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A couple wants to spend the evening together but disagree on where. Two outcomes are stable: both at the opera, or both at the football. Either one, once reached, holds.

Pure-strategy Nash equilibrium

A Nash equilibrium is a set of choices where no player can do better by changing their own move alone. Find it by looking for cells that are a best response for everyone at once.

From earlier lessons

One quick one before we start.

A strategy that is never a best response should be...

Find the stable cells

Tap a cell you think is stable, then reveal each player’s best responses. A cell glows when it is a best response for both, and that is a Nash equilibrium.

Definition, Nash equilibrium

A profile $s^*$ is a Nash equilibrium if, for every player $i$ , $u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*)$ for all $s_i$ . In words: given what everyone else does, no one has a profitable unilateral move.

This game has two pure equilibria, both-opera and both-football. Both players prefer coordinating to missing each other, but they disagree about which cell to land on.

Work it out

Mark best responses, then read off the equilibria

Work one player at a time. You compare down each column. If they pick opera, your opera pays $2$ against your football’s $0$ , so mark opera. If they pick football, your football pays $1$ against $0$ , so mark football. They compare across each row. If you pick opera, their opera pays $1$ against $0$ ; if you pick football, their football pays $2$ against $0$ .

Now look for cells carrying both marks. $(\text{opera}, \text{opera})$ has yours and theirs; so does $(\text{football}, \text{football})$ . Those two are the pure Nash equilibria. The off-diagonal cells pay $(0,0)$ and carry no marks at all: from either one, both players want to move.

Faded. Raise the mismatch payoff for missing each other from $0$ to $3$ in every off-diagonal cell, so the couple would rather be apart. Redo the marks: against their opera you now prefer football ( $3 > 2$ ), and against their football you prefer opera ( $3 > 1$ ). Their marks flip the same way, and the two equilibria move to the mismatched cells. The procedure did not change; only where the marks land.

Solo. Try the matrix from the dominance lesson (advertise pays $3$ and $6$ , staying quiet pays $1$ and $4$ ). Mark both players’ best replies and you will find exactly one cell with both marks. The next lesson is about what makes that cell uncomfortable.

Key idea

A Nash equilibrium is mutual best response, not group happiness. A game can have two, one, or none, and the stable outcome need not be the one everyone would pick together.

Check yourself

Check yourselfapply

How many pure-strategy Nash equilibria does the game above have?

Check yourselfconcept

Is a Nash equilibrium always the best outcome for the group?

Check yourselfbrings back: dominance

In the advertising game from the dominance lesson, both firms had a strictly dominant strategy. How many pure Nash equilibria does such a game have?

Sources