7.08d Nash equilibrium: identification and interpretation

80 questions

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AQA Further AS Paper 2 Discrete 2024 June Q10
7 marks Challenging +1.2
Bilal and Mayon play a zero-sum game. The game is represented by the following pay-off matrix for Bilal, where \(x\) is an integer.
Mayon
\(\mathbf{M_1}\)\(\mathbf{M_2}\)\(\mathbf{M_3}\)
\(\mathbf{B_1}\)\(-2\)\(-1\)\(1\)
Bilal \quad \(\mathbf{B_2}\)\(4\)\(-3\)\(1\)
\(\mathbf{B_3}\)\(-1\)\(x\)\(0\)
The game has a stable solution.
  1. Show that there is only one possible value for \(x\) Fully justify your answer. [6 marks]
  2. State the value of the game for Bilal. [1 mark]
AQA Further Paper 3 Discrete 2022 June Q4
6 marks Standard +0.3
Ben and Jadzia play a zero-sum game. The game is represented by the following pay-off matrix for Ben.
Jadzia
StrategyXYZ
A-323
Ben B60-4
C7-11
D6-21
  1. State, with a reason, which strategy Ben should never play. [1 mark]
  2. Determine whether or not the game has a stable solution. Fully justify your answer. [3 marks]
  3. Ben knows that Jadzia will always play her play-safe strategy. Explain how Ben can maximise his expected pay-off. [2 marks]
AQA Further Paper 3 Discrete 2024 June Q4
4 marks Standard +0.8
Daniel and Jackson play a zero-sum game. The game is represented by the following pay-off matrix for Daniel. Jackson
StrategyWXYZ
\multirow{4}{*}{Daniel}A3\(-2\)14
B51\(-4\)1
C2\(-1\)12
D\(-3\)02\(-1\)
Neither player has any strategies which can be ignored due to dominance.
  1. Prove that the game does not have a stable solution. Fully justify your answer. [3 marks]
  2. Determine the play-safe strategy for each player. [1 mark] Play-safe strategy for Daniel _______________________________________________ Play-safe strategy for Jackson ______________________________________________
OCR Further Discrete 2018 March Q7
8 marks Challenging +1.2
Each day Alix and Ben play a game. They each choose a card and use the table below to find the number of points they win. The table shows the cards available to each player. The entries in the cells are of the form \((a, b)\), where \(a =\) points won by Alix and \(b =\) points won by Ben. Each is trying to maximise the points they win.
Ben
\cline{2-4} \multicolumn{1}{c}{}Card XCard YCard Z
\cline{2-4} \multirow{3}{*}{Alix}
Card P(4, 4)(5, 9)(1, 7)
\cline{2-4} Card Q(3, 5)(4, 1)(8, 2)
\cline{2-4} Card R\((x, y)\)(2, 2)(9, 4)
\cline{2-4}
  1. Explain why the table cannot be reduced through dominance no matter what values \(x\) and \(y\) have. [2]
  2. Show that the game is not stable no matter what values \(x\) and \(y\) have. [2]
  3. Find the Nash equilibrium solutions for the various values that \(x\) and \(y\) can have. [4]
OCR Further Discrete 2017 Specimen Q4
11 marks Standard +0.8
The table shows the pay-off matrix for player \(A\) in a two-person zero-sum game between \(A\) and \(B\).
Player \(B\)
Strategy \(X\)Strategy \(Y\)Strategy \(Z\)
Player \(A\) Strategy \(P\)45\(-4\)
Player \(A\) Strategy \(Q\)3\(-1\)2
Player \(A\) Strategy \(R\)402
  1. Find the play-safe strategy for player \(A\) and the play-safe strategy for player \(B\). Use the values of the play-safe strategies to determine whether the game is stable or unstable. [3]
  2. If player \(B\) knows that player \(A\) will use their play-safe strategy, which strategy should player \(B\) use? [1]
  3. Suppose that the value in the cell where both players use their play-safe strategies can be changed, but all other entries are unchanged. Show that there is no way to change this value that would make the game stable. [2]
  4. Suppose, instead, that the value in one cell can be changed, but all other entries are unchanged, so that the game becomes stable. Identify a suitable cell and write down a new pay-off value for that cell which would make the game stable. [2]
  5. Show that the zero-sum game with the new pay-off value found in part (iv) has a Nash equilibrium and explain what this means for the players. [3]