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Repeated Bilateral Trade Against a Smoothed Adversary

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni, Federico Fusco, Stefano Leonardi

TL;DR

This work provides a complete characterization of the regret regimes for fixed-price mechanisms under different feedback models in the two cases where the learner can post either the same or different prices to buyers and sellers and proves that this rate is optimal.

Abstract

We study repeated bilateral trade where an adaptive $σ$-smooth adversary generates the valuations of sellers and buyers. We provide a complete characterization of the regret regimes for fixed-price mechanisms under different feedback models in the two cases where the learner can post either the same or different prices to buyers and sellers. We begin by showing that the minimax regret after $T$ rounds is of order $\sqrt{T}$ in the full-feedback scenario. Under partial feedback, any algorithm that has to post the same price to buyers and sellers suffers worst-case linear regret. However, when the learner can post two different prices at each round, we design an algorithm enjoying regret of order $T^{3/4}$ ignoring log factors. We prove that this rate is optimal by presenting a surprising $T^{3/4}$ lower bound, which is the main technical contribution of the paper.

Repeated Bilateral Trade Against a Smoothed Adversary

TL;DR

This work provides a complete characterization of the regret regimes for fixed-price mechanisms under different feedback models in the two cases where the learner can post either the same or different prices to buyers and sellers and proves that this rate is optimal.

Abstract

We study repeated bilateral trade where an adaptive -smooth adversary generates the valuations of sellers and buyers. We provide a complete characterization of the regret regimes for fixed-price mechanisms under different feedback models in the two cases where the learner can post either the same or different prices to buyers and sellers. We begin by showing that the minimax regret after rounds is of order in the full-feedback scenario. Under partial feedback, any algorithm that has to post the same price to buyers and sellers suffers worst-case linear regret. However, when the learner can post two different prices at each round, we design an algorithm enjoying regret of order ignoring log factors. We prove that this rate is optimal by presenting a surprising lower bound, which is the main technical contribution of the paper.
Paper Structure (27 sections, 9 theorems, 81 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 27 sections, 9 theorems, 81 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of results
  3. Technical challenges and our techniques
  4. The action space.
  5. The feedback structure.
  6. Our $T^{3/4}$ lower bound.
  7. Additional related works
  8. Warm-up: one-price setting
  9. Regret due to discretization.
  10. Posting a single price in full information.
  11. Posting a single price in partial information.
  12. A Omega(T 3/4) lower bound: two bits and two prices
  13. The construction of a hard family of adversaries
  14. Expected gain from trade.
  15. Two-bit feedback.
  16. ...and 12 more sections

Key Result

Lemma 1

Let $(S,B)$ be a $\sigma$-smooth random variable on $[0,1]^2$, then the induced expected gain from trade $\textsc{GFT}$ is $1/\sigma$-Lipschitz:

Figures (4)

  • Figure 1: The squares $Q_1,\dots,Q_6$ appearing in the proof of \ref{['thm:two-bit-one-price-lower']}.
  • Figure 2: Left/center: The six squares $Q_1, \dots, Q_6$ (in green) are the support of the base density $f$, and the four rectangles $R^1_{v,\varepsilon},\dots, R^4_{v,\varepsilon}$ (in red and blue) inside $Q_6$ are the regions where the density is perturbed with $g_{v,\varepsilon}$. Right: The corresponding qualitative plots of $p \mapsto \mathbb{E}[\textsc{GFT}(p,S,B)]$ (black, dotted) and $p \mapsto \mathbb{E}^{v,\varepsilon}[\textsc{GFT}(p,S,B)]$ (red, solid).
  • Figure 3: Left: The feedback graph of multi-apple tasting for $K=4$. Right: The map $\iota$.
  • Figure 4: Pictorial representation of \ref{['t:inverse-transformation-method-2']}. The way to interpret it is not event by event but in probability: the probability of a measurable set in $\mathcal{F}_Y$ can be computed in $\Omega$ equivalently via the pullback of $Y$, or of $\varphi\circ (X,U)$.

Theorems & Definitions (30)

  • Definition 1: HaghtalabRS21
  • Lemma 1: Lipschitzness
  • proof
  • Claim 1: Discretization error
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 20 more