There has been substantial recent concern that pricing algorithms might learn to ``collude.'' Supra-competitive prices can emerge as a Nash equilibrium of repeated pricing games, in which sellers play strategies which threaten to punish their competitors who refuse to support high prices, and these strategies can be automatically learned. In fact, a standard economic intuition is that supra-competitive prices emerge from either the use of threats, or a failure of one party to optimize their payoff. Is this intuition correct? Would preventing threats in algorithmic decision-making prevent supra-competitive prices when sellers are optimizing for their own revenue? No. We show that supra-competitive prices can emerge even when both players are using algorithms which do not encode threats, and which optimize for their own revenue. We study sequential pricing games in which a first mover deploys an algorithm and then a second mover optimizes within the resulting environment. We show that if the first mover deploys any algorithm with a no-regret guarantee, and then the second mover even approximately optimizes within this now static environment, monopoly-like prices arise. The result holds for any no-regret learning algorithm deployed by the first mover and for any pricing policy of the second mover that obtains them profit at least as high as a random pricing would -- and hence the result applies even when the second mover is optimizing only within a space of non-responsive pricing distributions which are incapable of encoding threats. In fact, there exists a set of strategies, neither of which explicitly encode threats that form a Nash equilibrium of the simultaneous pricing game in algorithm space, and lead to near monopoly prices. This suggests that the definition of ``algorithmic collusion'' may need to be expanded, to include strategies without explicitly encoded threats.

翻译：近期存在广泛担忧，即定价算法可能学会"合谋"。超竞争价格可能作为重复定价博弈的纳什均衡出现，其中卖方采用威胁惩罚拒绝维持高价的竞争对手的策略，而这些策略可以被自动学习。事实上，标准的经济学直觉认为，超竞争价格的出现要么源于威胁的使用，要么源于一方未能优化其收益。这种直觉是否正确？在算法决策中防止威胁是否能避免卖方为自身收益优化时出现超竞争价格？答案是否定的。我们证明，即使双方都使用不编码威胁且为自身收益优化的算法，超竞争价格仍可能出现。我们研究序贯定价博弈，其中先动者部署算法，后动者在由此形成的环境中进行优化。我们证明，若先动者部署任何具有无悔保证的算法，而后动者即使在这种静态环境中近似优化，也会产生类似垄断的价格。该结果对先动者采用的任何无悔学习算法以及后动者采用的任何定价策略均成立——只要该策略获得的利润不低于随机定价的收益，因此即使后动者仅在无法编码威胁的非响应性定价分布空间内优化，该结果仍然适用。事实上，存在一组策略，它们均未显式编码威胁，却构成算法空间中同时定价博弈的纳什均衡，并导致接近垄断的价格。这表明"算法合谋"的定义可能需要扩展，以包含未显式编码威胁的策略。