Abstract
We focus on a canonical Bayesian mechanism design setting: a seller wants to sell a single item to n bidders, whose values are drawn i.i.d. from a monotone-hazard-rate distribution. In the literature, three mechanisms receive particular attention: the revenue-optimal mechanism Myerson Auction (OPT), the welfare-optimal mechanism Second-Price Auction (SPA), and the most widely-used mechanism Anonymous Pricing (AP). In terms of revenue, we investigate how well the later two mechanisms can approximate Myerson Auction.
OPT vs. AP: over all \(n \in \mathbb {N}_{\ge 1}\), the supremum ratio is 1.27, and the worst-case distribution is exponential-like. This answers an open question of Giannakopoulos and Zhu (WINE 18), who proved an asymptotically tight bound of \(1 + \varTheta \big (\frac{\log \log n}{\log n}\big )\) for large \(n \in \mathbb {N}_{\ge 1}\). Thus, the approximability of AP is well understood.
OPT vs. SPA: for each \(n \ge 2\), this ratio is upper-bounded by \(\big (1 - (1 - 1 / e)^{n - 1}\big )^{-1} = 1 + 2^{-O(n)}\); an asymptotically matching lower bound can be reached by a truncated exponential distribution. This result settles an open problem asked of Allouah and Besbes (EC 18), who attained the supremum ratio of 1.40 over all \(n \ge 2\). Both bounds together supplement the seminal result of Bulow and Klemperer (Am. Econ. Rev. 96).
This work was supported by the Research Grant Council of Hong Kong (GRF Project no. 16215717 and 16243516).
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Notes
- 1.
When there are multiple alternative monopoly prices p’s, we would break ties by choosing the largest one.
- 2.
Note that the virtual value function \(\varphi \) may not be strictly increasing, i.e., may be two different values \(x < y\) both correspond to the same virtual value \(\varphi (x) = \varphi (y)\). For ease of notation, throughout the paper we always break ties by choosing the largest value, namely \(\varphi ^{-1}(z) = \max \{x \ge 0 \,|\, \varphi (x) \le z\}\).
- 3.
Here is another equivalent definition of the \(\textsc {MHR}\) property: \(y = \frac{f(x)}{1 - F(x)}\) is a non-decreasing function on support \(\textsf {supp}(F)\). Intuitively, an \(\textsc {MHR}\) distribution has a tail decaying (at least) exponentially fast.
- 4.
Distribution \(\textsc {TrExp}(p, q)\) corresponds to a linear (and thus concave) function \(G(x) = \ln \big (1 - F(x)\big ) = \frac{\ln q}{p} \cdot x\).
- 5.
In different orders, the winner of the item may be different, but the corresponding revenue is always the same.
- 6.
- 7.
More concretely, (a) given any \(a \in \big (1, a_{\max }\big ]\), we have \(s'(a) = \frac{H(a) - 1}{(H'(a))^2} \cdot H''(a) < 0\), where the inequality is due to \(H(x) < 0\) and \(H''(x) > 0\), for all \(x \in (1, +\infty )\); (b) \(s(a_{\max }) = a_{\max } - \frac{H(a_{\max })}{H'(a_{\max })} + \frac{1}{H'(a_{\max })} \overset{(\dagger )}{=} \frac{1}{H'(a_{\max })} < 0\), where \((\dagger )\) is due to the definition of \(a_{\max }\); (c) \(s(1^+) = 1 + \lim \limits _{a \rightarrow 1^+} \frac{1 - H(a)}{H'(a)} = 1 + \frac{1 - 0}{+\infty } = 1 > 0\).
- 8.
Actualy, one of the parameters has no effect on the ratio between Myerson Auction and Second-Price Auction.
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Jin, Y., Li, W., Qi, Q. (2019). On the Approximability of Simple Mechanisms for MHR Distributions. In: Caragiannis, I., Mirrokni, V., Nikolova, E. (eds) Web and Internet Economics. WINE 2019. Lecture Notes in Computer Science(), vol 11920. Springer, Cham. https://doi.org/10.1007/978-3-030-35389-6_17
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