Question 4 - A-Level Maths

AQA D2 2009 January — Question 4

Exam Board	AQA
Module	D2 (Decision Mathematics 2)
Year	2009
Session	January
Topic	Dynamic Programming

4

Two people, Raj and Cal, play a zero-sum game. The game is represented by the following pay-off matrix for Raj.
Cal
\cline { 2 - 5 } Strategy X Y Z
Raj I - 7 8 - 5
\cline { 2 - 5 } II 6 2 - 1
\cline { 2 - 5 } III - 2 4 - 3
\cline { 2 - 5 }
\cline { 2 - 5 }
Show that this game has a stable solution and state the play-safe strategy for each player.

Ros and Carly play a different zero-sum game for which there is no stable solution. The game is represented by the following pay-off matrix for Ros, where \(x\) is a constant.

	Carly
\cline { 2 - 4 }	Strategy		\(\mathbf { C } _ { \mathbf { 1 } }\)
\cline { 2 - 4 }\cline { 2 - 3 } \(\operatorname { Ros }\)	\(\mathbf { R } _ { \mathbf { 1 } }\)	5	\(\mathbf { C } _ { \mathbf { 2 } }\)
\cline { 2 - 4 }	\(\mathbf { R } _ { \mathbf { 2 } }\)	- 2	\(x\)
\cline { 2 - 4 }			4

Ros chooses strategy \(\mathrm { R } _ { 1 }\) with probability \(p\).

Find expressions for the expected gains for Ros when Carly chooses each of the strategies \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\).
Given that the value of the game is \(\frac { 8 } { 3 }\), find the value of \(p\) and the value of \(x\).

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