If I were submitting a modal agent, I’d probably use a PrudentBot that uses PA to look for mutual cooperation but PA+N (for some large N) to look for defection against DefectBot.
Or breakout the stronger systems, e.g., ZFC possibly with some inaccessible cardinals thrown in for good measure.
Or breakout the stronger systems, e.g., ZFC possibly with some inaccessible cardinals thrown in for good measure.
That would be even better in practice, but it wouldn’t be expressible in the modal formalism.