Why a 3x3 Nash Tool Helps
A 3x3 normal form game can hide several stable outcomes. Each player has three choices. Each cell has two payoffs. The first payoff belongs to the row player. The second payoff belongs to the column player. A Nash equilibrium occurs when neither player wants to move alone. This calculator reads all nine payoff pairs. It then compares every row choice against each column choice. It also compares every column choice against each row choice.
What the Calculator Checks
The pure search is direct. For each column, it finds the best row payoff. For each row, it finds the best column payoff. A cell is stable when both payoffs are best responses at the same time. The mixed search is deeper. It tests possible equal sized supports. A support is the group of strategies used with positive probability. The tool solves indifference equations for each support. It then checks probabilities, outside strategies, and tolerance limits.
Useful Inputs
You can rename all row and column strategies. You can enter positive, zero, or negative payoff values. You can adjust tolerance for rounding and near ties. You can also choose decimal places for cleaner reports. This helps when payoffs come from estimates, market models, auctions, voting games, pricing games, or classroom exercises.
Reading the Output
Pure equilibria appear first. Mixed candidates appear with probability vectors. Expected payoffs are shown for both players. Best response notes explain why a result passed. Dominated strategy warnings help you inspect weak parts of the game. A dominated strategy is not always removed automatically, because mixed equilibria can be sensitive in degenerate games.
Practical Advice
Use exact payoffs when possible. Keep player objectives consistent. Higher values should always mean better outcomes. Try several tolerance settings if your game has ties. Very small changes can create or remove mixed solutions. Use the CSV file for spreadsheets. Use the PDF report for sharing results with students, teammates, or clients. Always review the original game before making decisions.
Limits to Know
The mixed solver is intended for finite, two player games. It checks nondegenerate support patterns first. If your model has many ties, compare pure results, mixed candidates, and best response tables together with care today.