Is there any well-studied representation of a N player game with 2 strategies per player as a matrix?
Intuitively, I think that each strategy can be represented as a binary digit, and each strategy profile can be represented as a binary number with the LSB representing the strategy employed by the first player (player 0).
For example, a profile with player 2 playing strategy 1, player 1 playing 0, and player 0 playing 1 would be $101$.
The payoff function would then be a function mapping $\{x \: | \: 0 \leq x \leq 2^N - 1\}$ to $\mathbb{R}^N$, where $N$ is the number of players.
This can be encoded into a matrix, where each column is the payoff for playing the strategy provided by the column number in binary.
The example of the prisoner's dilemma is easy to illustrate:
$ \; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;A\\ B \begin{bmatrix} (3, 3) & (10, 1) \\ (1, 10) & (2, 2) \end{bmatrix} $
where each index is $(x, y)$, where $x$ describes the payoff to A, and $y$ describes the payoff to B.
The matrix to describe this would look like:
$ P = \begin{bmatrix} 3 & 10 & 1 & 2 \\ 3 & 1 & 10 & 2 \end{bmatrix} $
Is this described in literature? If not, is there an alternative? And what are the spectral properties of such a matrix?
