Ch. 3: Auctions in Corporate Finance 123
the target. This result is a consequence of the fact that as the toeholds become more
asymmetric, the bidder with the higher toehold bids more aggressively, i.e., further away
from the value. For the bidder with a smaller toehold, this implies that the target is
worth less conditional on winning. Exposed to this “winner’s curse”, the bidder with
the smaller toehold therefore bids lower. Since in the second-price auction the winner
pays the lower of the two bids, the expected sale price is adversely affected when the
toeholds become asymmetric.
Bulow, Huang and Klemperer (1999)next consider first-price auction and derive
the equilibrium bid functions using methods similar to those described above for the
second-price auction. In this case, we haveφ ̃j(ti)= t
( 1 −θi)/( 1 −θj)
i. The probability
that bidderiwith signaltiand toeholdθiwins the auction in this case is given by
1 −θj
( 1 −θi)+( 1 −θj), which is increasing inθi. It is easily checked that forθi<θj, the prob-
ability of bidderiwinning the auction is lower in the second-price auction than in the
first-price auction. Since in both auctions the probability is exactly 1/2 whenθi=θj,
this implies that the winning probability falls more steeply with a decrease in a bidder’s
toehold the second-price auction than in the first-price auction.
The incentive for bidders with toeholds to bid high in the first-price auction are not
as strong as in the second-price auction. This is because in the in the first-price auction
(unlike the second-price auction), bidding high does affect the bidder’s cost, although a
higher toehold does lower that cost since fewer shares need to be purchased.
Unlike the second-price auction, the expected sale price can increase in the first-price
auction as the toeholds become more asymmetric. Revenue comparisons indicate that
withsymmetrictoeholds, the expected sale price is higher in the second-price auction.
This is because as the winner’s curse problem is mitigated with symmetric toeholds,
both bidders can bid more aggressively and essentially set a higher reserve price for
their stakes in the second-price auction. With asymmetric toeholds, as we saw above,
the second-price auction generates low expected sale prices due to the winner’s curse.^35
4.5. Bidder heterogeneity and discrimination in takeover auctions
Bidder asymmetry is common in the context of corporate control contests and can take
several forms. Asymmetry in initial stakes or toeholds, discussed in the previous section,
is one form of bidder asymmetry. Bidder asymmetry can also arise when bidders draw
their signals from different distributions, or when (in a common value environment) the
bidder signals have asymmetric impact on the value function.
In Section2.6, we saw that when bidders are asymmetric, the optimal mechanism
may not allocate the object to the bidder with the highest valuation. For example, in
the independent private value context, an allocation rule that discriminates against a
(^35) The analysis of toeholds can be extended to models that include the private value model and the common
value model ofBulow, Huang and Klemperer (1999)as special cases.Dasgupta and Tsui (2004)analyze
auctions where bidding firms hold toeholds in each other in the context of such a model.