Recent research suggests that even straightforward pricing programs can push consumer costs upward. Picture a small town with two vendors selling the same widget. Buyers favor the lowest ticket, so each merchant undercuts the other until profits vanish. Fed up, they slip into a smoke-filled tavern and agree to raise their prices together. That secret bargain would be classic collusion — illegal under U.S. antitrust rules. The merchants decide not to risk prosecution, and shoppers keep enjoying low sticker prices.
The legal playbook that has guided American regulators for more than a century rests on that simple idea: outlaw explicit agreements to fix prices, and competitive forces will keep prices down. The market has grown more complicated, though, since many sellers now set prices with software rather than gut instinct. In online marketplaces and other modern settings, firms often deploy what researchers call learning algorithms: programs that adjust prices repeatedly as fresh data arrives about demand, competitors and inventory. These systems can be far simpler than the “deep learning” engines behind today’s generative AIs, and they may behave in surprising ways.
That surprise poses a problem for enforcement. Traditional antitrust investigations look for communications or clear signs that firms agreed to fix prices. With algorithms, there is no meeting in a back room. “The algorithms definitely are not having drinks with each other,” said Aaron Roth, a computer scientist at the University of Pennsylvania. Still, experiments and theory over the last several years have shown that pricing software can produce outcomes that look a lot like collusion without any overt pact.
A widely cited 2019 paper illustrated the point. Researchers set two identical copies of a simple learning algorithm against one another in a simulated market and let them experiment with various tactics for raising profits. Over repeated rounds, each algorithm discovered a pattern of retaliation: when one cut price, the other punished the move by slicing its price sharply in return, often by “some huge, disproportionate amount.” The dynamic created a mutual threat of a ruinous price war, which kept actual prices high.
Aaron Roth suspects the trouble won’t be easy to contain. “The message of our paper is it’s hard to figure out what to rule out,” he said. Implicit threats of this type are central to many human collusion cases, and they can emerge from learning dynamics without any programmer deciding that they should. That prospect has led some researchers to ask whether regulators might require sellers to run only pricing algorithms that are by design incapable of signaling threats.
A recent paper coauthored by Roth and four colleagues argues that such a rule would not be a silver bullet. They proved that algorithms that look harmless on the surface — ones that simply optimize for their own profit without an explicit retaliation routine — can nonetheless generate poor outcomes for buyers. “You can still get high prices in ways that kind of look reasonable from the outside,” said Natalie Collina, a graduate student working with Roth who co-authored the new study.
Scholars do not all agree about how to interpret this result. Judgment hinges on what counts as a “reasonable” strategy, and that question can be contested. “Without some notion of a threat or an agreement, it’s very hard for a regulator to come in and say, ‘These prices feel wrong,’” said Mallesh Pai, an economist at Rice University. To probe where and how algorithm-driven pricing breaks down, a number of researchers have turned to game theory, the branch of economics and computer science that models strategic interaction in mathematical terms.
“What we’re trying to do is create collusion in the lab,” said Joseph Harrington, a University of Pennsylvania economist who wrote an influential review paper on regulating algorithmic collusion and was not involved in the new research. “Once we do so, we want to figure out how to destroy collusion.” The lab Herrington describes can be as simple as the repeated play of rock-paper-scissors. In those games, a learning rule is any procedure that selects moves based on past outcomes; players might try different tactics over time and, if they play well, converge to an equilibrium in which no one has an incentive to change strategy unilaterally.
In rock-paper-scissors, the equilibrium strategy is to randomize evenly among the three options. A learning algorithm excels when the opponent departs from that random mix. If, after many rounds, you realize your opponent favored rock more than half the time, you could have done better by choosing paper every time instead of sticking with your actual history. Game theorists call that missed opportunity regret.
Scientists built a class of simple learning rules that guarantee zero regret. A stronger notion, called no-swap-regret, promises that whatever the opponent did, you could not have improved your payoff by swapping every occurrence of any one action for another — for example, replacing all your scissors with paper. In the year 2000, theorists proved that when two no-swap-regret algorithms face each other in any repeated game, their joint behavior converges to a special kind of equilibrium that would also be optimal in a single-shot version of the game. A useful feature of that result is that it removes the power of threats: single-round games do not allow credible punishment strategies, so threat-based collusion cannot sustain.
Jason Hartline, a computer scientist at Northwestern University, and two graduate students put this classic theorem to work in a pricing setting in a 2024 paper. They translated the 2000 logic into a market model where sellers reset prices each round and showed that if both firms deploy no-swap-regret algorithms, the long-run outcome will be competitive prices. In that framework, tacit collusion cannot persist through algorithmic play.
