  There have been two major lines of research aimed at capturing
resource-bounded players in game theory. The first, initiated by Rubinstein,
charges an agent for doing costly computation; the second, initiated by Neyman,
does not charge for computation, but limits the computation that agents can do,
typically by modeling agents as finite automata. We review recent work on
applying both approaches in the context of decision theory. For the first
approach, we take the objects of choice in a decision problem to be Turing
machines, and charge players for the ``complexity'' of the Turing machine
chosen (e.g., its running time). This approach can be used to explain
well-known phenomena like first-impression-matters biases (i.e., people tend to
put more weight on evidence they hear early on) and belief polarization (two
people with different prior beliefs, hearing the same evidence, can end up with
diametrically opposed conclusions) as the outcomes of quite rational decisions.
For the second approach, we model people as finite automata, and provide a
simple algorithm that, on a problem that captures a number of settings of
interest, provably performs optimally as the number of states in the automaton
increases.
