For $1,200 a month, Patricia Torres and her family were renting a bedroom, a share of time in the bathroom, one vegetable drawer and one shelf in the fridge, and two cupboards over the stove. They rented not so much a home as a fraction of one.

Karen Calderon had even less: a single room in a homeless shelter where she was not allowed so much as a hot plate to cook for her family.

Adrian Caratowsa had a studio he’d remade as his own, repainting the walls and wallpapering over the kitchen cabinets. But every day for five years, he walked out into a neighborhood he found depressing.

For each, San Francisco’s housing crisis had meant living without essential elements of home. A large affordable housing development rising downtown promised what they did not have: 95 complete homes, one-, two- and three-bedroom apartments with privacy, a sense of peace, a place to cook.

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The development, Natalie Gubb Commons, was reserved for households with incomes up to 50 percent of the local median. The applications were open for three weeks last fall, and 6,580 households applied for a chance to rent there, or nearly 70 for each unit.

As the development neared completion, Congress was closing in on a tax overhaul whose central goal — a much lower corporate tax rate — would make it harder to finance projects like this one, in a time of acute housing need. By one estimate, the bill would erase 235,000 affordable rental homes nationwide over the coming decade — one in six of those that might have been built or renovated without tax code changes.

With affordable housing scarce and likely to grow more so, San Francisco planned to draw winners for Natalie Gubb Commons at random.

Subsidized housing is often rationed this way, by lottery. Many apply, few win, most are disappointed. The process is meant to be more fair than first-come, first-served. But lotteries make literal a deeper unfairness. For homeowners, the mortgage interest deduction is available to anyone who qualifies. For poor renters, there is never enough housing assistance to go around.

Lotteries that allocate scarce resources are not set up to distinguish the neediest from the merely needy. Rather, they reward random chance, which is a distinctly different notion of what’s “fair.”

Ms. Calderon and her husband had been looking for a home for two years, since the last landlord learned of the third child on the way and told them they had to go. They could afford a studio, a one-bedroom maybe, but no one Ms. Calderon called would rent such a small space to a family of five. So they slept in their car, with a newborn, for months. They entered one homeless shelter, then another.

Ms. Calderon asked everyone she knew with a home how they did it. The market seemed a mystery, requiring some inside connection. She knew no one who’d won a home, but she was eager to try, exchanging her application for an eight-digit lottery number.

“I think about the number all the time — when I get up, during the day, and at night when I’m going to bed,” Ms. Calderon said in Spanish early last November, before those 6,580 numbers were fed into a computer, randomized and ranked. “Sometimes it wakes me up at night, thinking that they’re going to tell me, ‘Here are the keys to your apartment.'”

‘A place where I feel like I have hope’

Mr. Caratowsa walked by Natalie Gubb Commons a week before the lottery, as he had several times before. Scaffolding still obscured two sides of the development. The west-facing facade had come into view, eight stories of industrial-chic gray.

“Imagine the view you’d have from this window,” Mr. Caratowsa said, glancing up. He pictured himself on one of the upper floors, facing Fremont Street. He had long coveted the neighborhood for all the success that could be seen from that window: the luxury condos; the tech offices; the Salesforce Tower, the tallest skyscraper in the city.

Mr. Caratowsa, 31, had been living in another affordable building in the Tenderloin district, where the city’s homeless and addiction services are concentrated. He’d been on disability most of his adult life, living with years of pain after complications from a series of operations in his early 20s. He’d been H.I.V.-positive since childhood, and a subsidy from the San Francisco AIDS Foundation helped pay his rent.

He could see himself absorbing the city’s energy here, getting off disability, getting to work. He knows that he does not look like most of the office workers here, covered in his tattoos that commemorate travels and pain he has overcome. But he liked to stroll toward the Financial District downtown, where everyone appeared to be going places.

“To walk downtown, seeing people in suits, going to work, they have their coffee in their hand — to me it’s like when a little girl wants to be a princess and she wears her mom’s high heels,” Mr. Caratowsa said. “That’s me being in the Financial District. It’s just a place where I feel like I have hope.”

Amid all the wealth in this neighborhood, a one-bedroom at Natalie Gubb Commons would rent for around $1,000 to $1,200 a month, a three-bedroom up to $1,700. Apartments next door were three times as much.

