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Analytical solution of kth price auction


We provide an exact analytical solution of the Nash equilibrium for the kth price auction by using inverse of distribution functions. As applications, we identify the unique symmetric equilibrium where the valuations have polynomial distribution, fat tail distribution and exponential distributions.

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The reviewers had a key role in the conception of this article and we are grateful for the reviewers for the very helpful comments.

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Correspondence to Martin Mihelich.

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Lemma 6.1

Consider a real valued bounded function \(\hat{Q}:{\mathbb {R}}\mapsto [0,1]\). For positive integer r and positive real number m, let \(A_r(u,z){:=} {\hat{Q}}(z)(u-z)^r z^m\)and \(H_r(u){:=}\int _0^u A_r(u,z) dz\). Then the \((r+1)^{th}\)derivative of \(H_r(u)\)is

$$\begin{aligned} H_r^{(r+1)}(u) = r!\hat{Q}(u)u^m. \end{aligned}$$


By the Leibniz rule (Rudin et al. 1964) of differentiating an integral, if \(H(u){:=} \int _{l_0(u)}^{l_1(u)} A(u,z) dz\), under assumption of integrability of \(\partial _u A(u,z)\), it holds:

$$\begin{aligned} H'(u) = \left[ A(u,l_1(u))l_1'(u) - A(u,l_0(u))l_0'(u)\right] +\int _{l_0(u)}^{l_1(u)} \partial _u A(u,z) dz. \end{aligned}$$

It is easy to check the integrability of \(\partial _u{\hat{Q}}(z)(u-z)^r z^m\), thus taking \(l_0(u)=0, l_1(u) = u\) in the previous equation:

$$\begin{aligned} H_r'(u) = A_r(u,u) +\int _0^u \partial _u A_r(u,z) dz. \end{aligned}$$

For \(r=0\), \(A_0(u,u) = \hat{Q}(u)u^m\) and \(\partial _u A_0(u,z)=0\). For \(r\geqslant 1\), \(A_r(u,u) = 0\) and \(\partial _u A_r(u,z) = \hat{Q}(z)r(u-z)^{r-1}z^m = rA_{r-1}(u,z)\). Therefore (19) implies

$$\begin{aligned} H_r'(u) = rH_{r-1}(u)\quad \text {for}\quad r\geqslant 1\quad \text {and}\quad H_0'(u) = \hat{Q}(u)u^m. \end{aligned}$$

Thus for \(t\leqslant r\):

$$\begin{aligned} H_r^{(t)}(u) = r(r-1)\cdots (r-t+1)H_{r-t}(u), \end{aligned}$$

so \(H_r^{(r)}(u) = r!H_0(u)\) and therefore \(H_r^{(r+1)}(u) = r!H_0'(u) = r!\hat{Q}(u)u^m.\)\(\square \)

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Mihelich, M., Shu, Y. Analytical solution of kth price auction. Int J Game Theory 49, 875–884 (2020).

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  • Vickrey auctions
  • kth Price auctions
  • Inverse distribution functions