Mixed strategy

A fixed / pure strategy is easy for an adaptive player to beat. A mixed strategy is just a probability distribution across a set of pure strategies.

For a game where agent $i$ has two choices $a_1$ and $a_2$, a mixed strategy for $i$ is the distribution: $M{S}_i=\{P(a_1),P(b_2)\}$ Given this strategy, when $i$ comes to play, they pick action $a_1$ with probability $P(a_1)$, and the same for $a_2$ with probability $P(a_2)$.

To determine the mixed strategy, $i$ can compute the best values of $P(a_1)$ and $P(a_2)$. These will be the values which give $i$ the highest expected payoff given the options that $j$ can choose and the joint payoffs that result.