Analytical solution of kth price auction

Abstract

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.

Appendix

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

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

(18)

Proof

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:

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

(19)

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$