2 The table gives the pay-off matrix for a zero-sum game between two players, A my and Bea. The values in the table show the pay-offs for A my.
| Bea |
| \cline { 3 - 5 } | | Strategy X | Strategy Y | Strategy Z |
| \cline { 2 - 5 } | Strategy P | 4 | - 2 | 0 |
| \cline { 2 - 5 }
A my | Strategy Q | - 1 | 5 | 4 |
| \cline { 2 - 5 } | | | | |
| \cline { 2 - 5 } |
A my makes a random choice between strategies \(\mathbf { P }\) and \(Q\), choosing strategy \(P\) with probability \(p\) and strategy Q with probability \(1 - \mathrm { p }\).
- 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 B ea chooses strategy Y and when she chooses strategy \(Z\).
- Using graph paper, draw a graph to show A my's expected pay-off against p for each of Bea's choices of strategy. Using your graph, find the optimal value of pfor A my.
A my and Bea play the game many times. A my chooses randomly between her strategies using the optimal value for p.
- Showing your working, calculate A my's minimum expected pay-off per game. W hy might A my gain more points than this, on average?
- W hat is B ea's minimum expected loss per game? How should B ea play to minimise her expected loss?