games
Mixed strategy
A player can pick by chance instead of always using the same action.
Instead of always picking the same action, a player can choose by chance. This grid is matching pennies. You win 1 when the coins match. The other player wins 1 when they differ. A loss counts as −1. The scores below are those wins and losses added up over 100 plays. Heads is action A. Tails is the rest.
Set the probability each player uses. See when neither wants to change.
In this game, always showing the same face is not stable. The other player can win by showing the opposite of the face they expect.
The stable point is a mix: each of you shows Heads on 50% of plays, so the other person scores the same with either face.
At that mix, neither of you can raise your score over 100 plays by changing your chance of Heads.
A planned mix is also how a stable rock-paper-scissors player plays: not one fixed action every time.