9781118041581

value. Holding vi$348 thousand, the firm bids $324 thousand when it is one

of two bidders, but $342 thousand when it is one of eight bidders.^10

In fact, there is a very simple rule describing equilibrium bidding strategies

with any number of bidders and any common probability distribution:

In a sealed-bid auction, the equilibrium bidding strategy of the typical risk-neutral

buyer is to submit a bid, bi, equal to the expected value of the highest of the n  1

other buyer values, conditional on these values being lower than vi. Formally stated,

biE(vƒv’ vi), where vdenotes the largest of the other bidders’ personal values.

Since this bidding rule is something of a mouthful, a concrete example is use-

ful. Consider, once again, the office building auction—this time with two buyers.

Suppose firm 1’s value is v 1. Since firm 1 knows its opponent’s value is uniformly

distributed between $300 thousand and $360 thousand, the distribution of this

value, v, conditional onvbeing smaller than v 1 , is uniform between $300 thou-

sand and v 1. Therefore, the conditional expected value of vis simply (.5)(300) 

.5v 1 —the average of $300 thousand and v 1. This is the firm’s best bid. But this is

exactly the equilibrium bidding strategy depicted in Equation 16.3. More gen-

erally, if there are n 1 other bidders whose personal values are distributed uni-

formly between 300 and v 1 , the expectation of the greatest of these values is

300/n [(n 1)/n]v 1. This confirms Equation 16.4’s equilibrium strategy.^11

Although there is no easy intuitive argument to explain the preceding bid-

ding rule, one comment is in order. The key to crafting an optimal bid is to

assume for a moment that yours is the highest value. (If this is not the case, the

buyer will be outbid by a rival anyway.) Accordingly, the buyer bases its bid on

the expected value of v, conditional on vbeing smaller than v 1.

To sum up, we have examined two approaches to finding the firm’s opti-

mal bidding strategy. The first approach takes the distribution of opposing bidsas

given and asks what is the firm’s profit-maximizing bid strategy in response.

The equilibrium approach starts from a prediction concerning the underlying

values of the firmsand identifies bid strategies such that each buyer is profit max-

imizing against the bidding behavior of its competitors. By the way, the BCB dis-

tribution in Figure 16.1 is derived from equilibrium bidding behavior among

three bidders, each with a value distributed uniformly between $300 thousand

684 Chapter 16 Auctions and Competitive Bidding

(^10) Here is how to check Equation 16.4’s equilibrium strategy. With this strategy, each buyer’s bids
are in the range L to (L/n) [(n 1)/n]U. Therefore, bid biwins with probability [(bi L)/
((n 1)(U L)/n))]n ^1. Thus, the firm’s expected profit can be written in the form of k(vibi)
(biL)n ^1 , after collecting miscellaneous constant terms into the coefficient k. Therefore,
marginal profit is k[(n 1)(vibi)(biL)n ^2 (biL)n ^1 ]. Setting this equal to zero and
canceling out the common factor, (biL)n ^2 yields Equation 16.4. Thus, we have confirmed
that using the proposed strategy is an equilibrium.
(^11) For uniformly distributed values, the equilibrium bidding strategy can be expressed in a neat
formula. This is not the case for many other distributions, such as the normal distribution.
However, tables of conditional expected values for many distributions are readily available.
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