A2 June 2024 Q7

EdexcelCurrent spec13 marksGame Theory

7.

Player B
Option XOption YOption Z
Player AOption R\(3\)\(2\)\(-3\)
Option S\(4\)\(-2\)\(1\)
Option T\(-1\)\(3\)\(6\)

A two person zero-sum game is represented by the pay-off matrix for player A, shown above.

(a) Verify that there is no stable solution to this game. (2)

Player A intends to make a random choice between options R, S and T, choosing option R with probability \(p_1\), option S with probability \(p_2\) and option T with probability \(p_3\)

Player A wants to find the optimal values of \(p_1\), \(p_2\) and \(p_3\) using the Simplex algorithm.

Player A formulates the following objective function for the corresponding linear programme.

\[\text{Maximise } P = V \qquad \text{where } V = \text{the value of the game} + 3\]
(b) Determine an initial Simplex tableau, making your variables and working clear. (5)

After several iterations of the Simplex algorithm, a possible final tableau is

b.v.\(V\)\(p_1\)\(p_2\)\(p_3\)\(r\)\(s\)\(t\)\(u\)Value
\(p_3\)\(0\)\(0\)\(0\)\(1\)\(\dfrac{1}{10}\)\(-\dfrac{3}{80}\)\(-\dfrac{1}{16}\)\(\dfrac{33}{80}\)\(\dfrac{33}{80}\)
\(p_2\)\(0\)\(0\)\(1\)\(0\)\(-\dfrac{1}{10}\)\(\dfrac{13}{80}\)\(-\dfrac{1}{16}\)\(\dfrac{17}{80}\)\(\dfrac{17}{80}\)
\(V\)\(1\)\(0\)\(0\)\(0\)\(\dfrac{1}{2}\)\(\dfrac{5}{16}\)\(\dfrac{3}{16}\)\(\dfrac{73}{16}\)\(\dfrac{73}{16}\)
\(p_1\)\(0\)\(1\)\(0\)\(0\)\(0\)\(-\dfrac{1}{8}\)\(\dfrac{1}{8}\)\(\dfrac{3}{8}\)\(\dfrac{3}{8}\)
\(P\)\(0\)\(0\)\(0\)\(0\)\(\dfrac{1}{2}\)\(\dfrac{5}{16}\)\(\dfrac{3}{16}\)\(\dfrac{73}{16}\)\(\dfrac{73}{16}\)
(c)
(i) State the best strategy for player A.
(ii) Calculate the value of the game for player B. (3)

Player B intends to make a random choice between options X, Y and Z.

(d) Determine the best strategy for player B, making your method and working clear. (3)
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