In August, I became part of an exercise in market economics to save my older brother’s life.

We participated in an innovative program that creates exchanges of goods with immeasurable value—healthy kidneys—among strangers who never would have connected otherwise.

A little over a year earlier, my brother had rushed to the intensive-care unit of the Alfred Hospital while on a work trip in Melbourne, Australia, after experiencing extreme difficulty breathing.

Pesh had just turned 40. Until then, he’d experienced no major health issues. But doctors in Melbourne explained that he only had one functioning kidney and it was working at 2%. I was devastated to learn his new reality and flew to Australia right away to be with him. (I work at The Wall Street Journal in New York.) After two weeks of dialysis, he returned home to Houston, where he went on the waiting list for a kidney transplant.

Most kidneys in transplants come from people who have died. It can take years for those kidneys to become available. Thirteen patients die each day waiting for a transplant, the National Kidney Foundation says. Today 95,000 people are on the waiting list for a kidney donation, according to the government-funded Organ Procurement and Transplantation Network.

In the nearly 21,000 kidney transplants that took place last year, nearly a third were from live donors. Doctors consider a kidney from a living donor much better: It gets to work quicker and lasts longer.

But for years, living-donor transplants were hindered because recipients needed a close biological match to make use of the kidney.

Now, thanks to Nobel Prize-winning economistsAlvin Roth and Lloyd Shapley, whose work in game theory has been applied to the field of organ donation, there’s a new way to get a kidney faster: a paired donation. Say a donor for Patient A is a poor match for him. She can donate her kidney to a matching Patient B, and that patient’s non-matching donor can reciprocate with a kidney for Patient A. This process is repeatable, so that a chain of dozens of patients can be saved by strangers who enter the exchange organized by one or more hospitals.

How the Kidney Exchange Pool Works Four donors, including the author of this story, participated in an exchange where their kidneys were paired with matching patients, including the author's brother, who needed transplants. Actual recipients of kidney Intended recipients for kidney donation Donor 2 Donor 4 Donor 3 Yogita Patel Saw Facebook post for Patient 3 but couldn’t donate directly; entered exchange Wanted to donate but didn’t have an intended recipient; entered exchange Donated for her brother, Pesh Donated for his wife, Patient 2 Patient 4 Patient 2 Patient 3 Pesh Patel Received Yogita’s kidney Received kidney from Donor 4 Entered the exchange pool and received a kidney from Donor 2 Yogita’s brother; received kidney from Donor 3 Actual recipients of kidney Intended recipients for kidney donation Donor 3 Donor 2 Donor 4 Yogita Patel Wanted to donate but didn’t have an intended recipient; entered exchange Donated for her brother, Pesh Saw Facebook post for Patient 3 but couldn’t donate directly; entered exchange Donated for his wife, Patient 2 Pesh Patel Patient 2 Patient 3 Patient 4 Entered the exchange pool and received a kidney from Donor 2 Received kidney from Donor 4 Received Yogita’s kidney Yogita’s brother; received kidney from Donor 3 Actual recipients of kidney Intended recipients for kidney donation Donor 2 Donor 4 Yogita Patel Donor 3 Donated for his wife, Patient 2 Saw Facebook post for Patient 3 but couldn’t donate directly; entered exchange Wanted to donate but didn’t have an intended recipient; entered exchange Donated for her brother, Pesh Patient 3 Pesh Patel Patient 2 Patient 4 Received kidney from Donor 4 Received Yogita’s kidney Yogita’s brother; received kidney from Donor 3 Entered the exchange pool and received a kidney from Donor 2 Intended recipients for kidney donation Actual recipients of kidney Pesh Patel Yogita Patel Donated for her brother, Pesh Yogita’s brother; received kidney from Donor 3 Donor 2 Patient 2 Received Yogita’s kidney Donated for his wife, Patient 2 Patient 3 Donor 3 Received kidney from Donor 4 Saw Facebook post for Patient 3 but couldn’t donate directly; entered exchange Donor 4 Patient 4 Wanted to donate but didn’t have an intended recipient; entered exchange Entered the exchange pool and received a kidney from Donor 2

Even before my brother learned he could wait up to eight years for a kidney from a deceased donor, I hadn’t hesitated telling him I would get tested to see if I was eligible to donate to him. But actually following through felt like dealing with unfinished business.

