Delegate Pricing to Algorithms: When Slow and Steady Wins the Race

2026-10-08

AI Summary

This study investigates how firms strategically select learning algorithm rates in continuous-time pricing games to maximize path-wise profits. By constructing a meta-game framework that integrates gradient dynamics with discounted utility theory, we analyze the equilibrium effects of learning rate selection. Our findings reveal a "slower is faster" paradox under strategic complementarity: even when competitive pricing constitutes a dominant strategy, deliberately adopting slow learning algorithms enables firms to extract surplus rents by prolonging the convergence period, thereby constituting an optimal design. This work elucidates the mechanisms through which learning speed influences transitional profits and establishes principles for optimal algorithm design across diverse strategic environments.

Abstract

We study the strategic design of learning algorithms in a canonical continuous-time pricing game. We introduce a meta-game in which firms select learning rates for a gradient dynamic, evaluating payoffs as the discounted sum of profits accrued along the entire learning path. We uncover a fundamental dichotomy driven by stage-game incentives: if the stage game exhibits strategic substitutability, firms unambiguously prefer the fastest possible algorithms to rapidly exploit a gradually adjusting opponent. Under strategic complementarity, however, an excessively fast algorithm accelerates the rival's competitive response, destroying transitional profit margins. Even though the competitive price is a strictly dominant action in the underlying stage game, we prove that firms optimally design sluggish algorithms, extracting surplus during a prolonged convergence to the competitive equilibrium.

Problem

• algorithmic pricing
• learning algorithms
• meta-game
• strategic complementarity
• learning rate design

Innovation

• Algorithmic Pricing
• Meta-game
• Learning Rate Design
• Strategic Complementarity
• Gradient Dynamics