Normal-Form Games and Best Responses

Look directly at the two-player payoff table. Each of the four cells lists both players' payoffs.

In each column, mark the first player's best answer. In each row, mark the second player's best answer. The cell where both marks land gets a star. That cell is a pure-strategy Nash equilibrium. Neither player wants to move away alone.

For example, say you and a 🧑 friend each pick heads or tails on a coin. If your friend picks heads, one answer is best for you. Your pick likewise settles your friend's best answer. The pairing where both answers point at each other is the equilibrium.

s^* \text{ is Nash} \iff u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*) \;\; \forall s_i, \forall i

Part of these learning paths

The Game Theory Core

Examples & Exercises

In the Prisoner's Dilemma, find every best response and show the pure-strategy Nash equilibrium is unique.

\begin{matrix} & C & D \\ C & (3,3) & (0,5) \\ D & (5,0) & (1,1) \end{matrix}

Prisoner's Dilemma — selected cell

(D, D)

Show proof

For player 1: against $C$ , $D$ wins $5 > 3$ ; against $D$ , $D$ wins $1 > 0$ .
So $D$ is player 1's strictly dominant strategy.
Symmetrically, $D$ strictly dominates for player 2.
Player 1's best response in each column is always $D$ (bottom row); player 2's best response in each row is always $D$ (right column).
The only cell where both hold is $(D, D)$ — the unique pure-strategy Nash equilibrium.
It is Pareto dominated by $(C, C)$ 's $(3, 3)$ , but the unilateral incentive to deviate never disappears.

Find all pure-strategy Nash equilibria of the Stag Hunt game and compare their Pareto ranking.

\begin{matrix} & S & H \\ S & (4,4) & (0,3) \\ H & (3,0) & (3,3) \end{matrix}

Stag Hunt — selected cell

(S, S)

Show proof

Player 1's best responses: $S$ against the $S$ column ( $4 > 3$ ), $H$ against the $H$ column ( $3 > 0$ ). Symmetrically for player 2: $S$ against the $S$ row, $H$ against the $H$ row.
The cells where both best-respond are $(S, S)$ and $(H, H)$ — two pure-strategy Nash equilibria. $(S, S)$ Pareto-dominates $(H, H)$ ( $(4,4)$ vs $(3,3)$ ).
Yet $(H, H)$ is still an equilibrium: if the other hunts hare, hare is my best reply too.
Which equilibrium materializes is a matter of beliefs — the seed of the equilibrium selection problem.

8. Game Theory