As you’re watching the Raiders and Cowboys (or, for late eaters, the Giants and Broncos) butt heads this Thursday, ponder this: NFL teams don’t go for it enough. That can teach us a lot about more consequential fields, including foreign policy. The attachment to punting, at the expense of winning, shows how conventional wisdom discourages leaders from doing what is best for their teams, firms, and country.

Consider the controversy that’s still raging in Boston: the New England Patriots’ decision to go for the win on fourth down against the Indianapolis Colts on Nov. 15. The Patriots led the Colts by six with about two minutes left. It was fourth and 2. Going for it and failing meant giving the ball back to star quarterback Peyton Manning, in his home stadium, 28 yards from the endzone. Punt and the Colts would have to drive some 60 yards to get the touchdown. Punting, we all know, is the safe thing to do there. Coach Bill Belichick went for it.

Tom Brady’s pass to Kevin Faulk was ruled short of the first-down marker, and the Colts then scored and won. Fans and commentators denounced Belichick’s gamble. Boston Globe columnist Dan Shaughnessy called it an unrivaled gaffe. Former Patriots safety Rodney Harrison said it was the worst decision he had seen Belichick make. The old Colts coach, Tony Dungy, said you go with the percentages and punt.

But the percentages don’t say that. Belichick didn’t gamble, according to analysts who use NFL history to judge how coaching decisions impact the probability of winning. Advanced​NFLStats​.com, for example, found that going for it in that situation gives a team a 79 percent chance of winning. (The quick breakdown: Teams pick up fourth and twos 60 percent of the time. Convert, and you can run out the clock and win. If you don’t make it, the opponent gets a touchdown just over half the time.) Punting, by contrast, gives a team only a 70 percent chance of winning. And those are averages — with the Colts’ elite offense playing against a worn-out Pats’ defense, the case for punting is worse.

Among football-stats hobbyists, that conclusion is uncontroversial. They are puzzled why teams punt so often. An economist, David Romer, even analyzed past NFL fourth downs and found that teams have systematically given away points by kicking. He reasoned that if the NFL’s competitive pressures don’t cause teams to maximize scoring, markets might not cause firms to maximize profit.

Social convention explains this deviation from what seems rational. Five Super Bowl rings give Belichick the job security to play the odds and let critics howl. Less-secure coaches have a different calculus. Critics can ruin their career. To protect their reputation, they might punt even knowing that it lowers the odds of winning. Punting reinforces the convention — if Tony Dungy does it, it must make sense — making it tougher for new Belichicks to emerge.

Examples in business abound. Economic downturn might ruin a bank carrying excessive debt given its equity. The bank’s interest is to reduce debt. But bankers opposing risky investments might look foolish while a boom continues, encouraging them to let sleeping dogs lie. The managers’ interests don’t match the firm’s.

Economists call this the principal-agent problem. It can occur in any situation where someone is making decisions for someone else, including government. Democracy uses elections to tie agents’ interests to their principals’. But rational politicians can’t serve the national interest if the public misperceives it.

The business of fighting terrorism is particularly prone to this problem. As with most national security dangers, information about terrorism comes mostly from public institutions that combat it. Their bureaucratic interest is often to promote worst-case estimates. Furthermore, terrorism’s attributes make people overly fearful of its consequences. Psychology shows that traumatic events like the Sept. 11 attacks sear themselves into our consciousness, causing us to overestimate their likelihood. We also tend to fear mysterious, uncontrollable risks like biological terrorism far more than more mundane and deadly ones, like driving.

Overestimation of terrorism’s danger spreads as people learn from their friends. Some doubt that judgment but echo it anyway, because conformity is polite. Pundits, who fear that heretical views will damage their careers, do the same. This dishonesty strengthens the conventional wisdom.

This process explains why the public overestimates al Qaeda’s menace. In its history, it has killed about one-tenth the number of Americans that die annually from the flu. Even in the Taliban’s Afghanistan, it never came close to acquiring nuclear or biological weapons. Though friendly militias have harbored al Qaeda in western Pakistan since late 2001, the group has not launched another successful attack against U.S. territory. Opinion polls suggest that the jihadist movement that spawned al Qaeda is waning — unsurprisingly, for an unappealing ideology that considers even most Muslims legitimate targets for murder.

The exaggerated fear of al Qaeda creates a sellers’ market for policies advertised as confronting terrorism, like surging troops in Afghanistan. Without threat inflation, the war flunks a simple cost-benefit analysis. Nearly 500 coalition troops have died fighting this year. By some estimates, sending another 40,000 troops to Afghanistan will bring the war’s cost to about $100 billion a year, and that’s not even counting long-term expenses like veteran’s care, and investments foregone at home. Given a realistic threat assessment, it is hard to see how a drawdown in Afghanistan would allow al Qaeda to do enough damage to justify these costs. That’s true even if you grant hawks their questionable assumptions that our troops don’t just inflame the Pashtun separatism that fuels the Taliban and that Afghan Taliban commanders would again harbor al Qaeda.

The Belichick controversy suggests a possible solution. By drawing attention to analysis that challenged conventional wisdom, the debate convinced some fans, including me, that the coach was right. Growing dissent could encourage more coaches to go for it. Likewise, iconoclasm in foreign policy debate, abetted by events, can change public opinion and empower doves. The purpose of political science ought to be to provide its own version of AdvancedNFLStats, directed toward the national interest. Belichick has long provided a blueprint for how to win football games — given help, his rational approach can also provide insight into how protect America.