# Algorithmic Collusion Without Threats

arXiv: https://arxiv.org/abs/2409.03956 · HTML: https://mlsanity.com/paper/2409.03956/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 6 Sep 2024.

## Builds on (most cited first, 5 of 5)

- [Strategizing against No-regret Learners](https://mlsanity.com/paper/1909.13861/) (2019)
- [Selling to a No-Regret Buyer](https://mlsanity.com/paper/1711.09176/) (2017)
- [Auctions Between Regret-Minimizing Agents](https://mlsanity.com/paper/2110.11855/) (2021)
- [Regulation of Algorithmic Collusion](https://mlsanity.com/paper/2401.15794/) (2024)
- [Efficient Stackelberg Strategies for Finitely Repeated Games](https://mlsanity.com/paper/2207.04192/) (2022)

## Built on by (most cited first, 7 of 7)

- [Markets with Heterogeneous Agents: Dynamics and Survival of Bayesian vs. No-Regret Learners](https://mlsanity.com/paper/2502.08597/) (2025)
- [Breaking Algorithmic Collusion in Human-AI Ecosystems](https://mlsanity.com/paper/2511.21935/) (2025)
- [Oblivious Learning and Collusive Pricing](https://mlsanity.com/paper/2606.05363/) (2026)
- [Algorithmic Collusion and the Complexity of Information-Value-Free Equilibria](https://mlsanity.com/paper/2609.22757/) (2026)
- [Revisiting the Bertrand Paradox via Equilibrium Analysis of No-regret Learners](https://mlsanity.com/paper/2602.21620/) (2026)
- [Mean-based algorithms: A lower bound and regret](https://mlsanity.com/paper/2606.04931/) (2026)
- [Persistent Partners Raise Prices Among Learning Agents](https://mlsanity.com/paper/2609.35402/) (2026)

## Abstract (arXiv)

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.