Game Theory Explains How Algorithms Can Drive Up Prices

The archetypal version of this story appeared in Quanta Magazine.

Imagine a municipality with 2 widget merchants. Customers similar cheaper widgets, truthful the merchants indispensable vie to acceptable the lowest price. Unhappy with their meager profits, they conscionable 1 nighttime successful a smoke-filled tavern to sermon a concealed plan: If they rise prices unneurotic alternatively of competing, they tin some marque much money. But that benignant of intentional price-fixing, called collusion, has agelong been illegal. The widget merchants determine not to hazard it, and everyone other gets to bask inexpensive widgets.

For good implicit a century, US instrumentality has followed this basal template: Ban those backroom deals, and just prices should beryllium maintained. These days, it’s not truthful simple. Across wide swaths of the economy, sellers progressively trust connected machine programs called learning algorithms, which repeatedly set prices successful effect to caller information astir the authorities of the market. These are often overmuch simpler than the “deep learning” algorithms that powerfulness modern artificial intelligence, but they tin inactive beryllium prone to unexpected behavior.

So however tin regulators guarantee that algorithms acceptable just prices? Their accepted attack won’t work, arsenic it relies connected uncovering explicit collusion. “The algorithms decidedly are not having drinks with each other,” said Aaron Roth, a machine idiosyncratic astatine the University of Pennsylvania.

Yet a wide cited 2019 insubstantial showed that algorithms could larn to collude tacitly, adjacent erstwhile they weren’t programmed to bash so. A squad of researchers pitted 2 copies of a elemental learning algorithm against each different successful a simulated market, past fto them research antithetic strategies for expanding their profits. Over time, each algorithm learned done proceedings and mistake to retaliate erstwhile the different chopped prices—dropping its ain terms by immoderate huge, disproportionate amount. The extremity effect was precocious prices, backed up by communal menace of a terms war.

Implicit threats similar this besides underpin galore cases of quality collusion. So if you privation to warrant just prices, wherefore not conscionable necessitate sellers to usage algorithms that are inherently incapable of expressing threats?

In a caller paper, Roth and 4 different machine scientists showed wherefore this whitethorn not beryllium enough. They proved that adjacent seemingly benign algorithms that optimize for their ain nett tin sometimes output atrocious outcomes for buyers. “You tin inactive get precocious prices successful ways that benignant of look tenable from the outside,” said Natalie Collina, a postgraduate pupil moving with Roth who co-authored the caller study.

Researchers don’t each hold connected the implications of the finding—a batch hinges connected however you specify “reasonable.” But it reveals however subtle the questions astir algorithmic pricing tin get, and however hard it whitethorn beryllium to regulate.

“Without immoderate conception of a menace oregon an agreement, it’s precise hard for a regulator to travel successful and say, ‘These prices consciousness wrong,’” said Mallesh Pai, an economist astatine Rice University. “That’s 1 crushed wherefore I deliberation this insubstantial is important.”

No Regrets

The caller insubstantial studies algorithmic pricing done the lens of crippled theory, an interdisciplinary tract astatine the borderline of economics and machine subject that analyzes the mathematics of strategical competitions. It’s 1 mode to research the failures of pricing algorithms successful a controlled setting.

“What we’re trying to bash is make collusion successful the lab,” said Joseph Harrington, a University of Pennsylvania economist who wrote an influential reappraisal insubstantial connected regulating algorithmic collusion and was not progressive successful the caller research. “Once we bash so, we privation to fig retired however to destruct collusion.”

To recognize the cardinal ideas, it helps to commencement with the elemental crippled of rock-paper-scissors. A learning algorithm, successful this context, tin beryllium immoderate strategy that a subordinate uses to take a determination successful each circular based connected information from erstwhile rounds. Players mightiness effort retired antithetic strategies implicit the people of the game. But if they’re playing well, they’ll yet converge to a authorities that crippled theorists telephone equilibrium. In equilibrium, each player’s strategy is the champion imaginable effect to the other’s strategy, truthful neither subordinate has an inducement to change.

In rock-paper-scissors, the perfect strategy is simple: You should play a random determination each round, choosing each 3 possibilities arsenic often. Learning algorithms radiance if 1 subordinate takes a antithetic approach. In that case, choosing moves based connected erstwhile rounds tin assistance the different subordinate triumph much often than if they conscionable played randomly.

Suppose, for instance, that aft galore rounds you recognize that your opponent, a geologist, chose stone much than 50 percent of the time. If you’d played insubstantial each round, you would person won much often. Game theorists notation to this achy realization arsenic regret.

