Under N.F.L. Settlement, Making Pie Without Knowing Total Slices As intriguing football matchups go, Sunday's Super Bowl has nothing on one looming down the turnpike in federal court in Philadelphia - with Judge Anita B. Brody the ultimate referee. Brody, considering the N.F.L."s recent settlement with 4,500 retirees over work-related brain injuries, has asked both sides to demonstrate that their $765 million bargain will fulfill its promise to compensate every currently retired player who has or will develop a neurological condition such as dementia or Parkinson's disease. Lawyers for the plaintiffs and the N.F.L. said independent actuaries and medical experts had endorsed the terms of the settlement. But the lawyers refuse to share any of their data with the public to help substantiate how they arrived at the $765 million figure, and there is growing displeasure among plaintiffs who have not been allowed to see the data, either. Numbers can speak for themselves, though, and they bring a clear warning: The $765 million could run out faster than either side apparently believes. When one forecasts how many of the roughly 13,500 currently retired players may develop these conditions over the next 65 years, compensating them as the settlement directs could very well require close to $1 billion, and perhaps more. No one can divine how many players will develop these conditions. But the best data available comes straight from the N.F.L., and it becomes instructive after some basic guidelines. The settlement essentially promises that when a player receives a diagnosis of an eligible condition (primarily dementia), he will get a lump-sum payment that varies with his age and career length. For example, a 12-year player with Alzheimer's disease at 62 receives $950,000, while a one-year player with Parkinson's at 57 receives $260,000. No player has to prove that football caused his condition, a futile task in individual cases; those who get ill get paid. More housecleaning: Expenses and additional wrinkles leave the total payment pool capped at $712.5 million. Families of players who died with a condition after 2006 can receive retroactive payments immediately. And contrary to urban myth, N.F.L. players generally live as long as other American men. Those typical American men set a baseline for future settlement payments. The best research on nationwide dementia prevalence - rates that some experts consider conservatively low - estimates that, for example, 1.3 percent of American men aged 60 to 64 have the condition. For those aged 70 to 74, the rate is 6.8 percent, and so on. Those rates would predict that 220 living N.F.L. retirees have dementia now, and about 1,500 will in the future. Total payments, adjusting for career length: $262 million. Given the national rates for Parkinson's and amyotrophic lateral sclerosis (often known as Lou Gehrig's disease), which call for higher payments than dementia, the expected total over 65 years rises to $350 million - just among regular American men. So what happens if and when players develop conditions more often than the general population, which the settlement is designed to address? The exact numbers are unknown. But the best available data on dementia among former players suggests payments could soar. ¶ A 2009 University of Michigan survey commissioned by the league found that 1.9 percent of N.F.L. retirees aged 30 to 49 reported a dementia-type diagnosis, 19 times the national average. The rate was 6.1 percent for retirees 50 and above, five times higher than normal. ¶ A 2013 government study found that N.F.L. players were four times more likely than other American men to die with Lou Gehrig's disease or Alzheimer's, the most common form of dementia. N.F.L. players showed no higher risk for Parkinson's. ¶ The league's 88 Plan, which reimburses medical costs for retirees with dementia, Parkinson's or Lou Gehrig's disease, currently has 165 living enrollees, according to the league. That represents the minimum estimate of the true total living with these conditions: Older retirees who played fewer than four seasons are generally not eligible, and many ill players do not enroll for various reasons. Not all considerations in the forecasting model can be detailed here. But if one makes all retired players eligible, follows their expected aging patterns across the next 65 years and then applies disease rates that mirror current and historical 88 Plan data, total settlement payments project between $400 million and $600 million. This would be the floor. Players have much more incentive to seek and report early-onset diagnoses when the reward is not paperwork-heavy expense reimbursements, but instant six- or even seven-figure checks. This near certainty could easily add at least $200 million in payments, increasing the total to $600 million to $800 million - which, not coincidentally, begins to approach what the more crude Michigan and government studies foresee. Lawyers for both the players and the league declined an offer to scrutinize these projections. No model is perfect, including this one. The method was reviewed by Dr. Murali Doraiswamy, the director of the neurocognitive disorders program at Duke University Medical Center, and Richard Davis, chairman of the statistics department at Columbia University. "Aside from taking into account uncertainties associated with the disease rates," Davis said, "this modeling approach gives a reasonable estimate of the expected payments." The $600 million to $800 million figure is just for players currently alive. The roughly 50 players who died with chronic traumatic encephalopathy - a degenerative disease caused by repeated brain trauma - call for payments totaling about $40 million. The hundred or more families of players who died since 2006 with only dementia could come forward for another $30 million, depending on the players" age at diagnosis. The lesson here is that heeding what we have already seen with N.F.L. players leads to payments totaling about $700 million to $900 million. If the rate of early-onset disease rises among the heavier, faster players from the 1990s and 2000s - no one knows yet if it will - the numbers could rise well past $1 billion. The final total might come in low at $600 million. But it could be close to twice that. We do not know, and cannot. But making the fund $712.5 million and not a penny more, as proposed, runs a reasonable risk of leaving a thousand future players uncovered. These estimates might even be conservative. Better medical care is expected to leave all Americans living into the advanced ages when dementia more often develops. The estimates ignore all retired players under 30, as well as the roughly 1,000 replacement players from the 1987 strike season who did not play long but remain eligible for relatively low payments. And the model credits a player with a season of service only if he appeared in a game; the settlement rules are less strict and could lead to some higher payments. Of course, some assumptions could be off in the other direction. Some - perhaps many - players will become ineligible for payments either by formally opting out of the settlement or by not registering during the strict 180-day window. The reserve in the settlement fund will appreciate through investments that should outpace inflation. And although no cure for dementia is on the immediate horizon, 65 years is a long time for science to come up with a treatment that makes most payments unnecessary. But with the promise of payments to an unknown number of players, any hard cap must derive from what is already known about the aging and disease patterns of these men and then allow for reasonable worst-case scenarios. Some deviations could be wonderful news. Others could be disastrous. If the N.F.L. demands the certainty of a cap, it can set a higher one. If the plaintiffs demand the certainty of guaranteed player awards, they can set lower ones. Or both sides could concede to the unknown. Let the program run for five years so that information far better than either side has now could be collected and considered.