Visualize best response functions and find mixed-strategy equilibria in real time
A Nash equilibrium is a standoff: neither player can do better by switching alone. Find it where the two players' best responses cross.
Pick a game and follow the green curve (Row's best response) and the blue curve (Col's). The yellow dots where they cross are the Nash equilibria.
Best response curves for each player. Nash equilibria occur where the curves intersect.
Row's best response never changes: Defect is a dominant strategy, so the green curve is a single edge of the square. Col's best response never changes: Defect is a dominant strategy, so the blue curve is a single edge of the square.
A Nash equilibrium is a strategy profile where no player can improve their payoff by unilaterally changing their strategy. Named after John Nash, who proved in 1950 that every finite game has at least one Nash equilibrium (possibly in mixed strategies).
In mixed strategies, players randomize between their options with specific probabilities. The key insight: at a mixed equilibrium, each player is indifferent between their strategies — their opponent's mix makes every option equally good.
The green curve is Row's best response to Col's mixing probability q. The blue curve is Col's best response to Row's mixing probability p. Nash equilibria are the yellow dots where the curves intersect — neither player wants to deviate.
Try this: Switch between games and notice how the best response structure changes. Matching Pennies has only a mixed equilibrium at (0.5, 0.5). The Coordination Game has two pure equilibria plus a third, interior mixed equilibrium at (0.5, 0.5), where each player's mix leaves the other indifferent between A and B.
Edit the payoffs and watch equilibria appear and vanish. Can you build a game with exactly three Nash equilibria? (Hint: two pure and one mixed.)
| Column Player | |||
|---|---|---|---|
| Cooperate | Defect | ||
| Row Player | Cooperate | , | , |
| Defect | , | , | |
Spot a mistake or have an idea?