Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Strategically-Robust Learning Algorithms for Bidding in First-Price Auctions

Rachitesh Kumar, Jon Schneider, Balasubramanian Sivan

TL;DR

This work proposes a novel concave formulation for pure-strategy bidding in first-price auctions, and uses it to analyze natural Gradient-Ascent-based algorithms for this problem, and proves that their algorithms are the first to simultaneously achieve both optimal regret and strategic-robustness.

Abstract

Learning to bid in repeated first-price auctions is a fundamental problem at the interface of game theory and machine learning, which has seen a recent surge in interest due to the transition of display advertising to first-price auctions. In this work, we propose a novel concave formulation for pure-strategy bidding in first-price auctions, and use it to analyze natural Gradient-Ascent-based algorithms for this problem. Importantly, our analysis goes beyond regret, which was the typical focus of past work, and also accounts for the strategic backdrop of online-advertising markets where bidding algorithms are deployed -- we provide the first guarantees of strategic-robustness and incentive-compatibility for Gradient Ascent. Concretely, we show that our algorithms achieve $O(\sqrt{T})$ regret when the highest competing bids are generated adversarially, and show that no online algorithm can do better. We further prove that the regret reduces to $O(\log T)$ when the competition is stationary and stochastic, which drastically improves upon the previous best of $O(\sqrt{T})$. Moving beyond regret, we show that a strategic seller cannot exploit our algorithms to extract more revenue on average than is possible under the optimal mechanism. Finally, we prove that our algorithm is also incentive compatible -- it is a (nearly) dominant strategy for the buyer to report her values truthfully to the algorithm as a whole. Altogether, these guarantees make our algorithms the first to simultaneously achieve both optimal regret and strategic-robustness.

Strategically-Robust Learning Algorithms for Bidding in First-Price Auctions

TL;DR

This work proposes a novel concave formulation for pure-strategy bidding in first-price auctions, and uses it to analyze natural Gradient-Ascent-based algorithms for this problem, and proves that their algorithms are the first to simultaneously achieve both optimal regret and strategic-robustness.

Abstract

Learning to bid in repeated first-price auctions is a fundamental problem at the interface of game theory and machine learning, which has seen a recent surge in interest due to the transition of display advertising to first-price auctions. In this work, we propose a novel concave formulation for pure-strategy bidding in first-price auctions, and use it to analyze natural Gradient-Ascent-based algorithms for this problem. Importantly, our analysis goes beyond regret, which was the typical focus of past work, and also accounts for the strategic backdrop of online-advertising markets where bidding algorithms are deployed -- we provide the first guarantees of strategic-robustness and incentive-compatibility for Gradient Ascent. Concretely, we show that our algorithms achieve regret when the highest competing bids are generated adversarially, and show that no online algorithm can do better. We further prove that the regret reduces to when the competition is stationary and stochastic, which drastically improves upon the previous best of . Moving beyond regret, we show that a strategic seller cannot exploit our algorithms to extract more revenue on average than is possible under the optimal mechanism. Finally, we prove that our algorithm is also incentive compatible -- it is a (nearly) dominant strategy for the buyer to report her values truthfully to the algorithm as a whole. Altogether, these guarantees make our algorithms the first to simultaneously achieve both optimal regret and strategic-robustness.
Paper Structure (34 sections, 14 theorems, 117 equations, 1 figure, 3 algorithms)

This paper contains 34 sections, 14 theorems, 117 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Major Contributions
  3. Significance
  4. Additional Related Work
  5. Model
  6. Notation.
  7. Multiple Buyers and Tie Breaking
  8. Concave Formulation
  9. Known Value Distribution
  10. Regret Guarantees
  11. Strategic Robustness
  12. Unknown Value Distribution
  13. Regret Guarantee
  14. Strategic Robustness
  15. Incentive Compatibility
  16. ...and 19 more sections

Key Result

Lemma 1

For each $v \in [0, 1]$, define $s^*(v)$ to be the bid $b_j \in \{b_0, b_1, \dots, b_K\}$ which maximizes the quantity $(v - b_j) \cdot \sum_{i=0}^j d_i$, choosing the smaller one in case of equality. Then $s^* \in \mathop{\mathrm{argmax}}\nolimits_{s(\cdot)} u(s|F, \pmb d)$. Moreover, $s^*$ is non-

Figures (1)

  • Figure 1: The change of variables that transforms a value threshold $v_i$ to the corresponding bidding probability $p_i$, and vice versa. The indicated area represents the concave integral term $G(1-p_i) = \int_{1-p_i}^1 F^-(u) \cdot du$ in the utility function $u(\pmb{p}|F,\pmb d)$.

Theorems & Definitions (33)

  • Lemma 1
  • Theorem 1
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Remark 2
  • Proposition 3
  • Theorem 3
  • Theorem 4
  • ...and 23 more