Game theory Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Game theory questions. See exactly how to solve problems on game-theory, play-safe, maximin-minimax, saddle-point.

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Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
In a two-person zero-sum game the row player chooses a row and the column player chooses a column of the pay-off matrix M=(−3−30−41−23−4−4)M=\begin{pmatrix} -3 & -3 & 0 \\ -4 & 1 & -2 \\ 3 & -4 & -4 \end{pmatrix}, where each entry is the number of points the row player wins from the column player. Write down the smallest entry in row 1 (the row minimum).

Worked solution

  1. Write down the pay-off matrix

    M=(−3−30−41−23−4−4)M=\begin{pmatrix} -3 & -3 & 0 \\ -4 & 1 & -2 \\ 3 & -4 & -4 \end{pmatrix}

    Each entry is the row player's gain; the row player picks a row and the column player picks a column.

  2. Look along row 1

    row 1: −3, −3, 0\text{row }1:\ -3,\ -3,\ 0

    The row player, committed to row 1, receives the least of these entries.

  3. State the row minimum

    min⁡jM1j=−3\min_j M_{1j}=-3

    This is the smallest entry in the row.

Answer
min⁡=−3\min=-3
Question 2
2 markseasy
In a two-person zero-sum game the row player chooses a row and the column player chooses a column of the pay-off matrix M=(−2−35145−3−4−2)M=\begin{pmatrix} -2 & -3 & 5 \\ 1 & 4 & 5 \\ -3 & -4 & -2 \end{pmatrix}, where each entry is the number of points the row player wins from the column player. Write down the smallest entry in row 2 (the row minimum).

Worked solution

  1. Write down the pay-off matrix

    M=(−2−35145−3−4−2)M=\begin{pmatrix} -2 & -3 & 5 \\ 1 & 4 & 5 \\ -3 & -4 & -2 \end{pmatrix}

    Each entry is the row player's gain; the row player picks a row and the column player picks a column.

  2. Look along row 2

    row 2: 1, 4, 5\text{row }2:\ 1,\ 4,\ 5

    The row player, committed to row 2, receives the least of these entries.

  3. Recall what the row minimum represents

    min⁡jM2j\min_j M_{2j}

    It is the pay-off the row player is guaranteed from that row.

  4. State the row minimum

    min⁡jM2j=1\min_j M_{2j}=1

    This is the smallest entry in the row.

Answer
min⁡=1\min=1
Question 3
2 markseasy
In a two-person zero-sum game the row player chooses a row and the column player chooses a column of the pay-off matrix M=(5−3−13−20−243)M=\begin{pmatrix} 5 & -3 & -1 \\ 3 & -2 & 0 \\ -2 & 4 & 3 \end{pmatrix}, where each entry is the number of points the row player wins from the column player. Write down the smallest entry in row 3 (the row minimum).

Worked solution

  1. Write down the pay-off matrix

    M=(5−3−13−20−243)M=\begin{pmatrix} 5 & -3 & -1 \\ 3 & -2 & 0 \\ -2 & 4 & 3 \end{pmatrix}

    Each entry is the row player's gain; the row player picks a row and the column player picks a column.

  2. Look along row 3

    row 3: −2, 4, 3\text{row }3:\ -2,\ 4,\ 3

    The row player, committed to row 3, receives the least of these entries.

  3. Recall what the row minimum represents

    min⁡jM3j\min_j M_{3j}

    It is the pay-off the row player is guaranteed from that row.

  4. State the row minimum

    min⁡jM3j=−2\min_j M_{3j}=-2

    This is the smallest entry in the row.

Answer
min⁡=−2\min=-2
Question 4
2 markseasy
In a two-person zero-sum game the row player chooses a row and the column player chooses a column of the pay-off matrix M=(45−152151−3)M=\begin{pmatrix} 4 & 5 & -1 \\ 5 & 2 & 1 \\ 5 & 1 & -3 \end{pmatrix}, where each entry is the number of points the row player wins from the column player. Write down the smallest entry in row 1 (the row minimum).

Worked solution

  1. Write down the pay-off matrix

    M=(45−152151−3)M=\begin{pmatrix} 4 & 5 & -1 \\ 5 & 2 & 1 \\ 5 & 1 & -3 \end{pmatrix}

    Each entry is the row player's gain; the row player picks a row and the column player picks a column.

  2. Look along row 1

    row 1: 4, 5, −1\text{row }1:\ 4,\ 5,\ -1

    The row player, committed to row 1, receives the least of these entries.

  3. State the row minimum

    min⁡jM1j=−1\min_j M_{1j}=-1

    This is the smallest entry in the row.

Answer
min⁡=−1\min=-1
Question 5
2 markseasy
In a two-person zero-sum game the row player chooses a row and the column player chooses a column of the pay-off matrix M=(30000−335−4)M=\begin{pmatrix} 3 & 0 & 0 \\ 0 & 0 & -3 \\ 3 & 5 & -4 \end{pmatrix}, where each entry is the number of points the row player wins from the column player. Write down the largest entry in column 1 (the column maximum).

Worked solution

  1. Write down the pay-off matrix

    M=(30000−335−4)M=\begin{pmatrix} 3 & 0 & 0 \\ 0 & 0 & -3 \\ 3 & 5 & -4 \end{pmatrix}

    Each entry is the row player's gain; the row player picks a row and the column player picks a column.

  2. Look down column 1

    col 1: 3, 0, 3\text{col }1:\ 3,\ 0,\ 3

    The column player, committed to column 1, may lose as much as the greatest of these entries.

  3. Recall what the column maximum represents

    max⁡iMi1\max_i M_{i1}

    It is the most the column player can lose from that column.

  4. State the column maximum

    max⁡iMi1=3\max_i M_{i1}=3

    This is the largest entry in the column.

Answer
max⁡=3\max=3

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