What do the entries of a pay-off matrix represent?

The pay-off to the row player for each pair of strategies.

What does a negative entry mean?

The row player loses (and the column player gains).

How do you find Colin's pay-off matrix from Rowan's?

Negate every entry and swap rows and columns (transpose).

How does Rowan find his play-safe strategy?

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

How does Colin find his play-safe strategy?

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

The smallest of the column maxima.

What is the value of a stable game?

The common value of maximin and minimax, i.e. the saddle point entry.

What is a saddle point?

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

Why can neither player improve at a stable solution?

Changing strategy alone cannot improve their pay-off, because the entry is the best they can guarantee.

If maximin \neq minimax, where does the value of the game lie?

Between the maximin and the minimax; there is no stable solution.

Pay-off matrices and play-safe strategiesAQA 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 where one player's gain is exactly the other player's loss.
What do the entries of a pay-off matrix represent?: The pay-off to the row player for each pair of strategies.
What does a negative entry mean?: The row player loses (and the column player gains).
How do you find Colin's pay-off matrix from Rowan's?: Negate every entry and swap rows and columns (transpose).
How does Rowan find his play-safe strategy?: Take the minimum of each row and choose the row with the largest of these (maximin).
How does Colin find his play-safe strategy?: Take the maximum of each column and choose the column with the smallest of these (minimax).
Define maximin.: The largest of the row minima.
Define minimax.: The smallest of the column maxima.
When does a game have a stable solution?: When maximin equals minimax.
What is the value of a stable game?: The common value of maximin and minimax, i.e. the saddle point entry.
What is a saddle point?: An entry that is the least in its row and the greatest in its column.
Why can neither player improve at a stable solution?: Changing strategy alone cannot improve their pay-off, because the entry is the best they can guarantee.
If maximin $\neq$ minimax, where does the value of the game lie?: Between the maximin and the minimax; there is no stable solution.

Exam questions on Pay-off matrices and play-safe strategies

Two companies, Rowan Ltd and Colbert Ltd, each choose one of three advertising strategies at the same time. The pay-off matrix for Rowan, in percentage points of market share gained by Rowan (a negative entry is a loss for Rowan and a gain for Colbert), is $\begin{pmatrix} 3 & -1 & 2 \\ 1 & 0 & 4 \\ -2 & 5 & 1 \end{pmatrix}$ . Rows are Rowan's strategies and columns are Colbert's.
Show that this game does not have a stable solution.2 marks
A supermarket chain, Firm F, and its rival choose pricing strategies at the same time. The pay-off matrix for Firm F, in percentage points of market share gained by Firm F, is $\begin{pmatrix} 4 & 2 & 5 \\ 3 & 1 & 6 \\ 1 & 0 & 2 \end{pmatrix}$ . Rows are Firm F's strategies and columns are the rival's. The game is zero-sum.
Interpret the value of the game in the context of the problem.2 marks
Xavier and Yara play a game. At the same time, Xavier chooses 1, 2 or 3 and Yara chooses 2 or 3. If the sum of the two numbers is even, Yara pays Xavier the product of the two numbers, in pounds. If the sum is odd, Xavier pays Yara the sum of the two numbers, in pounds.
Construct the pay-off matrix for Xavier, with a row for each of Xavier's choices and a column for each of Yara's choices.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).