Bidding Strategy under Complete Information
Let’s go back to the example of complete information that we saw in the FPA. Now the auction is carried out under the Dutch auction rule. There are two bidders, and the bidders have values V1=$10 and V2=$8. The seller still accepts bids in dollars and pennies only.
At the beginning of the auction a bidder plans when she should press the button to stop the price. Both bidders work this out in their minds. The price at which she plans to stop the price is her “bid” in the auction.
There is no dominant strategy for either bidder. Bidding $8.50 is not a dominant strategy for bidder 1 since she will be better off bidding $6.00 if the other bidder is bidding $5.99. (Recall, again, that bidders do not see each other’s bids. There is also no guarantee that they will actually submit these bids in equilibrium, any more than there is a guarantee that the weather office will be right about the amount of rain tomorrow.) Similarly, there is no dominant bidding strategy for bidder 2. Hence, we calculate the Nash equilibrium as our predicted outcome.
Bids b1 = $7.99 by bidder 1 and b2 = $ 8.00 by bidder 2 are in Nash equilibrium. To see this, all you have to do is revisit the FPA argument and see how that same argument applies here, too.
Strategic Equivalence
Are you surprised by the similarity between the bidding strategies in the FPA and Dutch auction? Actually, the FPA and the Dutch auction are what is called “strategically equivalent” in that the bidders face the exact same strategic decision-making situation. In case you did not notice, the SPA and English auctions are also strategically equivalent. In both SPA and EA all you really did through your bid (i.e., your strategy) is state the maximum price that you are willing to pay for the object.
Dutch Auction under Incomplete Information
Given that a bidder faces the same strategic situation in the FPA and DA her equilibrium bids are going to be the same as well. It is just a rehashing of the same argument. So, your Nash equilibrium bidding strategy in a Dutch auction simply calculates the expected value of your highest rival bidder’s bid for the situation where you are the winning bidder:
b1(v1) = E[ Y1:n-1 | Y1:n-1 ≤ x1].