9781118041581

This important proposition is called the revenue equivalence theorem. We

already know that (1) the English and Vickrey auctions are equivalent and (2)

the Dutch and sealed-bid auctions are equivalent. Thus, it remains to show that

the English and sealed-bid auctions generate the same expected revenues—a

result that follows from our earlier characterization of equilibrium bidding

strategies. In the English auction, each buyer bids up to its true value (if nec-

essary). The bidding stops when the next-to-last bidder drops out, at a price

approximately equal to the second highest value among the bidders. Thus, the

seller’s expected revenue is simply

where v2nddenotes the second highest buyer value. In the sealed-bid auction,

each buyer uses the equilibrium bidding strategy

where vis the largest of the other bidders’ personal values. Note that the win-

ner’sbid is set at a price that is equal, on average, to the next-best—that is, sec-

ond-highest—valuation. Thus, under either the English or sealed-bid auction,

the average purchase price is the same—equal to the expectation of the second-

highest buyer value. Obviously, as the number of bidders increases, the

expected price rises under either auction.

A UNIFORM EXAMPLE Suppose there are n buyers with reservation prices

independently and uniformly distributed between lower and upper bounds L

and U, respectively. In other words, any value between L and U is considered

equally likely. It is a statistical fact that the expected value of the largest of the

n independent values (call this vmax) is given by

[16.5]

In words, the expected value of the greatest of n independent, uniformly dis-

tributed values lies n/(n 1) of the way toward the upper bound. For instance,

if there are three bidders and values range from 100 to 200, the expectation of

the greatest value lies three-quarters of the way between 100 and 200, that is,

at 175. As the number of bidders increases and the factor n/(n  1)

approaches 1, E(vmax) increases toward U.

In turn, the expectation of the second-highest value (v2nd) is given by

[16.6]

In words, E(v2nd) lies (n 1)/(n 1) of the way toward the upper bound.

This is illustrated by the position of E(v2nd) in Figure 16.3. This point indicates

E(v2nd)a

2

n 1

bLa

n 1

n 1

bU.

E(vmax)a

1

n 1

bLa

n

n 1

bU.

biE(vƒvvi),

E(PE)E(v2nd),

688 Chapter 16 Auctions and Competitive Bidding

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