(Nam Y. Huh/Associated Press)

In a letter to the players, Goodell cited the players’ refusal to participate in the appeals process or provide evidence to persuade him to reduce the punishments, according to the league’s announcement.

“Throughout this entire process, including your appeals, and despite repeated invitations and encouragement to do so, none of you has offered any evidence that would warrant reconsideration of your suspensions. Instead, you elected not to participate meaningfully in the appeal process,” Goodell told the players in his letter.

Saints linebacker Jonathan Vilma was suspended for the entire 2012 season for his role in the bounty scheme, which, the league concluded, paid players cash bonuses for hits that injured opponents.

Former Saints defensive lineman Anthony Hargrove was suspended for eight games. Defensive lineman Will Smith was suspended for four games and former Saints linebacker Scott Fujita was suspended for three games.

The players and the NFL Players Association have maintained that the league has provided no evidence that the players participated in a pay-to-injure scheme. Vilma has filed a defamation lawsuit against Goodell.

Also suspended for an entire year was Saints’ head coach Sean Payton. Gregg Williams, the New Orleans defensive coordinator at the time, has been suspended indefinitely. Other Saints officials received suspensions of various lengths.

The NFL said it has looked into whether the Redskins had a similar program under Williams when he led the Washington defense but has not found evidence it existed. Redskins players and former assistant coaches have said the Redskins did employ such a system.

The union tried unsuccessfully to challenge Goodell’s authority to impose the suspensions via a pair of arbitration cases. It issued a statement Tuesday that said “the NFLPA has never and will never condone dangerous or reckless conduct in football and to date, nothing the league has provided proves these players were participants in a pay-to-injure program. We will continue to pursue all options.”

Some people in pro football have said they expect the players to take additional legal steps to reduce or overturn the suspensions. But any legal challenges by the players could face significant obstacles, experts have said, unless the players can prove that Goodell and the league did not adhere to the terms of collective bargaining agreement.

According to the league’s announcement, Goodell left open the possibility of adjusting the players’ punishments if new information surfaces.