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。该博弈包含 $n \ge 2$ 名参与者，他们同时从集合 $\{1/k, 2/k, \ldots, 1\}$ 中选择价格，其中 $k\in\mathbb{N}$。设定最低价格的参与者获得全部需求；若多名参与者报出相同最低价，则均分需求。我们研究伯特兰悖论——经典理论预测低价，但实际市场常维持高价。为理解此差异，我们分析了一个重复博弈模型，其中企业使用无悔学习算法设定价格。我们的目标是刻画在不同无悔学习保证下可能出现的均衡结果。我们特别关注以下问题：无外部悔学习者能否收敛至不良的高价结果，以及更强保证（如无交换悔）如何影响竞争性低价行为的形成。我们通过理论分析探讨这些问题及相关议题，并辅以实验验证理论，同时揭示了无交换悔学习者表现出的意外现象。