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A penalty kick lasts a fraction of a second. The keeper cannot wait to see where the ball goes, so kicker and keeper are really choosing at the same time. If you always aim the same way, you will be read. What do you do?

Mixed strategies and mixed Nash

Some games have no resting point in fixed moves. Matching Pennies is the clean case: you want to match, they want to differ, so whatever you settle on, one of you wants to jump. The way out is to stop choosing a move and start choosing the odds.

From earlier lessons

Two quick ones before we start.

The other player chooses Left. To find your best reply in a payoff table, compare...

A best response is the move that...

No fixed move is stable

Two players each hide a coin as Heads or Tails. You win if the faces match; they win if the faces differ. Look for a cell where neither wants to move.

Show the best responses. Yours and theirs never land on the same cell, so there is no pure-strategy Nash equilibrium. Chase this in circles and you will chase forever.

Predict, then play

Instead of a move, pick a probability. The slider sets how often you play Heads. The two lines show what the opponent expects to earn from Heads and from Tails against your mix. A sharp opponent will pick their better line and beat you there.

Why did that happen?

Take a guess, then reveal the reason

In a fully mixed 2x2 equilibrium, you choose odds that leave the opponent with no reason to prefer either move, and they choose odds that do the same to you. This indifference calculation pins down both probabilities. Outside a fully mixed 2x2 equilibrium, players need not make every opposing action indifferent.

The math

A mixed strategy for player $i$ is a probability distribution over their pure strategies. In Matching Pennies, yours is a single number: the probability $p$ you play Heads.

Definition, mixed-strategy Nash equilibrium

A profile of mixed strategies is a Nash equilibrium if each player’s mix is a best response to the others’. A player who mixes between several moves must be indifferent among them: each move in the mix earns the same expected payoff, or they would drop the worse one.

Suppose the opponent plays Heads with probability $q$ . Your expected payoff to Heads is $q(1) + (1-q)(-1) = 2q - 1$ , and to Tails it is $q(-1) + (1-q)(1) = 1 - 2q$ . For you to be willing to mix, these must be equal:

2q - 1 = 1 - 2q \quad\Rightarrow\quad q = \tfrac{1}{2}.

The calculation above uses the opponent’s probability $q$ to make you indifferent. By symmetry, making the opponent indifferent sets your own probability $p = \tfrac{1}{2}$ . Each player’s probability is therefore determined by the other player’s indifference condition.

Work it out

Matching Pennies, then a game where 50/50 is wrong

Matching Pennies. Set the opponent’s Heads probability to $q$ . Your Heads pays $2q-1$ , your Tails pays $1-2q$ . Equal at $q=\tfrac12$ , so each of you plays $\tfrac12$ , and the value of the game is $0$ .

Battle of the Sexes. Opera/Opera pays $(2,1)$ , Football/Football pays $(1,2)$ , mismatches pay $(0,0)$ . To make the column player indifferent, set your Opera probability $p$ so that their Opera payoff equals their Football payoff: $p\cdot 1 = (1-p)\cdot 2$ , giving $p = \tfrac{2}{3}$ . So the mixing answer is not always 50/50. Here you lean toward your preferred outcome two-thirds of the time.

Solo. In Chicken, two drivers head at each other. If both swerve they each get $6$ ; if both go straight they crash for $0$ ; if one holds firm while the other swerves, the bold driver gets $7$ and the timid one $2$ . Pick your probability of going Straight to make the other driver indifferent between Swerve and Straight. Setting $6 - 4q = 7 - 7q$ gives Straight with probability $\tfrac{1}{3}$ .

A note from the real world

Reality check

Equilibrium says mix to a precise fraction. Real people do not. In first plays of new games they lean on hunches and simple reasoning, not the exact indifference point, which is why crowds are predictable in ways theory says they should not be. The Behavioral unit picks this up with models of limited reasoning. Professionals under pressure, though, get close: penalty takers and keepers mix at rates strikingly near the theory (Palacios-Huerta, 2003).

Key idea

In a fully mixed 2x2 equilibrium, each player’s probability makes the other player indifferent. That is why neither side can gain by changing its own mix.

Check yourself

Check yourselfapply

At the mixed-strategy equilibrium of Matching Pennies, what is your expected payoff?

Check yourselfanalyse

In Battle of the Sexes you play Opera two-thirds of the time. Why two-thirds and not one-half?

Check yourselfbrings back: dominance

Before mixing, you should strip out any strictly dominated move. Why does that never remove part of a mixed equilibrium?

Sources

Osborne, M. J. (2004). An Introduction to Game Theory. Oxford University Press.
Palacios-Huerta, I. (2003). Professionals play minimax. Review of Economic Studies, 70(2), 395-415.