Any member of a bike share program has come up against one of two frustrating scenarios: a full dock at the end of a trip, or an empty dock at the start.

Bike-share professionals call this the “rebalancing problem,” in which riders overload a system’s most popular takeoff points and destinations, rendering docks useless. The question, which has attracted the attention of dozens of mathematicians and scientists, is what kind of algorithm for redistribution might most efficiently remedy the issue, keeping every dock well-stocked but not fully stocked.

Motivate, the company that runs New York’s Citi Bike program, has tried a number of different strategies to move bikes around: trucks, sprinter vans, bike-drawn trailers, valet staff at the busiest drop-off locations. More than 120 people are on the task. For Washington’s Capital Bikeshare system, rebalancing has accounted for more than half of the system’s operating costs.

In May, Citi Bike tried something else: Pay members to redistribute the bikes themselves. Nine months later, 2,000 people have opted into that pilot program, called Bike Angels. Guided by online maps, they account for more than 10 percent of rebalancing on busy days—between 350 and 600 of the 3,500 bikes that must be moved to make the system function smoothly.