用无悔学习理论重新分析伯特兰价格战，揭示高定价为何仍能维持。
Revisiting the Bertrand Paradox via Equilibrium Analysis of No-regret Learners
- 用无悔学习模型模拟企业重复博弈定价行为。
- 无外部后悔学习者可能收敛到高价均衡，违背经典预测。
- 更强的无交换后悔保证可促进竞争性低价出现，适合研究市场机制设计的人看。
我们研究具有非增需求函数的离散伯特兰定价博弈，其中 $n \≥ 2$ 个参与者从集合 $\{1/k, 2/k, \ldots, 1\}$ 同时选择价格，$k\in\mathbb{N}$。设定最低价格的玩家获得全部需求；若多个玩家并列最低价，则平分需求。我们探讨伯特兰悖论——经典理论预测低价，但现实市场常维持高价——之间的差距。通过分析企业使用无悔学习者进行重复博弈的模型，我们旨在刻画不同无悔学习保证下可能出现的均衡结果。重点关注的问题包括：无外部后悔学习者能否收敛至不利的高价均衡？更强的学习保证（如无交换后悔）如何影响竞争性低价行为的涌现？通过理论分析和实验验证，我们支持了理论结论，并揭示了无交换后悔学习者中令人惊讶的现象。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。