2×2 Game Theory Mixed-Strategy Nash Equilibrium & Expected Payoff Value Calculator

Find indifference mixed-strategy probabilities (p*, q*) and expected game payoff.

Equilibrium Expected Payoff (Game Value)
2 payoff
Row Player Optimal Top Probability (p*) 33.333 %
Column Player Optimal Left Probability (q*) 50 %
Maximin Safety Floor 1 payoff
Inputs
⚡ Scenarios:
payoff a
Row player payoff when (Top, Left) is played (a).
payoff b
Row player payoff when (Top, Right) is played (b).
payoff c
Row player payoff when (Bottom, Left) is played (c).
payoff d
Row player payoff when (Bottom, Right) is played (d).
Results & Graph
Equilibrium Expected Payoff (Game Value)
2 payoff
Row Player Optimal Top Probability (p*)
33.333 %
Column Player Optimal Left Probability (q*)
50 %
Maximin Safety Floor
1 payoff

Equilibrium Expected Payoff (Game Value)

Graph 1 • 2×2 Mixed Nash Equilibrium • Domain Architecture & Live Telemetry Maps Payoff Top Left A, Payoff Top Right B, Payoff Bottom Left C through 2×2 Mixed Nash Equilibrium Formula.
2×2 Game Theory Mixed-Strategy Nash Equilibrium & Expected Payoff Value Calculator • Custom Domain Schematic (Game Theory, Ratings & Stochastic Processes) payoffTopayoffTo2×2 Mixed Nash Equilib Payoff Top Left A: 4 payoff a • Payoff Top Right B: 0 payoff b EQUILIBRIUM EXPECTED PAYOF 2 payoff • Row Player Optimal Top: 33.333 % • Column Player Optimal : 50 % Eq: q* = (d − b) / (a − b − c + d);
Graph 2 • Equilibrium Expected Payoff (Game Value) Across Payoff Top Left A Sweep (40% → 160%) Evaluates q* = (d − b) / (a − b − c + d); ExpectedRowPayoff = a·p·q + b·p(1−q) + across 7 operating points (current input highlighted).
0 payoff 1.26 payoff 2.51 payoff 1.33 1.60 1.64 2.40 1.85 3.20 2 4 2.12 4.80 2.21 5.60 2.29 6.40
Graph 3 • Joint Sensitivity: Payoff Top Left A × Payoff Top Right B Compares Equilibrium Expected Payoff (Game Value) when Payoff Top Right B is shifted +25% (solid) vs. −25% (dashed).
0 payoff 3.44 payoff 6.88 payoff --- Payoff Top Right B @ 75% 1 -30% 1 -15% 1 +0% 6.25 +15% 6.25 +30%
Summary: Mixed-strategy indifference occurs at Row playing Top p* = 33.3% and Column playing Left q* = 50%, yielding an equilibrium expected payoff of 2.
Summary: Mixed-strategy indifference occurs at Row playing Top p* = 33.3% and Column playing Left q* = 50%, yielding an equilibrium expected payoff of 2.
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Formula & Mathematical Methodology: 2×2 Mixed Nash Equilibrium Formula

q* = d − ba − b − c + d
ExpectedRowPayoff = a·p·q + b·p(1−q) + c(1−p)q + d(1−p)(1−q)
q* = (d − b) / (a − b − c + d); ExpectedRowPayoff = a·p·q + b·p(1−q) + c(1−p)q + d(1−p)(1−q)
  • payoffTopLeftA: Payoff Top Left A (payoff a)
  • payoffTopRightB: Payoff Top Right B (payoff b)
  • payoffBottomLeftC: Payoff Bottom Left C (payoff c)
  • payoffBottomRightD: Payoff Bottom Right D (payoff d)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: 2×2 Mixed Nash Equilibrium

Find indifference mixed-strategy probabilities (p*, q*) and expected game payoff.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: q* = (d − b) / (a − b − c + d); ExpectedRowPayoff = a·p·q + b·p(1−q) + c(1−p)q + d(1−p)(1−q).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes 2×2 Mixed Nash Equilibrium via q* = (d − b) / (a − b − c + d); ExpectedRowPayoff = a·p·q + b·p(1−q) + c(1−p)q + d(1−p)(1−q).

Frequently Asked Questions

How does the 2×2 Game Theory Mixed-Strategy Nash Equilibrium & Expected Payoff Value Calculator compute the solution?

It evaluates q* = (d − b) / (a − b − c + d); ExpectedRowPayoff = a·p·q + b·p(1−q) + c(1−p)q + d(1−p)(1−q) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Game Theory, Ratings & Stochastic Processes

2×2 Game Theory Mixed-Strategy Nash Equilibrium & Expected Payoff Value Calculator: inputs, method, and how to read the result

Find indifference mixed-strategy probabilities (p*, q*) and expected game payoff.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Payoff Top Left A: Row player payoff when (Top, Left) is played (a).
  • Payoff Top Right B: Row player payoff when (Top, Right) is played (b).
  • Payoff Bottom Left C: Row player payoff when (Bottom, Left) is played (c).
  • Payoff Bottom Right D: Row player payoff when (Bottom, Right) is played (d).

What the calculation includes

With the default values shown, Equilibrium Expected Payoff (Game Value) is 2 payoff. The calculation follows 2×2 Mixed Nash Equilibrium Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Payoff Top Left A from 4 payoff a to 5 payoff a changes Equilibrium Expected Payoff (Game Value) from 2 payoff to 2.143 payoff. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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