We develop a tractable model for studying strategic interactions between learning algorithms. We uncover a mechanism responsible for the emergence of algorithmic collusion. We observe that algorithms periodically coordinate on actions that are more profitable than static Nash equilibria. This novel collusive channel relies on an endogenous statistical linkage in the algorithms' estimates which we call spontaneous coupling. The model's parameters predict whether the statistical linkage will appear, and what market structures facilitate algorithmic collusion. We show that spontaneous coupling can sustain collusion in prices and market shares, complementing experimental findings in the literature. Finally, we apply our results to design algorithmic markets.

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