Gregg Williams with Jonathan Vilma (left) and Roman Harper(Chris Graythen/Getty Images)

Football is an extremely physical sport by nature, filled with violent collisions between players who are big, strong and fast. Defenders are conditioned to hit often and hit hard, in hopes of separating opponents from the ball and maybe their senses, too.

All of that is well within the rules and the spirit of the game. But rewarding players with bonus money for certain hits encourages dirty, reckless play, which is why the NFL forbids the practice. Now that the league has uncovered a New Orleans Saints bounty system, it should issue harsh penalties to make an example of the team.

The Saints are accused of paying players $1,500 for "knockouts" and $1,000 for "cart-offs," with payouts doubling or tripling during the playoffs. The team is also accused of targeting players such as former quarterbacks Brett Favre and Kurt Warner.

"The payments here are particularly troubling because they involved not just payments for 'performance' but also for injuring opposing players," Commissioner Roger Goodell said in a statement released Friday. "The bounty rule promotes two key elements of NFL football: player safety and competitive integrity.

"It is our responsibility to protect player safety and the integrity of our game, and this type of conduct will not be tolerated," he said. "We have made significant progress in changing the culture with respect to player safety and we are not going to relent. We have more work to do and we will do it."

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The first order of business is dealing with Gregg Williams, who served as the Saints' defensive coordinator from 2009 to 2011 โ€” the period in question โ€” before leaving in January for the same job with the St. Louis Rams. He took "full responsibility" in a statement he released Friday. "It was a terrible mistake, and we knew it was wrong while we were doing it," he said. "Instead of getting caught up in it, I should have stopped it."

Considering the NFL's recent moves to make the game safer and less violent, Goodell should ban Williams from coaching next season โ€” if he's ever allowed to coach again.

New Orleans head coach Sean Payton, who the league said didn't participate in the program but didn't stop it, either, should face a hefty fine and suspension. And the franchise should forfeit some draft picks, much as the New England Patriots did when they were found guilty of spying in 2007.

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Faced with a slew of lawsuits from former players with possible brain injuries, this is the last thing the NFL needs. The league should make Williams, Payton and the Saints pay in a big way.

Deron Snyder's Loose Ball column appears regularly on The Root. Follow him on Twitter and reach him at BlackDoor Ventures, Inc.