The Algorithms Behind Pricing Your Ride

Professor David Brown and co-authors developed a dynamic pricing model for spatially distributed demand-based services, such as ride sharing

May 9, 2023
Big Data
The Algorithms Behind Pricing Your Ride _ David Brown_Duke Fuqua School of Business

Technology makes it easy for companies like Uber and Lyft to add for-hire cars to city streets. But these transportation network companies (TNCs) face a daunting challenge. Every minute of every day they must provide an optimal supply of vehicles and coordinate their whereabouts to maximize their revenue in an ever-changing environment.

Ride-share companies have found at least one answer to this intricate price-setting problem—they calibrate the rates they charge in a forward-thinking way. Instead of considering the value of each current customer in isolation, their pricing algorithms take into strong consideration the short- and long-term positioning of all their cars in the system.

David Brown, a professor at Duke’s Fuqua School of Business, has developed a dynamic pricing policy model that could help these ride-share companies use their resources even more efficiently.  Brown explains his findings in the recent paper Dynamic Pricing of Relocating Resources in Large Networks which was published in Management Science and co-authored with Santiago R. Balseiro, an associate professor of business at Columbia’s Graduate School of Business, and Fuqua Ph.D. graduate, Chen Chen, now an assistant professor at New York University Shanghai.

“Think of the drivers or cars as the resources of companies like Lyft or Uber. The flow of those resources is very important from their standpoint, because that’s the supply of their service,” Brown said.  “If the distribution of drivers throughout the region is not properly balanced, then they will have a lot of dropped requests and, as a result, will lose revenue.”

The challenge in metros  

Consider the reality ride-share companies face in New York City, one of the world’s top five markets for TNCs. About 80,000 such vehicles, including taxis, roam the Big Apple’s sprawling boroughs and airports. (Seventy percent of U.S. ride-share trips are in nine cities, according to Schaller Consulting, which tracks the industry.) Their numbers rose by almost 60 percent between 2013 and 2017. These cars account for 29 percent of all Manhattan traffic, reports the city’s Taxi and Limousine Commission, and they have slowed traffic by nearly 25 percent since 2010, according to Wired Magazine. As of 2017, the most recent year for which data is available, ride-share vehicles spend more than 40 percent of their time empty.

Besides juggling hundreds of drivers simultaneously, the TNCs’ pricing problems are inextricably interrelated with satisfying the goals of other key parties—passengers and governments. Riders want to pay fair market value for cars that promptly arrive when they need them and take them where they need to go in a timely fashion. City leaders hope to ease traffic snarls and pollution while keeping voters happy.

Meanwhile, the app-based ride services companies must satisfy demand and retain drivers with reasonable wages and working conditions—all while achieving the highest possible revenue goals and steering clear of painful regulatory battles.

Developing a new model

The model developed by Brown and co-authors mimics most big cities: A large, far-flung metropolitan area divided into hubs and spokes that resemble the classic hub-and-spoke model. The hubs could be in one central area; multiple downtown places where people congregate; or remote airports. Spokes would typically be less frequented places like suburbs at the edges of the city.

In this environment, ride-share companies constantly conduct a price dance with potential customers. If they price a trip too high, possible riders will buy elsewhere or not travel, and driver utilization and revenue will plummet. If they charge too little, riders’ cars will be far busier, but revenue will be unacceptably low.

“The problem,” Brown and his co-authors write, “is to find a dynamic pricing policy that maximizes the provider’s average revenue over an infinite horizon. With many locations, this problem is difficult to solve, as optimal pricing policies may depend on the locations of all resources in the system.”

“These companies want to think—where’s that pricing sweet spot? It’s not a myopic [short-sighted] decision,” Brown said. “It affects more than the revenue they collect from that ride. It affects where that driver, an important resource, goes. The way companies coordinate pricing over time greatly affects the distribution and location of drivers throughout the system.”

Brown’s dynamic pricing model presumes hubs are, in a real sense, almost physically connected due to the heavy volume of traffic between them.

“Our analysis led us to use static pricing for requests between hubs,” Brown said. “For the spokes, we found instead that we need to use carefully controlled dynamic pricing that depends on how many drivers are located in the spokes.”

With the price of rides between hubs essentially fixed and the price of rides to and from spokes dynamic, the model that Brown and his co-authors created adjusts hub-to-spoke and spoke-to-hub fares based on the supply of drivers in the spokes, because it presumes hubs will always have a sufficient supply or oversupply of drivers.

Brown said companies should want to have a lot of leeway in how their drivers are positioned.

“If you’re oversupplied with drivers in a spoke, you’ll want to offer a somewhat higher price for a hub-to-spoke request to discourage that trip from happening,” Brown said. “On the other hand, if there’s only one driver in a spoke, you’ll want to offer a somewhat lower price for a hub-to-spoke request to make that trip more likely. Both decisions would hurt revenue in the short term but lead to better outcomes in the long run: it’s all in response to the flow of drivers to remote regions. Getting this tradeoff right is complex and was a significant technical challenge.”

The need for dynamic-spoke pricing surprised Brown and his co-authors. They thought strategic pricing at spokes would be unimportant, because spokes only contributed a modest fraction of rides and, therefore, a small fraction of revenue.

