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Strategic Cost Management · Game Theory

Dominance Rule and Reduction of Games in Game Theory

Updated 11 October 2026 · Fact-checked

The dominance rule shrinks a payoff matrix before you solve it. The row player deletes any row that is worse than another row in every column. The column player deletes any column that is worse, for the column player, than another column in every row. Repeat until the game is small, then solve it.

Understand Dominance Rule and Reduction of Games

A payoff matrix shows what the row player (Maximiser) gains, and what the column player (Minimiser) loses, for each pair of strategies. Big matrices such as 3x3 or 4x4 are slow to solve. Dominance lets you throw away strategies that a rational player would never choose.

A strategy is dominated if another strategy does at least as well against every choice of the opponent. A sensible player never loses by dropping a dominated strategy. So you delete it and work with a smaller matrix.

The two players want opposite things. The row player wants big numbers, so a row with larger entries is better. The column player wants small numbers, since the entries are the column player's losses, so a column with smaller entries is better. Students mix this up more than anything else.

After one deletion, a new dominance may appear in the smaller matrix. Keep going until nothing more can be removed. Then check for a saddle point, or use the 2x2 mixed strategy method, or the graphical method if the game has become 2xn or mx2.

There is also dominance by an average (convex combination). A row can be dominated by the average of two other rows, even if no single row dominates it. This is tested less often but appears in harder problems.

Key rules to remember

Row dominance
Row i is dominated by row k if aᵢⱼ ≤ aₖⱼ for every column j
Row player maximises, so the smaller row is deleted. Delete row i.
Column dominance
Column j is dominated by column l if aᵢⱼ ≥ aᵢₗ for every row i
Column player minimises, so the larger column is deleted. Delete column j.
Dominance by average
Row i is dominated if aᵢⱼ ≤ p·aₖⱼ + (1 − p)·aₘⱼ for every column j, where 0 < p < 1
For columns, reverse the inequality: column j is dominated if its entries are ≥ the weighted average of two other columns in every row.
Order of reduction
Reduce → check saddle point → else mixed strategy or graphical method
Dominance only simplifies the game. It does not give the value by itself.

How to solve Dominance Rule and Reduction of Games questions

Use this method for any question that asks you to reduce a payoff matrix and find the optimum strategies and value of the game. Payoffs are taken as the row player's gain.

  1. 1Write the matrix. Label the row player as Maximiser and the column player as Minimiser.
  2. 2First check for a saddle point: find the row minima and take the maximum (maximin), find the column maxima and take the minimum (minimax). If they are equal, you can solve directly.
  3. 3Compare rows in pairs. If every entry of one row is less than or equal to the matching entry of another row, delete the smaller row.
  4. 4Compare columns in pairs. If every entry of one column is greater than or equal to the matching entry of another column, delete the larger column.
  5. 5After each deletion, look again for new dominance in the reduced matrix. Repeat until no more rows or columns can be deleted.
  6. 6If no pair dominates, test whether a row or column is dominated by the average of two others.
  7. 7Solve the reduced game: saddle point, 2x2 formulas, or graphical method for 2xn and mx2.
  8. 8State the optimum strategy for each player (deleted strategies get probability zero) and the value of the game.

Quickest way: Scan rows for smaller, columns for larger

When to use it: Use it in the exam for 3x3 and 4x4 matrices when time is short.

  1. Run your eye down two rows at a time. If one row is lower in every column, cross it out.
  2. Run across two columns at a time. If one column is higher in every row, cross it out.
  3. Redraw the smaller matrix neatly. Do not delete mentally across several steps without redrawing.
  4. Repeat the scan once more on the new matrix.
  5. Stop as soon as the matrix is 2x2 or has a saddle point, then apply the standard formula.

Common mistakes in Dominance Rule and Reduction of Games

  • Deleting the larger column because it has bigger numbers

    Students apply the row rule to columns without remembering that the column player minimises.

    Fix: For columns, delete the column with larger entries in every row. Write 'Column player minimises' beside the matrix.

  • Deleting a row that is better in some columns but worse in others

    Students compare totals or averages instead of every column.

    Fix: Dominance needs the inequality to hold in every column (or every row). One exception means no dominance.

  • Stopping after one round of deletion

    Students assume dominance is a single step.

    Fix: After each deletion, scan the smaller matrix again. New dominance often appears.

  • Forgetting to give zero probability to deleted strategies

    The answer is written only for the reduced 2x2 matrix.

    Fix: Write the full strategy for the original game, such as (0, 2/5, 3/5), including the deleted rows or columns as zero.

  • Using dominance when a saddle point already exists, or ignoring a saddle point after reducing

    Students treat dominance as a compulsory first step and skip the saddle check.

    Fix: Check for a saddle point at the start and again after reduction. If it exists, the game is solved with pure strategies.

  • Treating equal rows as a mistake or deleting both

    Students are unsure what 'at least as good' means.

