Regular type distributions in mechanism design and $\rho$-concavity

Abstract

Some of the best-known results in mechanism design depend critically on Myerson’s (Math Oper Res 6:58–73, 1981) regularity condition. For example, the second-price auction with reserve price is revenue maximizing only if the type distribution is regular. This paper offers two main findings. First, a new interpretation of regularity is developed—similar to that of a monotone hazard rate—in terms of being the next to fail. Second, using expanded concepts of concavity, a tight sufficient condition is obtained for a density to define a regular distribution. New examples of regular distributions are identified. Applications are discussed.

Appendix: Parameterized distributions

This Appendix outlines the derivations underlying Tables 1 and 2. The main tool is the following smooth criterion for $\rho$-concavity.

Lemma A.1.

Let $f>0$ be twice continuously differentiable on some interval $X\subseteq \mathbb R$, with a discrete set $X_{1}$ over which $f^{\prime }(x)=0$. Then, for finite $\rho$, the function $f$ is $\rho$-concave if and only if $r_{f}(x)\equiv -(\ln f(x))^{\prime \prime }/(\ln f(x))^{\prime 2}\ge \rho$ for all $x\in X\setminus X_{1}$.

Proof.

A straightforward calculation shows that $r_{f}(x)=1-f(x)f^{\prime \prime }(x)/f^{\prime }(x)^{2}$. Hence, $r_{f} (x)\ge \rho$ if and only if $f(x)f^{\prime \prime }(x)\le (1-\rho )f^{\prime }(x)^{2}$, provided $f^{\prime }(x)\ne 0$. By continuity, this proves the assertion. $\square$

Lemma A.1 reduces the determination of the global concavity parameter to a straightforward minimization problem. More specifically, to find the tightest parameter $\rho$ for which a given $f$ is $\rho$-concave, one calculates the minimum (or infimum) of $r_{f}(x)$ on $X$. Table 1 shows the results for selected examples. A particular case is the Pearson distribution. Its density function solves the differential equation $f^{\prime }(x)=f(x)(x-x^{M})/\chi (x)$ for $x^{M}\in \mathbb R$ and $\chi (x)=b_{0}+b_{1}x+b_{2}x^{2}$, where $b_{0},b_{1},b_{2}\in \mathbb R$. We focus on distributions with unbounded support and such that $\chi (x^{M})<0$. Then, $f$ is $b_{2}$-concave. Thus, both $J_{f}$ and $K_{f}$ are increasing provided that $b_{2}>-\frac{1}{2}$.

Table 2 shows examples of distributions that do not allow a strongly $(-\frac{1}{2})$-concave density function. Unless noted otherwise, all parameter values are strictly positive. The only regular example is the mirror-image Pareto distribution. The entries of the form $J_{f}^{\prime } \ngeq 0$, $K_{f}^{\prime }\ngeq 0$, or “mixed” have been established by direct calculation at boundary values. In some cases, numerical calculations have been used. The entries with $K_{f}^{\prime }>0$ are typically straightforward, e.g., when the density function is everywhere declining. Some cases, however, need additional arguments. For example, both the log-normal distribution and the F distribution (Finner and Roters 1997) possess a log-normal distribution function, even though the corresponding density functions are not log-normal. In other cases (inverse gamma distribution and inverse Chi-squared), the density function is log-concave for low values and decreasing for high values, which again is sufficient for $K_{f}^{\prime }>0$.