Visualize best response functions and find mixed-strategy equilibria in real time
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.
Best response curves for each player. Nash equilibria occur where the curves intersect.
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 but no interior mixed equilibrium where both players mix with probability strictly between 0 and 1.
Modify the payoffs below and watch how equilibria appear and vanish. Can you create a game with exactly three Nash equilibria? (Hint: two pure and one mixed.)
| Column Player | |||
|---|---|---|---|
| Cooperate | Defect | ||
| Row Player | Cooperate | , | , |
| Defect | , | , | |