Strategic Cost Management · Game Theory

Mixed Strategy and 2x2 Games: Oddments Method

Q: When do I use a mixed strategy in game theory?

Use it when the game has no saddle point, which means maximin does not equal minimax. Each player then plays each strategy with a probability so the opponent cannot exploit a fixed choice.

Q: What is the oddments method?

It is a shortcut for 2x2 games. Take absolute differences of the entries in each row and column, swap them across the rows and across the columns, and divide by their total to get the probabilities.

Q: Is the arithmetic method different from the oddments method?

They are the same idea written differently. Both use differences of payoffs to fix the probabilities. The algebraic method gives the same answer by solving two equations.

Q: How do I find the value of a 2x2 game?

If there is no saddle point, use V = (ad − bc) ÷ [(a + d) − (b + c)]. Or take the expected payoff of either row using the column player's probabilities. Both must agree.

Updated 11 October 2026 · Fact-checked

A mixed strategy game has no saddle point, so each player randomises between strategies. For a 2x2 game, find the probabilities that make the opponent indifferent. Use the oddments method: subtract the entries in each row and column, ignore signs, swap them, and divide by the total to get probabilities and the value of the game.

Understand Mixed Strategy and 2x2 Games

In a two-person zero-sum game, you first test for a saddle point. Find the row minima and take their maximum (maximin). Find the column maxima and take their minimum (minimax). If the two are equal, the game has a saddle point and a pure strategy. If maximin is less than minimax, there is no saddle point.

Without a saddle point, a pure strategy is a bad idea. If you always play the same row, your rival learns it and responds to hurt you. So each player plays each strategy with a probability. This is a mixed strategy. The aim is to pick probabilities so the opponent cannot gain by changing their own choice.

For a 2x2 game, the logic is simple. You choose the probabilities so that your expected gain is the same whichever column the rival plays. That is why the method works: the rival is indifferent, so the expected payoff is fixed at the value of the game. For a game with no saddle point, maximin < V < minimax.

Three methods give the same answer: the algebraic method (set up two equations), the arithmetic method (take differences) and the oddments method. The arithmetic method and oddments method are the same shortcut. Before using any of them, reduce the game by the dominance rule if possible. If a larger game reduces to 2x2, solve it as 2x2.

Key rules to remember

Saddle point test: Maximin = max of row minima; Minimax = min of column maxima; saddle point exists if Maximin = Minimax; If they are not equal, use a mixed strategy.
2x2 game with payoffs a b / c d (to the row player): p1 (probability of row 1) = |d − c| ÷ (|a − c| + |d − b|), p2 = 1 − p1; The matrix is row 1 = a, b and row 2 = c, d. Applies only when no saddle point exists.
Column player's probability of column 1: q1 = |b − d| ÷ (|a − b| + |c − d|), q2 = 1 − q1; Same condition: no saddle point.
Value of the game: V = (ad − bc) ÷ [(a + d) − (b + c)]; Valid for a 2x2 game with no saddle point. Check by finding the expected payoff of either row using the q values.
Oddments check: Row probabilities sum to 1; column probabilities sum to 1; Each probability is an oddment divided by the sum of the oddments.

How to solve Mixed Strategy and 2x2 Games questions

Use this order for any 2x2 game, or a larger game that reduces to 2x2.

1Write the payoff matrix from the row player's view. Mark the row minima and column maxima.
2Find maximin and minimax. If equal, state the saddle point and value, and stop.
3If not equal, check dominance. Delete dominated rows or columns to reach a 2x2 matrix.
4Oddments: subtract the two entries in each row and write the difference, ignoring sign. Do the same for each column.
5Swap the row differences: the difference of row 1 becomes the oddment for row 2, and vice versa. Do the same for the columns.
6Probability of each strategy = its oddment ÷ sum of the two oddments for that player.
7Find the value by taking the expected payoff of any row using the column probabilities, and confirm it with the other row.
8State the optimal mixed strategies for both players and the value, and say who gains.

Quickest way: Oddments in one pass

When to use it: Any 2x2 game with no saddle point, where the exam asks for strategies and value.

Check the saddle point first. It takes 20 seconds and saves a wrong answer.
Write the row differences to the right of the matrix and the column differences below it, ignoring signs.
Cross over the differences: row 1 gets row 2's difference, column 1 gets column 2's difference.
Divide by the sum to get probabilities. They must add to 1.
Find V = (ad − bc) ÷ [(a + d) − (b + c)] and verify with one row.

