120 S. Dasgupta and R.G. Hansen
Burkart (1995)^32 considers a two-bidder and independent private values model. The
private values are best interpreted as synergies. The auction form is a second-price auc-
tion, which in this context is strategically equivalent to an ascending auction (Lemma 1
in the paper). From standard arguments, it follows that (i) it is a dominant strategy for
the bidder with no toeholds to bid exactly her valuation, and (ii) it is adominatedstrat-
egy for the bidder with positive toehold to bid below her valuation. A general result
is that any bidder with positive initial stake will bid strictly above her valuation. The
model is then specialized to the case in which one bidder—call her bidder 1—has an
initial stake ofθwhile the other bidder—bidder 2—has no initial stake.
Since bidder 2 will bid her value, we haveb 2 (v 2 )=v 2. Thus, bidder 1’s problem is
to chooseb 1 to maximize
Maxb 1 Π 1 (v 1 ,b 1 ,θ)= (29)∫b 10[
v 1 −( 1 −θ)v 2]
f 2 (v 2 )dv 2 +θb 1(
1 −F 2 (b 1 ))
.
The first-order condition is
(
v 1 −( 1 −θ)b 1)
f 2 (b 1 )+θ(
1 −F 2 (b 1 ))
−θb 1 f 2 (b 1 )= 0.
Re-arranging, we getb 1 =v 1 +θ (30)1 −F 2 (b 1 )
f 2 (b 1 )>v 1.If one assumed that the hazard function 1 −fF^2 ( 2 ·)(·)is increasing, then a number of results
follows immediately. First, bidder 1’s equilibrium bid is increasing in her valuation and
the size of her toehold. Therefore, the probability that bidder 1 wins the auction is also
increasing in her toehold. It is also clear that the auction outcome can be inefficient:
since bidder one bids more aggressively than bidder 2, it is clearly possible thatv 1 <
v 2 <b 1 (v 1 ), i.e., bidder 1 has the lower valuation but wins the auction. This result is
similar to the inefficiency in the standard auctions where the seller sets a reserve price.
In fact, the intuition for the overbidding result is exactly that of an optimal reserve price
from the point of view of a seller. Indeed, with a toehold, a bidder is a part-owner and
we should not be surprised to find that she wants to “set a reserve price” in excess of
her own value.
It is interesting to note that winning can be “bad news” for bidder 1. Supposev 1 = 0
with probability 1. Then bidder 1 still bids a positive amount (equal to bidder 2’s value)
but since her bid exceeds the value of the synergy, she always overpays when she wins
the auction. By continuity, the same conclusion holds forv ̄ 1 (the upper bound of the
support of the distribution of bidder 1’s synergy) sufficiently small, and for bidder 2’s
valuation in some interval[v′ 2 ,b 1 (v ̄ 1 )].^33
(^32) Singh (1998)has essentially similar results.
(^33) Using Burkart’s private value setting with two bidders,Betton, Eckbo and Thorburn (2005)also show
optimal overbidding when the bidder has a lock-up agreement with the target. Moreover, they show optimal
underbidding when the bidder has a breakup fee agreement with the target.