I'm reading up on Game Theory.
So far, I feel like I have a pretty good understanding on two-player games and their properties.
Consider a two person game where the payoff matrices are $A_{m\times n}$ and $B_{m\times n}$. A pair of strategies $(x, y)$ is a Nash equilibrium if and only if
- $\forall 1\leq i \leq m, x_i > 0 \implies (Ay)_i = max_k(Ay)_k$
- $\forall 1\leq j \leq m + n, y_j > 0 \implies (x^TB)_j = max_k(x^TB)_k$
Where $(x, y)$ are probability distributions.
As said above, I have a pretty good understanding why this is true.
Unfortunately, the definition above describes an equilibrium for the two-player case.
Is there a general definition for the n-player case? I can't seem to find one.
A mixed action profile $\alpha ^{\ast }$ is a mixed equilibrium (or mixed Nash equilibrium) of game $G$ if, for all $i\in I$, \begin{equation*} \alpha _{i}^{\ast }\in \arg \max_{\alpha _{i}\in \Delta \left( A_{i}\right) }u_{i}\left( \alpha _{i},\alpha _{-i}^{\ast }\right) \end{equation*}
Where $\alpha_i$ is a mixed action for player $i$, $\alpha = \prod_i \alpha_i$ is a profile of mixed actions for each player, $\alpha_{-i} = \prod_{j\ne i} \alpha_j$ is the profile of mixed actions for all players other than $i$ and $\Delta(A_i)$ is the set of all probability distributions over $A_i$, which is the set of player $i$'s pure strategies.
I think that's everything -- is that clear? Basically, it's very much like the 2-player definition you give, except every player is best responding to all other players. You may think this is stated slightly differently than the version you have. This just reflects stylistic differences between pure game theory and algorithmic game theory. Consider it an exercise to translate the above pure game theory definition into an equivalent one that looks more like the algorithmic presentation.