Auction Theory, 2nd Edition

218 Sequential Sales

The equilibrium calls on bidder 1 to bid β

k

(x) in the kth stage but con-

sider what happens if he decides to bid slightly higher—say, β

k

(x +). Now if

Y

k

< x, he would have won with his equilibrium bid β

k

(x) anyway, so the only

consequence of his bidding higher is that he pays more than he would have. His

expected payment increases by

F

k



x |Y

k−1

=y

k−1



×

[

β

k

(

x +

)

−β

k

(

x

)]

(15.7)

The expression in (15.7) thus represents the expected loss from bidding

β

k

(

x +

)

as opposed to β

k

(

x

)

. On the other hand, if x < Y

k

< x +,he

would have lost the kth auction with his equilibrium bid, whereas bidding

higher results in his winning. Now there are two ...