# definition of payoff matrix, how is it defined?

If $$I$$ and $$J$$ are the strategy sets for 2 players respectively.

Let A = $$[a_{ij}]$$ and B = $$[b_{ij}]$$ be the random payoff matrices of player 1 and player 2 respectively.

Let X be defined as $$$$X=\left\{x \in \mathbb{R}^{m} \mid \sum_{i \in I} x_{i}=1, x_{i} \geq 0, \forall i \in I\right\}$$$$ representing the sets of mixed strategies of player 1, and Y be defined as: $$$$y=\left\{y \in \mathbb{R}^{n} \mid \sum_{j \in J} y_{j}=1, y_{j} \geq 0, \forall j \in J\right\}$$$$representing the sets of mixed strategies of player 2.

Then, for each (x, y) ∈ X × Y the payoff of player 1 given by $$x^TAy$$.

I cannot seem to wrap my head around the last sentence. Is $$x^TAy$$ some sort of expected value for the payoffs? Please help, thanks.

Pick a term in $$x^T A y$$ like $$x_i a_{ij} y_j$$ this is the payoff for player 1 when 1 plays $$i$$ and 2 plays $$j$$ which is $$a_{ij}$$ then weighted by the probability of that happening which is the product of the two independent probabilities $$x_i$$ for 1 to play $$i$$ and $$y_j$$ for 2 to play $$j$$.
Add these all up and you get an expected value for the payoff to 1 when 1 is using mixed strategy $$x$$ and 2 is using mixed strategy $$y$$.