2 x 2 matrix game and expected value

Question

I have found the optimal strategy for the row player and column player. How do I find the expected value of the game for the row player and determine whether the game is favourable to the row player or column player (or neither)?

Answer 1

The expected value for given mixed (random, independent) strategies is given by $\vec{r}^TM\vec{c}$, where the column vector $\vec{r}$ is the row player's mixed strategy (expressed as a vector of probabilities); $\vec{c}$ is the column player's strategy, and $M$ is the payoff matrix.

Typically payoffs are from the point of view of the row player, so a positive expected value is favorable to the row player; negative, to the column player.