Why must a preference relation for an agent be over a set of alternatives for which they can choose from instead of any set? [migrated]

A preference relation is of course a reflexive, total and transitive binary relation on a set, with the additional requirement that the agent be able to select one alternative from the set. I don't understand why this last requirement is needed?

In particular, I once heard from someone I consider much smarter than myself that for a player i in a normal form game, it doesn't really make sense to compare u_i(s) and u_i(t) for arbitrary strategy profiles s, t because player i can't necessarily deviate from s to t and vice versa.

But why wouldn't it make sense for players to assign preferences to the possible outcomes of a game, even though they don't directly get to choose amongst the outcomes, only contribute to it?

Furthermore:

1. If u_i(s) and u_i(t) are not in general comparable, how does it make sense to take linear combinations of these for expected utilities?

2. Notions of equivalence then lose preservation of players preferences over Nash equilibria, which is relevant when considering equilibrium selection.

-

migrated to math.stackexchange.com by Turukawa♦May 4 '12 at 4:36

This question belongs on our site for people studying math at any level and professionals in related fields.

Nick, could you elaborate on question 2? –  Peteris Jan 19 '12 at 22:27