Recently when I started reading Nash's paper, I found a little question about the linearity of payoff functions. Is it an assumption? Or did I miss some idea about the payoff function and its extension with my lack of basic knowledge?
Here is the original paper, and the problem is on page 287, line 8.
That line only states that there is a unique extension of the payoffs to profiles of mixed strategies that is linear. This extension is then used in the definition of equilibrium points. So implicitly, Nash is using expected utility as the underlying decision theory.