simple thought experiment: You are carrying a gun. Someone else decides they want to do something dangerous with a gun. (shoot some people; commit a gun-crime, etc.). They know they are about to become a target because everyone else is usually also self-preserving. They decide to shoot anyone with the means to slow them down. That primarily includes everyone else with a gun; anyone else strong enough to overpower them, and anyone able to alert authorities on them.

Who do they shoot first? anyone else with a gun. Likely a not safe position to carry a gun

First big problem is obviously that things are only proven given some starting premises, and in this case those premises are highly questionable. Carrying a gun has plenty of costs that might outweigh the benefits.

Obviously it costs money, and peoples' reactions to you may be a cost, but I think the most interesting, and possibly biggest, cost may be the mortal one. Gun accidents are rare but they happen, especially if you're going to be carrying your gun around loaded, so in order to check whether it's worth it to carry a gun, one of the things you might want to estimate is the risk of accidents. Even more interesting to me is the risk that if I become temporarily suicidal, having a gun might increase my probability of suicide, and right now I don't want my future self to commit suicide (unless terminally ill etc.).

I only read a synopsis of their book, but it's massively incorrect to take their statements as "game theory says" anything about carrying a gun in the real world. In their incredibly wrong payoff model, gun ownership does dominate. But that payoff model is simply is simply insane.