Whose pay-offs does the pay-off matrix show?

The row player's (Rose's), unless directed otherwise. Colin's pay-off is the negative.

How does Rose find her play-safe strategy?

Take the minimum of each row, then choose the row with the largest of these (maximin).

How does Colin find his play-safe strategy?

Take the maximum of each column, then choose the column with the smallest of these (minimax).

When does a game have a stable solution?

If and only if maximin = minimax.

What is a saddle point?

An entry that is the least in its row and the greatest in its column.

What is the value of the game?

The pay-off to Rose at the stable solution (equal to the maximin and the minimax).

Is the maximin ever greater than the minimax?

No. Maximin \le minimax always.

What does maximin < minimax tell you?

There is no stable solution, so players need mixed strategies.

Why is a play-safe strategy 'safe'?

It guarantees the best of the worst-case outcomes, whatever the opponent does.

Can a game have more than one saddle point?

Yes, but all saddle points have the same value.

Zero-sum games, play-safe strategies and saddle pointsEdexcel A-Level Further Maths: Flashcards

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What is a zero-sum game?

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What is a zero-sum game?: One player's gain is exactly the other player's loss, so the pay-offs always sum to zero.
Whose pay-offs does the pay-off matrix show?: The row player's (Rose's), unless directed otherwise. Colin's pay-off is the negative.
How does Rose find her play-safe strategy?: Take the minimum of each row, then choose the row with the largest of these (maximin).
How does Colin find his play-safe strategy?: Take the maximum of each column, then choose the column with the smallest of these (minimax).
When does a game have a stable solution?: If and only if maximin $=$ minimax.
What is a saddle point?: An entry that is the least in its row and the greatest in its column.
What is the value of the game?: The pay-off to Rose at the stable solution (equal to the maximin and the minimax).
Is the maximin ever greater than the minimax?: No. Maximin $\le$ minimax always.
What does maximin $<$ minimax tell you?: There is no stable solution, so players need mixed strategies.
Rose plays a row and column giving $-3$ . What does Colin gain?: $+3$ (the game is zero-sum).
Why is a play-safe strategy 'safe'?: It guarantees the best of the worst-case outcomes, whatever the opponent does.
Can a game have more than one saddle point?: Yes, but all saddle points have the same value.

Exam questions on Zero-sum games, play-safe strategies and saddle points

Rose and Colin play a zero-sum game. Rose chooses a row and Colin chooses a column, and the pay-off matrix shows Rose's winnings: $\begin{pmatrix} 1 & 7 & 2 \\ 4 & 5 & 6 \\ 2 & 0 & 3 \end{pmatrix}$ .
Show that the game has a stable solution and state its value.2 marks
Rose and Colin play a different zero-sum game, with Rose choosing the rows and Colin the columns. Rose's pay-off matrix is $\begin{pmatrix} -1 & 3 & -2 \\ 2 & -3 & 4 \end{pmatrix}$ .
Explain why this game has no stable solution.2 marks
Two online retailers, Aria and Bolt, each choose one of three advertising strategies. The pay-off matrix shows Aria's gain in market share (percentage points) for each pair of strategies, where Aria chooses the rows and Bolt the columns, and $x$ is a constant: $\begin{pmatrix} 4 & 2 & 6 \\ 4 & x & 5 \\ 1 & 7 & 0 \end{pmatrix}$ . Any gain for Aria is an equal loss for Bolt.
Given that $x=1$ , find each retailer's play-safe strategy and show that the game has no stable solution.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).