|I play||You play||I get||You get|
|North / East||Up||30||60|
|North / West||Up||50||20|
This, however, leads to a bit of a contradiction. "If I play North and you play Up, then I will play West" entails some behavioral assumptions that, while very compelling, seem to be violated if I play North. If I play North, regardless of what you play, your assumptions about my behavior have been violated; it is, in fact, a bit hard to reason about what will happen if I play North. If I'm just bad at reasoning, you should probably play Down. If you're mistaken about the true payoffs — perhaps the numbers I've listed above are dollars, and my actual preferences place some value on your payoff as well — then it might make sense to play Up, depending on what you think my actual payoffs might be. Perhaps I'm mistaken about your payoffs (in which case you should probably choose Down).
In mechanism design, it is important to distinguish between an "outcome" and a "strategy profile" insofar as leaves on different parts of the decision tree may give the same outcome, but in the approach to game theory that does not separate those, you don't gain much from allowing for irrational behavior; given any sequence of behavior, you can choose payoffs for the resulting outcome that make that behavior rational. The easiest way to handle this problem philosophically, then, is to treat it as being embedded in a game of incomplete information, in which agents are all rational but not quite sure about others' payoffs (or possibly even their own). In the approach to game theory that I've been trying to take lately, though, where we look at probability distributions of actions and types where agents may have direct beliefs about other agents' actions, "rationality" becomes a constraint that is satisfied at certain points in the action/type space and not at others, and it's just as easy to suppose players are "almost rational" as that they are "almost certain" of the payoffs. I wonder whether this would be useful; it might clean up some results in global games, by which I mostly mean results related to Carlsson and van Damme (1993): "Global Games and Equilibrium Selection," Econometrica, 61(5): 989–1018.