Say either two fans are catching the ball simultaneously, or one fan catches it but another punches them enough times to loosen their grip or snatches it out of their grip, etc.
Do ballpark officials just watch the melee and then certify the ball to whomever eventually wins the brawl? Is it almost like a football reception, where you have to have possessed the ball in your hands for a few seconds for it to count?
I think except in very rare cases, like a player’s 300th career homer or something, MLB doesn’t officially “certify” the ball, whoever comes up with it just pockets it and the ballpark staff play no role (except to the extent that they would intervene in any other case of fights breaking out in the stands).
How they handle it on those rare occasions, though, I’m not sure. I imagine they might ask the people sitting nearby.
If you get violent you could end up in stadium jail, or real jail if bad enough. As for the ball, they don’t care but tradition is you give it to the smaller child.
I was going to say, stadiums have security to stop that stuff. You don’t “let a fight play out” because I imagine the park would then be liable for such a thing. If someone beat you up while security stood there and watched, you probably have grounds for a lawsuit.
A lot of stadiums will certify foul balls and home run balls for you if you ask the usher.
In the early 70s, a friend got/retrieved two home run balls in the same game. Jarry Park bleachers - home of the Montreal Expos at the time. Lousy weather, and not many fans in the bleachers that day. I don’t think “certification” was on his mind at the time, and I don’t think any superstars hit the homers, and there was no indication that the batters wanted the balls back.
They have authenticators at every game now.
Sometimes the courts have to get involved:
At many stadiums you can buy actual game played balls in the team’s gift shop. They are serialized so you can see what happened to the ball while in play. A cousin bought 2 (at $9 each) at a Seattle Mariners game. One lasted 3 pitches while Eugenio Suarez was at bat. The second lasted 6 pitches, one to Julio Rodriguez, then 5 to Ty France. Don’t know if they actually track each ball used in a game or if they just come up with something that sounds good.
The authenticators are only there to look for potentially historic moments like someone’s first hit, or a triple play, or whatnot. For really big moments, like someone approaching 500 home runs, the game balls are all marked with serial numbers in invisible ink so the ball can be accurately identified. But ordinarily, MLB won’t officially authenticate fouls or home runs into the stands, although per Munch apparently some teams will “certify” them.
If they aren’t marked then they can’t tell the difference between a foul ball into the stands in the 5th inning and a notable but not greatly significant home run in the 6th. So the value of ‘certification’ is going to be very limited without balls marked in anticipation of a major record being set.
I dunno. I regularly get emails from
The Royals’ team store with an excel sheet of balls for sale, and they’re all of the “Gavin Sheets foul ball off Zach Greinke” variety. If it’s a hit, it’ll say which career number hit it was, if it was a strikeout ball, which career strikeout, etc.
Were those balls that ended up in the stands? I’d be kind of suspicious, except a team record of the ball makes it worth more than the ball from a random guy who says he caught a homer. Either one might not be right but the team’s claim will be more believable.
I’m under the impression they’re balls collected by staff - either from the fountains, batter’s eye, etc.
Screw that “tradition” I’ve been going to baseball games for nearly 60 years and damned if I’m giving up my first ever game ball (if I ever get one) to some kid.
Consider more figurative interpretations of ‘smaller child’.
If there is a dispute over a ball in the stands park officials will hold the ball until the end of the game. After the game, the disputants go down to the infield and do battle for the spoils of the game. A person may name a champion to fight for them. It’s a long honored tradition.
I know this is a joke, but I would actually stay after the game to watch this. They’d have to reopen the alcohol carts though. 
This will get the aggressor ejected and arrested, and he will not be allowed to keep his ill-gotten gains. MLB is marketed as a family friendly activity, so MLB takes a dim view of any behavior that might cause parents to think twice about bringing their kids to a game. In fact, while watching a game last year on TV, I saw an incident that fits the OP’s description. One of the security guards simply handed the ball to a kid. The fans cheered, and doing something like that is good PR.