid but raises the risk that a rival stops first.

Unlike the English–second-price equivalence, the Dutch–first-price equivalence does not require private values. Because no rival's action is observed during a Dutch auction until it ends, the equivalence also holds with common values or risk-averse bidders.

Key Results¶

• The Dutch and first-price sealed-bid auctions are strategically equivalent under any assumptions on values and risk attitudes.
• With i.i.d. $\operatorname{Unif}(0,1)$ private values and $n$ risk-neutral bidders, the symmetric equilibrium stopping price is $\frac{n-1}{n}\theta$ and expected revenue is $\frac{n-1}{n+1}$.
• There is no weakly dominant stopping rule: the best stopping price depends on the rivals' stopping prices.
• Under the conditions of the revenue equivalence theorem, the Dutch auction gives the same expected revenue as the English and second-price auctions.

Examples¶

Two bidders. Values are i.i.d. $\operatorname{Unif}(0,1)$ and both use the first-price equilibrium $b(\theta)=\theta/2$ as stopping rule. A bidder with $\theta=0.8$ stops the clock at $0.4$. She wins if the rival's stopping price $\theta_2/2$ is below $0.4$, i.e. with probability $0.8$, and expects $(0.8-0.4)(0.8)=0.32$$0.32$.

Waiting longer. If she instead plans to stop at $0.3$, she wins with probability $0.6$ and expects $(0.8-0.3)(0.6)=0.30<0.32$$0.32$. Stopping earlier, at $0.5$, gives $(0.3)(1)=0.30<0.32$ as well.

Prerequisites¶

• First-Price Auction
• Normal-Form Game
• Strategy in an Extensive-Form Game

Related Concepts¶

• English Auction
• Second-Price Auction
• Revenue Equivalence Theorem
• Common-Value Auction

Sources¶

• Game Theory · Week 6

Aliases: descending auction, descending-price auction, descending clock auction, open descending auction Keywords: descending clock, clock auction, strategic equivalence with first-price auction, stop the clock Symbols: theta

Last updated: 27 October 2026 11:52