Asymmetric first-price auctions with uniform distributions: analytic solutions to the general case

Kaplan, Todd R.; Zamir, Shmuel

doi:10.1007/s00199-010-0563-9

Cited by 78 publications

(39 citation statements)

References 25 publications

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“…We can then define C L ( b ) and C H ( b ) as the respective inverse bid functions with supports

(][{\underset{̲}{b}}_{L}, {\overset{b}{true}}_{L})

and

(][{\underset{̲}{b}}_{H}, {\overset{b}{true}}_{H})

, where these intervals denote the range of bids for which someone wins the auction with positive probability as in Kaplan and Zamir (). It is straightforward to extend Lemma 4 of Kaplan and Zamir () to the LLH and LHH cases to show that both inverse functions share the same upper support which we define as

\overset{b}{true}

. However, it is not necessary for both types to share the same lower support in the LLH composition as the two low‐cost types may be incentivized to bid below

{\underset{̲}{c}}_{H}

when competing against each other.…”

Section: Equilibrium Bidding Strategies and Ex Ante Predictionsmentioning

confidence: 99%

“…However, it is not necessary for both types to share the same lower support in the LLH composition as the two low‐cost types may be incentivized to bid below

{\underset{̲}{c}}_{H}

when competing against each other. Following Appendices A.2 and A.3 of Kaplan and Zamir (), the upper boundary conditions for each type are

C_{H} (\overset{b}{true}) = \bar{b}

and

C_{L} (\overset{b}{true}) = {\overset{c}{true}}_{L} = 20

. From these conditions, we can derive

\overset{b}{true}

in each case.…”

Section: Equilibrium Bidding Strategies and Ex Ante Predictionsmentioning

confidence: 99%

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An Experimental Investigation of Procurement Auctions with Asymmetric Sellers

Aloysius

Deck

et al. 2016

Production and Operations Management

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E lectronic reverse auctions are a commonly used procurement mechanism. Research to date has focused on suppliers who are ex ante symmetric in that their costs are drawn from a common distribution. However, in many cases, a seller's range of potential costs depends on their own operations, location, or economies of scale and scope. Thus, understanding how different bidder types impact auction outcomes is key when designing an auction. This study reports the results of the first controlled laboratory experiment designed to compare prices between first-price and second-price procurement auctions for homogeneous goods when seller cost types are asymmetric and the number of bidders varies. The results indicate that first-price auctions generate lower prices regardless of market composition. The results also reveal that first-price auctions are at least weakly more efficient than second-price auctions despite the theoretical prediction that the reverse should hold in asymmetric auctions. Post hoc analysis of individual bidders' behavior in first-price auctions revealed evidence that bidders systematically underbid when their cost realizations were close to the lower bound. Furthermore, bidders adjust their behavior based on the type of the other bidders in the market in a manner inconsistent with theory. Consequently, adding a third bidder to a two-bidder market is not advantageous to the buyer unless that third bidder is a low-cost type.

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“…We can then define C L ( b ) and C H ( b ) as the respective inverse bid functions with supports

(][{\underset{̲}{b}}_{L}, {\overset{b}{true}}_{L})

and

(][{\underset{̲}{b}}_{H}, {\overset{b}{true}}_{H})

\overset{b}{true}

. However, it is not necessary for both types to share the same lower support in the LLH composition as the two low‐cost types may be incentivized to bid below

{\underset{̲}{c}}_{H}

when competing against each other.…”

Section: Equilibrium Bidding Strategies and Ex Ante Predictionsmentioning

confidence: 99%

“…However, it is not necessary for both types to share the same lower support in the LLH composition as the two low‐cost types may be incentivized to bid below

{\underset{̲}{c}}_{H}

when competing against each other. Following Appendices A.2 and A.3 of Kaplan and Zamir (), the upper boundary conditions for each type are

C_{H} (\overset{b}{true}) = \bar{b}

and

C_{L} (\overset{b}{true}) = {\overset{c}{true}}_{L} = 20

. From these conditions, we can derive

\overset{b}{true}

in each case.…”

Section: Equilibrium Bidding Strategies and Ex Ante Predictionsmentioning

confidence: 99%

An Experimental Investigation of Procurement Auctions with Asymmetric Sellers

Aloysius

Deck

et al. 2016

Production and Operations Management

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show abstract

“…Notice that, in our setting, the supports of effective cost distributions are naturally different for contractors with different backlog levels. We adjust the standard argument (see Kaplan and Zamir, ), accounting for the all‐pay component in order to obtain the boundary condition for our optimization program. More specifically, without loss of generality, assume that

{\overset{φ}{true}}_{1} \leq {\overset{φ}{true}}_{2}

.…”

Section: Equilibrium Characterizationmentioning

confidence: 99%

Dynamic auction environment with subcontracting

Jeziorski

Krasnokutskaya

2016

The RAND J of Economics

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This article provides evidence on the role of subcontracting in the auction‐based procurement setting with private cost variability and capacity constraints. We demonstrate that subcontracting allows bidders to modify their costs realizations in a given auction as well as to control their future costs by reducing backlog accumulation. Restricting access to subcontracting raises procurement costs for an individual project by 12% and reduces the number of projects completed in equilibrium by 20%. The article explains methodological and market design implications of subcontracting availability.

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“…For example, Kaplan and Zamir (2012) provide explicit forms of the equilibrium bid functions in an asymmetric first-price auction with two buyers whose values v 1 and v 2 are uniformly distributed in [v 1 , v 1 ] and [v 2 , v 2 ], respectively. However, an explicit expression of the BNE is mathematically hard to obtain and so far it is available only for simple models, subject to highly restrictive assumptions.…”

Section: First-price Auctions: Theoretical Advancesmentioning

confidence: 99%

“…These were fromGriesmer et al (1967). Also, seeKaplan and Zamir (2012) for the general solution to the uniform case. 20 See Section 7.2.5 for examples byMaskin and Riley (2000a) where the revenue between a first-price auction and a second-price auction can be ordered in either way.…”

mentioning

confidence: 99%