Game theory

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

Verify that there is no stable solution to this game.

Let player A play option X with probability p

Use a graphical method to find the optimal value of p and hence find the best strategy for player A in this game.

A third option, Z, is added to player A’s options. Option Z has the pay-offs shown in the matrix below.

Player A now intends to make a random choice between options X, Y and Z, choosing option X with probability x, option Y with probability y and option Z with probability z

Player A decides to use the Simplex algorithm to find the optimal values of x, y and z

Determine an initial Simplex tableau to solve this 33 game, making your variables clear.

In the optimal Simplex tableau for this 3×3 game, x = 0 and y =

By considering the values of the two games, determine how much better the 33 game (with option Z) is for player A compared with the 2×3 game (without option Z). You must make your reasoning clear.

A two person zero‑sum game is represented by the pay‑off matrix for player A shown above. It is given that k is an integer.

Show that Q is the play‑safe option for player A regardless of the value of k.

Given that Z is the play‑safe option for player B,

determine the range of possible values of k. You must make your working clear.

Explain why player B should never play option X. You must make your reasoning clear.

Player A intends to make a random choice between options Q, R and S, choosing option Q with probability p₁, option R with probability p₂ and option S with probability p₃

Player A wants to find the optimal values of p₁, p₂ and p₃ using the Simplex algorithm.

Given that k –4, player A formulates the following objective function for the corresponding linear program.

Maximise P = V, where V = the value of the original game + 4

Formulate the constraints of the linear programming problem for player A. You should write the constraints as equations.

Write down an initial Simplex tableau, making your variables clear.

The Simplex algorithm is used to solve the linear programming problem. It is given that in the final Simplex tableau the optimal value of p = , the optimal value of p = and all the slack variables are zero.

Determine the value of k, making your method clear.

Alexis and Becky are playing a zero‑sum game.

Alexis has two options, Q and R. Becky has three options, X, Y and Z.

Alexis intends to make a random choice between options Q and R, choosing option Q with probability p₁ and option R with probability p₂ Alexis wants to find the optimal values of p₁ and p₂ and formulates the following linear programme, writing the constraints as inequalities.

Maximise P = V

where V = 3 + the value of the game to Alexis

subject=to=$V 6p₁ + p₂

Complete the pay‑off matrix for Alexis in the answer book.

Use a graphical method to find the best strategy for Alexis.

Calculate the value of the game to Alexis.

Becky intends to make a random choice between options X, Y and Z, choosing option X with probability q₁, option Y with probability q₂ and option Z with probability q₃

Determine the best strategy for Becky, making your method and working clear.

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

Explain, with justification, why this matrix may be reduced to a 3 × 3 matrix by removing option S from player A’s choices.

Verify that there is no stable solution to the reduced game.

Player A intends to make a random choice between options Q, R and T, choosing option Q with probability p₁, option R with probability p₂ and option T with probability p₃ Player A wants to find the optimal values of p₁, p₂ and p₃ using the Simplex algorithm. Player A formulates the following linear programme, writing the constraints as inequalities.

Maximise P=V, where V= the value of original game + 3 subject to V ≤4p₁+ 7p₂+ 6p₃ V 8p₁+p₃ V 6p₁+ 4p₂+ 3p₃ p₁+p₂+p₃ 1 p₁ 0, p₂ 0, p₃ 0, V 0

Explain why V cannot exceed any of the following expressions

Explain why it is necessary to use the constraint p₁+p₂+p₃ ≤1

The Simplex algorithm is used to solve the linear programming problem. Given that the optimal value of p = and the optimal value of p = 0

calculate the value of the game to player A.

Player B intends to make a random choice between options X, Y and Z, choosing option X with probability q₁, option Y with probability q₂ and option Z with probability q₃

Determine the optimal strategy for player B, making your working clear.

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

Verify that there is no stable solution to this game.

Player A intends to make a random choice between options P, Q and R, choosing option P with probability p₁, option Q with probability p₂ and option R with probability p₃

Player A wants to find the optimal values of p₁, p₂ and p₃ using the Simplex algorithm. Player A formulates the following linear programming problem for the game, writing the constraints as inequalities.

Maximise P = V

Correct the errors made by player A in the linear programming formulation of the game, giving reasons for your answer.

Write down an initial Simplex tableau for the corrected linear programming problem.

The Simplex algorithm is used to solve the corrected linear programming problem.

The optimal values are p₁ = 0.6, p₂ = 0 and p₃= 0.4

Calculate the value of the game to player A.

Determine the optimal strategy for player B, making your reasoning clear.