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IPO Auctions: English, Dutch, ... French, and Internet

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34 BIAIS AND FAUGERON-CROUZET<br />

The retail investors <strong>and</strong> the informed investor with a bad signal obtain 0-expected<br />

gains in equilibrium <strong>and</strong> it is straightforward to see that they cannot obtain positive<br />

profits by deviating.<br />

Proof of Proposition 5. Consider a c<strong>and</strong>idate equilibrium, whereby tacit collusion<br />

would prevail, <strong>and</strong> in which the <strong>IPO</strong> price would always equal v0. In such<br />

an equilibrium the investors would bid as much as possible while not pushing the<br />

price above v0 <strong>and</strong> still deterring the other investor from competing away market<br />

shares. To avoid pushing the price above v0, they place bids for (slightly less than)<br />

S<br />

N shares at pN <strong>and</strong> for S(1 − 1/N) atv0. The expected profit from such tacit<br />

collusion after observing a good signal is<br />

S<br />

[E(v | g) − v0].<br />

N + 1 − k<br />

If, she expects the others to follow this strategy, the informed investor placing a<br />

bid for S shares at p1 = v1 expects to gain<br />

S[E(v | g) − v1].<br />

Hence, she prefers to follow that strategy rather than to implicitly collude if<br />

E(v | g) − v1 ><br />

1<br />

[E(v | g) − v0].<br />

N + 1 − k<br />

REFERENCES<br />

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Biais, B., Bossaerts, P., <strong>and</strong> Rochet, J. C. (2002). An optimal <strong>IPO</strong> mechanism, Rev. Econ. Stud.,<br />

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in the UK, mimeo, London Business School.<br />

Chemmanur, T. (1993). The pricing of <strong>IPO</strong>s: A dynamic model with information production, J. Finance<br />

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