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An $α$-regret analysis of Adversarial Bilateral Trade

Yossi Azar, Amos Fiat, Federico Fusco

TL;DR

This paper considers gain from trade which is harder to approximate than social welfare and shows that it is impossible to achieve sublinear $\alpha$-regret for any $\alpha<2$, but with full feedback sublinear $2$- Regret is achievable.

Abstract

We study sequential bilateral trade where sellers and buyers valuations are completely arbitrary (i.e., determined by an adversary). Sellers and buyers are strategic agents with private valuations for the good and the goal is to design a mechanism that maximizes efficiency (or gain from trade) while being incentive compatible, individually rational and budget balanced. In this paper we consider gain from trade which is harder to approximate than social welfare. We consider a variety of feedback scenarios and distinguish the cases where the mechanism posts one price and when it can post different prices for buyer and seller. We show several surprising results about the separation between the different scenarios. In particular we show that (a) it is impossible to achieve sublinear $α$-regret for any $α<2$, (b) but with full feedback sublinear $2$-regret is achievable (c) with a single price and partial feedback one cannot get sublinear $α$ regret for any constant $α$ (d) nevertheless, posting two prices even with one-bit feedback achieves sublinear $2$-regret, and (e) there is a provable separation in the $2$-regret bounds between full and partial feedback.

An $α$-regret analysis of Adversarial Bilateral Trade

TL;DR

This paper considers gain from trade which is harder to approximate than social welfare and shows that it is impossible to achieve sublinear -regret for any , but with full feedback sublinear - Regret is achievable.

Abstract

We study sequential bilateral trade where sellers and buyers valuations are completely arbitrary (i.e., determined by an adversary). Sellers and buyers are strategic agents with private valuations for the good and the goal is to design a mechanism that maximizes efficiency (or gain from trade) while being incentive compatible, individually rational and budget balanced. In this paper we consider gain from trade which is harder to approximate than social welfare. We consider a variety of feedback scenarios and distinguish the cases where the mechanism posts one price and when it can post different prices for buyer and seller. We show several surprising results about the separation between the different scenarios. In particular we show that (a) it is impossible to achieve sublinear -regret for any , (b) but with full feedback sublinear -regret is achievable (c) with a single price and partial feedback one cannot get sublinear regret for any constant (d) nevertheless, posting two prices even with one-bit feedback achieves sublinear -regret, and (e) there is a provable separation in the -regret bounds between full and partial feedback.
Paper Structure (36 sections, 10 theorems, 72 equations, 6 figures, 1 table)

This paper contains 36 sections, 10 theorems, 72 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview of our Results
  3. Technical challenges
  4. From experts to prices
  5. A magic estimator
  6. Lower bounds
  7. Related work
  8. Online Learning and Economics
  9. The notion of $\alpha$-regret
  10. Online learning with partial feedback
  11. Bilateral Trade
  12. Conference Version and follow-up Work
  13. Preliminaries
  14. Single Price vs. Two Prices --- Seller price and Buyer price
  15. Single price
  16. ...and 21 more sections

Key Result

Theorem 1

In the full-feedback model, for all $\varepsilon \in (0,1]$ and horizons $T$, the minimax $(2-\varepsilon)$-regret satisfies $R_T^{2-\varepsilon,\star} \ge \frac{1}{8} \varepsilon T.$

Figures (6)

  • Figure 1: Lower bound construction that "hides" the optimal price.
  • Figure 2: The proof of Theorem \ref{['thm:lower-full']} makes use of two (appropriately scaled and shifted) copies of the lower bound from Theorem \ref{['thm:lower-full-2-eps']} (See Figure \ref{['fig:regret1']}). In this example the left hand copy choose right and then left, while the right hand copy happened to choose left and then right. The (seller,buyer) bids at time $t$ are then chosen independently at random from $(s_t^L,b_t^L)$ and $(s_t^R,b_t^R)$.
  • Figure 3: Example of sets used in the lower bound of \ref{['thm:lower-two-bits-one-price']} and how the grid is hidden. This example has $\Delta =1/10$, $\delta=1/{3}0$, so each section of size $\Delta$ is partitioned into 3 sections of size $\delta$. The $(sell,buy)$ pairs in $S_4$ are as described in Equality \ref{['eq:SI']} (not all such pairs are shown, there are $1/\delta =30$ such pairs in $S_4$ ). Note that the gain from trade is $\Delta$ if the bids are $(4\Delta,5\Delta)$ and if a price in between is posted. Also note that that seller and buyer valuations are equal for $(4\Delta+\delta, 4\Delta+\delta)$ and for $(4\Delta+2\delta, 4\Delta+2\delta)$.
  • Figure 4: Visualization of the family of distributions used in the lower bound construction of \ref{['thm:lower-two-bits-two-prices']}.
  • Figure 5: Feedback regions
  • ...and 1 more figures

Theorems & Definitions (28)

  • Claim 1: Discretization error
  • proof
  • Theorem 1: Lower bound on $(2-\varepsilon)$-regret
  • proof
  • Theorem 2: Upper bound on $2$-regret with full feedback
  • proof
  • Lemma 1: Property of Random Walks
  • proof
  • Theorem 3: Lower bound on $2$-regret with full feedback
  • proof
  • ...and 18 more