Many big tournaments, including the world championships, have found a middle ground: They use an official known as an observer or adviser, who acts more like a mediator than a judge or executioner.

The observers are not empowered to make calls on their own, as a referee would. Rather, they can mediate if two teams cannot come to an agreement.

“Maybe two players are going up to catch a disc, and the offensive player feels they were fouled, while the defender feels they kept their proper distance,” said Nob Rauch, the president of the World Flying Disc Federation, which runs the world championships. “The observer makes sure the players advance to a quick conclusion. It’s a way to keep the game on track.”

Tom Crawford, the chief executive of U.S.A. Ultimate, which runs the sport in the United States, said that self-policing could remain a big part of the game.

“One of the things the I.O.C. found intriguing about ultimate is that it kind of embodies the Olympic ideal,” he said. “The athletes treat each other with dignity and respect. It’s a really unique ethos. It is baked into the sport; the I.O.C. saw that right away.”

Opponents of adding full-fledged referees also contend that the players are simply best equipped to make the calls. Athletes take an online test to make sure that they know the rules.