124 S. Dasgupta and R.G. Hansen
stronger bidder may provide a higher expected profit to the seller. Thus, standard auc-
tions are no longer optimal in the presence of various forms of bidder heterogeneity.
To increase the expected sale price when bidders are asymmetric, the seller has es-
sentially two alternative responses. Both involve “levelling the playing field”. When the
asymmetry is due to differences in toeholds or access to information, the target’s board
may decide to restore symmetry by allowing the disadvantaged bidder increase his toe-
hold cheaply or provide access to additional information.^36 Alternatively, the board may
decide to design the auction rules in a way that discriminates against the strong bidder.
An especially simple way to discriminate is to impose an order of moves on the bid-
ders. Since bidding games are price-setting games, there is usually a “second-mover
advantage” associated with bidding games (seeGal-Or, 1985, 1987). Thus, to discrimi-
nate against the strong bidder, the seller could ask this bidder to bid first. This bid could
then be revealed to a second bidder, who wins the auction if she agrees to match the first
bid. Otherwise, the first bidder wins. In the context of takeover bidding, this “matching
auction” has been studied byDasgupta and Tsui (2003), who note that since courts are
more concerned about shareholder value than whether the playing field is level or not,
it is unlikely that the matching auction will run into trouble because it does not treat the
bidders symmetrically.^37
To see that the matching auction can generate a higher expected sale price than the
second-price auction in the independent private value setting, let us return to the private
values model introduced in Section4.4. Assume that the private values of both bidders
are drawn from the uniform[ 0 , 1 ]distribution. From equation(30), we get the bid of
bidder 1 who has a toehold ofθto be
b 1 (t 1 )=t 1 +θ
1 +θ.
Thus, the expected bid from bidder 1 isP 1 =∫ 1
0∫(t 1 +θ)/( 1 +θ)
0 t^2 dt^2 dt^1 =1
63 θ^2 + 3 θ+ 1
( 1 +θ)^2
and that from bidder 2 isP 2 =
∫ 1
0∫( 1 +θ)t 2 −θ
0t 1 +θ
1 +θdt^1 dt^2 =1
61 − 2 θ^2 + 2 θ
1 +θ. Thus, the
expected sale price in the second-price auction is
PS=P 1 +P 2 = (39)
( 2 θ+ 1 )( 2 + 2 θ−θ^2 )
6 ( 1 +θ)^2.
Now consider the matching auction. Given a bidb 1 from bidder 1, bidder 2 will match
if and only ift 2 >b 1. Thus, bidder 1 choosesb 1 to maximize
∫b 1
0(
t 1 −( 1 −θ)b 1)
dt 2 +θ( 1 −b 1 )b 1.(^36) Betton and Eckbo (2000)note that when a rival (second) bidder enters the auction with a toehold, the
toehold is of roughly the same magnitude as the initial bidder’s toehold (about 5%). This is consistent with
the “leveling the playing field” argument ofBulow, Huang and Klemperer (1999).
(^37) Herzel and Shepro (1990)note: “Opinion in several cases in the Delaware Chancery court has noted that
the duty and loyalty [of managers] runs to shareholders, not bidders. As a result, ‘the board may tilt the
playing field if it is in the shareholder interest to do so’ ”.