# Fixed Price Approximability of the Optimal Gain From Trade

arXiv: https://arxiv.org/abs/1710.08394 · HTML: https://mlsanity.com/paper/1710.08394/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2017.

## Built on by (most cited first, 8 of 8)

- [Bilateral Trade: A Regret Minimization Perspective](https://mlsanity.com/paper/2109.12974/) (2021)
- [A Regret Analysis of Bilateral Trade](https://mlsanity.com/paper/2102.08754/) (2021)
- [An α-regret analysis of Adversarial Bilateral Trade](https://mlsanity.com/paper/2210.06846/) (2022)
- [An Online Learning Theory of Brokerage](https://mlsanity.com/paper/2310.12107/) (2023)
- [A Parametric Contextual Online Learning Theory of Brokerage](https://mlsanity.com/paper/2407.01566/) (2024)
- [Contextual Online Bilateral Trade](https://mlsanity.com/paper/2602.12903/) (2026)
- [A Tight Regret Analysis of Non-Parametric Repeated Contextual Brokerage](https://mlsanity.com/paper/2503.02646/) (2025)
- [Trading Volume Maximization with Online Learning](https://mlsanity.com/paper/2405.13102/) (2024)

## Abstract (arXiv)

Bilateral trade is a fundamental economic scenario comprising a strategically acting buyer and seller, each holding valuations for the item, drawn from publicly known distributions. A mechanism is supposed to facilitate trade between these agents, if such trade is beneficial. It was recently shown that the only mechanisms that are simultaneously DSIC, SBB, and ex-post IR, are fixed price mechanisms, i.e., mechanisms that are parametrised by a price p, and trade occurs if and only if the valuation of the buyer is at least p and the valuation of the seller is at most p. The gain from trade is the increase in welfare that results from applying a mechanism; here we study the gain from trade achievable by fixed price mechanisms. We explore this question for both the bilateral trade setting, and a double auction setting where there are multiple buyers and sellers. We first identify a fixed price mechanism that achieves a gain from trade of at least 2/r times the optimum, where r is the probability that the seller's valuation does not exceed the buyer's valuation. This extends a previous result by McAfee. Subsequently, we improve this approximation factor in an asymptotic sense, by showing that a more sophisticated rule for setting the fixed price results in an expected gain from trade within a factor O(log(1/r)) of the optimal gain from trade. This is asymptotically the best approximation factor possible. Lastly, we extend our study of fixed price mechanisms to the double auction setting defined by a set of multiple i.i.d. unit demand buyers, and i.i.d. unit supply sellers. We present a fixed price mechanism that achieves a gain from trade that achieves for all epsilon > 0 a gain from trade of at least (1-epsilon) times the expected optimal gain from trade with probability 1 - 2/e^{#T epsilon^2 /2}, where #T is the expected number of trades resulting from the double auction.