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

Arnab Maiti, Junyan Liu, Kevin Jamieson, Lillian J. Ratliff

TL;DR

A repeated-game model in which firms set prices using no-regret learners is analyzed, to characterize the equilibrium outcomes that can arise under different no-regret learning guarantees, and to address questions such as whether no-external-regret learners can converge to undesirable high-price outcomes.

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.

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

TL;DR

A repeated-game model in which firms set prices using no-regret learners is analyzed, to characterize the equilibrium outcomes that can arise under different no-regret learning guarantees, and to address questions such as whether no-external-regret learners can converge to undesirable high-price outcomes.

Abstract

We study the discrete Bertrand pricing game with a non-increasing demand function. The game has players who simultaneously choose prices from the set , where . 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.
Paper Structure (23 sections, 13 theorems, 108 equations, 34 figures)

This paper contains 23 sections, 13 theorems, 108 equations, 34 figures.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Our contributions
  4. Related Works
  5. Results
  6. Symmetric costs setting
  7. Asymmetric costs setting
  8. Experiments
  9. Symmetric cost setting
  10. Asymmetric cost setting
  11. Conclusion
  12. Omitted proofs
  13. Additional Experiments and Plots
  14. Numerical experiments for constant demand
  15. Numerical experiments for linear demand
  16. ...and 8 more sections

Key Result

Theorem 2.1

Consider any non-increasing demand function $f:\mathcal{P}\to[0,1]$ and the symmetric-cost setting, where all players have the same marginal cost $c$. There exists a CCE $\mathcal{D}$ such that, for each player $i\in\{1,2\}$, the following holds: In other words, for any non-increasing demand function, there exist two no-external-regret learners that can sustain high transaction prices while compe

Figures (34)

  • Figure 1: Numerical experiments for constant demand with $c_1=0$.
  • Figure 2: Experiments using the Hedge-based no-swap-regret learner with different combinations of costs and learning rates. The plots show the frequency of transaction prices over $T=10^7$ rounds, averaged across 100 random seeds.
  • Figure 3: Numerical experiments for asymmetric CCE and constant demand.
  • Figure 4: Numerical experiments for symmetric CCE, constant demand and $c=0.0$.
  • Figure 5: Numerical experiments for symmetric CCE, constant demand and $c=0.5$.
  • ...and 29 more figures

Theorems & Definitions (28)

  • Theorem 2.1
  • proof : Proof Sketch
  • Theorem 2.2
  • proof : Proof Sketch
  • Theorem 2.3
  • proof : Proof Sketch
  • Theorem 2.4
  • Theorem 2.5
  • proof : Proof Sketch
  • Theorem 2.6: Informal
  • ...and 18 more