Game Theory Payoff Matrix Solver
Game Theory Payoff Matrix Solver computes Nash Equilibria for a 2×2 two-player game. Enter payoffs in the format (Player 1, Player 2), e.g., “3,1” for top-left cell.
Nash Equilibrium Calculation
A Nash Equilibrium occurs when no player can improve their payoff by unilaterally changing their strategy. For a 2×2 game:
Payoff Matrix:
\[ \begin{array}{c|cc} & \text{Player 2: Strategy 1} & \text{Player 2: Strategy 2} \\ \hline \text{Player 1: Strategy 1} & (a_{11}, b_{11}) & (a_{12}, b_{12}) \\ \text{Player 1: Strategy 2} & (a_{21}, b_{21}) & (a_{22}, b_{22}) \\ \end{array} \]Pure Strategy Nash Equilibrium: A pair \((i, j)\) where \(a_{ij} \geq a_{kj}\) for Player 1 and \(b_{ij} \geq b_{il}\) for Player 2.
Mixed Strategy Nash Equilibrium: Solve for \(p\) (Player 1’s probability of Strategy 1):
\[ p \cdot a_{11} + (1-p) \cdot a_{21} = p \cdot a_{12} + (1-p) \cdot a_{22} \]And \(q\) (Player 2’s probability of Strategy 1):
\[ q \cdot b_{11} + (1-q) \cdot b_{12} = q \cdot b_{21} + (1-q) \cdot b_{22} \]