English ≡ Second-Price IPV Equivalence
Theorem (MSZ Thm 12.7; Krishna Prop 2.2). Under independent private values, the English (ascending-price open-outcry) auction is strategically equivalent in weakly dominant strategies to the sealed-bid second-price (Vickrey) auction ([[mechanism_design.auction.basic.second_price_mechanism]]): the dominant strategy "stay in until the price reaches my own valuation" yields the same allocation and payment as truthful bidding in the second-price auction.
This is a weaker equivalence than the Dutch–first-price equivalence ([[mechanism_design.auction.bayesian.dutch_first_price_equivalence]]) — extensive-form strategy spaces differ — but the dominant-strategy outcomes coincide.
Setup
Two formats of the open ascending-price auction are usually considered:
- English (Krishna): the price clock rises continuously; bidders drop out publicly; the last remaining bidder wins at the drop-out price of the second-last.
- Japanese / Milgrom-Weber: bidders post a "stay-in" indicator that flips to "drop-out" at a price each chooses; the last remaining bidder wins at the drop-out price of the second-last. This is the cleanest model for the equivalence.
In both formats, under IPV, the natural strategy is to stay in as long as the current price is below one's own valuation $t_i$ and drop out when it reaches $t_i$. The result is:
- The bidder with the highest valuation wins.
- The price paid equals the second-highest valuation (the drop-out price of the last competitor).
These are exactly the allocation and payment of the second-price auction at truthful bidding $b_i = t_i$.
Proof
route
- Dominance in English/Japanese. Staying in until $t_i$ is weakly dominant: dropping out earlier risks losing an item that could be bought below $t_i$; staying in longer risks paying more than $t_i$. This is the open-outcry analogue of the Vickrey dominance argument ([[mechanism_design.auction.basic.second_price_dsic]]).
- Outcome coincidence. Under these dominant strategies, the winner is $\mathrm{argmax}_i\, t_i$ and the payment is $\max_{j \ne \mathrm{winner}}\, t_j$ — identical to the second-price auction at truthful bidding.
Outside the IPV setting (e.g. with affiliated values), the open-outcry English auction can reveal information about opponents' signals during the bidding process, which the sealed-bid second-price cannot. The two formats are then no longer equivalent (Krishna Chapter 6); this gap is the basis of linkage and winner's curse analyses.
Why it matters
The English-second-price equivalence is the practical bridge between the open-outcry auction format used in art and asset markets and the theoretical sealed-bid model used in mechanism-design. It justifies analysing the Vickrey auction as a stand-in for English under IPV, and motivates the interdependent values extension where the equivalence breaks.
Lean port (deferred)
Planned Lean module: EconCSLib/Auction/English.lean.
Planned declarations:
EnglishAuction/JapaneseAuction(extensive-form representation)- The "stay-in-until-valuation" strategy and its weak dominance.
EnglishAuction.equivalence_second_price_mechanism_under_ipv(strategic equivalence theorem under IPV).
Dependencies on yet-to-formalise pieces: open-outcry extensive form,
public drop-out information sets, and an IPV-environment specification.
The second-price side is fully in
EconCSLib/Auction/Vickrey.lean. Tracked in the MSZ Ch.12 auction gap
review.
References
- [MSZ Chapter 12, Thm 12.7] Maschler, Solan, and Zamir, Game Theory. English-second-price equivalence under IPV.
- [Krishna, Chapter 2, Section 2.4] Vijay Krishna, Auction Theory, 2nd ed.. Japanese auction and its equivalence to the second-price sealed-bid auction.
- [Krishna, Chapter 6] Vijay Krishna, Auction Theory, 2nd ed.. Breakdown of equivalence under interdependent values.
- [Milgrom-Weber 1982] Paul Milgrom and Robert Weber, "A theory of auctions and competitive bidding", Econometrica 50(5):1089–1122. Japanese auction formulation.