On the Optimal Fixed-Price Mechanism in Bilateral Trade

We study the problem of social welfare maximization in bilateral trade, where two agents, a buyer and a seller, trade an indivisible item. 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 DSIC and budget balanced mechanisms in bilateral trade. 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 fixed-price 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 $1 - 1/e + 0.0001\approx 0.632$. 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 one, we assume that the designer has access to more than one but still finitely many samples from the value distributions. We propose a new family of sample-based fixed-price mechanisms called 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 and observe that they significantly outperform the single sample mechanism.