Tight Regret Bounds for Fixed-Price Bilateral Trade

TL;DR

The paper studies regret minimization for fixed-price bilateral trade under Global Budget Balance, covering independent, correlated, and adversarial value models across full, semi, and partial feedback settings.It introduces a novel fractal elimination paradigm to achieve sublinear regret with one-bit feedback in the independent-values setting, and develops new lower-bound constructions to handle correlation constraints.The main results show a tight near-optimal regime: for independent values under GBB with one-bit feedback, $ ilde{O}(T^{2/3})$ regret with matching $ ilde{ heta}(T^{2/3})$ lower bound; for correlated/adversarial values, a new $ ilde{ heta}(T^{3/4})$ lower bound matches the existing $ ilde{O}(T^{3/4})$ upper bound up to polylog factors.Together with prior works, these findings provide a comprehensive understanding of regret minimization for fixed-price bilateral trade under global budget constraints, and introduce technical tools potentially useful beyond this problem setting.

Abstract

We examine fixed-price mechanisms in bilateral trade through the lens of regret minimization. Our main results are twofold. (i) For independent values, a near-optimal $\widetildeΘ(T^{2/3})$ tight bound for $\textsf{Global Budget Balance}$ fixed-price mechanisms with two-bit/one-bit feedback. (ii) For correlated/adversarial values, a near-optimal $Ω(T^{3/4})$ lower bound for $\textsf{Global Budget Balance}$ fixed-price mechanisms with two-bit/one-bit feedback, which improves the best known $Ω(T^{5/7})$ lower bound obtained in the work [BCCF24] and, up to polylogarithmic factors, matches the $\widetilde{\mathcal{O}}(T^{3 / 4})$ upper bound obtained in the same work. Our work in combination with the previous works [CCCFL24mor, CCCFL24jmlr, AFF24, BCCF24] (essentially) gives a thorough understanding of regret minimization for fixed-price bilateral trade. En route, we have developed two technical ingredients that might be of independent interest: (i) A novel algorithmic paradigm, called $\textit{fractal elimination}$, to address one-bit feedback and independent values. (ii) A new $\textit{lower-bound construction}$ with novel proof techniques, to address the $\textsf{Global Budget Balance}$ constraint and correlated values.

Figures (8)

• Figure 1: Diagrams of the previous $\Omega(T^{5 / 7})$ lower-bound construction BCCF24 and our new $\Omega(T^{3 / 4})$ lower-bound construction (\ref{['thm:GBB-correlated:LB']}).
• Figure 2: A Hasse diagram of various feedback models. An arrow "$\mathcal{F} \to \mathcal{F}'$" indicates that feedback $\mathcal{F}$ implies feedback $\mathcal{F}'$, i.e., being more informative. (For instance, we have $(S^{t}, B^{t}) \to (S^{t}, Y^{t})$ given that $Y^{t} = {\mathbb 1}[Q^{t} \le B^{t}]$.) Note that $(S^{t}, Y^{t})$ is symmetric to $(X^{t}, B^{t})$, while $(S^{t}, Z_{}^{t})$ is symmetric to $(Z_{}^{t}, B^{t})$.
• Figure 3: Diagram of a specific stage $\ell \in [0 : L]$ of the subroutine FractalElimination (\ref{['alg:GBB-independent:fractal']}). Here, $\mathbin{ \vcenter{\hbox{{\m@th$\otimes$} }}}$'s in general refer to candidates $\{a_{k, k}\}_{k \in [1 : K]}$, and $\otimes$'s in particular refer to candidates $\{a_{k, k}\}_{k \in [1 : K]}$ eliminated in the current stage $\ell \in [0 : L]$. Also, the red horizontal/vertical lines $a_{\sigma, \sigma}$---$a_{\tau, \sigma}$ and $a_{\tau, \sigma}$---$a_{\tau, \tau}$ refer to actions taken in \ref{['alg:GBB-independent:fractal:1']}, and the six blue/green$\square$'s (with $a_{\tau, \sigma}$ counted twice) refer to actions taken in \ref{['alg:GBB-independent:fractal:left-1', 'alg:GBB-independent:fractal:right-1']}. When FractalElimination proceeds from the current stage $\ell \in [0 : L]$ to the next stage $\ell + 1 \in [1 : L + 1]$, the considered segment $[\sigma : \tau]$ (and its associated red triangle) shrinks to two smaller segments $[\sigma' : \tau']$ and $[\sigma" : \tau"]$ (and their associated blue/green triangles).
• Figure 4: Diagrams for the proof of \ref{['lem:GBB-independent:exploration']}, including (\ref{['fig:GBB-independent:2-1']}) the optimal action $a^{*} = (p^{*}, q^{*})$, which is on the diagonal $\{(p, q) \;|\; p = q \in [0, 1]\}$, the candidate $a_{\lambda, \lambda}$ closest to this optimal action $a^{*}$, and (\ref{['fig:GBB-independent:2-2']}) the action $(P^{t}, Q^{t}) = a_{i, \sigma}$, for some $i \in [\sigma : \tau]$, taken in the considered round $t \in [T]$.
• Figure 5: Diagram for proving \ref{['thm:GBB-independent:LB']}. Herein, $\mathcal{I}$ denotes the informative action subset, and each take of an action outside $\mathcal{G}'_{1}$ (resp. $\mathcal{G}'_{2}$) incurs $\Theta(\delta)$ regret on the hard instance $\mathcal{D}^{1}$ (resp. $\mathcal{D}^{2}$).

Theorems & Definitions (56)

• Remark 1: Budget Balance
• Remark 2: Benchmarks
• Remark 3: Feedback Models
• Remark 4
• Remark 5: A Vanilla Estimation Scheme
• Remark 6: Information Reuse
• Remark 7: Feedback Models
• Definition 2: Benchmarks
• Theorem 8: GBB One-Bit-Feedback Upper Bound for Independent Values
• Lemma 9: Gains from Trade for Independent Values