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