Some games have no pure-strategy equilibrium: in matching pennies, whatever one player does, the other wants to switch. With Best Responses and Nash Equilibrium and Expected Value, we can find the equilibrium that always exists: players randomize.
Key idea
In a mixed-strategy equilibrium each player randomizes so that the other player is indifferent between her choices. So each player's probabilities are found from the other player's payoffs. John Nash proved in 1950 that every game with finitely many players and choices has at least one equilibrium, possibly in mixed strategies.
Why it works
Suppose Row mixes between Up and Down. That only makes sense if Up and Down give her the same expected payoff; if one were better, she would play it for sure. Her expected payoffs depend on Column's mix $q$, not her own. So Column's $q$ must be the one that makes Row indifferent, and by the same reasoning Row's $p$ must make Column indifferent. A useful consequence: changing Row's own payoffs changes Column's equilibrium mix, not Row's.
Worked example: An asymmetric game
Payoffs (Row, Column): Up–Left (3, 3), Up–Right (6, 4), Down–Left (2, 11), Down–Right (9, 6). Find the mixed equilibrium.
- No pure equilibrium: Row prefers Up against Left and Down against Right; Column prefers Right against Up and Left against Down.
- Row's $p$ (Up) makes Column indifferent: Left gives $3p + 11(1 - p)$, Right gives $4p + 6(1 - p)$. Equal when $11 - 8p = 6 - 2p$, so $p = 5/6$.
- Column's $q$ (Left) makes Row indifferent: Up gives $3q + 6(1 - q)$, Down gives $2q + 9(1 - q)$. Equal when $6 - 3q = 9 - 7q$, so $q = 3/4$.
Answer: Row plays Up with probability $5/6$; Column plays Left with probability $3/4$.
Worked example: A penalty kick
Scoring chances: kick left & dive left 30%, kick left & dive right 80%, kick right & dive left 90%, kick right & dive right 30%. How often does the kicker go left?
- The kicker's mix $x$ must make the goalie indifferent: diving left allows $30x + 90(1 - x)$, diving right $80x + 30(1 - x)$.
- Equal when $90 - 60x = 30 + 50x$, so $x = 60/110 = 6/11$.
- The kicker goes left a bit more than half the time. Right kicks are deadlier when the goalie guesses wrong (90% against 80%), so if the kicker went right half the time the goalie would simply guard the right. Aiming left a little more often keeps the goalie guessing.
Answer: $6/11$ of the time.
Common mistake
Finding a player's mixing probabilities from that player's own payoffs. In the first example, using Row's own payoffs to set Row's $p$ would give $3/4$, which is really Column's equilibrium mix. Check: after you find $p$, plug it into the other player's two expected payoffs. They must come out equal.
More on randomizing, including zero-sum games, in Mixed Strategies and Zero-Sum Games.