“We found good performance relied heavily on how we priced at these locations, especially when there are many of them,” Brown said.  “For the opposite reason, it surprised us that we could get away with static prices between hubs.”

On an overall basis, however, the pricing model views cars not as a constant flow or fluid, as other researchers have presumed, but as “lumpy”, a term which means their revenue come in chunks at irregular intervals.

“Fluid models are well-studied and lead to static prices. This works well when supply greatly outstrips demand, but that isn’t realistic,” Brown said.

Brown said a fixed pricing policy throughout the entire network of cars would be a mistake. He and his co-authors found that companies would likely lose a great deal of revenue if they follow a static pricing approach.

To demonstrate this, Brown and his co-authors collected data from RideAustin, a now defunct local ride-sharing non-profit that operated in Austin, Texas, that covered 1.5 million transactions over a 10-month period. Each transaction included detailed information about every ride such as the coordinates of origins and destinations, start and end times, and each total fare. They found that their pricing policies collected significantly more revenue than prices inspired by fluid-based models that had been developed in earlier work.

“It is essential to adjust prices dynamically based on the locations of resources to attain good performance,” Brown and his co-authors write. “This is good both for the company and the drivers, as they will spend less time idle.”

The complexity of the optimization problem

Creating a dynamic pricing policy in the ride-share industry required Brown and his co-authors to work through a large optimization problem, something that is also known as a stochastic dynamic program.

“It allows you to think about pricing decisions for every possible configuration of drivers throughout a big city, a number that could be greater than the number of atoms in the universe,” Brown said.  “To tackle this problem, you must give up the idea of solving it and settle for approximately solving it with novel techniques. The actual optimal solution would be fantastically difficult to calculate.”

The complexity of this problem actually benefitted Brown and his co-authors.

 “As the problem gets bigger, our approach becomes closer and closer to optimal, even though obtaining truly optimal prices gets progressively more difficult. That was our key theoretical finding. It represents the beauty of the approach.”

The challenge of optimizing numbers of drivers

Ride-share providers keep their pricing algorithms secret.

“I’d love to see exactly what they’re doing under the hood,” Brown said. “They’re fairly tight lipped about it—for obvious reasons.”

Former New York City Mayor Bill de Blasio accused companies like Lyft and Uber of racing for profits and dominant market share to such a degree they overwhelmed his city with cars.

“The Uber business model is to flood the market with as many cars and drivers as possible,” de Blasio complained.

Brown doubts that was the companies’ intention.

“I’m sure they’d be happy to take revenue from Yellow Cab, but beyond that if there’s a huge number of cars available, drivers would be idle a large fraction of the time,” Brown said, “If that was the case, why would anybody want to be an Uber driver? Plus, a huge number of drivers would create congestion.”

Nonetheless, New York City in an effort to improve traffic flow and protect the Yellow Cab industry capped the number of ride-share cars in 2018. Brown wonders if government intervention was necessary.

“Even if a city doesn’t explicitly cap the number, implicitly caps are going to be created by competition,” Brown said.

Four ‘dynamic’ research conclusions

Brown suggested the following takeaways for companies considering dynamic pricing:

  1.  Vary prices in time, space, and in response to how the supply (drivers) is distributed over an area when considering ridesharing and other problems that involve shared resources that move.
  2.  Keep prices static when supply (drivers) greatly exceeds demand. But when total supply and demand are roughly balanced, dynamic pricing is essential. Supply and demand will achieve this state when they are in equilibrium. This is especially true in places like New York City that cap the number of Lyft and Uber drivers.
  3.   Remember that due to this problem’s tremendous complexity, finding truly optimal dynamic prices is impractical. Nonetheless, mathematically sound, easy-to-compute dynamic prices that work in large metro areas can be devised.
  4. Adjust prices for requests that involve remote spokes, but keep prices static for requests between high-density hubs. Dynamic prices are necessary at spokes, even though they only represent a small fraction of the flow.

Other business implications

This research on how price shifts in response to demand has relevance in other fields. The business models of companies like Zipcar, DoorDash, Uber Eats, bike-sharing companies like Lime, and car rental providers also revolve around continuously relocating resources. Many consumer goods companies must also manage ever shifting product assortments online. Even Ticketmaster uses dynamic pricing when faced with overwhelming ticket demand for artists like Bruce Springsteen.

Brown is now part of a team of researchers at several institutions, including Duke’s Nicholas School of the Environment, who are developing optimization methods to benefit the energy industry. This work, which is supported by the Department of Energy’s Advanced Research Projects Agency-Energy, involves studying how vertically integrated utility companies like Duke Energy should optimize their operations.

“How to properly balance the mix of electricity from various generation sources, hydro storage, and renewable resources dynamically over time in response to weather and fluctuating consumer demand for electricity is an enormous challenge,” Brown said.

“Although this is a very different problem, there are connections between it and pricing in ride-sharing in terms of the algorithms that work. Ultimately, both problems are about matching supply and demand, albeit in different settings and pulling different levers. I am continually fascinated by the fact that problems in disparate applications often have deep, structural similarities.”




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