Auction theory wrap up!

My previous article looked at a very common type of auction, the open outcry auction. In this article let’s have a quick engagement with other auctions economists study in the static, independent values framework. The last few words may sound slightly unfamiliar. ‘Static’ simply refers to the fact that we are looking at single period models as opposed to ‘dynamic’ models and the term ‘independent values’ is alluding to the idea that the agents valuation of the good being sold by the seller are independent of each other’s valuation.

The auction we should naturally look at after last week’s article is the sealed bid second price auction. This auction format can rarely be seen in reality; however it is a very important special case for a more general class of mechanisms called the Vickrey-Clarke-Groves mechanisms.  In the second price sealed bid auction, the following happens. Every voter submits a bid to the seller and after observing the bids, the seller sells the good to the highest bidder but at the second highest price.

It is very well known that the sealed bid second price auction is actually strategically equivalent to the open outcry auction we discussed in my last article, and the weakly dominant strategy is to report your true value as the bid you submit.

Imagine yourself back in the auction for Van Gogh’s Starry Night. This time the auctioneer is holding a sealed bid auction and he has announced that the highest bidder will get the object at the second highest price. What’s your best strategy? We need to keep track of 3 variables (your true value for the object, the highest bid of the other buyers, and your bid) and Consider 3 scenarios, 1) your bid and the other buyers bids are higher than your true value for the painting. 2) The highest bid of the other bidders is higher than your bid and your value. 3) Your bid and value is higher than the highest bid of the other buyers. In all of these scenarios the reader should be able to verify that it is better or equally good to equate your true value and the submitted bids.

However, truth-telling is not a feature in all auctions. The next auction we look at is the sealed bid first price auction. Here, the auction proceeds in the same manner as its second price counterpart, but the highest bidder wins the auction and pays his own price.  Under this auction it is immediately clear that one should not bid his own value, because by following this strategy you guarantee yourself a zero profit! With the first price rule in place, one would like to bid slightly lower than his/her own value of the good, in order to potentially make a profit.

There is something in common for these different types of auctions, and it is the much celebrated ‘revenue equivalence theorem’ result. Loosely, this theorem claims that all auctions that award the object to the bidder with the highest value in equilibrium and give the bidder with the lowest valuation zero profits (note that the above auctions all qualify!), generates the same revenue (for the seller) in expectation.

We have looked at truth-telling auctions, namely the sealed bid second price and, open outcry auctions. Also, we have considered how the first price seal bid auction would not induce a truth-telling equilibrium. However, all of these auctions, by the revenue equivalence theorem yield the same profits in expectation to the seller. Now, if we add time and take the analysis to the dynamic setting, we can allow for a richer model and consider more realistic situations. However, things get more complicated very quickly, and its exploration is probably for the keenest of readers!