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)
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.