That discount is possible through a mix of resources. Mercy Housing, the project’s nonprofit developer, effectively got the land free as part of a city requirement that the neighborhood’s redevelopment include affordable housing. The market-rate developer next door was subsidizing the project, along with city funds. Revenue from the state’s cap-and-trade emissions program helped. And Mercy used the backbone of nearly every affordable housing project in America, federal Low-Income Housing Tax Credits.

Affordable-housing projects were already becoming harder to execute as federal support dwindled and construction costs rose. Then, after Donald J. Trump was elected president, the value of Low-Income Housing Tax Credits began to fall. Changes to the tax code suddenly looked likely, and a lower corporate tax rate would mean companies would have less reason to put equity into affordable housing in exchange for credits to offset their tax liability.

What investors were willing to pay for the credits plummeted by February 2017, once the likelihood of a tax overhaul had set in.

Since Mr. Trump’s election, that fall in prices has translated to about $45,000 less per housing unit in San Francisco from investors in the common 4 percent tax credits, according to Novogradac & Company, an accounting firm that tracks the affordable housing market. And because tax-credit investors aren’t providing that equity, the city must fill gaps in projects with other public dollars.

The federal government more directly subsidizes low-income renters through public housing and vouchers for use in the private market. But voucher funding has stagnated, and the stock of public housing has shrunk as units have deteriorated. The tax credit program now serves the largest number of households.

Put together, these programs amount to only a partial housing policy.

“This country does have a national housing policy,” said Kate Hartley, director of the San Francisco Mayor’s Office of Housing and Community Development. “And that is that we provide our greatest subsidies to relatively affluent, housed people.”

Homeowners get the mortgage interest deduction, which has cost the federal government more than these programs for the poor combined. Other tax changes mean that fewer homeowners will now claim the mortgage deduction, and its benefits will be even more heavily concentrated among the rich.

But the tax bill’s implications for poor renters will be more profound. The odds are likely to grow worse than these: Last year, 53 households applied per each new affordable unit at The Meridian in Los Angeles; 84 for every home at Parcel 25 in Boston; 391 for each unit at Stargell Commons in Alameda, Calif.; 979 for every home at Our Lady of Lourdes Apartments in New York.

The lotteries that follow, said Kirk McClure, a professor of urban planning at the University of Kansas, “are a simple manifestation — and apparently an unavoidable one — of the fact that we have never in America made affordable housing a right.”

‘Please don’t give up’

For years, San Francisco conducted public lotteries in a bingo drum. But the raffle tickets were always getting stuck in the drum’s crevices. The city also tried a big box. People couldn’t see what was happening inside, however, and tickets got stuck under the lid.

This exercise — rolling the drum, shaking the box, inspecting for trapped tickets and repeating — lasted hours on a building the size of Natalie Gubb Commons. Ms. Torres would bus around town, picking up applications, dropping off applications. Lines would wrap around some developers’ offices on deadline day.

Mr. Caratowsa never had faith that the lotteries were fair. After one, he cut his own strips of paper at home, stuffed them in a shoe box and shook it. As he suspected, papers stuck together. In a world where housing was assigned randomly, he was offended that nothing was truly random.

Last year, San Francisco moved the whole process online. Renters can now more easily apply, which means that more do, and the odds have grown longer. But the system is more humane. The parts of the process where it has been most awkwardly apparent that people in need are competing are now less visible. The city still holds public lotteries, but they are primarily pep talks.

“Please don’t give up, please keep trying, and please know that sometimes people have to apply several times before they get offered a unit,” Maria Benjamin, who runs several of the city’s affordable housing programs and the lotteries, announced before the Natalie Gubb Commons lottery on a Tuesday morning in November.

About 50 people, a tiny share of the total applicants, turned up in the basement auditorium of San Francisco’s central library. On a 95-unit building, Ms. Benjamin told them, any number in the top 200 had a chance, as many applicants fail to qualify.

At the front of the auditorium, a laptop sat in place of the bingo drum. Kenneth Biby, the property manager at Natalie Gubb Commons, clicked a button, sending the lottery numbers to a randomizing website. Ten minutes later, city staff printed results for each person who’d signed in at the door. The staff called them by first name, their results kept private. Of everyone present, there were two decent draws: 19 and 150.

