Friday, October 16, 2015

rational agents in a dynamic game

I've mentioned this before, but I'll repeat from the beginning: Consider a game in which I pick North or South, you pick Up or Down, and then I pick East or West, with each of us seeing the other's actions before choosing our own.  If I pick South, I get 30 and you get 0, regardless of our other actions.  If I pick North and you pick Down, you get 40 and I get 10 regardless of our other actions.  If we play North and Up, then I get 50 and you get 20 if I play West, and I get 30 and you get 60 if I play East.
I playYou playI getYou get
North / EastUp3060
North / WestUp5020
We each have access to all of this information before playing, so you can see what will happen; if I play North and you play Up, I will play West, which gives you 20, while if I play North and you play Down, you get 40.  We therefore know that you will play Down, so I get 10 if I play North, and I get 30 if I play South, so we can work out that I will play South.

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.

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