I am doing a bit of research into games, and want to know if something like this is possible:
Imagine I have a game with N players, each with either the moves $\{a\}$, $\{b\}$, $\{a, b\}$,
This can be encoded into binary, where player 1 is the least significant bit, and player N is the most significant bit.
The payoff function then be:
$P: \mathbb{R} \rightarrow \mathbb{R}^N$
I was wondering if there is a function, like the payoff function, that is
$U: \mathbb{R} \rightarrow \mathbb{R}$
The purpose of this function would be to output $0$ if the input is a Nash equilibrium, and anything else if the input is not a Nash equilibrium.
My thought experiment is this: is there a way for this function to be only self-referencing? Meaning it does not reference the initial payoff function.
If yes, what would the function's equation be? Would it be differentiable? Would it be unique for every different game?