A Regret Analysis of Bilateral Trade

TL;DR

This work inaugurates a regret-minimization view of bilateral trade, contrasting the Myerson–Satterthwaite impossibility with a spectrum of near-optimal fixed-price mechanisms learned online. It introduces two feedback models—full revelation and realistic posted pricing—and provides tight regret characterizations across stochastic, adversarial, and informational regimes. The authors design constructive algorithms (Follow the Best Price and Scouting Bandits) achieving sublinear regret where possible, and they prove matching lower bounds via Embedding and Simulation lemmas that connect to partial-monitoring lower bounds. The results reveal sublinear regret rates of $ ilde{Θ}(√T)$ in full feedback, $ ilde{Θ}(T^{2/3})$ in realistic feedback under iv+bd, and linear regret in several relaxed settings or adversarial scenarios, thereby clarifying the feasibility landscape for online bilateral trade with no priors. The findings have practical implications for designing simple, budget-balanced, incentive-compatible mechanisms that learn optimal pricing in online markets and highlight the critical role of feedback quality and valuation independence on achievable performance.

Abstract

Bilateral trade, a fundamental topic in economics, models the problem of intermediating between two strategic agents, a seller and a buyer, willing to trade a good for which they hold private valuations. Despite the simplicity of this problem, a classical result by Myerson and Satterthwaite (1983) affirms the impossibility of designing a mechanism which is simultaneously efficient, incentive compatible, individually rational, and budget balanced. This impossibility result fostered an intense investigation of meaningful trade-offs between these desired properties. Much work has focused on approximately efficient fixed-price mechanisms, i.e., Blumrosen and Dobzinski (2014; 2016), Colini-Baldeschi et al. (2016), which have been shown to fully characterize strong budget balanced and ex-post individually rational direct revelation mechanisms. All these results, however, either assume some knowledge on the priors of the seller/buyer valuations, or a black box access to some samples of the distributions, as in D{ü}tting et al. (2021). In this paper, we cast for the first time the bilateral trade problem in a regret minimization framework over rounds of seller/buyer interactions, with no prior knowledge on the private seller/buyer valuations. Our main contribution is a complete characterization of the regret regimes for fixed-price mechanisms with different models of feedback and private valuations, using as benchmark the best fixed price in hindsight. More precisely, we prove the following bounds on the regret: $\bullet$ $\widetildeΘ(\sqrt{T})$ for full-feedback (i.e., direct revelation mechanisms); $\bullet$ $\widetildeΘ(T^{2/3})$ for realistic feedback (i.e., posted-price mechanisms) and independent seller/buyer valuations with bounded densities; $\bullet$ $Θ(T)$ for realistic feedback and seller/buyer valuations with bounded densities; $\bullet$ $Θ(T)$ for realistic feedback and independent seller/buyer valuations; $\bullet$ $Θ(T)$ for the adversarial setting.

Figures (4)

• Figure 1: The best posted price is $1/4$ (resp., $3/4$) in the $+\varepsilon$ (resp., $-\varepsilon$) case. By posting $1/4$, the player suffers a $\Omega( \varepsilon )$ regret in the $-\varepsilon$ case, and the same is true posting $3/4$ if in $+\varepsilon$ case.
• Figure 2: The only three regions where it makes sense for the learner to post prices are $a_1, a_2, a_3$. Prices in $a_1$ reveal information about the sign of $\pm\varepsilon$ suffering a $\Omega(1)$ regret; prices in $a_2$ are optimal if the distribution of the seller is the red one $(+\varepsilon)$ but incur $\Omega(\varepsilon)$ regret if it is the blue one $(-\varepsilon)$; the converse happens in $a_3$.
• Figure 3: Under realistic feedback, the two densities $f$ and $g$ are indistinguishable. The optimal price $p^\star$ for $f$ gives constant regret under $g$ and $q^\star$ does the converse.
• Figure 4: All prices, except for $x$, have high regret. However, under realistic feedback, finding $x$ in a finite time is as harder than finding a needle in a haystack.

Theorems & Definitions (40)

• Lemma 3.1: First Decomposition Lemma
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof : Prook sketch
• Lemma 4.1: Second decomposition Lemma
• proof
• Theorem 4.1
• Lemma 4.2