[Paper Review] Liquid Welfare Guarantees for No-Regret Learning in Sequential Budgeted Auctions

Giannis Fikioris, Éva Tardos|arXiv (Cornell University)|Oct 14, 2022

Auction Theory and Applications25 references4 citations

TL;DR

This paper establishes liquid welfare guarantees for no-regret learning in sequential first-price auctions with budget-constrained bidders. Under a behavioral assumption that each player's utility is within a γ factor of the optimal utility achievable via value shading, the authors prove a price of anarchy bound of γ + 1/2 + O(1/γ) for additive valuations, which is asymptotically tight and significantly better than the unbounded loss in second-price auctions under the same conditions.

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.

Motivation & Objective

• To analyze the efficiency of sequential first-price auctions when bidders are budget-constrained and use no-regret learning strategies.
• To establish provable bounds on liquid welfare under a general behavioral model where bidders achieve utility within a γ factor of the optimal shading strategy.
• To contrast the performance of first-price auctions with that of second-price auctions under similar learning assumptions, particularly regarding welfare loss.
• To design a learning algorithm that ensures bidders achieve utility within O(1) of the optimal utility even under adversarial value and bid sequences, assuming linearly growing budgets.
• To extend the results to submodular valuations and analyze the resulting price of anarchy bound.

Proposed method

• The authors model bidders as using a multiplicative shading factor λ to adjust their bids relative to their true values, with the key assumption that each bidder's utility is within a γ factor of the utility achievable by shading with the same λ.
• They derive a lower bound on individual utility based on whether the bidder is budget-constrained or not, using the relationship between payment, value, and the shading factor λ.
• The proof partitions bidders into those who exceed their budget (X) and those who do not (Y), and applies concentration bounds to ensure high-probability guarantees across all bidders.
• By optimizing over the shading factor λ, the authors derive a price of anarchy bound that depends on γ, minimizing the ratio between optimal and actual liquid welfare.
• The analysis uses a competitive ratio framework and regret bounds to relate actual performance to the optimal liquid welfare achievable in hindsight.
• The learning algorithm is designed to maintain utility within O(1) of the optimal even under adversarial inputs, with performance degradation when the budget grows sublinearly.

Figure 1 : Price of Anarchy plots for Theorems 4.2 and 7.1 for $\gamma\in[1,10]$ .

Experimental results

Research questions

• RQ1 What is the worst-case liquid welfare loss in sequential first-price auctions when bidders use no-regret learning strategies with a bounded competitive ratio γ?
• RQ2 How does the performance of first-price auctions compare to second-price auctions under the same behavioral assumptions regarding shading and regret?
• RQ3 Can a learning algorithm be designed such that bidders achieve utility within O(1) of the optimal utility, even when values and other bidders’ bids are chosen adversarially?
• RQ4 What is the price of anarchy for liquid welfare when bidders have submodular valuations rather than additive valuations?
• RQ5 How does the bound on liquid welfare depend on the competitive ratio γ, and is the derived bound asymptotically tight?

Key findings

• The paper establishes a price of anarchy bound of γ + 1/2 + O(1/γ) for liquid welfare in sequential first-price auctions with additive valuations under the γ-competitive regret assumption.
• The bound is asymptotically tight, as the paper proves a lower bound of γ on the liquid welfare loss under the same behavioral assumption.
• For the special case γ = 1, the price of anarchy is at most approximately 2.41, and the paper proves a lower bound of 2 for this case.
• The learning algorithm proposed ensures that each bidder achieves utility within a γ = O(1) factor of the optimal utility, provided the budget grows linearly with time.
• When the budget grows sublinearly, the competitiveness guarantee of the learning algorithm deteriorates, though it remains bounded.
• For submodular valuations, the price of anarchy bound degrades to γ + 1 + O(1/γ), which is slightly worse than the additive case.