Objectives:

• Find the value for 2 x n games and analyse strategies.

• To understand and apply dominance to reduce pay-off matrices.

• To graphically represent pay-offs for 2 x n games.

Mixed Strategies

A Beautiful Mind

Survival

3 men are on a hill, only 1 can come down alive.

They are armed with pistols and honourable.

They will take it in turns to shoot at 1 of their adversaries or fire into the air.

The 1st man has a 1/6 probability of shooting and killing.

The 2nd man a ½ probability.

The 3rd man a 5/6 probability.

Q1) Who would you prefer to be?

Q2) Who is most likely to die first?

Q3)Who is most likely to remain alive?

Q4) If you were the 1st man what would your tactics be?

Q5) Would you like to change your choice for Q1?

Bilborough College Maths – Decision 2 game theory : mixed strategies (Adrian) 19/04/13

A less risky version

• Model the survival game using a six-faced die numbered 1, 2, 3, 4, 5, 6.

• Player A needs 6 to kill.• Player B needs ≥ 4 to kill.• Player C needs ≥ 2 to kill.• Players have option of throwing or passing

each time their turn comes around.• Play in groups of 3 and tally results.

Game Player A : 1,2,S Player B: 1,2,S Player C: 1,2,S

1

2

3

4

5

6

7

8

9

10

11

12

TOTAL 1= , 2= , S= 1= , 2= , S= 1= , 2= , S=

Game Player A : 1,2,S Player B: 1,2,S Player C: 1,2,S

1

2

3

4

5

6

7

8

9

10

11

12

TOTAL 1= , 2= , S= 1= , 2= , S= 1= , 2= , S=

Objectives:

• Find the value for 2 x n games and analyse strategies.

• To understand and apply dominance to reduce pay-off matrices.

• To graphically represent pay-offs for 2 x 2 games.

Mixed Strategies

A Beautiful Mind

Pay-off matrix for player A

2 -1 3

0 2 -2

A’s expected pay-off

Finding the value

2-3p = 5p -2

Value (v) = (-1) x + 2 x (1 - ) =

v = 3 x + (-2) x (1 - ) =

P = V =

Activity

Exercise 5B

Pages 86-87

Q1,3,4

Nash Equilibrium

Golden Balls

Activity

Finish

Multi guess worksheet

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

2-3p = 5p -2

Value (v) = (-1) x + 2 x (1 - ) =

v = 3 x + (-2) x (1 - ) =

P = V =

Player B

2 -1 3 0 2 -2

-1 3 2 -2

How can we find the value of the game with pay-off matrix -2 0 ?

1 -2

-3 2