Yes, some would answer, since the cognitive and physical processes that take place in my mind/body/whatever while I make decisions or choose actions can be simulated by a computer program.
Even if we accept this assertion as true, I find the argument unsatisfactory. For me, the subject matter of game theory and decision theory is the idea of rationality, not the lousy shadow of rationality that evolved in my neighbourhood. When I come across all sort of dumb things people do, my response is `So what ?’ Why should we redo our theory of rationality to accommodate the petty goings-on of a bunch of talking monkeys on a mostly harmless planet in the middle of nowhere ?
So, the fact that human beings happen to be walking computers doesn’t mean players are also computers.
And yet, even if like me you don’t think of your players as models for human beings, I still believe you may want to consider computability of their strategies. My reason is that game theory traditionally studies not only rational behavior but also the more vague notion of rational reasoning:
Imagine each player instead of making each decision as the necessity for it arises, makes up his mind in advance for all possible contingencies; i.e. that the player begins to play with a complete plan which specifies what choices he will make in every possible situation …. We call such a plan a strategy. (von-Neumann and Morgenstern, Section 11.1)
Back to Rabin’s game that I talked about yesterday. Even if we accept the existence of the function that appears in the second item as a mathematical entity, such a function presents a `plan’ that cannot, even in principle, be executed, described, or reasoned about.
So I think a case can be made that by restricting our attention to computable functions we don’t arbitrarily restrict the player’s strategy set. On the contrary, the standard game theoretic formulation, by allowing non-computable functions, expands it to mathematical creatures that actually don’t capture our ideal concept of strategy