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About the article
Published Online: 2013-09-27
Published in Print: 2013-01-01
This result contrasts with a claim in Maskin and Riley (1985) for the case in which the only deviation from a symmetric setting is given by unequal high valuations [this claim is reproduced in Klemperer (1999)]. However, for this case, Maskin and Riley (1983) agree with our ranking between the FPA and the SPA.
In fact, the latter result is known, and in more general settings, since Maskin and Riley (2000a).
As we mentioned above, these results hold if the probability of a low value is the same for each bidder. In Doni and Menicucci (2012a), we study the case in which the probabilities of a low value for the two bidders are different. We also partially extend our results to a setting in which each bidder’s valuation has a three-point support.
Doni and Menicucci (2012b) study a procurement setting in which the auctioneer privately observes the qualities of the products offered by the suppliers and needs to decide how much of the own information on qualities should be revealed to suppliers before a (first score) auction is held. Our results on the comparison between the FPA and the SPA contribute to determine the best information revelation policy for the auctioneer.
Cheng (2010) studies the environments such that each bidder’s equilibrium bidding function is linear.
Cheng (2011) employs the same setting of Maskin and Riley (1983) in order to show that in some special cases the asymmetry increases the expected revenue in the FPA, unlike in the examples studied in Cantillon (2008).
A very similar idea appears in Lebrun (2002), in the auction he denotes with .
For instance, 1H bids according to the uniform distribution on with α < 1 and close to 1.
In the case that (which occurs if and only if ), 2L bids and , thus and for each .
In a setting with continuously distributed valuations, Maskin and Riley (2000a) identify an analogous BNE and provide the intuition we describe here and after Proposition 2. In addition, Maskin and Riley (1983) identify the BNE we describe in Proposition 1 for the case of . Thus Proposition 1 is a new result for the case in which and  is violated.
Obviously, an analogous result holds if is kept fixed and is allowed to vary.
Lebrun (1998) considers a setting with continuously distributed valuations and assumes that the valuation distribution of one bidder changes into a new distribution which dominates the previous one in the sense of reverse hazard rate domination (the support is unchanged). He show that, as a consequence, for each bidder the new bid distribution first-order stochastically dominates the initial bid distribution and thus the expected revenue increases.
In particular, for any small deviation from the symmetric setting, that is when and are close to zero, but and/or .
Since they assume , Maskin and Riley (1983) do not consider the various cases covered in our Proposition 4, and they do not have the results in our Lemma 1 and Propositions 5 to 7.
Proposition 1 still holds even though violates our assumption . However, when the Vickrey tie-breaking rule is needed also if .
This similarity should not be overstated, since the uniform distribution on gives zero probability to , unlike the uniform distribution on . See Section 4.3 for a discussion on the relationship between the results in our model and in the rest of the literature.
If we set , then , which violates the assumption , but nevertheless is the c.d.f. of the equilibrium mixed strategy of bidder 2 when .
This effect appears also in Example 3 in Maskin and Riley (2000a).
Maskin and Riley (2000a) prove the same result under slightly stronger assumptions.
In fact, we can prove that a small shift reduces as it has a zero first-order effect on the bidding of types , but induces to bid less aggressively.
A similar result is obtained if we fix and set , , with . For the case of a large ,  is satisfied and thus . If instead is close to 1, then  reveals that . On the other hand, Kirkegaard (2012b) proves that if is such that is convex and logconcave, is such that and is not much larger than 1.
Kirkegaard (2012b) provides an economic interpretation of this order linked to the relative steepness of the demand function of bidder 1 with respect to the demand function of bidder 2.
We are grateful to one referee for suggesting the main ideas in this paragraph.
We thank one referee for suggesting to investigate the effect of reserve prices.
Kirkegaard (2012a) shows that if  and  are satisfied, then the FPA is superior to the SPA for any common reserve price smaller than , that is such that it allows participation of some type of the weak bidder. Asymmetric auctions with reserve prices are analyzed, using numerical techniques, also in Gayle and Richard (2008), Li and Riley (2007), and Marshall and Schulenberg (2003). In these papers, introducing a reserve price tipically either makes the SPA superior to the FPA (even though the reverse result holds when there is no reserve price) or reduces the revenue advantage of the FPA over the SPA. The latter results are consistent with our results in this section.
In fact, a small shift reduces as in the case of binary supports (see footnote 22), mainly because it induces type to bid less aggressively.
We do not provide here the proof for the case in which  is satisfied since the BNE in that case is similar to a BNE in Maskin and Riley (2000a): see footnote 11. Doni and Menicucci (2012a) provide a complete proof.
Notice that given .
Precisely, if , then and given first-order stochastically dominate, respectively, and given .
In case that we can prove that  is violated.
Actually,  can be replaced in Theorem 1 in Kirkegaard (2012a) with a weaker condition, inequality  in Kirkegaard (2012a), but our argument establishes that also such a condition is violated. Furthermore, our argument does not require that and have binary supports, consistently with the results we describe in Section 5.2 for the case in which supports are three-point sets.