    Fix: If two rows are identical, delete only one. Dominance allows equal entries as long as no entry is better.

Worked examples

Example 1

Reduce the following game, where the payoffs are the gains of player A (rows A1, A2, A3) against player B (columns B1, B2, B3), and find the value of the game.

A1: 3, 5, 4
A2: 5, 6, 7
A3: 4, 5, 6

Show the solution
  1. Check for rows. Compare A1 with A2: 3 ≤ 5, 5 ≤ 6, 4 ≤ 7. A1 is dominated by A2, so delete A1.
  2. Compare A3 with A2: 4 ≤ 5, 5 ≤ 6, 6 ≤ 7. A3 is dominated by A2, so delete A3.
  3. Only A2 remains: 5, 6, 7.
  4. Now columns. B tries to minimise, so B picks the smallest entry in row A2, which is 5 under B1. B2 and B3 are dominated by B1 (6 ≥ 5, 7 ≥ 5). Delete them.
  5. The reduced game is a single cell: A2 against B1 with payoff 5.
  6. Verify with the saddle test on the original matrix. Row minima are 3, 5, 4, so maximin = 5. Column maxima are 5, 6, 7, so minimax = 5. They match.

Answer: A plays A2 and B plays B1. A1 and A3 and B2 and B3 are never used. The value of the game is 5 in favour of A, with a saddle point at (A2, B1).

Example 2

Using dominance, reduce and solve this game. Payoffs are the gains of player A (rows A1, A2, A3) against B (columns B1, B2, B3).

A1: 1, 7, 2
A2: 6, 2, 7
A3: 5, 1, 6

Show the solution
  1. Check for a saddle point. Row minima: 1, 2, 1, so maximin = 2. Column maxima: 6, 7, 7, so minimax = 6. They are unequal, so no saddle point.
  2. Rows: compare A3 with A2: 5 ≤ 6, 1 ≤ 2, 6 ≤ 7. A3 is dominated by A2. Delete A3.
  3. The matrix is now A1: 1, 7, 2 and A2: 6, 2, 7.
  4. Columns: compare B3 with B1: 2 ≥ 1 for A1 and 7 ≥ 6 for A2. B3 is larger in both rows, so B3 is dominated by B1. Delete B3.
  5. The reduced 2x2 game is A1: 1, 7 and A2: 6, 2, with columns B1, B2.
  6. No saddle point here: row minima 1 and 2, maximin = 2. Column maxima 6 and 7, minimax = 6.
  7. Apply the 2x2 formulas. Let A play A1 with probability p and A2 with 1 − p. Equate the expected payoffs against B1 and B2: 1p + 6(1 − p) = 7p + 2(1 − p). That is 6 − 5p = 2 + 5p, so 10p = 4 and p = 2/5.
  8. So A plays A1 with 2/5 and A2 with 3/5. Value = 6 − 5(2/5) = 4.
  9. Let B play B1 with probability q and B2 with 1 − q. Equate against A1 and A2: 1q + 7(1 − q) = 6q + 2(1 − q). That is 7 − 6q = 2 + 4q, so 10q = 5 and q = 1/2.
  10. Check the value: 7 − 6(1/2) = 4. It agrees.

Answer: A's optimum strategy is (2/5, 3/5, 0) over (A1, A2, A3). B's optimum strategy is (1/2, 1/2, 0) over (B1, B2, B3). The value of the game is 4 in favour of A.

Exam tips

  • Write 'Row player maximises, column player minimises' beside the matrix before you start. It prevents the commonest slip.
  • Show each dominance comparison in one line, with the inequality, for example '3 ≤ 5, 5 ≤ 6, 4 ≤ 7, so delete A1'. Marks are given for the reasoning.
  • Redraw the reduced matrix after every deletion so the examiner can follow you.
  • Always give the final strategies for the original game, with zero for deleted rows and columns, and state the value of the game.
  • Verify the value of the game from both players' sides when you use the 2x2 formulas. A mismatch shows an arithmetic error.

Practice questions from Game Theory

Dominance Rule and Reduction of Games: frequently asked questions

What is the dominance rule in game theory?

It says a player never uses a strategy that is no better than another strategy against every choice of the opponent. You delete such strategies from the payoff matrix. This reduces the game to a smaller one that is easier to solve.

Which columns do I delete under the dominance rule?

Delete the column with larger entries in every row, because the column player wants to minimise the payoff. This holds when the matrix shows the row player's gains. If the matrix shows the column player's gains, reverse the roles.

Does dominance always give the final answer?

No. It only reduces the matrix. After reduction you still need a saddle point check, the 2x2 mixed strategy formulas or the graphical method to find the strategies and the value of the game.

Can a row be dominated by the average of two other rows?

Yes. If every entry of a row is less than or equal to the matching weighted average of two other rows, that row can be deleted. Check this when no single row or column dominates but the matrix is still too large.