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.

翻译：我们研究具有非递增需求函数的离散伯特兰定价博弈。该博弈有 $n \ge 2$ 个参与者，他们同时从集合 $\{1/k, 2/k, \ldots, 1\}$（其中 $k\in\mathbb{N}$）中选择价格。设定最低价格的玩家获得全部需求；若多个玩家价格并列最低，则均分需求。我们研究伯特兰悖论——经典理论预测价格偏低，而现实市场常维持高价。为理解这一差距，我们分析一个重复博弈模型，其中企业使用无遗憾学习算法定价。我们的目标是刻画不同无遗憾学习保证下可能出现的均衡结果。特别关注以下问题：无外部遗憾学习者是否会收敛到不利的高价结果，以及无交换遗憾等更强保证如何塑造竞争性低价行为的涌现。我们通过理论分析回答这些及相关问题，并辅以支持理论且揭示无交换遗憾学习者中意外现象的实验。