Ch. 3: Auctions in Corporate Finance 125
From the first-order condition, one readily getsb 1 (t 1 )=( 1 / 2 )t 1 +( 1 / 2 )θ. Thus, the
expected sale price in the matching auction is
PM= (40)
1
4
+
1
2
θ.Comparing(39) and (40), it can be verified thatPM>PSif and only ifθ> 0 .2899.
Thus, if the toeholds are sufficiently asymmetric, asking the strong bidder to move first
increases the expected sale price.
The matching auction’s properties in the context of a common value model with
independent signals similar toBulow, Huang and Klemperer (1999)have been explored
byDasgupta and Tsui (2003). The authors show that there exists a perfect Bayesian
Nash equilibrium in which bidder 1 with stakeθ 1 bids
b 1 (t 1 )=v (41)(
t 1 ,F 2 −^1 (θ 1 ))
and bidder 2 matches if and only ift 2 F 2 −^1 (θ 1 ).^38 Here, bidderi’s signal is drawn
from the distributionFi(ti). Notice that the expected sale price is then
PM=Et 1 (42)(
v(
t 1 ,F 2 −^1 (θ 1 )))
.
Notice that (i) conditional on her bid, losing is better than winning for bidder 1, since
her payoff in the former event isθ 1 v(t 1 ,F−^1 (θ 1 )), and her payoff in the latter event is
at mostv(t 1 ,F−^1 (θ 1 ))−( 1 −θ 1 )v(t 1 ,F−^1 (θ 1 )), and (ii) as a consequence, winning is
“bad news” for bidder 1, i.e., if she wins, there would be a negative effect on the stock
price. In contrast, winning is always “good news” for the second bidder.
It is also immediate that the expected sale price increases in the first bidder’s toehold.
In contrast, bidder 2’s stake has no effect on the expected sale price. The probability of
bidder 1 winning the auction isF 2 −^1 (θ 1 )and is therefore increasing inθ 1. However, the
common value feature of the model is apparent in that if bidder 1’s toehold is 0, then her
probability of winning is also 0; moreover, in this case, she bidsv(t 1 , 0 ), i.e., the lowest
possible value conditional on her own signal. This is because the bidder who moves first
is subjected to an extreme winner’s curse problem.
How can the matching auction improve the expected sale price compared to the stan-
dard auctions? Recall that in the second-price auction with asymmetric toeholds, the
smaller toehold bidder is exposed to an extreme winner’s curse problem. The matching
auction is a way to shield the low toehold bidder from this extreme winner’s curse by
asking her to move second. This, of course, imposes a winner’s curse on the first bidder.
However, if the asymmetry is large, the first bidder with a higher toehold will act more
like a seller, and this the sale price will not suffer as much.Dasgupta and Tsui (2003)
show that, for the case of a value function that is symmetric and linear in the signals
(i.e.,v(t 1 ,t 2 )=t 1 +t 2 ) that are drawn from the uniform distribution, the matching
(^38) For a derivation and a complete characterization of the equilibrium, seeDasgupta and Tsui (2003).