1. \ B A \ 12 130 22 Draw a graph to show the expected pay-off for A. What is the value of the game. How often should A choose strategy 1? If A adopts a mixed strategy what should B do?

2. 3 0 -1 2

4. 3 0 -1 2

5. Objectives: Find the value for 2 x n games for Player A and Player B and analyse strategies. To understand and apply dominance to reduce pay-off matrices. To graphically represent pay-offs for 2 x n games. To begin to consider how to find mixed strategies for both players in mxn games. Mixed Strategies Nash EquilibriumGolden Balls

6. Pay-off matrix for player A 2 -1 3 0 2 -2

7. A’s expected pay-off

8. Finding the value 2-3p = 5p -2 Value (v) = (-1) x + 2 x (1 - ) = v = 3 x + (-2) x (1 - ) = P = V =

9. How can we find the value of the game with pay-off matrix -2 0 ? 1 -2 -3 2 How about B’s strategy?

10. \ B A \ 12 121 22 Draw a graph to show the expected pay-off for A. If A adopts a mixed strategy what should B do? What is the value of the game. How often should A choose strategy 1?

11. 2 1 -1 2

13. 2 1 -2 1 -1 2 -1 -2

15. Dominance 1 2.5 2 5 7 -0.5 8 -3 6 -1 4 -5 0.2 0.6 0.1 0.3 0.5 0.6 0.1 0.4 0.4 Bilborough College Maths – Decision 2 Game Theory: value of 2 x n games (Adrian) 27 th March 2012

16. Activity Topic assessment Nash Equilibrium A Beautiful Mind

17. plenary “pure and mixed strategies”

18. Activity Exercise 5B Pages 86-87 Q3,4 Extension: Q5 Nash Equilibrium