122 S. Dasgupta and R.G. Hansen
Integrating, and using the boundary conditionbi( 0 )=bj( 0 )(seeBulow, Huang and
Klemperer, 1999, for a proof), we get
φj(ti)= 1 −( 1 −ti)θi/θj. (36)Since the probability that bidderiwins the object is∫ 1
0∫φj(ti)
0 dt dti=θi
θi+θj,itis
clear that bidderiis more likely to win the auction as her stake increases and that
of bidderjdecreases. Remarkably, a bidder’s probability of winning goes to 0 as her
stake becomes arbitrarily small, given that the other bidder has a positive stake. The
intuition for this result is that while bidderiwith zero stake has no incentive to bid
abovev(ti,φj(ti))given the equilibrium bidding strategy ofj, as we shall see below,
bidderjwithtj=φj(ti)and a positive stake will strictly bid above this value.^34
Now, equation(34)can be integrated to give
bi(ti)= (37)∫ 1
tiv(t, φj(t))(^1 −t)1
θj−^1 dt
∫ 1
ti(^1 −t)θ^1
j−^1 dt,
where the boundary conditionbi( 1 )=bj( 1 )=v( 1 , 1 )is used (seeBulow, Huang and
Klemperer, 1999).
From(36), we then get
bi(ti)= (38)∫ 1
tiv(t,^1 −(^1 −t)θi/θj)( 1 −t)
1
θj−^1 dt
∫ 1
ti(^1 −t)1
θj−^1 dt.
Bidderj’s bid function is derived similarly. From(37), it is clear that forti <1,
bi(ti)>v(ti,φj(ti)). Thus, when bidderiwins the auction, she is paying more than the
target is worth to her. Moreover, bidderi’s bid is increasing in her stakeθi, i.e., a higher
stake makes the bidder act more like a seller and causes her to bid higher.
Bulow, Huang and Klemperer (1999)extend the analysis in two main directions.
First, they consider the effect of a more asymmetric distribution of the toeholds and find
that subject to an overall constraint on the toeholds of the two bidders that is sufficiently
small, a more uneven distribution of toeholds leads to lower expected sale price for
(^34) Klemperer (1998)demonstrates in the context of the “Wallet Game” how a very small asymmetry in a
common value model can give rise to very asymmetric equilibria. This is a consequence of the fact that in
the standard Wallet Game, there are in fact a continuum of asymmetric equilibria. A small toehold—like a
small bonus to one of the players in the Wallet Game—introduces a slight asymmetry that can have a major
impact on the equilibrium, i.e., one of the bidders essentially having a zero probability of winning. With a
slight advantage, the stronger player bids slightly more aggressively, but that increases the winner’s curse on
the weaker player. The latter then bids less aggressively, which reduces the winner’s curse on the stronger
player, who then bids still more aggressively, and so on. With slight entry or bidding costs, this prevents the
weaker player/players from entering the auction, so that very low prices result.Klemperer (1998)provides
several illustrative examples from Airwaves Auctions.