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Randomized Truthful Auctions with Learning Agents

Gagan Aggarwal, Anupam Gupta, Andres Perlroth, Grigoris Velegkas

TL;DR

A notion of {\em auctioneer regret} comparing the revenue generated to the revenue of a second price auction with truthful bids is defined, which shows that the ratio of the learning rates of the bidders can affect the convergence of the bidders.

Abstract

We study a setting where agents use no-regret learning algorithms to participate in repeated auctions. \citet{kolumbus2022auctions} showed, rather surprisingly, that when bidders participate in second-price auctions using no-regret bidding algorithms, no matter how large the number of interactions $T$ is, the runner-up bidder may not converge to bidding truthfully. Our first result shows that this holds for \emph{general deterministic} truthful auctions. We also show that the ratio of the learning rates of the bidders can \emph{qualitatively} affect the convergence of the bidders. Next, we consider the problem of revenue maximization in this environment. In the setting with fully rational bidders, \citet{myerson1981optimal} showed that revenue can be maximized by using a second-price auction with reserves.We show that, in stark contrast, in our setting with learning bidders, \emph{randomized} auctions can have strictly better revenue guarantees than second-price auctions with reserves, when $T$ is large enough. Finally, we study revenue maximization in the non-asymptotic regime. We define a notion of {\em auctioneer regret} comparing the revenue generated to the revenue of a second price auction with truthful bids. When the auctioneer has to use the same auction throughout the interaction, we show an (almost) tight regret bound of $\smash{\widetilde Θ(T^{3/4})}.$ If the auctioneer can change auctions during the interaction, but in a way that is oblivious to the bids, we show an (almost) tight bound of $\smash{\widetilde Θ(\sqrt{T})}.$

Randomized Truthful Auctions with Learning Agents

TL;DR

A notion of {\em auctioneer regret} comparing the revenue generated to the revenue of a second price auction with truthful bids is defined, which shows that the ratio of the learning rates of the bidders can affect the convergence of the bidders.

Abstract

We study a setting where agents use no-regret learning algorithms to participate in repeated auctions. \citet{kolumbus2022auctions} showed, rather surprisingly, that when bidders participate in second-price auctions using no-regret bidding algorithms, no matter how large the number of interactions is, the runner-up bidder may not converge to bidding truthfully. Our first result shows that this holds for \emph{general deterministic} truthful auctions. We also show that the ratio of the learning rates of the bidders can \emph{qualitatively} affect the convergence of the bidders. Next, we consider the problem of revenue maximization in this environment. In the setting with fully rational bidders, \citet{myerson1981optimal} showed that revenue can be maximized by using a second-price auction with reserves.We show that, in stark contrast, in our setting with learning bidders, \emph{randomized} auctions can have strictly better revenue guarantees than second-price auctions with reserves, when is large enough. Finally, we study revenue maximization in the non-asymptotic regime. We define a notion of {\em auctioneer regret} comparing the revenue generated to the revenue of a second price auction with truthful bids. When the auctioneer has to use the same auction throughout the interaction, we show an (almost) tight regret bound of If the auctioneer can change auctions during the interaction, but in a way that is oblivious to the bids, we show an (almost) tight bound of

Paper Structure

This paper contains 18 sections, 14 theorems, 55 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Related Work
  4. Model
  5. Deterministic Truthful Auctions
  6. The Value of Randomized Truthful Auctions: The Asymptotic Case
  7. Revenue Maximization in the Finite Time Horizon Setting
  8. Conclusion
  9. Multiplicative Weights Update (MWU)
  10. Further Related Work
  11. Omitted Details from \ref{['sec:setting']}
  12. Omitted Details from \ref{['sec:deterministic auctions']}
  13. Omitted Details from \ref{['sec:randomized-truthful-auctions']}
  14. Omitted Details from \ref{['sec:no-regret-learning-auctioneer']}
  15. Extensions
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let two agents draw their valuations from the uniform distribution over $U[B_\Delta]$ and participate in $T$ repeated auctions using mean-based learners. Let $b_1^T, b_2^T$ be the bid distributions after $T$ rounds. Let $\mathrm{Rev}(b_1,b_2;r)$ denote the revenue of the second-price auction with re where $c > 0$ is some constant that does not depend on $T.$

Theorems & Definitions (38)

  • Definition 2.1: Randomized Truthful Auction
  • Definition 2.2: Mean-Based Property braverman2018selling
  • Theorem 3.1: SPA with Reserve Is Not Revenue Optimal
  • Definition 4.1: Strictly IC Auctions
  • Lemma 4.2: Convergence in Strictly IC Auctions
  • Definition 4.3: Mixture of Auctions
  • Claim 1: Mixture of IC and Strictly IC Auction
  • Theorem 4.4
  • Corollary 4.5
  • Remark 1: Randomized Auctions vs. SPA with Reserve
  • ...and 28 more