Auction Theory, 2nd Edition

122 Asymmetries and Other Complications

so

∂

1

∂z

> 0 for all z < x and

∂

1

∂z

< 0 for all z > x. Thus, it is optimal for bidder 1

to choose z =x or, in other words, to bid β(x).

Now consider bidder 2 and suppose that bidder 1 follows the strategy β.

Again, clearly bidder 2 will never choose b > E

[

X

1

]

. If bidder 2 bids b and

wins, her expected profit is

E

[

X

1

|β

(

X

1

)

< b

]

−b = E



X

1

|X

1

<β

−1

(b)



−b

= β



β

−1

(b)



−b

= 0

The profit if bidder 2 loses is, of course, 0 and thus bidder 2’s expected profit

from bidding any b ≤E

[

X

1

]

is exactly 0. Thus, we have shown that bidder 2 is

indifferent on the support of her mixed strategy. This establishes that bidder 2’s

strategy is also