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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn7367Paper 3 · Discrete optionGame theoryZero-sum games

Game theory · Subtopic 1 of 1

Game Theory

In a two-person zero-sum game, one player's gain is the other's loss. Play-safe: row player maximises their row minimum, column player minimises their column maximum. If these match, there's a saddle point (stable solution). Otherwise use a mixed strategy, found by equating expected pay-offs or graphically.

Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 50 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find play-safe strategies and test for a stable solution.
Reduce a pay-off matrix using dominance.
Find optimal mixed strategies and the value of the game.

Pause and think

In rock–paper–scissors, what's the best strategy if your opponent is smart?

Section 2 of 14

Core idea

Start hereOut-guessing an opponent

A penalty taker chooses left or right; the goalkeeper guesses. If either becomes predictable, the other exploits it. Game theory finds the strategy that guarantees the best result against a perfectly smart opponent.

The big idea

Each player guarantees their best worst case: the row player maximises row minima, the column player minimises column maxima. If these agree there is a saddle point; if not, randomising (a mixed strategy) achieves the game's value.

1. Stable solutions

Worked exampleSaddle point

Pay-offs to A: $R_1(3,-1,2)$ , $R_2(4,1,3)$ , $R_3(-2,0,1)$ .

Goal: Stable solution?
Strategy: Compare maximin and minimax.

$\text{row minima: }-1,1,-2\Rightarrow1$
A's play-safe value ( $R_2$ ).
$\text{column maxima: }4,1,3\Rightarrow1$
B's play-safe value ( $C_2$ ).
$1=1$
Equal: saddle point.

AnswerA plays

R_2

, B plays

C_2

; value

1

✓ CheckNeither player gains by switching alone ✓.

2. Mixed strategies

Equalising expected pay-offs

$E(C_1)=4p-2(1-p)=6p-2$
A plays $R_1$ with prob $p$ ; game $R_1(4,-1)$ , $R_2(-2,3)$ .
$E(C_2)=-p+3(1-p)=3-4p$
Against B's second column.
$6p-2=3-4p\Rightarrow p=\tfrac12$
Top of the lower envelope.
$V=6(\tfrac12)-2=1$
Value of the game.

A mixes

R_1

R_2

equally and guarantees an average of

1

Worked exampleB’s mix

Find B's optimal probability $q$ of playing $C_1$ .

Goal: $q$ .
Strategy: Equate A's returns from each row.

$R_1:\ 4q-(1-q)=5q-1$
A's expected pay-off from $R_1$ .
$R_2:\ -2q+3(1-q)=3-5q$
From $R_2$ .
$5q-1=3-5q\Rightarrow q=\tfrac25$
Equalise.

Answer

q=\frac25

; value

5(\frac25)-1=1

✓.

✓ CheckSame value from both players' viewpoints ✓.

Solving a zero-sum game

Drawing diagram…

Simplify before solving.

Common misconceptionColumn dominance For B (who wants small pay-offs to A), a column with every entry smaller dominates.

Key takeawayZero-sum Pay-offs are to the row player; B's pay-off is the negative.

Pause and think

What's the value of rock–paper–scissors?

Summary

Maximin vs minimax.
Equal → saddle point.
Dominance reduces.
Mixed: equalise expectations.

Zero-sum game: One player's gain is the other's loss.
Saddle point: Entry that is its row minimum and column maximum.
Mixed strategy: Choosing options at random with set probabilities.

HistoryVon Neumann. John von Neumann proved the minimax theorem in 1928: every finite zero-sum game has a value if mixed strategies are allowed.

Fun factPenalty kicks. Studies of professional penalty kicks found players mix left and right almost exactly as game theory predicts.

Section 3 of 14

Key formulae

Play-safe

\text{row: }\max(\text{row minima});\quad\text{column: }\min(\text{column maxima})

Stable

\max(\text{row min})=\min(\text{col max})

Mixed (2×2)

p\,a_{11}+(1-p)a_{21}=p\,a_{12}+(1-p)a_{22}

TrickDominance. For the row player, delete a row if another row is at least as good in every column. For the column player, delete a column if another is at most as large in every row (pay-offs are to the row player).

Section 4 of 14

Worked examples

Example 1 · Saddle point0 / 3 steps

Rows:

R_1(3,-1,2)

R_2(4,1,3)

R_3(-2,0,1)

. Find the stable solution.

Example 2 · Mixed0 / 3 steps

Rows

R_1(4,-1)

R_2(-2,3)

. Find A's optimal mix.

Section 5 of 14

Explore the graph

Finding the optimal mix

Loading the interactive graph…

Try this

$x$ is $p$ , the probability A plays $R_1$ . Lines: A's expected pay-off against $C_1$ and $C_2$ .
A maximises the lower envelope: the peak at $p=\frac12$ , value $1$ .

Maximise the minimum.

In the real worldAuctions and pricing. Game theory guides spectrum auctions and pricing strategies when competitors react to each other.

Section 6 of 14

Common mistakes

IncorrectColumn player maximises

CorrectMinimises the row player's pay-off

Zero-sum: B's gain is A's loss.

IncorrectUsing the upper envelope

CorrectLower envelope, highest point

A guarantees the worse of B's replies.

IncorrectDominance the wrong way for columns

CorrectColumns with smaller entries dominate

Smaller is better for B.

Opponent predicts my move

Mixed strategy: even I don't know it.

Section 7 of 14

Exam tip

Exam tipSaddle check first. Always test for a stable solution before attempting a mixed strategy.

Section 8 of 14

Guided practice

SA-DM-GT-G011 markL1 FoundationNon-calculatorSaddle points

Pay-offs to A:

R_1(3,-1,2)

R_2(4,1,3)

R_3(-2,0,1)

. Find A's play-safe value (maximin).

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-DM-GT-0012 marksL2 StandardNon-calculatorSaddle points

For the same game, find B's play-safe value (minimax of column maxima).

Salam Academy Original Practice · Proposed mark scheme

SA-DM-GT-0022 marksL3 ExamNon-calculatorMixed strategies

Pay-offs to A:

R_1(4,-1)

R_2(-2,3)

. Find the probability

p

with which A should play

R_1

Salam Academy Original Practice · Proposed mark scheme

SA-DM-GT-0031 markL2 StandardNon-calculatorMixed strategies

Find the value of that game to A.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-DM-GT-0042 marksL3 ExamNon-calculatorMixed strategies

In the same

2\times2

game, find the probability

q

with which B should play the first column.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Maximin vs minimax.
Equal: stable, saddle point.
Dominance reduces the matrix.
Mixed: equate expectations.

Section 12 of 14

Quick quiz

Q1 · SA-DM-GT-Q011 markL1 FoundationNon-calculatorDominance

Row

R_2(4,1,3)

and row

R_1(3,-1,2)

(pay-offs to the row player). Which is true?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 7367 A-level Further Mathematics
Where: Discrete · Game theory
Expected: Pay-off matrices, play-safe strategies, saddle points, dominance, optimal mixed strategies graphically and by LP.
Calculator: Permitted on all papers.
Typical style: AQA mixes short multiple-choice questions with longer structured ones.
Open in: 7367 · Paper 3

Section 14 of 14

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).