Nash Convergence of Mean-Based Learning Algorithms in First Price Auctions (2110.03906v4)

Abstract: Understanding the convergence properties of learning dynamics in repeated auctions is a timely and important question in the area of learning in auctions, with numerous applications in, e.g., online advertising markets. This work focuses on repeated first price auctions where bidders with fixed values for the item learn to bid using mean-based algorithms -- a large class of online learning algorithms that include popular no-regret algorithms such as Multiplicative Weights Update and Follow the Perturbed Leader. We completely characterize the learning dynamics of mean-based algorithms, in terms of convergence to a Nash equilibrium of the auction, in two senses: (1) time-average: the fraction of rounds where bidders play a Nash equilibrium approaches 1 in the limit; (2)last-iterate: the mixed strategy profile of bidders approaches a Nash equilibrium in the limit. Specifically, the results depend on the number of bidders with the highest value: - If the number is at least three, the bidding dynamics almost surely converges to a Nash equilibrium of the auction, both in time-average and in last-iterate. - If the number is two, the bidding dynamics almost surely converges to a Nash equilibrium in time-average but not necessarily in last-iterate. - If the number is one, the bidding dynamics may not converge to a Nash equilibrium in time-average nor in last-iterate. Our discovery opens up new possibilities in the study of convergence dynamics of learning algorithms.

An Analysis of Nash Convergence in Mean-Based Learning Algorithms for Repeated First Price Auctions

The paper "Nash Convergence of Mean-Based Learning Algorithms in First Price Auctions" provides a rigorous examination of the convergence properties of learning algorithms in repeated first price auctions. This research is especially relevant in the context of online advertising markets, where first price auctions have become predominant, exemplified by Google's shift from second price to first price auctions in 2019.

Overview

The paper focuses on repeated first price auctions where bidders have fixed valuations for items and employ mean-based learning algorithms to adjust their bids. Mean-based algorithms include prominent no-regret algorithms such as Multiplicative Weights Update (MWU) and Follow the Perturbed Leader (FTPL). The authors aim to determine whether these algorithms lead to Nash equilibrium (NE) in various scenarios characterized by the number of bidders with the highest valuation.

Key Results

The authors establish a comprehensive characterization of the convergence dynamics of mean-based algorithms, offering insights into two notions of convergence to NE:

1. Time-Average Convergence: This implies that the fraction of auction rounds where bidders play an NE approaches 1 over time.
2. Last-Iterate Convergence: This specifies that the mixed strategy profile of bidders converges to an NE as the number of rounds increases.

Main Findings

• When the number of highest-value bidders $|M^1|$ is at least three: The bidding dynamics almost surely converge to an NE, both in the time-average sense and in the last-iterate sense. This suggests that the presence of multiple bidders with the highest valuation supports robust convergence to strategic stability.
• When $|M^1| = 2$: The dynamics still converge to an NE in time-average terms. However, convergence in the last-iterate sense depends on the interaction dynamics between the two top-value bidders and may not always occur.
• When $|M^1| = 1$: The paper provides counterexamples demonstrating potential non-convergence to NE, highlighting situations where a single high-value bidder can result in complex and unstable dynamics.

Theoretical Implications

The research advances the understanding of how automated learning strategies can influence bidder behavior in competitive environments. The nuanced insights regarding conditions under which mean-based algorithms reach equilibrium provide valuable guidance for designing such algorithms in auction contexts. Additionally, the work emphasizes the necessity of considering both time-average and last-iterate convergence notions, adding depth to analyses of strategic interactions in game-theoretical settings.

Practical Implications and Future Research

Practically, this research is critical for auction platforms and ad exchanges, where strategic stability and predictable outcomes ensure efficient market operations. Given the growing prevalence of machine-run auctions, understanding algorithmic convergence properties becomes essential to avoid suboptimal bidding strategies that could impact market efficiency.

Future research might delve into extending these results to scenarios with variable valuations or more intricate bidding environments, further exploring the potential of learning algorithms in dynamically complex auction settings. Furthermore, investigating the rate of convergence might offer additional insights into the applicational viability of these algorithms in commercial settings.

By establishing conditions for algorithmic stability in repeated auctions, the paper contributes significantly to the literature on learning in games, particularly within economically pertinent contexts such as digital advertising markets.