In an odd coincidence, at the age of 12 I’d informed my family that I would never donate a kidney. One thing I knew for sure was that if things didn’t work out for the recipient, you don’t get it back. That felt like a raw deal for the donor. What we didn’t know then: One of my brother’s kidneys hadn’t fully developed, and it was only a matter of time before the other one gave out.

Now, with my brother in need, we had to cross off my mother and older sister as candidates for health reasons. Five family friends who came forward were excluded or weren’t able to move on with testing. A late kidney test ruled out the penultimate candidate, our dad: Donating to save my brother’s life could put his own health in jeopardy.

Yogita and Pesh were part of an eight-person swap at Houston Methodist hospital in August. Photo: Dipa Patel

My brother broke the news in a series of texts that our dad had been eliminated, and I was now candidate No. 1. I would already be visiting the following week and could complete my donor testing at Houston Methodist Hospital. This wasn’t how I wanted to celebrate my 37th birthday, but I knew I couldn’t delay further.

Over two days, I had countless vials of my blood drawn, a chest X-ray, CT scan and other tests. I had hours of meetings with a dietitian, a kidney specialist and a social worker. I had a psychological evaluation to gauge how I would fare emotionally, especially after surgery.

Hearing about the technical points and risks associated with the laparoscopic surgery scared me. I found a small degree of comfort knowing that I would get priority status should I later need a transplant myself. My donor-screening team praised me for years of Pilates and a vegetarian diet. Finally, some recognition for the healthy choices that made me such a strong candidate.

But I wouldn’t be giving my kidney to my brother. I learned early on we weren’t the same blood type. Before I even knew if I was healthy enough to give up a kidney, I agreed to take part in a paired donation to increase his chances of a speedy transplant.

Houston Methodist first aims to create a donor chain without other hospitals—its longest involved 12 people. If appropriate matches fail to come together, the hospital expands the search regionally, then ultimately to a national registry that helped create a 30-pair swap the hospital took part in eight years ago.

When my donor coordinator informed me that I was healthy enough to have a vital organ removed from my body and go on with my life as normal, that was enough to absorb. Wrapping my head around the idea that a stranger, not my brother, would receive the lifesaving gift I’d vowed never to share was more complicated.

Yogita the morning of the operation. Photo: Dipa Patel

There is really no way, in the U.S. at least, to put a price on a kidney. Buying and selling human organs is illegal here, and in most countries around the world.

But if my kidney were a million dollars I could spare, giving it to my ailing sibling with zero expectations would make complete sense. Now that the pool of candidates had dwindled to just me, it felt more like paying someone I’d never met a million dollars to save my brother.

Keeping my brother alive was my incentive to donate. But critical to the swap system are people who want to give a kidney just because. As they aren’t linked to a particular recipient, they open up many more possibilities to create matches and longer chains of transplants.

Those people often come forward because of social media. Two donors made their way into our eight-person exchange by way of Facebook.

Chris Faucett, the 27-year-old whose kidney eventually would go to my brother, was drawn in after his girlfriend saw a plea put up by a friend for a 34-year-old anonymous patient. Mr. Faucett was set on donating after not being able to give his kidney to his sick great aunt.

The anonymous patient had a story of sudden diagnosis startlingly similar to my brother’s and was initially reluctant to publicize his need. His new kidney eventually came from a 33-year-old woman who was encouraged by the hospital to join the exchange after an attempt to donate to a Facebook group contact fell through.

When both of her fallopian tubes were damaged because of ectopic pregnancies, a preteen wish to donate a kidney took on an even greater meaning. “I guess I felt like if I can’t have children, I want to be someone who can still give life,” the woman said recently. Her entry made a transplant possible for 61-year-old Ronald Garrett, who would have otherwise faced up to a seven-year wait. (Some of the participants declined to be identified.)

