Games with Simultaneous Moves I :Discrete Strategies

1
Games with Simultaneous Moves IDiscrete
Strategies
2
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

3
Games with Simultaneous Moves

• Simultaneous moves?
• Strategy vs. action
• Discrete/Continuous strategy

4

• Games in normal (strategic) form.
• Games Table/ Game Matrix/ Payoff Table

Column Column Column
Left Middle Right
Row Top 3, 1 2, 3 10, 2
Row High 4, 5 3, 0 6, 4
Row Low 2, 2 5, 4 12, 3
Row Bottom 5, 6 4, 5 9, 7
5

• Zero-sum Game

Column Column Column
Run Pass Blitz
OFFENSE Run 2 5 13
OFFENSE Short Pass 6 5.6 10.5
OFFENSE Medium Pass 6 4.5 1
OFFENSE Long Pass 10 3 -2
6
Nash Equilibrium

• What is Equilibrium?
• Cell-by-cell-inspection or enumeration
• Best Response Analysis

Column Column Column
Left Middle Right
Row Top 3, 1 2, 3 10, 2
Row High 4, 5 3, 0 6, 4
Row Low 2, 2 5, 4 12, 3
Row Bottom 5, 6 4, 5 9, 7
7
Column Column Column
Left Middle Right
Row Top 3, 1 2, 3 10, 2
Row High 4, 5 3, 0 6, 4
Row Low 2, 2 5, 4 12, 3
Row Bottom 5, 6 4, 5 9, 7
8

• 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.

9

• A N.E. does not have to be jointly best for the
  players.
• EX Prisoners dilemma

WIFE WIFE
Confess (Defect) Deny (Cooperate)
HUSBAND Confess (Defect) 10yr, 10yr 1yr, 25yr
HUSBAND Deny (Cooperate) 25yr, 1yr 3yr, 3yr
10

• 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

11
Dominance
WIFE WIFE
Confess (Defect) Deny (Cooperate)
HUSBAND Confess (Defect) 10yr, 10yr 1yr, 25yr
HUSBAND Deny (Cooperate) 25yr, 1yr 3yr, 3yr

• Dominant strategy is an action clearly best for a
  player, no matter what the others might be doing.

• Confess is a dominant strategy for the husband,
  while Confess is also a dominant strategy for
  the wife. NE (Confess, Confess) ?(10yr, 10yr)

12

• (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.

13

• Both (all) Players Have Dominant Strategies
• One Player has a Dominant Strategy

FED FED
Low interest rate High interest rate
CONGRESS Budget balance 3, 4 1, 3
CONGRESS Budget deficit 4, 1 2, 2
14

• 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
15
MiniMax Method for Zero-Sum Games

• MiniMax/Maximin

Column Column Column
Run Pass Blitz
OFFENSE Run 2 5 13
OFFENSE Short Pass 6 5.6 10.5
OFFENSE Medium Pass 6 4.5 1
OFFENSE Long Pass 10 3 -2
16
Three Players
TALIA chooses C
TALIA chooses NC
NINA NINA
C NC
EMILY C 5, 5, 5 3, 6, 3
EMILY NC 6, 3, 3 4, 4, 1
NINA NINA
C NC
EMILY C 3, 3, 6 1, 4, 4
EMILY NC 4, 1, 4 2, 2, 2
17
Three Players
TALIA chooses D
TALIA chooses ND
NINA NINA
D ND
EMILY D 2, 7, 2 3, 2, 2
EMILY ND 5, 1, 2 2, 5, 2
NINA NINA
D ND
EMILY D 3, 3, 3 1, 2, 3
EMILY ND 2, 1, 1 2, 2, 1
18
Multiple Equilibria in Pure Strategies

• Coordination Games
• Battle of the Sexes

SALLY SALLY
Starbucks Banyan
HARRY Starbucks 2, 1 0, 0
HARRY Banyan 0, 0 1, 2
19

• Focal Point
• Convergence of expectation
• Assurance Games

DEAN DEAN
Swerve (Chicken) Straight (Tough)
JAMES Swerve (Chicken) 0, 0 -1, 1
JAMES Straight (Tough) 1, -1 -2, -2
20
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

21

• Homework, Ch4
• question 3, 6, 11, and 12(a b)

22
Tree form into a strategic form
L
(3, 1)
U
(2, 2)
R
2
1
D
L
(1, 3)
R
(4, 1)
2
LL LR RL RR
U
D
LL LR RL RR
U 3, 1 3, 1 2, 2 2, 2
D 1, 3 4, 1 1, 3 4, 1
23

• Minimax/Maximin vs. Nash Equilibrium Revisited