Two of the world’s richest and most influential men, who happen to run rival social media networks, appeared to goad each other into agreeing to a fight, setting their platforms abuzz at the notion of a “cage match” between the billionaires.

Elon Musk, who frequently posts jokes and provocations on Twitter, which he bought for $44 billion last year, responded this week to tweets about a potential competing service by Meta, run by Mark Zuckerberg, by saying, “I’m up for a cage match if he is.”

On Instagram, the social network owned by Meta, Mr. Zuckerberg responded with a screenshot of Mr. Musk’s tweet and the caption, “Send Me Location.”

It went on from there, in a prominent demonstration of the defiant, combative tone of “bro” culture in Silicon Valley, though sparring on social media has rarely, if ever, spilled into the physical world.

A Meta spokeswoman said Mr. Zuckerberg’s response “speaks for itself.” Twitter’s media office auto-responded to a request for comment with a poop emoji.

Last summer, Mr. Zuckerberg said he had taken up Brazilian jujitsu. He has since posted several times about his training, including images with professional fighters like Dave Camarillo and Mikey Musumeci. Last month, Mr. Zuckerberg competed in his first Brazilian jujitsu event, in Woodside, Calif., where he defeated an Uber engineer and won two medals.

Mr. Musk said, “I almost never work out.” He proposed that the fight be held at an arena in Las Vegas.

More than in other sectors, tech leaders have been known to embrace conflict in public forums, said Katy Cook, author of “The Psychology of Silicon Valley,” who described the industry as a “male-dominated, emotionally primitive” environment where leaders are rewarded for displays of hypermasculinity.

“When you get a taste of power via some of those ways of behaving, it tastes good for a lot of people,” Dr. Cook said.

The two moguls have a history of butting heads — rhetorically, not physically — over big topics in their industry. Mr. Zuckerberg invited Mr. Musk to a dinner at his home in 2014 to try to convince him that he was too alarmist about the threats of artificial intelligence. Mr. Musk did not change his mind, and has continued to warn of the dangers of A.I. Mr. Musk has said that Instagram “makes people depressed” and that Facebook, also owned by Meta, “gives me the willies.”

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