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 for A)*
p* = (a₂₂ − a₂₁) / (a₁₁ − a₁₂ − a₂₁ + a₂₂)
This is the p that makes Player 2 indifferent between X and Y, so the row payoffs a₂₁ and a₂₂ (Player 1’s B row) are what appear on top. Set the two column payoffs equal and solve: a₁₁p + a₂₁(1−p) = a₁₂p + a₂₂(1−p).
Formula for q (Player 2’s mixing probability for X)*
q* = (a₂₂ − a₁₂) / (a₁₁ − a₁₂ − a₂₁ + a₂₂)
Same idea from the other side. It makes Player 1 indifferent between A and B, so the Y-column payoffs a₁₂ and a₂₂ are what appear on top. Getting these two numerators the wrong way round is the classic slip, and the giveaway is that the resulting “equilibrium” leaves one player with a strictly better reply.
Expected Payoff at Mixed Nash Equilibrium
V = a₁₁·p*·q* + a₁₂·p*·(1−q*) + a₂₁·(1−p*)·q* + a₂₂·(1−p*)·(1−q*)
Worked example: matching pennies
Both players show a coin. Player 1 wins a dollar if the faces match, loses a dollar if they do not, so the matrix is a₁₁ = 1, a₁₂ = −1, a₂₁ = −1, a₂₂ = 1. The denominator is 1 + 1 + 1 + 1 = 4. Then p* = (1 − (−1)) / 4 = 0.5 and q* = (1 − (−1)) / 4 = 0.5, and the value of the game is 0. Neither side can do better than a coin flip, which is why matching pennies is the standard example of a game with no pure equilibrium.
A caution about Rock-Paper-Scissors: the full game is 3×3, not 2×2, and the 50/50 answer people quote belongs to matching pennies. If you cut Rock-Paper-Scissors down to Rock versus Scissors you no longer have a mixing problem at all, because Rock never loses. The denominator comes out zero and this calculator will say so.
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.
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