3rd parties are already so screwed, I don't even know where to start fixing the problems. For example, last I checked in Illinois third-party candidates needed _way_ more petition signatures to even get on a ballot than Republican or Democratic ones (though the legislation is careful to not call out the Republican or Democratic parties by name; it's written in terms of vote share in the previous election). That sort of thing is all over the place.

As a mathematical exercise, trying to extend the game to more than 2 parties is interesting, for sure. Real-world relevance is unclear, unfortunately.