When is a column dominated?

When another column has every entry less than or equal to it (Colin prefers smaller pay-offs for Rowan).

Why remove dominated strategies?

A sensible player never uses them, so they reduce the game to a smaller one.

After removing a dominated row, what should you check?

Whether any column has now become dominated, and repeat.

What is a mixed strategy?

Choosing each strategy with a given probability, at random.

When are mixed strategies needed?

When the game has no stable solution.

For a 2\times2 game, how do you find Rowan's optimal p?

Equate his expected pay-offs against Colin's two columns and solve.

What is the value of the game with mixed strategies?

Rowan's expected pay-off when both players use their optimal mixes.

In the graphical method for a 2\times n game, which point gives Rowan's optimal p?

The highest point of the lower boundary of the expected pay-off lines.

For an m\times2 game, what does Colin look for?

The lowest point of the upper boundary of the lines in q.

Dominance and mixed strategiesAQA A-Level Further Maths: Flashcards

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When is a row dominated?

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When is a row dominated?: When another row has every entry greater than or equal to it (Rowan prefers larger pay-offs).
When is a column dominated?: When another column has every entry less than or equal to it (Colin prefers smaller pay-offs for Rowan).
Why remove dominated strategies?: A sensible player never uses them, so they reduce the game to a smaller one.
After removing a dominated row, what should you check?: Whether any column has now become dominated, and repeat.
What is a mixed strategy?: Choosing each strategy with a given probability, at random.
When are mixed strategies needed?: When the game has no stable solution.
Rowan plays row 1 with probability $p$ . What is the probability of row 2?: $1-p$ .
For a $2\times2$ game, how do you find Rowan's optimal $p$ ?: Equate his expected pay-offs against Colin's two columns and solve.
What is the value of the game with mixed strategies?: Rowan's expected pay-off when both players use their optimal mixes.
In the graphical method for a $2\times n$ game, which point gives Rowan's optimal $p$ ?: The highest point of the lower boundary of the expected pay-off lines.
What does a line above the optimal point tell you?: Colin never plays that column.
For an $m\times2$ game, what does Colin look for?: The lowest point of the upper boundary of the lines in $q$ .
$\begin{pmatrix} 5 & 1 \\ 2 & 4 \end{pmatrix}$ : optimal $p$ and value?: $p=\frac13$ for row 1, value $3$ .

Exam questions on Dominance and mixed strategies

Rowan and Colin play a zero-sum game. The pay-off matrix for Rowan is $\begin{pmatrix} 3 & 2 & 4 \\ 1 & 5 & 2 \\ 4 & 3 & 5 \end{pmatrix}$ . Rows are Rowan's strategies and columns are Colin's. Rowan wants to maximise his pay-off and Colin wants to minimise it.
Explain why Colin will never play column 3.2 marks
Rowan and Colin play a zero-sum game with pay-off matrix for Rowan $\begin{pmatrix} 5 & 1 \\ 2 & 4 \end{pmatrix}$ . The game has no stable solution. Rowan chooses row 1 with probability $p$ and row 2 with probability $1-p$ , to maximise his smallest expected pay-off.
Find Colin's optimal strategy.2 marks
Rowan and Colin play a zero-sum game with pay-off matrix for Rowan $\begin{pmatrix} 2 & 5 & 1 \\ 4 & 1 & 6 \end{pmatrix}$ . The game has no stable solution. Rowan chooses row 1 with probability $p$ and row 2 with probability $1-p$ . Colin has three strategies, columns 1, 2 and 3.
Write down Rowan's expected pay-off, in terms of $p$ , when Colin plays each of columns 1, 2 and 3.3 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).