Common mistakes in Mixed Strategy and 2x2 Games

Applying oddments without checking for a saddle point.
Students jump to the formula because the question says 'mixed strategy'.
Fix: Always compute maximin and minimax first. Use mixed strategy only when they differ.
Not swapping the differences.
The row-1 difference looks like it belongs to row 1.
Fix: The probability of row 1 uses the row-2 difference. Test it: the larger payoff gap gets the smaller weight in the other row.
Keeping the negative sign in differences.
Subtraction is done in a fixed order.
Fix: Take absolute differences for oddments. Probabilities cannot be negative.
Giving probabilities that do not sum to 1, or giving only one player's strategy.
Time pressure and forgetting that the question asks for both players.
Fix: Add the probabilities to confirm 1. Always write the strategies of both players and the value.
Skipping dominance in larger games.
Students try to solve a 3x3 matrix with a 2x2 method.
Fix: Remove dominated rows and columns first. Only then apply the 2x2 method.
Writing a value outside the range of maximin and minimax.
Arithmetic slips in the value formula.
Fix: For a game with no saddle point, maximin < V < minimax. Check the value against this range and confirm using both rows.

Worked examples

Example 1

Two firms A and B compete. The payoff to A (market share gain in %) is: A1 against B1 = 6, A1 against B2 = 2; A2 against B1 = 1, A2 against B2 = 7. Find the optimal strategies and the value of the game.

Show the solution

Row minima: A1 = 2, A2 = 1. Maximin = 2.
Column maxima: B1 = 6, B2 = 7. Minimax = 6. They differ, so no saddle point.
Row differences: A1 = |6 − 2| = 4; A2 = |1 − 7| = 6. Column differences: B1 = |6 − 1| = 5; B2 = |2 − 7| = 5.
Swap row differences: oddment for A1 = 6, A2 = 4. Sum = 10. So p(A1) = 6/10 = 0.6 and p(A2) = 0.4.
Swap column differences: oddment for B1 = 5, B2 = 5. Sum = 10. So q(B1) = 0.5 and q(B2) = 0.5.
Value = (6×7 − 2×1) ÷ [(6 + 7) − (2 + 1)] = (42 − 2) ÷ (13 − 3) = 40 ÷ 10 = 4.
Check with A1: 6×0.5 + 2×0.5 = 4. Check with A2: 1×0.5 + 7×0.5 = 4. The value is consistent and lies between 2 and 6.

Answer: A plays A1 with probability 0.6 and A2 with 0.4. B plays B1 and B2 with probability 0.5 each. Value of the game = 4% in favour of A.

Example 2

The payoff matrix to X (₹ in lakh) is: X1 against Y1 = 3, X1 against Y2 = −1; X2 against Y1 = −2, X2 against Y2 = 4. Find the best mixed strategies and the value.

Show the solution

Row minima: X1 = −1, X2 = −2. Maximin = −1.
Column maxima: Y1 = 3, Y2 = 4. Minimax = 3. No saddle point.
Row differences: X1 = |3 − (−1)| = 4; X2 = |−2 − 4| = 6. Swap: X1 = 6, X2 = 4. Sum = 10. p(X1) = 0.6, p(X2) = 0.4.
Column differences: Y1 = |3 − (−2)| = 5; Y2 = |−1 − 4| = 5. Swap: Y1 = 5, Y2 = 5. q(Y1) = 0.5, q(Y2) = 0.5.
Value = (3×4 − (−1)(−2)) ÷ [(3 + 4) − (−1 + −2)] = (12 − 2) ÷ (7 + 3) = 10 ÷ 10 = 1.
Check using X1: 3×0.5 + (−1)×0.5 = 1. Check using X2: (−2)×0.5 + 4×0.5 = 1.

Answer: X plays X1 with probability 0.6 and X2 with 0.4. Y plays Y1 and Y2 with probability 0.5 each. Value of the game = ₹1 lakh in favour of X.

Exam tips

Show the saddle point test in every answer. Marks are often given for proving no saddle point exists.
Write both players' strategies and the value. Many students lose marks by giving only one.
In Section A, a quick check helps: if maximin equals minimax, the answer is pure. If not, there is no saddle point and maximin < V < minimax.
Verify the value with two rows. It takes a minute and catches arithmetic errors.
For larger matrices, apply dominance first and state which rows or columns were deleted.

Practice questions from Game Theory

Mixed Strategy and 2x2 Games: frequently asked questions

When do I use a mixed strategy in game theory?

Use it when the game has no saddle point, which means maximin does not equal minimax. Each player then plays each strategy with a probability so the opponent cannot exploit a fixed choice.

What is the oddments method?

It is a shortcut for 2x2 games. Take absolute differences of the entries in each row and column, swap them across the rows and across the columns, and divide by their total to get the probabilities.

Is the arithmetic method different from the oddments method?

They are the same idea written differently. Both use differences of payoffs to fix the probabilities. The algebraic method gives the same answer by solving two equations.

How do I find the value of a 2x2 game?

If there is no saddle point, use V = (ad − bc) ÷ [(a + d) − (b + c)]. Or take the expected payoff of either row using the column player's probabilities. Both must agree.