Best Response Calculator

Find best responses across payoff matrices with confidence. Test beliefs, compare payoffs, and reveal equilibria. Build strategy intuition using clear tables, charts, and exports.

Enter payoff matrix data

Use payoff pairs in row,column format. Example: 8,6.

Row player meaning
The row player chooses rows and reacts to column moves.
Column player meaning
The column player chooses columns and reacts to row moves.

Strategy labels

Payoff matrix inputs

Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.
Enter row payoff first, then column payoff.

Belief weights for expected payoff analysis

Row player beliefs about column actions

Weights do not need to sum to one. The calculator normalizes them automatically.

Column player beliefs about row actions

Expected payoffs and regrets are computed from these normalized belief weights.

Example data table

This example matches the default sample loaded on first visit. Payoffs are shown as row, column.

Strategy C1 C2 C3
R1 8,6 2,5 4,3
R2 6,4 7,7 3,6
R3 5,8 1,2 9,4

Suggested beliefs: columns = (0.30, 0.40, 0.30), rows = (0.25, 0.50, 0.25).

Formula used

Best response to a pure action

Row player: BRR(j) = arg maxi uR(i, j)

Column player: BRC(i) = arg maxj uC(i, j)

Pure Nash equilibrium test

A cell (i, j) is a pure Nash equilibrium when row strategy i is a best response to column j, and column strategy j is a best response to row i.

Expected payoff under beliefs

Row expected payoff: EUR(i | q) = Σ qj · uR(i, j)

Column expected payoff: EUC(j | p) = Σ pi · uC(i, j)

Regret under beliefs

Regret = best expected payoff available under the same beliefs minus the chosen strategy's expected payoff. Zero regret means the strategy is belief-optimal.

How to use this calculator

  1. Choose a 2×2, 3×3, or 4×4 payoff matrix size.
  2. Rename row and column strategies to match your game.
  3. Enter each cell payoff as row,column, such as 8,6.
  4. Add belief weights for the opponent’s likely actions.
  5. Submit the form to see best responses, expected payoffs, regrets, and pure Nash equilibria.
  6. Read the highlighted matrix to locate row-best, column-best, and Nash cells quickly.
  7. Use the Plotly graph to compare expected payoffs by strategy.
  8. Export the analysis as CSV or PDF for reporting, teaching, or review.

FAQs

1) What does this calculator measure?

It evaluates best responses in a payoff matrix, highlights pure Nash equilibria, computes expected payoffs under beliefs, and shows regret for each strategy.

2) How should I enter each payoff?

Use the format row,column. For example, enter 8,6 if the row player earns 8 and the column player earns 6 in that cell.

3) How are ties handled?

If two or more actions share the same maximum payoff, the calculator treats all tied actions as valid best responses and lists each one.

4) What are belief weights used for?

Belief weights estimate how likely the opponent is to choose each action. They are normalized automatically before expected payoffs are calculated.

5) What is a pure Nash equilibrium here?

A pure Nash equilibrium is a payoff cell where both players are already choosing best responses to one another at the same time.

6) What does universal best response mean?

It identifies strategies that remain best responses against every pure action of the opponent. This helps spot highly robust actions quickly.

7) Why does the calculator show regret?

Regret shows how much expected payoff is lost by choosing a weaker strategy under the same beliefs. Zero regret means the action is optimal.

8) Can I use decimal or negative payoffs?

Yes. The calculator accepts positive, negative, and decimal values, which makes it useful for gains, costs, utilities, and penalty-based models.

Related Calculators

2×2 Normal-Form Nash Equilibrium Finder Calculatornash equilibrium solverdominant strategy findermixed strategy calculatorprisoner dilemma calculatorshapley value calculatorutility maximization solver

Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.