When do you need a mixed strategy?

When the game has no stable solution (maximin \ne minimax).

How do you find Rose's expected pay-off against a column?

Add (probability of each row \times the entry in that column).

In a 2\times n game, what does each line show?

Rose's expected pay-off against one column, as the probability p of row 1 varies from 0 to 1.

Where does Rose choose p in a 2\times n game?

At the highest point of the lower boundary (maximising the lowest line).

Where does Colin choose q in an n\times2 game?

At the lowest point of the upper boundary (minimising the highest line).

How do you find the optimal p for a 2\times2 game?

Make the expected pay-offs against both columns equal and solve.

How do you find Colin's strategy after finding Rose's?

Use only the two columns whose lines met at the peak; solve that 2\times2 game for q.

What is the value of the game?

The expected long-run pay-off to Rose per play when both use optimal strategies.

What does a positive value of the game mean?

Rose gains on average each play and Colin loses the same amount.

Mixed strategies: the graphical methodEdexcel A-Level Further Maths: Flashcards

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What is a mixed strategy?

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What is a mixed strategy?: Choosing each pure strategy at random with fixed probabilities.
When do you need a mixed strategy?: When the game has no stable solution (maximin $\ne$ minimax).
How do you find Rose's expected pay-off against a column?: Add (probability of each row $\times$ the entry in that column).
In a $2\times n$ game, what does each line show?: Rose's expected pay-off against one column, as the probability $p$ of row 1 varies from $0$ to $1$ .
Where does Rose choose $p$ in a $2\times n$ game?: At the highest point of the lower boundary (maximising the lowest line).
Where does Colin choose $q$ in an $n\times2$ game?: At the lowest point of the upper boundary (minimising the highest line).
How do you find the optimal $p$ for a $2\times2$ game?: Make the expected pay-offs against both columns equal and solve.
How do you find Colin's strategy after finding Rose's?: Use only the two columns whose lines met at the peak; solve that $2\times2$ game for $q$ .
What probability do unused strategies get?: Zero. State it in your answer.
What is the value of the game?: The expected long-run pay-off to Rose per play when both use optimal strategies.
For $\begin{pmatrix} 5 & 1 \\ 2 & 4 \end{pmatrix}$ , what are Rose's $p$ and the value?: $p=\frac13$ and the value is $3$ .
What does a positive value of the game mean?: Rose gains on average each play and Colin loses the same amount.

Exam questions on Mixed strategies: the graphical method

Rose and Colin play a zero-sum game with Rose's pay-off matrix $\begin{pmatrix} 3 & -1 \\ -2 & 4 \end{pmatrix}$ , where Rose chooses the rows. Rose plays row 1 with probability $p$ and row 2 with probability $1-p$ .
Find Colin's optimal strategy.2 marks
Rose and Colin play a zero-sum game with Rose's pay-off matrix $\begin{pmatrix} 2 & -1 & 4 \\ -3 & 3 & -2 \end{pmatrix}$ , where Rose chooses the rows. Rose plays row 1 with probability $p$ and row 2 with probability $1-p$ .
Find Colin's optimal strategy.2 marks
A zero-sum game has Rose's pay-off matrix $\begin{pmatrix} 4 & -1 \\ 1 & 3 \\ -2 & 5 \end{pmatrix}$ , where Rose chooses the rows. Colin plays column 1 with probability $q$ and column 2 with probability $1-q$ .
Find Colin's optimal strategy.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).