No-swap-regret algorithms are not the only options available to sellers. Game theorists can ask how one should respond when facing a no-swap-regret opponent. The answer can be odd: pick a fixed probability for each allowable price, then play one price at random each round regardless of what the rival does. The exact probabilities that maximize profit will depend on the payoff structure of the particular pricing game, but the strategy’s defining feature is that it is nonresponsive — it never reacts to the other player’s moves.
In the summer of 2024, Collina and her coauthor Eshwar Arunachaleswaran calculated those profit-maximizing probabilities for a two-player pricing game. The result surprised them. The best response assigns very high probability to extremely high prices and spreads the remaining probability across a broad swath of lower prices. “To me, it was a complete surprise,” Arunachaleswaran said.
At first glance a nonresponsive plan looks innocuous. It cannot carry a threat because it does not change when the rival cuts price. Yet that inaction can induce the learning algorithm across the table to raise its prices in search of better rewards. The nonresponsive player then benefits by occasionally undercutting and capturing sales at elevated margins.
Collina and Arunachaleswaran initially suspected the situation would be unstable in practice: the no-swap-regret algorithm should detect its exploitation and switch to a different rule. Their follow-up work with Roth and two other coauthors showed that their intuition was incorrect. The pair of strategies they studied constituted an equilibrium: both firms earned roughly equal profits, and those profits were as high as they could be so long as neither player abandoned its algorithm. No single firm had an incentive to move elsewhere, leaving buyers stuck with elevated prices. Moreover, the effect did not hinge on a knife-edge set of probabilities; many different nonresponsive choices produced the same high-price result when paired with a no-swap-regret opponent. The market outcome resembled cartel behavior but emerged with no agreement, no threats and no explicit collusion.
That finding leaves regulators in a bind. Banning nonresponsive play would be difficult to justify: a static random pricing plan can look like simple business error, not a deliberate scheme to limit competition. At the same time, outlawing all algorithms that are not no-swap-regret would narrow sellers’ options dramatically. Roth says he has no clean fix. “The message of our paper is it’s hard to figure out what to rule out,” he added in discussion of the broader research program.
Hartline offers a tighter prescription: restrict pricing software to the family of no-swap-regret algorithms that theorists have long favored. He and his coauthors proposed a practical test to detect the no-swap-regret property by observing an algorithm’s behavior under many hypothetical scenarios, a check that can be carried out without inspecting proprietary source code. The approach would make it possible for platforms or regulators to certify algorithms as safe in the sense relevant to the 2000 theorem.
Hartline allows that his proposal would not block every damaging pattern that can appear when algorithmic sellers face human rivals. He argued, though, that the high-price equilibria identified in the Roth group’s work do not meet his definition of collusion. “Collusion is a two-way thing,” he said. “It fundamentally must be the case that there are actions a single player can do to not collude.”
That exchange captures how unsettled the field remains. Some researchers treat the mathematical examples as proof that algorithmic pricing presents a new antitrust problem that needs legal and technical remedies. Others view the anomalies as corner cases that can be managed by limiting certain classes of algorithms and by giving platforms stronger oversight tools. Either position accepts that the mapping between algorithm design and market outcomes is subtle, and that standard enforcement doctrines were not written with adaptive software in mind.
Practical questions multiply quickly: how to detect when actual marketplace prices reflect strategic interaction between learning agents rather than ordinary competition; how to measure the role of stochastic behavior in real bids and offers; how to design regulatory tests that do not quash useful innovations in automated commerce. Mallesh Pai framed the challenge bluntly. “We still don’t understand nearly as much as we want,” Pai said.
Policymakers and platform managers are beginning to wrestle with these issues. Some propose disclosure or audit regimes for pricing software; others suggest standards that would allow certified algorithm types while banning opaque procedures that lack provable game-theoretic guarantees. On the research side, teams continue to conduct experiments that isolate different strategic forces, then translate those theoretical facts into diagnostic checks that could be used in practice.
The debate reaches beyond abstract models because the commercial incentives are real. Online sellers can deploy off-the-shelf tools that tune prices automatically, and a firm that makes persistent extra profit through a benign-looking algorithm can be tempted to keep using it. The tensions between profit-seeking code and consumer interest mean that regulators will need both sharper analytic tools and clearer legal tests if they want to identify harmful patterns that arise without explicit agreements.
For now, the literature is still coalescing around the most useful principles. Some paths forward would expand enforcement tools to target outcomes as well as agreements, while others would narrow algorithm design choices to those that carry provable safety properties. Neither route is free of trade-offs. As the connection between automated decision rules and market effects becomes clearer, the policy conversation will likely accelerate. “One way to have regret is just to be kind of dumb,” Roth said. “Historically, that hasn’t been illegal.”