9781118041581

Bidder Strategies 683

Let’s check that this strategy constitutes an equilibrium. Consider a typical

firm (say, firm 1) whose expected profit is

[16.1]

The competing bidsof its rival, firm 2, are distributed uniformly between $300

thousand and $330 thousand (since firm 2 is presumed to employ the equilib-

rium bidding strategy just shown). Thus, bid b 1 wins with probability (b 1 

300)/30. For instance, a bid of $300 thousand never wins, a bid of $315 thou-

sand wins with probability .5, and a bid of $330 thousand wins with certainty.

Substituting the probability expression into Equation 16.1 yields

[16.2]

The bidder’s marginal profit is MdE()/db 1 [v 1 2b 1 300]/30.

Setting this equal to zero implies

[16.3]

Note that the firm’s expected profit is zero for the extreme bids, $300 thousand

(which never wins), and v 1 (for which the firm makes no profit). The firm’s

optimal bid is halfway between these extremes. Thus, we have confirmed that

the suggested equilibrium strategy is indeed optimal for the typical firm and for

any value the firm might hold.

b 1 (.5)(300).5v 1.

E()(v 1 b 1 )a

b 1  300

30

b.

E()[v 1 b 1 ][Pr(b 1 wins)].

CHECK

STATION 3

Two firms compete in a sealed-bid auction, with each firm’s private value uniformly dis-

tributed between $0 and $50. (i) Suppose firm 2 is expected to use the (nonequilibrium)

bidding strategy, b 2 .6v 2. Confirm that firm 1’s optimal response is b 1 .5v 1. (In other

words, firm 1 still does best by bidding “one-half its value” as in Equation 16.3.) (ii)

Against b 2 .4v 2 , show that firm 1’s optimal sealed-bid strategy remains b 1 .5v 1.

How are equilibrium bidding strategies affected by changing the number

of competing firms? Let buyer values be uniformly and independently distrib-

uted between lower and upper bounds, denoted by L and U, respectively. (In

the preceding example, L is $300 thousand and U is $360 thousand.) Then

the common equilibrium strategy when n firms compete is simply

[16.4]

The firm’s equilibrium bid is a weighted average of the firm’s actual value

and the lowest possible bidder value. Note that, as the number of bidders

increases, the equilibrium bid rises and comes closer and closer to the firm’s

bia

1

nbLa

n 1

n bvi.

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