Abstract
We consider the design of platforms that facilitate trade between a single seller and a single buyer. The most efficient mechanisms for such settings are complex and sometimes even intractable, and we therefore aim to design simple mechanisms that perform approximately well. We devise a mechanism that always guarantees at least 1 / e of the optimal expected gain-from-trade for every set of distributions (assuming monotone hazard rate of the buyer’s distribution). Our main mechanism is extremely simple, and achieves this approximation in Bayes-Nash equilibrium. Moreover, our mechanism approximates the optimal gain-from-trade, which is a strictly harder task than approximating efficiency. Our main impossibility result shows that no Bayes-Nash incentive compatible mechanism can achieve better approximation than 2 / e to the optimal gain from trade. We also bound the power of Bayes-Nash incentive compatible mechanisms for approximating the expected efficiency.
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Notes
- 1.
A mechanism is budget balanced if the mechanism does not gain any profit nor requires any subsidies. A mechanism is individually rational if the utility of each player cannot decrease by participating in the mechanism. Formal definitions will be given later in the paper.
- 2.
We note that our mechanism satisfies two stronger and desired versions of the above economic properties: it is strongly budget balanced, i.e., the sum of payments is always exactly zero; it is also ex-post individually rational, i.e., agents cannot lose in every instance and not only in expectation.
- 3.
Note that the characterization of the “second-best” mechanism by [21] requires that both agents have Myerson-regular distributions, while we require the stronger MHR assumption for the buyer and require nothing for the seller.
- 4.
Concavity of the hazard rate is satisfied by some standard distributions (e.g., exponential, Weibull(2,1), etc.), and does not hold for some other distributions (e.g., uniform on [0, 1]).
- 5.
We consider the weaker version of interim IR, which makes the proof only harder.
- 6.
This follows from \(EFF_M\cdot GFT_{OPT}\ge GFT_M\cdot EFF_{OPT}\) which is equivalent by definition to the inequality \((GFT_M+E[S])\cdot GFT_{OPT}\ge GFT_M\cdot (GFT_{OPT}+E[S])\) that holds by \(GFT_{OPT}\ge GFT_M\).
- 7.
Most of the literature assumes a weaker condition, that the \(\varphi \) is non-decreasing. In our paper we often use the inverse function of \(\varphi \), and the notations become much simpler when \(\varphi \) is strictly increasing. Moreover, our main results consider MHR distributions that imply that \(\varphi \) is always strictly increasing.
- 8.
We note that \(\overline{\varphi ^{-1}}\left( x\right) \) is defined for every x, even when \(F_b\)’s support is \([\underline{b},\infty )\), since \(\varphi \left( x\right) \) is unbounded from above in that case. This can be seen by noticing that for every \(y\in \mathbb {R}\), choosing \(x>max\{\frac{1}{h\left( \underline{b}\right) }+y+1,\underline{b}\}\) yields \(\varphi \left( x\right) =x-\frac{1}{h\left( x\right) }\ge \frac{1}{h\left( \underline{b}\right) }+y+1-\frac{1}{h\left( \underline{b}\right) }>y\) by the MHR assumption.
- 9.
Recall that \(\varphi \) denotes the virtual valuation of the buyer, and the seller use the details of this distribution to determine what price to post.
- 10.
This theorem holds for a weaker notion of interim individual rationality (as in [21]); This clearly strengthens the result.
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Blumrosen, L., Mizrahi, Y. (2016). Approximating Gains-from-Trade in Bilateral Trading. In: Cai, Y., Vetta, A. (eds) Web and Internet Economics. WINE 2016. Lecture Notes in Computer Science(), vol 10123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54110-4_28
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