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Sure, I can only go off what I've read. Would you like to recommend a textbook which goes into making predictions when playing a game with irrational players?
Any mathematical textbook will do it. Try Springer.
It’s quite simple really. Games in game theory are represented with a payoff matrix which shows the utility for each player. Pure strategies are defined as rows or columns in the payoff matrix. The math of game theory doesn’t care about why a player chooses a particular strategy, only its payoff.
I would define a purely rational player at minimum as one chooses a dominant strategy, when one is available. You’re free to expand that to mixed strategies and games where (strong or weakly) dominant strategies do not exist. Irrational players would be anyone who is otherwise not a rational player.
This isn’t very interesting in basic game theory. It becomes a lot more relevant in cooperative game theory, which can have more than 2 players and players forming coalitions.