\ B A \

1 2

1 3 0

2 -1 2

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?

3 0-1 2

3 0-1 2

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 Equilibrium Golden Balls

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 =

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

1 -2 -3 2

How about B’s strategy?

\ B A \

1 2

1 2 1

2 -1 2

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?

2 1-1 2

2 1 -2 1 -1 2 -1 -2

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) 27th March 2012

Activity

Topic assessment

Nash Equilibrium

A Beautiful Mind

plenary

“pure and mixed strategies”

Activity

Exercise 5BPages 86-87

Q3,4Extension: Q5

Nash Equilibrium