Auction Theory, 2nd Edition

2.3 First-Price Auctions 13

In a second-price auction, each bidder submits a sealed bid of b

i

, and given

these bids, the payoffs are:



i

=



x

i

−max

j=i

b

j

if b

i

> max

j=i

b

j

0ifb

i

< max

j=i

b

j

We also assume that if there is a tie, so b

i

=max

j=i

b

j

, the object goes to each win-

ning bidder with equal probability. Bidding behavior in a second-price auction

is straightforward.

Proposition 2.1. In a second-price sealed-bid auction, it is a weakly dominant

strategy to bid according to β

II

(x) =x.

Proof. Consider bidder 1, say, and suppose that p

1

=max

j=1

b

j

is the highest

competing bid. By bidding x

1

, bidder 1 will win if x

1

> p

1

and not if x

1

< p

1

(if

x

1

=p

1

, bidder 1 is