Date post: | 17-Dec-2015 |
Category: |
Documents |
Upload: | kathryn-poole |
View: | 224 times |
Download: | 0 times |
Games with Simultaneous Moves I:Discrete Strategies
Outline
Games with simultaneous moves Nash Equilibrium Dominance Minimax in Zero-sum Game Three Players Multiple/Zero Equilibria in pure strategies Tree Forms into Strategic Forms
Games with Simultaneous Moves Simultaneous moves?
Strategy vs. action
Discrete/Continuous strategy
Games in normal (strategic) form. Games Table/ Game Matrix/ Payoff Table
Column
Left Middle Right
Row
Top 3, 1 2, 3 10, 2
High 4, 5 3, 0 6, 4
Low 2, 2 5, 4 12, 3
Bottom 5, 6 4, 5 9, 7
Zero-sum Game
Column
Run Pass Blitz
OFFENSE
Run 2 5 13
Short Pass 6 5.6 10.5
Medium Pass 6 4.5 1
Long Pass 10 3 -2
Nash Equilibrium What is Equilibrium? Cell-by-cell-inspection or enumeration Best Response Analysis
Column
Left Middle Right
Row
Top 3, 1 2, 3 10, 2
High 4, 5 3, 0 6, 4
Low 2, 2 5, 4 12, 3
Bottom 5, 6 4, 5 9, 7
Column
Left Middle Right
Row
Top 3, 1 2, 3 10, 2
High 4, 5 3, 0 6, 4
Low 2, 2 5, 4 12, 3
Bottom 5, 6 4, 5 9, 7
The N.E is (Low, Middle) →(5, 4)
A Nash Equilibrium in a game is a list of strategies, one for each player, such that no player can get a better payoff by switching to some other strategy that is available to her while all the other players adhere to the strategies specified for them in the list.
A N.E. does not have to be jointly best for the players.
EX: Prisoner’s dilemma
WIFE
Confess
(Defect)
Deny
(Cooperate)
HUSBAND
Confess
(Defect)10yr, 10yr 1yr, 25yr
Deny
(Cooperate)25yr, 1yr 3yr, 3yr
Nash Equilibrium as a system of beliefs
Nash Equilibrium is a set of strategies, such that(1)each player has correct beliefs about the strategies of the others(2)the strategy of each is the best for herself, given her beliefs about the strategies of the others
Dominance
Dominant strategy is an action clearly best for a player, no matter what the others might be doing.
WIFE
Confess
(Defect)
Deny
(Cooperate)
HUSBAND
Confess
(Defect)10yr, 10yr 1yr, 25yr
Deny
(Cooperate)25yr, 1yr 3yr, 3yr
“Confess” is a dominant strategy for the husband, while “Confess” is also a dominant strategy for the wife. NE: (Confess, Confess) →(10yr, 10yr)
(A,B,…..) are strategies for Player 1. A is a dominant strategy for Player 1. A dominates B/C/D…. B/C/D… is a dominated strategy for Player 1.
Both (all) Players Have Dominant Strategies One Player has a Dominant Strategy
FED
Low interest rate
High interest rate
CONGRESS
Budget balance 3, 4 1, 3
Budget deficit 4, 1 2, 2
Successive (Iterated) Elimination of Dominated Strategies
Left Middle Right
Up 5, 2 2, 3 3, 4
Medium 4, 1 3, 2 4, 0
Down 3, 3 1, 2 2, 2
MiniMax Method for Zero-Sum Games MiniMax/Maximin
Column
Run Pass Blitz
OFFENSE
Run 2 5 13
Short Pass 6 5.6 10.5
Medium Pass 6 4.5 1
Long Pass 10 3 -2
Three Players
NINA
C NC
EM
ILY
C 5, 5, 5 3, 6, 3
NC 6, 3, 3 4, 4, 1
TALIA chooses C
NINA
C NC
EM
ILYC 3, 3, 6 1, 4, 4
NC 4, 1, 4 2, 2, 2
TALIA chooses NC
Three Players
NINA
D ND
EM
ILY
D 2, 7, 2 3, 2, 2
ND 5, 1, 2 2, 5, 2
TALIA chooses D
NINA
D ND
EM
ILYD 3, 3, 3 1, 2, 3
ND 2, 1, 1 2, 2, 1
TALIA chooses ND
Multiple Equilibria in Pure Strategies Coordination Games Battle of the Sexes
SALLY
Starbucks Banyan
HARRY
Starbucks 2, 1 0, 0
Banyan 0, 0 1, 2
Focal Point Convergence of expectation Assurance Games
DEAN
Swerve
(Chicken)
Straight
(Tough)
JAMES
Swerve
(Chicken)0, 0 -1, 1
Straight
(Tough)1, -1 -2, -2
No Equilibrium in Pure Strategies Rock-Paper-Scissors
Rock Paper Scissors
Rock 0, 0 -1, 1 1, -1
Paper 1, -1 0, 0 -1, 1
Scissors -1, 1 1, -1 0, 0
N.E exists in mixed Strategies
Homework, Ch4 question 3, 6, 11, and 12(a & b)
Tree form into a strategic form
1U
D
L
R
L
R
(3, 1)
(2, 2)
LL LR RL RR
U
D
(1, 3)
(4, 1)
2
2
LL LR RL RR
U 3, 1 3, 1 2, 2 2, 2
D 1, 3 4, 1 1, 3 4, 1
Minimax/Maximin vs. Nash Equilibrium Revisited