Nash Equilibrium and Best-Response Correspondences

Draw both players' best-response curves on one plane. The blue curve and the orange curve appear together.

The blue curve is my best answer to the rival's mixing probability. The orange curve is the best answer in the other direction. Where they cross, each is a best response to the other. That crossing is a Nash equilibrium. Pure equilibria sit at the corners, and the mixed one is starred inside.

For example, say you and a 🧑 friend each choose left or right. Your best answer changes with your friend's choice. Your friend's best answer changes with yours too. The pairing where both answers fit each other is the equilibrium.

s^* \text{ Nash} \iff s_i^* \in BR_i(s_{-i}^*) \;\; \forall i

BR_1(q) \cap BR_2(p) \neq \varnothing

Part of these learning paths

The Game Theory Core

Examples & Exercises

Draw the best-response correspondences of the Stag Hunt game and show there are three Nash equilibria (two pure, one mixed).

u_1(S,q) = 4q, \quad u_1(H,q) = 3

q^* = 3/4, \quad p^* = 3/4

Stag Hunt — three intersections of the correspondences.

Show proof

Player 1: $u_1(S, q) = 4q$ , $u_1(H, q) = 3q + 3(1-q) = 3$ . From $4q = 3$ , $q^* = 3/4$ : $H$ is best for $q < 3/4$ , $S$ for $q > 3/4$ .
By symmetry player 2's indifference point is $p^* = 3/4$ .
Intersections: $(q,p) = (0,0)$ i.e. $(H,H)$ , $(1,1)$ i.e. $(S,S)$ , and the fully mixed $(3/4, 3/4)$ . At each, the correspondences "hold" each other — no unilateral deviation pays.

Show Matching Pennies has no pure-strategy Nash equilibrium and its unique equilibrium is $(p^*, q^*) = (1/2, 1/2)$ .

BR_1(q) = \begin{cases} 1 & q > 1/2 \\ [0,1] & q = 1/2 \\ 0 & q < 1/2 \end{cases}

Matching Pennies — the correspondences meet only at the center.

Show proof

Check the four cells: at $(H,H)$ player 2 wants $T$ ( $1 > -1$ ); at $(H,T)$ player 1 wants $T$ ; at $(T,H)$ player 1 wants $H$ ; at $(T,T)$ player 2 wants $H$ . No cell is a mutual best response.
In mixtures, $u_1(H,q) = 2q-1$ and $u_1(T,q) = 1-2q$ give $q^* = 1/2$ , and symmetrically $p^* = 1/2$ . In the correspondence diagram the two step functions cross exactly once, at $(1/2, 1/2)$ — the unique Nash equilibrium.

8. Game Theory