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A parametric family of two ranked objects auctions: equilibria and associated risk

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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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Correspondence to Estrella Alonso.

Appendix

Appendix

Proof Theorem 1

Case n>2 The profit for bidder i is:

$$ \begin{array}{l} B_{i}^{ ( \gamma _{1},\gamma _{2} ) }\bigl(\theta_{i},b_{i},b(\theta_{-i})\bigr) \\ \\ \quad {} =\left \{ \begin{array}{l} \theta_{i}- ( \gamma_{1}b_{i}+ ( 1-\alpha -\gamma_{1}+\gamma_{2} ) b ( \theta_{-i}^{ ( 1 ) } ) + ( \alpha -\gamma_{2} ) b ( \theta_{-i}^{ ( 2 ) } ) ) \\[2pt] \quad \mbox{if} \ b^{-1}(b_{i})>\theta_{-i}^{ ( 1 ) } \\[2pt] \alpha \theta_{i}- ( \gamma_{2}b_{i}+ ( \alpha -\gamma_{2} ) b ( \theta_{-i}^{ ( 2 ) } ) ) \quad \mbox{if}\ \theta_{-i}^{ ( 2 ) }<b^{-1}(b_{i})<\theta_{-i}^{ ( 1 ) } \\[2pt] 0 \quad \mbox{otherwise}\end{array} \right .\end{array}$$

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,

$$ b^{\ast }(\theta_{i})=\theta_{i}-\frac{\int_{0}^{\theta _{i}}F ( x )^{\frac{\alpha ( n-2 ) }{\gamma _{2}}} ( \frac{ ( n-1 ) \gamma_{2}-\gamma_{1}}{ ( n-1 ) }F ( x ) -\gamma_{2} )^{\frac{ ( ( n-1 ) \alpha -1 ) ( n-1 ) }{ ( n-1 ) \gamma _{2}-\gamma _{1}}-\frac{\alpha ( n-2 ) }{\gamma _{2}}}dx}{F ( \theta_{i} )^{\frac{\alpha ( n-2 ) }{\gamma _{2}}} ( \frac{ ( n-1 ) \gamma_{2}-\gamma_{1}}{ ( n-1 ) }F ( \theta_{i} ) -\gamma_{2} )^{\frac{ ( ( n-1 ) \alpha -1 ) ( n-1 ) }{ ( n-1 ) \gamma _{2}-\gamma _{1}}-\frac{\alpha ( n-2 ) }{\gamma _{2}}}}. $$

If γ 1=(n−1)γ 2, and γ 2≠0

$$ b^{\ast }(\theta_{i})=\theta_{i}-\frac{\int_{0}^{\theta _{i}}F ( x )^{\frac{\alpha ( n-2 ) }{\gamma _{2}}}e^{-\frac{ ( ( n-1 ) \alpha -1 ) F ( x ) }{\gamma _{2}}}dx}{F ( \theta_{i} )^{\frac{\alpha ( n-2 ) }{\gamma _{2}}}e^{-\frac{ ( ( n-1 ) \alpha -1 ) F ( \theta _{i} ) }{\gamma _{2}}}}. $$

If γ 2=0 and γ 1=0

$$ b^{\ast }(\theta_{i})=\theta_{i}. $$

If γ 2=0 and γ 1≠0, then

$$ b^{\ast }(\theta_{i})=\theta_{i}-\frac{\int_{0}^{\theta _{i}}F ( x )^{-\frac{ ( ( n-1 ) \alpha -1 ) ( n-1 ) }{\gamma _{1}}}e^{-\frac{\alpha ( n-2 ) ( n-1 ) }{\gamma _{1}F ( x ) }}dx}{F ( \theta_{i} )^{-\frac{ ( ( n-1 ) \alpha -1 ) ( n-1 ) }{\gamma _{1}}}e^{-\frac{\alpha ( n-2 ) ( n-1 ) }{\gamma _{1}F ( \theta _{i} ) }}}. $$

Case n=2 The profit for bidder i is:

$$ \begin{array}{l} B_{i}^{ ( \gamma _{1},\gamma _{2} ) }\bigl(\theta_{i},b_{i},b(\theta_{j})\bigr) \\ \\ \quad{} = \left\{ \begin{array}{l@{\quad }l} \theta_{i}- ( \gamma_{1}b_{i}+ ( 1-\alpha -\gamma_{1}+\gamma_{2} ) b ( \theta_{j} ) ) & \mbox{if} \ b^{-1}(b_{i})>\theta_{j} \\[3pt] \alpha \theta_{i}-\gamma_{2}b_{i} & \mbox{if}\ b^{-1}(b_{i})>\theta_{j} \\ 0 &\mbox{otherwise}. \end{array}\right. \end{array}$$

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:

$$ \bigl( \gamma_{2}- ( \gamma_{2}-\gamma_{1} ) F ( \theta_{i} ) \bigr) b^{\ast \prime }(\theta_{i})+ ( 1- \alpha ) f ( \theta_{i} ) b^{\ast }(\theta_{i})= ( 1- \alpha ) f ( \theta_{i} ) \theta_{i}. $$

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

$$ \bigl\{\omega_{i,p}^{A}(b_{1},\ldots,b_{n}), \psi_{i,p}^{A}(b_{1},\ldots,b_{n})\bigr \}_{i=1,\ldots,n}^{p=1,2} $$

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

$$ \theta_{i}=\arg \max_{t_{i}} \Biggl\{ \,\sum _{p=1}^{2} \bigl( \alpha_{p} \theta_{i}y_{i,p}^{A}(t_{i})-q_{i,p}^{A}(t_{i}) \bigr) \Biggr\} . $$

Then

$$ \sum_{p=1}^{2}\frac{d}{d\theta_{i}}q_{i,p}^{A}( \theta_{i})=\sum_{p=1}^{2} \alpha_{p}\theta_{i}\frac{d}{d\theta_{i}}y_{i,p}^{A}(\theta_{i}) . $$

Integrating both sides

$$ \sum_{p=1}^{2}q_{i,p}^{A}( \theta_{i})=\sum_{p=1}^{2} \alpha_{p}\int_{0}^{\theta _{i}}t \frac{d}{dt}y_{i,p}^{A}(t)dt+\sum _{p=1}^{2}q_{i,p}^{A}(0)\text{.} $$

Then

$$ P_{i}^{A}(\theta_{i})=\sum _{p=1}^{2}\alpha_{p}\int _{0}^{\theta _{i}}t\frac{d}{dt}y_{i,p}^{A}(t)dt+P_{i}^{A}(0) $$

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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