Expected Value in Game Theory Calculator
Calculate mixed strategy Nash equilibrium for 2×2 zero-sum games.
Find optimal mixed strategies and expected payoffs for both players.
2×2 Zero-Sum Game Theory
In a zero-sum game, one player’s gain is exactly the other’s loss. Both players simultaneously choose a strategy without knowing the other’s choice.
The Payoff Matrix
| Player 2: X | Player 2: Y | |
|---|---|---|
| Player 1: A | a₁₁ | a₁₂ |
| Player 1: B | a₂₁ | a₂₂ |
Values represent Player 1’s payoff (Player 2 receives the negative).
Pure Strategy Nash Equilibrium
A pair of strategies where neither player can benefit by switching unilaterally. Not every game has a pure strategy Nash equilibrium.
Mixed Strategy Nash Equilibrium
When no pure Nash equilibrium exists, players randomize. Player 1 plays A with probability p* and B with probability (1−p*).
Formula for p (Player 1’s mixing probability)*
p* = (a₂₂ − a₁₂) / (a₁₁ − a₁₂ − a₂₁ + a₂₂)
This makes Player 2 indifferent between X and Y.
Formula for q (Player 2’s mixing probability for X)*
q* = (a₂₂ − a₂₁) / (a₁₁ − a₁₂ − a₂₁ + a₂₂)
Expected Payoff at Mixed Nash Equilibrium
V = a₁₁·p*·q* + a₁₂·p*·(1−q*) + a₂₁·(1−p*)·q* + a₂₂·(1−p*)·(1−q*)
Classic Example: Rock-Paper-Scissors
In the 2×2 version (just Rock vs. Scissors): The unique Nash equilibrium is to mix 50/50.
Real Applications
- Economics: pricing wars, auctions, market competition
- Military strategy: attack/defend allocation
- Sports: penalty kicks in soccer (randomize to prevent prediction)
- Poker: bluffing strategy (mix bluffs and value bets)