Nothing really specific.. just some kind of of relatively tidy insight in classical terms. Game-trees, clearly articulated dominant strategies, surreal numbers.

You could model most games with anything simple that's convenient (trees, state machines, term-rewriting systems). Meta-games, dynamic protocols, and multi-agent systems are broadly related but also different animals where you might need sims, full-blown process calculi, weird new kinds of logic. Depending on where you land a natural model candidate might be messy, maybe you have to give up things like completeness or decidability. Maybe the closest fit for a formalism here is dynamic deontic logic: https://www.cse.chalmers.se/~gersch/jlap2012.pdf