Before becoming the Miami Dolphins’ executive vice president for football operations, Mike Tannenbaum joined the organization as a consultant. He traveled to Europe to research how professional soccer teams use science to optimize performance, and now the Dolphins monitor sleep patterns, revitalize players with customized smoothies and employ a director of sports performance.

Tannenbaum assumed his new position in February, but his mission has not changed. It is the same as it was during his seven years as the Jets’ general manager: ascertaining how to overtake the New England Patriots in the A.F.C. East.

In a league defined by parity, every year the Dolphins, the Bills and the Jets confront a predicament unique to them: They must contend with a team that always posts a winning record. Since the Tom Brady-Bill Belichick era began in earnest in 2001, the Patriots are 66-20 within the A.F.C. East, having captured 12 of the last 14 titles — by an average margin of 3.4 games — without ever finishing with a record inferior to that of a division rival.

At some point every year, New England appears vulnerable. And still, every off-season, the rest of the division winds up chasing the Patriots. Coaches are fired, front offices are overhauled and lavish free-agent contracts are dispensed, all to depose the overlord.