Theorem Dutch ≡ First-Price Strategic Equivalence
theorem staged

Dutch ≡ First-Price Strategic Equivalence

Theorem (MSZ Thm 12.3; Krishna Prop 2.1). Under independent private values, the Dutch (descending-price open-outcry) auction and the sealed-bid first-price auction ([[mechanism_design.auction.basic.first_price_mechanism]]) are strategically equivalent: they induce the same extensive-form game (up to information set relabelling) and hence the same set of equilibrium strategies and outcomes.

Setup

  • Dutch auction: the auctioneer announces a high opening price and decreases it continuously over time. The first bidder to stop the clock wins the item at the current price.
  • First-price sealed-bid auction: each bidder simultaneously submits a sealed bid; the highest bidder wins and pays their own bid.

In both formats, a bidder's only strategic choice is the price at which to commit. In Dutch this is the stopping price; in sealed-bid first-price this is the submitted bid. Under the standard observability assumption (a Dutch bidder learns nothing about opponents before deciding when to stop), the two extensive forms coincide.

Proof

route

The proof has two parts:

  1. Strategy spaces coincide. A bidder's pure strategy in the Dutch auction is a real number $b_i \in \mathbb{R}_{\ge 0}$ — the stopping price as a function of (no observed information). The same strategy space describes the first-price sealed-bid auction.
  2. Outcomes coincide. In both formats, given any pure-strategy profile $b : I \to \mathbb{R}_{\ge 0}$, the winner is $\mathrm{argmax}_i\, b_i$ and the payment is $\max_i b_i$. The allocation and payment maps are pointwise equal.

Together: identical extensive forms ⇒ identical strategic-game reductions ⇒ identical equilibria.

Why it matters

The Dutch-first-price equivalence is the simplest example of the broader revenue equivalence phenomenon: auction formats can differ in their dynamic structure while implementing the same allocation and payment rules, and hence the same revenue under any equilibrium.

This equivalence motivates the focus on the sealed-bid formulation in theoretical analysis: any result proved for sealed-bid first-price applies verbatim to Dutch, including the symmetric IPV equilibrium formula ([[mechanism_design.auction.bayesian.symmetric_first_price_equilibrium]]) and the revenue comparison with second-price ([[mechanism_design.myerson.revenue_equivalence]]).

Lean port (deferred)

Planned Lean module: EconCSLib/Auction/Dutch.lean.

Planned declarations:

  • DutchAuction (extensive-form representation)
  • DutchAuction.toFirstPriceMechanism (strategic equivalence map)
  • DutchAuction.equivalence_first_price_mechanism (strategic equivalence theorem, stated as equality of induced strategic games).

Dependencies on yet-to-formalise pieces: an extensive-form auction encoding that respects the continuous descending-price clock; the information-set definition that captures the "no observation before stopping" rule. The first-price sealed-bid side is already in EconCSLib/Auction/FirstPrice.lean. Tracked in the MSZ Ch.12 auction gap review.

References

  • [MSZ Chapter 12, Thm 12.3] Maschler, Solan, and Zamir, Game Theory. Dutch-first-price strategic equivalence.
  • [Krishna, Chapter 2, Proposition 2.1] Vijay Krishna, Auction Theory, 2nd ed.. Dutch auction and its equivalence to first-price.
  • [Vickrey 1961] William Vickrey, "Counterspeculation, Auctions, and Competitive Sealed Tenders", Journal of Finance 16(1):8–37. Original observation of the equivalence.

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