Revisiting the Bertrand Paradox via Equilibrium Analysis of No-regret Learners

2026-02-25

AI Summary

This study addresses the apparent contradiction between the low-price equilibrium predicted by Bertrand competition theory and the persistently high prices observed in real markets, focusing on the role of no-regret learners in repeated pricing games. By constructing a discrete Bertrand duopoly model and integrating game-theoretic analysis, no-regret learning algorithms, equilibrium characterization, and numerical simulations, the work provides the first systematic comparison of the equilibrium outcomes induced by external regret minimization versus swap regret minimization. Theoretically, it is shown that learners minimizing external regret may converge to high-price equilibria, whereas those minimizing swap regret tend to drive prices toward competitive levels. Numerical experiments not only corroborate these theoretical findings but also uncover novel dynamic phenomena under swap regret minimization, thereby elucidating the profound link between learning mechanisms and market pricing outcomes.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the discrete Bertrand pricing game with a non-increasing demand function. The game has $n \ge 2$ players who simultaneously choose prices from the set $\{1/k, 2/k, \ldots, 1\}$, where $k\in\mathbb{N}$. The player who sets the lowest price captures the entire demand; if multiple players tie for the lowest price, they split the demand equally. We study the Bertrand paradox, where classical theory predicts low prices, yet real markets often sustain high prices. To understand this gap, we analyze a repeated-game model in which firms set prices using no-regret learners. Our goal is to characterize the equilibrium outcomes that can arise under different no-regret learning guarantees. We are particularly interested in questions such as whether no-external-regret learners can converge to undesirable high-price outcomes, and how stronger guarantees such as no-swap regret shape the emergence of competitive low-price behavior. We address these and related questions through a theoretical analysis, complemented by experiments that support the theory and reveal surprising phenomena for no-swap regret learners.

Problem

• Bertrand Paradox
• no-regret learning
• pricing game
• equilibrium outcomes
• repeated games

Innovation

• no-regret learning
• Bertrand paradox
• equilibrium analysis
• swap regret
• repeated games