For Ms. Torres, who divided a kitchen with strangers, the latest rejection was one of many. She did not have her heart set on this property; she applied because she applied to everything. She drew a bad number, as she always did: this time, 824.

After more than two years of searching, she was growing tired. In December, another couple moved into her apartment. There were now seven people in a four-bedroom with one bath. Ms. Torres and her husband and 7-year-old daughter shared one room. They had to dash for the shower when the others were cooking.

They did not understand why a family like theirs couldn’t find a home of its own. Ms. Torres, 43, worked part time as a school monitor and a teacher’s aide. Her husband worked two jobs, as a security guard and a building concierge. But the overcrowded driveways on their block hinted at many households like theirs, all living in too few apartments.

After the lottery, on a visit to relatives in Portland, Ore., Ms. Torres considered possible schools for her daughter. Perhaps it was time to leave San Francisco, a city she loved. Moving would probably mean dividing the family; her husband planned to stay behind temporarily for his work.

“What am I waiting for?” Ms. Torres said. “How can I keep waiting? Why? So that it will get more and more expensive, and possibly I still won’t find anything?”

In the homeless shelter, Ms. Calderon’s family had already been given an extension, and they had to be out by early January. Natalie Gubb Commons would not work out, though. The family drew No. 905.

Ms. Calderon avoided the shelter as much as she could. She spent afternoons in the park with her children, returning to the building only at night. Ms. Torres studied for child development classes she was taking at a cafe that felt more comfortable than her own apartment. They had to find privacy in public spaces, a sense of home separate from where they slept.

Mr. Caratowsa kept returning downtown, envisioning himself amid the skyscrapers. He drew the best number: 71. He was elated, but cautious. Of Natalie Gubb’s 95 apartments, 46 were one-bedrooms. He needed people ahead of him to request larger units, or bow out.

An empty building like Natalie Gubb Commons is a math problem. Some units were designated for tenants below 40 percent of the median income, others below 50 percent. Each household must also earn at least twice the monthly rent, meaning that many who are poor enough to qualify for the building are too poor to cover the rent.

By February, as construction workers were hanging interior doors, Mercy was juggling 100 applicant files. Everything had to happen in lottery order: the initial interviews, the application approvals, the pick of units.

Mr. Caratowsa waited six weeks for his interview, and another month to learn his application had been approved. Then things happened too quickly. In March, Mercy was ready for him to sign a lease and move in. He had to give notice in the Tenderloin, and couldn’t afford April rent in two places in addition to a deposit. Anxious, he put in his 30 days’ notice and asked Mercy for an installment plan to cover the April rent. He waited.

Ms. Torres, by the spring, had given up applying for new places in San Francisco. On another trip to Portland, she toured four apartments that she liked, one- and two-bedrooms where she imagined how nice it would be to feel alone in her own home. She thought she and her daughter might move once the school year ended.

Ms. Calderon’s family left the homeless shelter in January and moved in with a co-worker of her husband’s, another home with too many families. Then in March she heard of a building where no one would mind the size of their family. An available $1,200 studio had a modern kitchen where she could cook for her children, and a patio where they could roller skate.

The family moved in this spring. The building, one block from the homeless shelter where they had been, is in the Tenderloin. But Ms. Calderon, with her children much happier now, thinks they might stay in this home forever.

In early April, Mercy offered Mr. Caratowsa the rest of the month for free. He had gotten his first pick: a seventh-floor apartment facing Fremont Street.

On his inspection, the day he received his keys, Mr. Caratowsa walked up to the living room window. He could see into the offices across the street. A highway off-ramp spilled traffic onto the street below, and another high-rise was under construction. But his apartment was surprisingly quiet.

“To be able to look down on this chaos and not hear anything — and then to open the window…” Mr. Caratowsa said, filling the room with the noise of cars honking and cement trucks spinning across the street. “I love the sound. This is so downtown.”

Mr. Caratowsa inspected the stove and the shower head, and noted exactly where he would put his chairs and his bed. By moving from one affordable building to another, he was leaving a coveted apartment back in the Tenderloin. That building last opened its wait list in 2011. This spring, 351 names were still on it.