The Fox Sports 1 baseball studio show is the television equivalent of the Oakland Raiders during their outlaw days under Al Davis: a home for reclamation projects.

Having hired Pete Rose as an occasional analyst during the season, Fox added Alex Rodriguez as a commentator through the American League Championship Series and World Series. Here was a rare chance to understand what it is like to be banned by Major League Baseball for a season (Rodriguez in 2014) and for life (Rose).

All Fox needed was to prop a cardboard cutout of Shoeless Joe Jackson on the studio’s faux field.

How much better than standard postseason chatter a real Rose-Rodriguez conversation would have been — a historic opportunity, in fact. It would supersede anything being said on TBS’ postseason show, which features Pedro Martinez and the nightly “Who’s Your Daddy?” award and Martinez’s “Thoorrrrrr!” screams to honor the New York Mets right-hander Noah Syndergaard. But the one-on-one of exiles was not to be.

Rose and Rodriguez were never even alone on the set. But late in the show, they cheerfully fist-bumped over Rodriguez’s assertion that he has a “Ph.D. in getting booed.” The host, Kevin Burkhardt, never steered the banter to the subject of banishment, choosing to keep it on ALCS business. Rose, loud and raucous, and Rodriguez, smartly dressed and brimming with baseball insight, did not share their opinions about who banned them, Bart Giamatti and Bud Selig, or the subjects of their respective punishment (gambling and illicit drug use).

Instead, they played their roles as if they had always been good citizens of baseball, a television fiction, but at times an entertaining and fast-moving one. In the hourlong show before Game 3 of the ALCS between the Kansas City Royals and Toronto Blue Jays, Rodriguez handled each question well, and was probably measuring a seat for his post-Yankee employment. He has an affinity for the camera, knows which way to turn as he starts his answers and did not stumble over his words.

You were trying to seduce us, Rodriguez!

Discussing Blue Jays starter Marcus Stroman, Rodriguez said: “When I stood in the box and he released the baseball, at about the 59-foot mark his ball sprayed like a spray can. The ball went everywhere. Very unpredictable.” He said Rogers Centre, the Blue Jays’ stadium, was so loud that he and Yankees third baseman Chase Headley could not hear each other talk when they were almost face to face.

Rose is a totally different television creature: all rough edges, a kibitzer, an interrupter, a needler bouncing on his cowboy boots — and someone who loves to razz Frank Thomas, whose comfort zone rests in uttering banalities that sound overly practiced. In one moment worth cringing about, Thomas turned to Rose and declared: “I’m fired up. That’s the Hit King, baby, that is the Hit King! How many, how many hits you got?”

Nothing pleases Rose more than to be reminded of his career hits record. Happily, he said, “I got more than anybody.”

Thomas, the Hall of Famer, and Rose, whose on-field achievements merit election, also engaged in a childish argument over whether the Blue Jays can turn their home run switches on and off. Thomas said they could; Rose disagreed, sometimes with what looked like contempt. Eventually, their spat was defused by none other than the smooth-talking Rodriguez, who, after being asked by Rose if he agreed with Thomas, said: “I haven’t turned the switch on yet. But I’ll tell you this. Toronto can annihilate good pitches.”

The crowded studio cast also included Max Scherzer, who pitched two no-hitters this season for the Washington Nationals, and Raul Ibanez, who finished his 19-season career last year with the Kansas City Royals. Each is likable and smart. They will be in studios until they’re old men.

Then, when it was over, it was back to playoff baseball — without any talk by the former and current exiles of anyone’s personal Elbas.