📗 A pure strategy is when a player uses only one action with probability 1.
📗 A mixed strategy Nash equilibrium is a Nash equilibrium for the game in which mixed strategies are allowed (called mixed extension of the original game).
➩ If the mixed strategies are mutual best responses, then they form a mixed strategy Nash equilibrium (see Math Note for the mathematical definition).
Math Note
📗 For a two-player general-sum game with rewards \(r\) (both players are maximizing), an action profile \(\left(a_{1}, a_{2}\right)\) is a pure strategy Nash equilibrium if \(r_{1}\left(a_{1}, a_{2}\right) \geq r_{1}\left(a', a_{2}\right)\) for every \(a'\) valid action of player \(1\); and \(r_{2}\left(a_{1}, a_{2}\right) \geq r_{2}\left(a_{1}, a'\right)\) for every \(a'\) valid action of player \(2\).
📗 An mixed strategy profile \(\left(\pi_{1}, \pi_{2}\right)\), where \(\pi_{i}\left(a\right)\) is the probability that player \(i\) chooses action \(a\), is a mixed strategy Nash equilibrium if \(\mathbb{E}\left[r_{1}\left(\pi_{1}, \pi_{2}\right)\right] \geq \mathbb{E}\left[r_{1}\left(\pi', \pi_{2}\right)\right]\) for every \(\pi'\) mixed strategy of player \(1\); and if \(\mathbb{E}\left[r_{2}\left(\pi_{1}, \pi_{2}\right)\right] \geq \mathbb{E}\left[r_{2}\left(\pi_{1}, \pi'\right)\right]\) for every \(\pi'\) mixed strategy of player \(2\).
➩ Here, the expectation is over the randomness of the mixed strategies, or \(\mathbb{E}\left[r_{i}\left(\pi_{1}, \pi_{2}\right)\right] = \displaystyle\sum_{a_{1}, a_{2}} r_{i}\left(a_{1}, a_{2}\right) \pi_{1}\left(a_{1}\right) \pi_{2}\left(a_{2}\right)\), where the sum is over all valid \(\left(a_{1}, a_{2}\right)\) actions of players \(1\) and \(2\).
➩ This condition is usually difficult to check: another equivalent condition is: \(\left(\pi_{1}, \pi_{2}\right)\) is a mixed strategy Nash equilibrium if \(\mathbb{E}\left[r_{1}\left(\pi_{1}, \pi_{2}\right)\right] \geq \mathbb{E}\left[r_{1}\left(a', \pi_{2}\right)\right]\) for every \(a'\) valid action of player \(1\); and if \(\mathbb{E}\left[r_{2}\left(\pi_{1}, \pi_{2}\right)\right] \geq \mathbb{E}\left[r_{2}\left(\pi_{1}, a'\right)\right]\) for every \(a'\) valid action of player \(2\).