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Robust Temporal Guarantees in Budgeted Sequential Auctions

Giannis Fikioris, Robert Kleinberg, Yoav Kolumbus, Yishay Mansour, Eva Tardos

TL;DR

This work proposes a very simple learning algorithm for budgeted sequential auctions where agents maximize their total number of wins and shows that it has surprisingly appealing properties.

Abstract

In modern advertising platforms, learning algorithms are deployed by budget-constrained bidders to maximize their accumulated value. These algorithms often offer classical utility guarantees like no-regret, i.e., the agent's utility is at least the utility achieved by some benchmark in which it is assumed that every other agent's bidding remains the same. These guarantees offer compelling properties: They are optimal against stationary competition distributions, and in unconstrained settings, the resulting empirical distribution of play induced by no-regret dynamics approximates a Coarse Correlated Equilibrium. However, no-regret algorithms are easily manipulable, and in budgeted settings, no stronger notion of regret (such as swap regret) is currently known that would limit such manipulation. We propose a very simple learning algorithm for budgeted sequential auctions where agents maximize their total number of wins and show that it has surprisingly appealing properties. We analyze this algorithm from two perspectives. First, we show that when an agent with a $ρ$ fraction of the total budget uses this algorithm, then she is guaranteed to win at least $ρT - O(\sqrt T)$ of the total $T$ rounds. This result holds for adversarial behavior by the other agents, as long as they respect their own budget restrictions. Second, we examine the scenario when all the agents follow our algorithm. By the first result, every agent's total wins are proportional to her budget, up to the additive $O(\sqrt T)$ term. In addition, we show that this result holds in a much stronger sense: after an initial period of $O(\sqrt T \log T)$ rounds, every agent gets the same guarantee over any time interval. For intervals of length $O(\sqrt T)$, we show that the deviation from the desired number of wins is an additive constant.

Robust Temporal Guarantees in Budgeted Sequential Auctions

TL;DR

This work proposes a very simple learning algorithm for budgeted sequential auctions where agents maximize their total number of wins and shows that it has surprisingly appealing properties.

Abstract

In modern advertising platforms, learning algorithms are deployed by budget-constrained bidders to maximize their accumulated value. These algorithms often offer classical utility guarantees like no-regret, i.e., the agent's utility is at least the utility achieved by some benchmark in which it is assumed that every other agent's bidding remains the same. These guarantees offer compelling properties: They are optimal against stationary competition distributions, and in unconstrained settings, the resulting empirical distribution of play induced by no-regret dynamics approximates a Coarse Correlated Equilibrium. However, no-regret algorithms are easily manipulable, and in budgeted settings, no stronger notion of regret (such as swap regret) is currently known that would limit such manipulation. We propose a very simple learning algorithm for budgeted sequential auctions where agents maximize their total number of wins and show that it has surprisingly appealing properties. We analyze this algorithm from two perspectives. First, we show that when an agent with a fraction of the total budget uses this algorithm, then she is guaranteed to win at least of the total rounds. This result holds for adversarial behavior by the other agents, as long as they respect their own budget restrictions. Second, we examine the scenario when all the agents follow our algorithm. By the first result, every agent's total wins are proportional to her budget, up to the additive term. In addition, we show that this result holds in a much stronger sense: after an initial period of rounds, every agent gets the same guarantee over any time interval. For intervals of length , we show that the deviation from the desired number of wins is an additive constant.
Paper Structure (17 sections, 29 theorems, 78 equations, 1 algorithm)

This paper contains 17 sections, 29 theorems, 78 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Algorithm against an Optimizer
  4. Discrepancy of wins
  5. Our Techniques
  6. Related Work
  7. Definitions and Learning Algorithm
  8. Learning Algorithm
  9. Primal Budget Pacing Algorithm against budgeted opponent
  10. Interval Discrepancy with Multiple Learners
  11. Convergence of bids to 1 +- sqrt(eta)
  12. Convergence of bids to O(eta) of each other
  13. Convergence of bids O(eta) close to 1
  14. Putting everything together
  15. Interval Discrepancy with Equal Budgets
  16. ...and 2 more sections

Key Result

Lemma 2.1

Fix an agent $i$ who is running our algorithm with $\eta \in (0, 1)$ in either a first or second price auction, and let $b_{i}$ be the initial bid. Then, for arbitrary behavior by the other agents, the algorithm's bids are always non-negative, and the agent's total payment by round $T$ is In addition, if $b_{i} \le \rho_i$ then the agent never runs out of budget.

Theorems & Definitions (49)

  • Lemma 2.1
  • proof
  • Remark
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3: Weaker version of \ref{['lem:adv:opt_lagrange_unnormalized']}
  • proof : Proof Sketch
  • proof : Proof of \ref{['thm:adv:main']} for normalized budgets
  • Theorem 4.1
  • ...and 39 more