Abstract
This paper deals with simultaneous auctions of two commonly ranked objects following the model studied in Menezes and Monteiro (J. Real Estate Finance Econ., 17(3):219–232, 1998). For these problems we introduce a parametric family of auction mechanisms which includes the three classic auctions (discriminatory-price auction, uniform-price auction and Vickrey auction) and we call it the \(\mathcal{DUV}\) family. We provide the unique Bayesian Nash equilibrium for each auction in \(\mathcal{DUV}\) and prove a revenue equivalence theorem for the parametric family. Likewise, we study the value at risk of the auctioneer as a reasonable decision criterion to determine which auctions in \(\mathcal{DUV}\) may be better taking into account the interests of the auctioneer. We show that there are auction mechanisms in \(\mathcal{DUV}\) which are better than the classic auction mechanisms with respect to this criterion.
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Acknowledgements
The authors gratefully acknowledge the support of the Spanish Ministry of Education and Science though the projects MTM 2008-06778-C02-01 and MTM 2008-06778-C02-02.
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Appendix
Appendix
Proof Theorem 1
Case n>2 The profit for bidder i is:
where \(\theta_{-i}^{ ( j ) }\) is a realization of \(\varTheta_{-i}^{(j)}\) that is the j-highest order statistic of {Θ −i }. Bidder i knows his type θ i , but θ −i is a realization of a random variable, so the expected profit for bidder i is given by:
Thus, b i is the best bid for bidder i if it maximizes the expected profit, given that his type is θ i . The derivative with respect to b i is:
As b −1(b i )=θ i ⇔b i =b(θ i ), replacing and setting the above equation equal to zero, we obtain the following differential equation:
Then, solving this lineal differential equation with the condition \(\lim_{\theta_{i}\rightarrow 0}b^{\ast }(\theta_{i})=0\) we obtain:
If γ 1≠(n−1)γ 2, and γ 2≠0,
If γ 1=(n−1)γ 2, and γ 2≠0
If γ 2=0 and γ 1=0
If γ 2=0 and γ 1≠0, then
Case n=2 The profit for bidder i is:
Bidder i knows his type θ i , but θ j is a realization of the random variable Θ j , so the expected profit for bidder i is given by:
Thus, b i is the best bid for bidder i if it maximizes the expected profit, given that his type is θ i . The derivative with respect to b i is:
As b −1(b i )=θ i ⇔b i =b(θ i ), replacing and setting the above equation equal to zero, we obtain the following differential equation:
This equation is a particular case of the previous differential equation with n=2 so the expression for the Bayesian Nash equilibrium is the same. □
Proof Theorem 2
Let A be an auction of 2 heterogeneous commonly-ranked objects and n bidders (n≥2). The values for bidder i is given by α 1 θ i ≤α 2 θ i and associated list is
where \(\omega_{i,p}^{A}(b_{1},\ldots,b_{n})\) is the expected payment of bidder i if he buys the p th object auctioned; \(\psi_{i,p}^{A}(b_{1},\ldots,b_{n})\) is the probability that bidder i will buy p th object auctioned, if the bids are b 1,…,b n . Let (b ∗(θ 1),…,b ∗(θ n )) be the unique Bayesian Nash equilibrium. Let \(\widetilde{A}\) be the direct mechanism of A with associated list:
then \(\widetilde{A}\) yields incentive compatibility.
If bidder i’s bid is t i and the others bid their own types then the bidder i’s expected profit in the auction mechanism \(\widetilde{A}\) is:
where we denote
to the expected payment of bidder i if he buys the p th object auctioned and to the expected probability that bidder i buys the p th object auctioned, if bidder i’s type is t i , respectively. The incentive compatibility implies
Then
Integrating both sides
Then
where \(P_{i}^{A}(\theta_{i})=\sum_{p=1}^{2}q_{i,p}^{A}(\theta_{i})\) is the total expected payment of bidder i. If \(A\in \mathcal{DUV}\) then α 1=1, α 2=α, \(P_{i}^{A}(0)=0\), \(y_{i,1}^{A}(t)=F^{n-1}( t ) \) and \(y_{i,2}^{A}(t)= ( n-1 ) ( F^{n-2} ( t ) -F^{n-1} ( t ) ) \). Then
and
\(\forall A\in \mathcal{DUV}\). □
Remark 5
The generalization of the previous theorem to m objects is immediate.
Proof Proposition 1
In the same manner that in single object auction (Krishna 2002) we obtain
where the virtual valuation \(J(\theta_{i})=\theta_{i}-\frac{1-F(\theta_{i})}{f(\theta_{i})}\geq 0\) is monotone in θ i then \(\theta_{i}^{\ast }\) verifies \(J(\theta_{i}^{\ast })=0\). □
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Alonso, E., Sanchez-Soriano, J. & Tejada, J. A parametric family of two ranked objects auctions: equilibria and associated risk. Ann Oper Res 225, 141–160 (2015). https://doi.org/10.1007/s10479-012-1297-9
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DOI: https://doi.org/10.1007/s10479-012-1297-9