Researchers person devised elemental learning algorithms that are ever guaranteed to permission you with zero regret. Slightly much blase learning algorithms called “no-swap-regret” algorithms besides warrant that immoderate your hostile did, you couldn’t person done amended by swapping each instances of immoderate determination with immoderate different determination (say, by playing insubstantial each clip you really played scissors). In 2000, crippled theorists proved that if you pit 2 no-swap-regret algorithms against each different successful immoderate game, they’ll extremity up successful a circumstantial benignant of equilibrium—one that would beryllium the optimal equilibrium if they lone played a azygous round. That’s an charismatic property, due to the fact that single-round games are overmuch simpler than multi-round ones. In particular, threats don’t enactment due to the fact that players can’t travel through.

In a 2024 paper, Jason Hartline, a machine idiosyncratic astatine Northwestern University, and 2 postgraduate students translated the classical results from the 2000 insubstantial to a exemplary of a competitory market, wherever players tin acceptable caller prices each round. In that context, the results implied that dueling no-swap-regret algorithms would ever extremity up with competitory prices erstwhile they reached equilibrium. Collusion was impossible.

However, no-swap-regret algorithms aren’t the lone pricing crippled strategies successful the satellite of online marketplaces. So what happens erstwhile a no-swap-regret algorithm faces a antithetic benign-looking opponent?

The Price Is Wrong

According to crippled theorists, the champion strategy to play against a no-swap-regret algorithm is simple: Start with a circumstantial probability for each imaginable move, and past take 1 determination astatine random each round, nary substance what your hostile does. The perfect duty of probabilities for this “nonresponsive” attack depends connected the circumstantial crippled you’re playing.

In the summertime of 2024, Collina and her workfellow Eshwar Arunachaleswaran acceptable retired to find those optimal probabilities for a two-player pricing game. They recovered that the champion strategy assigned strikingly precocious probabilities to precise precocious prices, on with little probabilities for a wide scope of little prices. If you’re playing against a no-swap-regret algorithm, this unusual strategy volition maximize your profit. “To me, it was a implicit surprise,” Arunachaleswaran said.

Nonresponsive strategies look superficially innocuous. They can’t convey threats, due to the fact that they don’t respond to their opponents’ moves astatine all. But they tin coax learning algorithms to rise their prices, and past reap profits by occasionally undercutting their competitors.

At first, Collina and Arunachaleswaran thought that this artificial script wasn’t applicable to the existent world. Surely the subordinate utilizing the no-swap-regret algorithm would power to a antithetic algorithm aft realizing that their rival was profiting astatine their expense.

But arsenic they studied the occupation further and discussed it with Roth and 2 different colleagues, they realized their intuition was wrong. The 2 players successful their script were already successful a authorities of equilibrium. Their profits were astir equal, and some were arsenic precocious arsenic imaginable arsenic agelong arsenic neither subordinate switched to a antithetic algorithm. Neither subordinate would person an inducement to alteration strategy, truthful buyers would beryllium stuck with precocious prices. What’s more, the precise probabilities weren’t that important. Many antithetic choices led to precocious prices erstwhile pitted against a no-swap-regret algorithm. It’s an result you’d expect from collusion, but without immoderate collusive behaviour successful sight.

It Pays to Be Dumb

So, what tin regulators do? Roth admits helium doesn’t person an answer. It wouldn’t marque consciousness to prohibition no-swap-regret algorithms: If everyone uses one, prices volition fall. But a elemental nonresponsive strategy mightiness beryllium a earthy prime for a seller connected an online marketplace similar Amazon, adjacent if it carries the hazard of regret.

“One mode to person regret is conscionable to beryllium benignant of dumb,” Roth said. “Historically, that hasn’t been illegal.”

As Hartline sees it, the occupation of algorithmic collusion has a elemental solution: Ban each pricing algorithms but the no-swap-regret algorithms that crippled theorists person agelong favored. There whitethorn beryllium applicable ways to bash this: In their 2024 work, Hartline and his colleagues devised a method for checking if an algorithm has a no-swap-regret spot without looking astatine its code.

Hartline acknowledged that his preferred solution wouldn’t forestall each atrocious outcomes erstwhile no-swap-regret algorithms vie with humans. But helium argued that scenarios similar the 1 successful Roth’s insubstantial aren’t cases of algorithmic collusion.

“Collusion is simply a two-way thing,” helium said. “It fundamentally indispensable beryllium the lawsuit that determination are actions a azygous subordinate tin bash to not collude.”

Either way, the caller enactment inactive leaves galore unfastened questions astir however algorithmic pricing tin spell incorrect successful the existent world.

“We inactive don’t recognize astir arsenic overmuch arsenic we want,” Pai said. “It’s an important question for our time.”

Original story reprinted with support from Quanta Magazine, an editorially autarkic work of the Simons Foundation whose ngo is to heighten nationalist knowing of subject by covering probe developments and trends successful mathematics and the carnal and beingness sciences.