Hey guys =]

I really need help with this question, I have some answers to make it easier for you to help if you want.

Thanks guys!

Here is the question...

**QUESTION**

The table gives the pay-off matrix for a zero-sum game between two players, Amy and Bea. The values in the table show the pay-offs for Amy.

Code:

Bea
_________| Strategy X | Strategy Y | Strategy Z |
Amy Strategy P | 4 | -2 | 0 |
Strategy Q | -1 | 5 | 4 |

Amy makes a random choice between strategies P and Q, choosing strategy P with probability 'p' and strategy Q with probability '1-p'

*i) Write down and simplify an expression for the expected pay-off for Amy when Bea chooses strategy X. Write down similar expressions for the cases when Cea chooses strategy Y and when she chooses strategy Z.*

*ii) Using graph paper, draw a graph to show Amy's expected pay-off against p for each of Bea's choices of strategy. Using your graph, find the optimal value of p for Amy*

Amy and Bea play the game many times. Amy chooses randomly between her strategies using the optimal value for p.

*iii) Showing your working, calculate Amy's minimum expected pay-off per game. Why might Amy gain more points than this on average?*

*iv) What is Bea's minimum expected loss per game? How should Bea play to minimise her expected loss?*

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**My Correct/Incorrect Answers**

i)E(X)=5p-1

E(Y)=5-7p

E(Z)=4-4p

ii) I drew my graph and found that her maximum value for p occurs when

E(Z)=E(X)

4-4p=5p-1

p=(5/9)

iii)Using the maximum value for p, found above. I used it in the following to find the Expected pay-offs:

E(X)=1.78

E(Y)=1.11

E(Z)=1.78

and found that the minimum expected pay-off is 1.11...

Amy may gain more points than this as Bea may not choose randomly and play strategies X and Z more.

iv) Bea's minimum expected loss is when...

E(X)=E(Y)

5p-1=5-7p

p=0.5

I plug this value for p into the following

E(X)=1.5

E(Y)=1.5

E(Z)=2

Therefore, Bea's minmum expected loss = 1.5

and Bea should play strategies X and Y with probability, p=0.5 .

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I hope this is easy to understand, my apologies for the messy table as I could not get it right here. Thankyou anyone who reads this and more thanks for anyone who helps =]

Thankyou guys =]

Anthmoo