Also important to the efficiency of a paired system are those who could give directly to a loved one but opt instead to join the exchange to help lengthen the chain. This is sometimes referred to as maximizing a donation.

Dr. A. Osama Gaber, director of the J.C. Walter Jr. Transplant Center at Houston Methodist, says part of his role includes convincing donors to give their kidney to someone they’ve never met instead of a loved one. It helps to focus on the idea of saving multiple lives. “You have to have people who believe in the common good,” Dr. Gaber said.

For college sweethearts celebrating their 25th wedding anniversary, a swap medically offered the best chance for the wife to avoid dialysis and continue with her normal routine as a mother and teacher. Her husband, who had recently lost his mother to lung cancer, had no question about his next steps.

“It was a given that I would do it,” he told me recently. They entered the exchange.

As months went by, our small network continued to come together behind the scenes. I occupied my mind with demands of a new job. But the wait was often agonizing. The delay left me particularly stressed by the increased risk living donors face later during pregnancy. I don’t have children yet. Though I wanted to help my brother, the idea that I could limit my own choices as a result felt like more than I had bargained for.

As I expressed my growing concerns, my brother quietly acknowledged to other family members some of his own doubts. “I wasn’t sure if it would always be something you’d hold over me,” he recently admitted. I wasn’t sure, either.

Twice, Pesh got word from his coordinator that it was nearly go time in what turned out to be false alarms. As the wait dragged on, he made two trips to the ICU because of complications tied to dialysis. Each episode left him fearing that the prospect of a donation was fading.

Finally in June, we got word that they had found matches for both of us. Our four-pair exchange took place on the same day, Aug. 7, with everyone having surgery at Houston Methodist.

When I awoke from surgery, the first thing I asked was if they were really going to do this, unaware that it had already happened. It felt good to start life with a single kidney not missing the one I had just lost.

Pesh with his kidney donor, Chris Faucett, and Mr. Faucett's father, two days after surgery. Photo: Dipa Patel

Days after, the hospital invited the donors, recipients and their families to a gathering to give participants a chance to meet the strangers with whom they now shared an intimate bond. I didn’t attend and neither did my recipient. When agreeing to the paired donation, I knew that I wouldn’t want to meet the person who ended up with my organ (unless it was someone famous). After making peace with the notion of donating, I just wanted to be done with the process.

Readjusting to life in New York was a challenge, in the first months especially. I’ve suffered severe backaches regularly since the operation. The three-inch slash across my lower abs and a constellation of dimpled scars that now define my left obliques are other permanent reminders.

Doctors say my remaining kidney will grow and pick up some of the slack for its departed mate. For the next two years I’ll take tests every six months to make sure it’s bearing the load as it should. Seeing my brother living a young person’s life again is all I need to reaffirm that changing my mind when the facts changed was absolutely the right decision.

The Mind Behind Kidney Exchanges

The matching of donors and recipients functions very much like a traditional marketplace, and one that has cried out for efficiency for years.

“We have real shortages of transplants. Some of it is fixable. It seems like a place where economists could help,” says Alvin Roth, who shared the 2012 Nobel Prize in economics for his work in matching markets that helped form the idea for kidney exchanges.

Dr. Roth’s pioneering work in applying game theory to market design addressed the kidney conundrum by taking the otherwise healthy donors forced to sit on the sidelines.

That thickened the market and created more opportunities for transplants through exchanges.

For all the benefits of paired donations, the transplant market still needs vast improvement, Dr. Roth and other experts say.

There aren’t enough kidneys to clear the waiting list that has for years hovered around 100,000. Experts believe even more people require a lifesaving kidney.

An economic solution could be the best hope, Dr. Roth says.

Because of tight restrictions over paying donors, he and others think lessening or removing financial disincentives—travel costs, time off work, postsurgery care—would encourage more people to consider donating.

“Markets need social support,” Dr. Roth says. “Understanding which markets get it and which don’t—it’s too important to leave to the philosophers.”

Write to Yogita Patel at yogita.patel@wsj.com