English Auction :

The price rises continuously and each bidder indicates an interest in purchasing at the current price
in a manner apparent to all by, say, raising a hand. Once a bidder finds the price to be too high, he
signals that he is no longer interested by lowering his hand. The auction ends when only a single
bidder is still interested. This bidder wins the object and pays the auctioneer an amount equal to the
price at which the second-last bidder dropped out.

Dutch Auction:

The Dutch auction is the open descending price counterpart of the English Auction. The auctioneer
begins by calling out a price high enough so that presumably no bidder is interested in buying the
object at that price. This price is gradually lowered until some bidder indicates her interest. The
object is then sold to this bidder at the given price.

First Price Auction:

Bidders submit bids in sealed envelopes; the person submitting the highest bid wins the object and
pays what he bid.

Second Price Auction:

Bidders submit bids in sealed envelopes; the person submitting the highest bid wins the object but
pays not what he bid, but the second highest bid.

Valuations

If each bidder knows the value of the object to himself at the time of bidding, the situation is called
one of privately known values or private values. Implicit in this situation is that no bidder knows with
certainty the values attached by other bidders and knowledge of other bidders’ values would not
affect how much the object is worth to a particular bidder. The assumption of private values is most
plausible when the value of the object to a bidder is derived from its consumption or use alone.

In many situations, how much the object is worth is unknown at the time of the auction to the bidder
himself. He may have only an estimate of some sort or some privately known signal–such as an
expert’s estimate or a test result–that is correlated with the true value. Indeed, other bidders may
possess information, say additional estimates or test results, that if known, would affect the value
that a particular bidder attaches to the object. Thus, values are unknown at the time of the auction
and may be affected by information available to other bidders. Such a specification is called one of
interdependent values and is particularly suited for situations in which the object being sold is an
asset that can possibly be resold after the auction. A special case of this is a situation in which the
value, though unknown at the time of bidding, is the same for all bidders–a situation described as
being one of a pure common value. A common value model is most appropriate when the value of
the object being auctioned is derived from a market price that is unknown at the time of the auction.
An archetypal example is the sale of a tract of land with an unknown amount of oil underground.
Bidders may have different estimates of the amount of oil, perhaps based on privately conducted
tests, but the final value of the land is derived from the future sales of the oil, so this value is, to a
first approximation, the same for all bidders.
Equivalent Auctions

Two games are strategically equivalent if they have the same normal form except for duplicate
strategies. Roughly this means that for every strategy in one game, a player has a strategy in the
other game, which results in the same outcomes.

The Dutch open descending price auction is strategically equivalent to the first-price sealed-bid
auction. In a first-price sealed bid auction, a bidder’s strategy maps his private information into a bid.
Although the Dutch auction is conducted in the open, it offers no useful information to bidders. The
only information that is available is that some bidder has agreed to buy at the current price; but that
causes the auction to end. Bidding a certain amount in a first-price sealed-bid auction is equivalent to
offering to buy at that amount in a Dutch auction, provided the item is still available. For every
strategy in a first-price auction there is an equivalent strategy in the Dutch auction and vice versa.

Second, when values are private, the English open ascending auction is also equivalent to the
second-price sealed-bid auction, but in a weaker sense than noted earlier. The English auction offers
information about when other bidders drop out, and by observing this, it may be possible to infer
something about their privately known information. With private values, however, this information is
of no use. In an English auction, it clearly cannot be optimal to stay in after the price exceeds the
value–this can only cause a loss–or to drop out before the price reaches the value–thus forgoing
potential gains. Likewise, in a second-price auction it is best to bid the value (this is discussed in more
detail later). Thus, with private values, the optimal strategy in both is to bid up to or stay in until the
value.

This equivalence between the English and second-price auction is weak in two senses. First, the two
auctions are not strategically equivalent. Second, and more important, the optimal strategies in the
two are the same only if values are private. With interdependent values, the information available to
others is relevant to a particular bidder’s evaluation of the worth of the object. Seeing some other
bidder drop out early may bring bad news that may cause a bidder to reduce his own estimate of the
object’s value. Thus, if values are interdependent, the two auctions need not be equivalent from the
perspective of the bidders.

Revenue versus Efficiency

The main questions that guide auction theory involve a comparison of the performance of different
auction formats as economic institutions. These are evaluated on two grounds and the relevance of
one or the other criterion depends on the context. From the perspective of the seller, a natural
yardstick in comparing different auction forms is the revenue, or the expected selling price, that they
fetch. From the perspective of society as a whole, however, efficiency–that the object end up in the
hands of the person who values it the most ex post–may be more important.

The Symmetric Model

There is a single object for sale and N potential buyers are bidding for the object. Bidder i assigns a
value of Xi to the object–the maximum amount a bidder is willing to pay for the object. Each Xi is
independently and identically distributed on some interval [0, ω] according to the increasing
distribution function F. It is assumed that F admits a continuous density f = F’ and has full support.

Bidder i knows the realization xi of Xi and only that other bidders’ values are independently
distributed according to F. Bidders are risk neutral– they seek to maximize their expected profits. All
components of the model other than the realized values are assumed to be commonly known to all
bidders. In particular, the distribution F is common knowledge, as is the number of bidders.

We emphasize that the distribution of values is the same for all bidders and we will refer to this
situation as one involving symmetric bidders.

A strategy for a bidder is a function βi : [0, ω] → R+, which determines his or her bid for any value.
We will typically be interested in comparing the outcomes of a symmetric equilibrium–an
equilibrium in which all bidders follow the same strategy–of one auction with a symmetric
equilibrium of the other. Given that bidders are symmetric, it is natural to focus attention on
symmetric equilibria.

Second-Price Auctions

In a second-price auction, each bidder submits a sealed bid of bi, and given these bids, the payoffs
are:

It should be noted that the argument in Proposition 2.1 relied neither on the assumption that
bidders’ values were independently distributed nor the assumption that they were identically so.
Only the assumption of private values is important and Proposition 2.1 holds as long as this is the
case.

With Proposition 2.1 in hand, let us ask how much each bidder expects to pay in equilibrium. Fix a
bidder, say 1, and let the random variable

denote the highest value among the N −1 remaining bidders. In other words, Y1 is the highest order
statistic of X2,X3, . . . , XN . Let G denote the distribution function of Y1. Clearly, for ally, G(y) =
F(y)^N−1. In a second-price auction, the expected payment by a First-Price Auctions bidder with value
x can be written as
In a first-price auction, the winner pays what he or she bid and thus the expected payment by a
bidder with value x is

which is the same as in a second-price auction (see (2.1)). Figure 2.3 depicts both the expected
payment and the expected payoff of a bidder with value x in either auction. Because the expected
revenue of the seller is just the sum of the ex ante (prior to knowing their values) expected payments
of the bidders, this also implies that the expected revenues in the two auctions are the same. Let us
see why.

The ex ante expected payment of a particular bidder in either auction is

where A = I or II. Interchanging the order of integration we obtain that