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Strategic Cost Management · Game Theory

Pure Strategy Game Theory: Maximin, Minimax and Saddle Point

Updated 11 October 2026 · Fact-checked

A pure strategy game has a saddle point when the maximin of the row minima equals the minimax of the column maxima. That common figure is the value of the game. Find the minimum of each row, take the largest; find the maximum of each column, take the smallest; then compare them.

Understand Pure Strategy: Maximin, Minimax and Saddle Point

Game theory studies decisions where your result depends on what a rival does. In a two-person zero-sum game, one player's gain is exactly the other's loss. The payoff matrix shows the gain of the row player (Player A) for each pair of choices. The column player (Player B) pays that amount.

Each player is cautious. Player A, who wants a high payoff, looks at the worst result of each row and picks the row whose worst result is best. This is the maximin principle. Player B, who wants to pay as little as possible, looks at the worst (largest) payment in each column and picks the column where it is smallest. This is the minimax principle.

If the maximin value equals the minimax value, the game has a saddle point. The saddle point cell is the smallest in its row and the largest in its column. Neither player gains by changing choice alone, so each plays one fixed course of action. This is a pure strategy, and the game is strictly determinable.

The common figure is the value of the game. If it is positive, A gains on average; if negative, B gains; if zero, the game is fair. If maximin is less than minimax, there is no saddle point and you need mixed strategies, which is a separate topic.

Key rules to remember

Maximin value
Maximin = max over rows of (minimum of each row)
Player A (maximiser) writes the row minima, then picks the largest of them.
Minimax value
Minimax = min over columns of (maximum of each column)
Player B (minimiser) writes the column maxima, then picks the smallest of them.
Saddle point condition
Maximin = Minimax = V
If equal, the game has a saddle point and V is the value of the game. Always maximin ≤ minimax.
Fair game
V = 0
A game is fair when the value is zero. It is strictly determinable when a saddle point exists.

How to solve Pure Strategy: Maximin, Minimax and Saddle Point questions

Use this method for any pure strategy question where the payoff matrix is given from A's point of view.

  1. 1Write the payoff matrix with Player A's strategies as rows and Player B's as columns. Confirm the payoffs are A's gains.
  2. 2Find the minimum of each row and write it in a column on the right.
  3. 3Find the largest of these row minima. This is the maximin, and it gives A's strategy.
  4. 4Find the maximum of each column and write it in a row below the matrix.
  5. 5Find the smallest of these column maxima. This is the minimax, and it gives B's strategy.
  6. 6Compare the two. If they are equal, a saddle point exists and that figure is the value of the game.
  7. 7State the optimal strategy of each player, the value of the game, and who gains. If the value is zero, call the game fair.
  8. 8If maximin ≠ minimax, say there is no saddle point and move to dominance or mixed strategy methods.

Quickest way: Circle the row minima and column maxima

When to use it: Use when the matrix is up to 4×4 and you only need the saddle point and the value.

  1. Mark the smallest number in each row with a small circle.
  2. Mark the largest number in each column with a small square.
  3. Any cell with both marks is a saddle point.
  4. Read the value from that cell, and the strategies from its row and column.
  5. Check that no other cell is marked with both. More than one saddle point is possible, all with the same value.

Common mistakes in Pure Strategy: Maximin, Minimax and Saddle Point

  • Taking the row maxima and column minima instead of the reverse.

    Students mix up which player wants high or low figures.

    Fix: Remember: A is the maximiser, so A looks at the worst case in each row, the minimum. B looks at the worst case in each column, the maximum.

  • Declaring a saddle point when maximin and minimax differ.

    Students stop after finding one of the two values.

    Fix: Always compute both and compare. A saddle point exists only when they are equal.

  • Choosing the smallest row minimum as maximin.

    The word 'min' pulls attention to the smallest figure.

    Fix: Maximin means the maximum of the row minima. Pick the largest value in the right-hand column.

  • Giving the value of the game with the wrong sign or to the wrong player.

    Students forget the matrix shows A's gain.

    Fix: A positive value means A gains and B loses. A negative value means B gains. Say so in the answer.

  • Not naming the strategies, only the value.

    Students treat it as a pure arithmetic question.

    Fix: Write 'A plays Strategy 2, B plays Strategy 3, value = 4'. Marks are given for the optimal strategies.

  • Reading the matrix from B's point of view when it is given as B's payoffs.

    The question wording is skipped.

    Fix: Check whose payoff is shown. If it is B's gain, swap the roles or take negatives before solving.

Worked examples

Example 1

Two firms A and B compete. The matrix shows A's gain (₹ lakh) for A's strategies A1, A2, A3 (rows) against B's strategies B1, B2, B3 (columns): A1: 3, 5, 4; A2: 6, 7, 8; A3: 2, 1, 9. Find the optimal strategies and the value of the game.

Show the solution
  1. Row minima: A1 = 3, A2 = 6, A3 = 1.
  2. Maximin = largest of 3, 6, 1 = 6, so A chooses A2.
  3. Column maxima: B1 = 6, B2 = 7, B3 = 9.
  4. Minimax = smallest of 6, 7, 9 = 6, so B chooses B1.
  5. Maximin = Minimax = 6, so a saddle point exists at (A2, B1).

Answer: A plays A2, B plays B1, and the value of the game is ₹6 lakh in favour of A. The game is strictly determinable and not fair.

Example 2

The payoff matrix to Player X (₹ thousand) for strategies X1, X2, X3 (rows) against Y1, Y2, Y3, Y4 (columns) is: X1: 2, -3, 4, 1; X2: -1, -2, -1, 0; X3: 3, 1, 2, 5. Find the saddle point, if any, and the value of the game.

Show the solution
  1. Row minima: X1 = -3, X2 = -2, X3 = 1.
  2. Maximin = largest of -3, -2, 1 = 1, so X chooses X3.
  3. Column maxima: Y1 = 3, Y2 = 1, Y3 = 4, Y4 = 5.
  4. Minimax = smallest of 3, 1, 4, 5 = 1, so Y chooses Y2.
  5. Maximin = Minimax = 1, so the saddle point is at (X3, Y2).

Answer: X plays X3, Y plays Y2, and the value of the game is ₹1,000 in favour of X.

Exam tips

  • Show the row minima and column maxima beside the matrix. Examiners award marks for these workings.
  • Always state the strategies of both players and the value, with the sign explained.
  • If a question gives a matrix with unknown x and asks for a range that keeps a saddle point, set maximin = minimax conditions and solve the inequalities.
  • In MCQs, check quickly whether the maximin equals the minimax before looking at any mixed strategy option.
  • If there is no saddle point, say so clearly and then apply dominance or the mixed strategy method.

Practice questions from Game Theory

Pure Strategy: Maximin, Minimax and Saddle Point: frequently asked questions

What is the difference between maximin and minimax?

Maximin is the largest of the row minima and is used by the maximising player. Minimax is the smallest of the column maxima and is used by the minimising player. Maximin can never exceed minimax.

Can a game have more than one saddle point?

Yes. A matrix can have several saddle points. All of them give the same value of the game, and any of them gives an optimal pair of strategies.

What does a pure strategy mean?

A pure strategy means a player always chooses one fixed course of action. It is optimal when the game has a saddle point.

What if there is no saddle point?

Then maximin is less than minimax. You first reduce the matrix using dominance and then use mixed strategies, where each player picks courses of action with set probabilities.