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Court statistics and conventional logic suggest that Tom Brady has only a slim chance of obtaining another hearing before the U.S. Court of Appeals for the Second Circuit or a hearing before the U.S. Supreme Court. The odds for a typical litigant in Brady’s shoes would likely be about 1%.

Brady is anything but the typical litigant, though, and his case has proven especially ripe for debate. Two federal judges (Chief Judge Robert Katzmann of the Second Circuit and U.S. District Judge Richard Berman) have ruled in Brady’s favor while Judges Barrington D. Parker, Jr. and Denny Chin of the Second Circuit have ruled for the NFL. Brady’s case presents contentious applications of law to facts, which are the very kinds of cases that fare better in obtaining new hearings. The fact that some of the country’s most prominent attorneys are involved in the case only amplifies its significance.

Brady’s case becomes more relevant to judges if it is regarded as emblematic of larger trends in law. His odds for a new hearing increase if the judges conclude that the four-game suspension to start the 2016 season is less about a bizarre equipment controversy and more about whether a union member has been treated unlawfully by management in a disciplinary matter.

On Tuesday, the American Federation of Labor and Congress of Industrial Organizations (AFL-CIO) and Kenneth Feinberg filed separate amicus curiae briefs. The AFL-CIO and Feinberg briefs urged the 13 active judges on the Second Circuit to rehear NFL Management Council et al. v. NFL Players Association et al. (aka Brady v. NFL). Both briefs contend that the ruling by Judges Parker and Chin undermines basic principles of law and should be reversed so as to avoid impact on future cases.

The limitations of amicus briefs

Before considering the arguments by the AFL-CIO and Feinberg, it is worth acknowledging that amicus briefs are of limited impact. Amicus briefs are filings by non-parties or “friends of the court” who purport to possess an interest in an appeal. Unlike filings by the actual parties in a case, judges have the discretion to completely disregard amicus briefs. For that reason, the amicus briefs filed on Tuesday—just like the amicus briefs filed last Wednesday by the New England Patriots and a group of scientists—should be viewed with appropriate caution. 13 active judges on the Second Circuit will have an opportunity to review all of the amicus briefs, but these judges are under no obligation to factor them into their decision-making.

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Still, amicus briefs filed on behalf of Brady can only help the quarterback. As discussed last Wednesday, the Patriots filing an amicus brief that expressed an interest adverse to the NFL was stunning and likely will be considered by the judges. The brief signed by the group of neutral scientists who contend Deflategate is completely explainable through Ideal Gas Law should also catch the judges’ attention, especially those who believe Brady has been treated unfairly. Similarly, the AFL-CIO and Feinberg briefs will supply more motivation for judges when it comes to granting Brady a new hearing.

Why the AFL-CIO weighing in matters

The AFL-CIO is the nation’s largest federation of labor unions. It represents 12.2 million workers—including teachers, plumbers, pilots and, through the NFLPA, professional football players—across 57 national and international labor organizations. As the AFL-CIO stresses in its amicus brief, the AFL-CIO has a “significant interest” in how unionized employees are treated by management in disciplinary matters. More specifically to Brady’s case, the AFL-CIO demands that the “proper standard for judicial review of decisions” be applied when those decisions are “rendered pursuant to arbitration procedures.” With Brady, his “arbitration” was in the unusual, but collectively bargained form of NFL commissioner Roger Goodell serving as the arbitrator.

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The AFL-CIO insists that Goodell, while serving as the arbitrator (rather than as NFL commissioner) disregarded “basic procedural fairness” and “acted arbitrarily” in upholding Brady’s four-game suspension. As has been written on numerous occasions, Goodell made several moves in his review of Brady’s matter that raise serious questions about Goodell’s capabilities as an arbitrator. For example, Goodell appeared to inexplicably change the rationale for Brady’s punishment. The Wells Report found that Brady was merely “generally aware” that other Patriots employees may have altered footballs. But Goodell later portrayed Brady as an active culprit who approved, consented and induced those employees to alter footballs. The AFL-CIO contends that Goodell’s actions as the arbitrator run afoul of case precedent dictating that arbitrators should not devise new theories of fault.

Normally the arbitrator in a labor-management dispute is both neutral and new to the dispute; Goodell was neither. The AFL-CIO’s arguments for Brady are not new, but the willingness of the AFL-CIO to raise them is notable. By depicting the Brady case as one related to the core principles of labor management relations, the AFL-CIO seeks to help Brady overcome the concern that his case is too exceptional to warrant a rehearing. If the judges view the Brady/NFL dispute as a self-contained controversy within the unique and flawed universe known as Article 46 of the NFL’s collective bargaining agreement, then the Brady case is less likely to be considered relevant to conventional labor management disputes and the judges will be less willing to rehear the case.

Why Kenneth Feinberg weighing in matters

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Feinberg may be the most influential voice yet to speak up on behalf of Brady. Feinberg is a legendary figure in legal circles, particularly among those connected to “fund distribution,” which refers to allocation of settlements to victims of large-scale harms. Feinberg has been entrusted with determining formulas that most fairly compensate victims of some of the worst incidents in recent history. He undertook this duty in assessing claims by victims of the September 11th terrorist attacks, the BP Deepwater Horizon disaster and the Boston Marathon bombings. In this capacity Feinberg acts as a neutral arbitrator. In 2014, I had the opportunity to interview Feinberg regarding his role in the Ed O’Bannon v. NCAA case and specifically his plans to form the Former College Athletes Association, a group that would negotiate name, image and likeness rights on behalf of former college athletes.

Feinberg has no personal or reputational stake in whether Brady wins or loses, nor does he have a connection to the NFL or the NFLPA. In truth, you might expect someone of Feinberg’s stature to view getting involved in Deflategate as a net negative: no one on either side of this controversy has looked particularly good, and Feinberg has an impeccable reputation to protect. But as Feinberg explains in his amicus brief, he believes that the enforcement of Brady’s suspension would trigger consequences well beyond sports. He contends it would destabilize the system of arbitration in the Unites States and that it would make it harder for Feinberg and others like to him to carry out their duties.

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Feinberg asserts that the public’s “trust and confidence in arbitration” would be “fundamentally eroded” if Goodell can impose his “own brand of industrial justice” and “act with bias.” In one of the harshest characterizations of Goodell to date, Feinberg describes Goodell as acting “simply beyond the bounds” of permissible conduct by arbitrators. Goodell’s decision, Feinberg contends, has “no credibility,” reflects deep “bias,” and imposed suspensions for equipment violations when in the past such violations only triggered fines or mere warnings. Feinberg cites other criticisms of Goodell, including that the commissioner was so “one-sided” that “it cannot be the result of good faith mistake.” These and other critiques, Feinberg contends, threaten the public’s belief in arbitration as a viable form of dispute resolution. This is a key argument because it communicates to the judges that Brady’s case goes well beyond the limited world of professional football and into the very prevalent world of arbitration in America.

Michael McCann is a legal analyst and writer for Sports Illustrated. He is also a Massachusetts attorney and the founding director of the Sports and Entertainment Law Institute at the University of New Hampshire School of Law. McCann also created and teaches the Deflategate undergraduate course at UNH. He serves on the Board of Advisors to the Harvard Law School Systemic Justice Project and is the distinguished visiting Hall of Fame Professor of Law at Mississippi College School of Law. He is also on the faculty of the Oregon Law Summer Sports Institute.