The government has tried before to fix the rent subsidy program. In the early 1990s, an experiment called Moving to Opportunity required some families to use their vouchers in more expensive neighborhoods. In 2000, a broader initiative raised the value of all vouchers in 49 metropolitan areas. Officials hoped the change would make it possible for families to find rental apartments in a broader range of neighborhoods. Instead, a recent study by the economists Peter Ganong of Harvard and Robert Collinson of New York University found that most families ended up paying higher rents in the same neighborhoods — and often for the same units.

“When you give somebody a voucher, it tends to be the case that they buy better-quality housing in the same neighborhood,” Mr. Ganong said. “That’s always been a disappointing fact within HUD.” He said it raised the question: “If they’re not finding better neighborhoods, why are we putting this money into housing?”

The problem, officials have concluded, is that the subsidies were much too small. In 2011, HUD started the Dallas experiment as a part of a settlement with housing activists. It has since added five smaller regions to the program.

In Dallas, the maximum subsidy for a three-bedroom apartment is now just $850 in the cheapest ZIP codes, but as much as $1,840 in the most expensive ZIP codes. In 33 of the 50 largest metropolitan areas, the subsidy for the wealthiest ZIP code would be more than twice as high as the subsidy in the poorest ZIP code. In New York, San Diego and Washington, it would be more than three times as high.

MaryAnn Russ, chief executive of the Dallas Housing Authority, the agency that administers the experiment, said the changes had provided a “tremendous benefit” to thousands of families in the Dallas area. In 2011, Dallas voucher recipients lived in 129 ZIP codes. Four years later, recipients live in 163 ZIP codes.

Also important was that the overall cost of the program did not increase.

But Ms. Russ said there were some unexpected problems. Officials predicted that cutting subsidies would force landlords in lower-income areas to cut rents. Instead, many of those landlords required subsidized tenants to pay more out of their own pockets. And moving has proved difficult for many tenants. Some are elderly, disabled or so poor that they have no savings for a security deposit.