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Liquid Welfare Guarantees for No-Regret Learning in Sequential Budgeted Auctions

Giannis Fikioris, Éva Tardos

TL;DR

This work studies liquid welfare in sequential budgeted first-price auctions under budget constraints and no-regret-like learning. It introduces a competitive-ratio benchmark against fixed shading multipliers, derives a $\gamma + \tfrac{1}{2} + O(1/\gamma)$ price of anarchy for additive valuations, and provides a learning algorithm via a Bandits with Knapsacks reduction. The results are shown to be near-tight and are extended to submodular valuations with a slightly weaker bound, while contrasting with negative results in sequential second-price settings. The findings offer practical guidance for budget-managed bidding in sequential first-price settings and highlight open questions about stronger learning models and submodular cases.

Abstract

We study the liquid welfare in sequential first-price auctions with budget-limited buyers. We focus on first-price auctions, which are increasingly commonly used in many settings, and consider liquid welfare, a natural and well-studied generalization of social welfare for buyers with budgets. We use a behavioral model for the buyers, assuming a learning style guarantee: the resulting utility of each buyer is within a $γ$ factor (where $γ\ge 1$) of the utility achievable by shading her value with the same factor at each round. Under this assumption, we show a $γ+1/2+O(1/γ)$ price of anarchy for liquid welfare assuming buyers have additive valuations. This positive result is in contrast to sequential second-price auctions, where even with $γ=1$, the resulting liquid welfare can be arbitrarily smaller than the maximum liquid welfare. We prove a lower bound of $γ$ on the liquid welfare loss under the above assumption in first-price auctions, making our bound asymptotically tight. For the case when $γ= 1$ our theorem implies a price of anarchy upper bound that is about $2.41$; we show a lower bound of $2$ for that case. We also give a learning algorithm that the players can use to achieve the guarantee needed for our liquid welfare result. Our algorithm achieves utility within a $γ=O(1)$ factor of the optimal utility even when a buyer's values and the bids of the other buyers are chosen adversarially, assuming the buyer's budget grows linearly with time. The competitiveness guarantee of the learning algorithm deteriorates somewhat as the budget grows slower than linearly with time. Finally, we extend our liquid welfare results for the case where buyers have submodular valuations over the set of items they win across iterations with a slightly worse price of anarchy bound of $γ+1+O(1/γ)$ compared to the guarantee for the additive case.

Liquid Welfare Guarantees for No-Regret Learning in Sequential Budgeted Auctions

TL;DR

This work studies liquid welfare in sequential budgeted first-price auctions under budget constraints and no-regret-like learning. It introduces a competitive-ratio benchmark against fixed shading multipliers, derives a price of anarchy for additive valuations, and provides a learning algorithm via a Bandits with Knapsacks reduction. The results are shown to be near-tight and are extended to submodular valuations with a slightly weaker bound, while contrasting with negative results in sequential second-price settings. The findings offer practical guidance for budget-managed bidding in sequential first-price settings and highlight open questions about stronger learning models and submodular cases.

Abstract

We study the liquid welfare in sequential first-price auctions with budget-limited buyers. We focus on first-price auctions, which are increasingly commonly used in many settings, and consider liquid welfare, a natural and well-studied generalization of social welfare for buyers with budgets. We use a behavioral model for the buyers, assuming a learning style guarantee: the resulting utility of each buyer is within a factor (where ) of the utility achievable by shading her value with the same factor at each round. Under this assumption, we show a price of anarchy for liquid welfare assuming buyers have additive valuations. This positive result is in contrast to sequential second-price auctions, where even with , the resulting liquid welfare can be arbitrarily smaller than the maximum liquid welfare. We prove a lower bound of on the liquid welfare loss under the above assumption in first-price auctions, making our bound asymptotically tight. For the case when our theorem implies a price of anarchy upper bound that is about ; we show a lower bound of for that case. We also give a learning algorithm that the players can use to achieve the guarantee needed for our liquid welfare result. Our algorithm achieves utility within a factor of the optimal utility even when a buyer's values and the bids of the other buyers are chosen adversarially, assuming the buyer's budget grows linearly with time. The competitiveness guarantee of the learning algorithm deteriorates somewhat as the budget grows slower than linearly with time. Finally, we extend our liquid welfare results for the case where buyers have submodular valuations over the set of items they win across iterations with a slightly worse price of anarchy bound of compared to the guarantee for the additive case.
Paper Structure (10 sections, 11 theorems, 56 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries and Model
  4. The behavioral assumption
  5. Liquid Welfare Guarantees
  6. Impossibility Results for Liquid Welfare
  7. Algorithm for Bounded Competitive Ratio
  8. Liquid Welfare Guarantees for Submodular Valuations
  9. Conclusion
  10. Calculations for the proof of Theorem \ref{['thm:guar']}

Key Result

Lemma 4.1

Fix a player $i$, her values $v_{i1},\ldots,v_{iT}$, and the price of the items $p_1,\ldots,p_T$. Let $\mathcal{O}_i\subseteq [T]$ be the items that player $i$ gets in the allocation that maximizes the total liquid welfare. Let $c(\lambda) = \frac{-1}{\ln(1-\lambda)}$. Then for any $\lambda\in(0,1)$

Figures (1)

  • Figure 1: Price of Anarchy plots for Theorems \ref{['thm:guar']} (additive) and \ref{['thm:subm']} (submodular) for $\gamma\in [1, 10]$.

Theorems & Definitions (26)

  • Lemma 4.1
  • proof : Proof.
  • Theorem 4.2
  • Remark 4.3
  • proof : Proof.
  • Theorem 5.1: DBLP:journals/corr/GaitondeLLLS22
  • proof : Proof.
  • Theorem 5.2
  • proof : Proof.
  • Theorem 5.3
  • ...and 16 more