Approximately Efficient Bilateral Trade

Deng, Yuan; Mao, Jieming; Sivan, Balasubramanian; Wang, Kangning

doi:10.48550/arxiv.2111.03611

Cited by 1 publication

(1 citation statement)

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“…Note that [29] also implies that optimal GFT is not achievable in bilateral trade. There has been increasing interest from the algorithmic mechanism design community to study the approximability of the optimal GFT [6,8,12,2,3,10,14]. It will be interesting to study the optimal approximation ratio obtainable for GFT maximization in both the full information and the limited information settings.…”

Section: Related Workmentioning

confidence: 99%

On the Optimal Fixed-Price Mechanism in Bilateral Trade

Cai¹,

Wu²

2023

Preprint

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We study the problem of social welfare maximization in bilateral trade, where two agents, a buyer and a seller, trade an indivisible item. The seminal result of Myerson and Satterthwaite [29] shows that no incentive compatible and budget balanced (i.e., the mechanism does not run a deficit) mechanism can achieve the optimal social welfare in bilateral trade. Motivated by this impossibility result, we focus on approximating the optimal social welfare. We consider arguably the simplest form of mechanisms -the fixed-price mechanisms, where the designer offers trade at a fixed price to the seller and buyer. Besides the simple form, fixed-price mechanisms are also the only dominant strategy incentive compatible and budget balanced mechanisms in bilateral trade [23].We obtain improved approximation ratios of fixed-price mechanisms in both (i) the full information setting, where the designer knows the value distributions of both the seller and buyer; and (ii) the limited information settings. In the full information setting, we show that the optimal fixedprice mechanism can achieve at least 0.72 of the optimal welfare, and no fixed-price mechanism can achieve more than 0.7381 of the optimal welfare. Prior to our result the state of the art approximation ratio was 1 − 1 e + 0.0001 ≈ 0.632 [24]. We further consider two limited information settings. In the first one, the designer is only given the mean of the buyer's value (or the mean of the seller's value). We show that with such minimal information, one can already design a fixed-price mechanism that achieves 0.65 of the optimal social welfare, which surpasses the previous state of the art ratio in the full information setting. In the second limited information setting, we assume that the designer has access to finitely many samples from the value distributions. Recent results show that one can already obtain a constant factor approximation to the optimal welfare using a single sample from the seller's distribution [3,16,24]. Our goal is to understand what approximation ratios are possible if the designer has more than one but still finitely many samples. This is usually a technically more challenging regime and requires tools different from the single-sample analysis. We propose a new family of sample-based fixed-price mechanisms that we refer to as the order statistic mechanisms and provide a complete characterization of their approximation ratios for any fixed number of samples. Using the characterization, we provide the optimal approximation ratios obtainable by order statistic mechanism for small sample sizes (no more than 10 samples) and observe that they significantly outperform the single sample mechanism.

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