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Let $V\sim F$ be some positive valued random variable such that $F$ has a monotone hazard rate (alternatively $F$ is log-concave). Let $p = \operatorname{argmax}q(1-F(q))$. We call $p$ the monopoly reserve price, when $F$ is MHR it's well known that $p$ is unique.

I'm interested the relationship between the monopoly price and the mean, specifically I want to know if $p \le \Bbb E[V]$, or else be given an MHR distribution such that $p > \Bbb E[V]$. The best bounds I've seen in the literature come from Lemma 3.4 of Giannakopoulos et al. (2021)1 which shows that $$\frac{p}{\mu} \le \frac{e}{e-1} = 1.58$$ by loosely using a result of Barlow et al. (1964)2.

For the canonical MHR distributions like exponential, uniform, etc. $p$ is either equal to or less than $\mu$ so I believe if the result fails is should take some sort of construction.


References

[1] Giannakopoulos, Y., Poças, D., Zhu, K. (2021). Optimal Pricing For MHR and $\lambda$-Regular Distributions. ACM Transactions on Economics and Computation. 9(1).

[2] Barlow, R. E., Marshall, A. W. (1964). Bounds for distributions with monotone hazard rate, i. Ann. Math. Stat. 35(3):1234-1257.

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  • $\begingroup$ You might have better luck on Mathematics StackExchange. $\endgroup$
    – prubin
    Commented Mar 1, 2021 at 23:03
  • 1
    $\begingroup$ Note: as outlined in the paper the upper bound of $e/(e-1)$ can also be constructed from Chawla et al. (2010). $\endgroup$
    – TheSimpliFire
    Commented Mar 2, 2021 at 9:50

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