Game Theory

1
Chapter 28

• Game Theory

2

• Three fundamental elements
• to describe a game
• Players,
• (pure) strategies or actions,
• payoffs.

3
Color Matching
B
b r
1, -1
-1, 1
b A r
-1, 1
1, -1
4

• Payoff matrices
• for Two-person games.
• Simultaneous(-move) games.
• Finite games Both the numbers of players and of
  alternative pure strategies are finite

5
The Prisoners Dilemma
B
Confess Deny
Confess A Deny
-3, -3
0, -5
-5, 0
-1, -1
6

• Dominant strategies, and dominated
  strategies.
• Method of iterated elimination of strictly
  dominated strategies.

7

• The Prisoners Dilemma
• shows also that a Nash equilibrium does not
  necessarily lead to a Pareto efficient outcome.
• Two-win games.

8

• A pair of strategies is a
• Nash equilibrium if As choice
• is optimal given Bs choice,
• and vice versa.
• Nash is a situation,
• or a strategy combination of
• no incentive to deviate unilaterally.

9
Battle of Sexes
Girl
Soccer Ballet
2, 1
0, 0
Soccer Boy Ballet
-1, -1
1, 2
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• Method of underlining relatively advantageous
  strategies.
• Double underlining gives Nash.
• There can be no, one, and multiple (pure) Nash
  equilibria.

11
Price Struggle
Pepsi
L H
L Coke H
3, 3
6, 1
1, 6
5, 5
12

• How if there is no Nash
• of pure strategies?
• Mixed strategies
• (by probability).
• Method of response functions.

13
Color Matching again
B
b r
q 1-q
1, -1
-1, 1
b p A r 1-p
-1, 1
1, -1
14

• With
• EUA 1 pq (-1)p (1-q)
• (-1) (1-p) q 1 (1-p) (1-q)
• pq - p pq - q pq 1 - p - q pq
• 4 pq - 2 p - 2 q 1
• 2 p (2 q - 1) (1 - 2 q ) ,
• we have
• 1 if q gt 1/2 ,
• p 0, 1 if q 1/2 ,
• 0 if q lt 1/2 .

15

• Similarly,
• 0 if p gt 1/2 ,
• q 0, 1 if p 1/2 ,
• 1 if p lt 1/2 .

q 1
N
1 p
0
16

• Method of response functions The intersections
  of response functions give Nash equilibria
• ((p, q) (1/2, 1/2) in example)
• Nash Theorem
• There is always a ( maybe mixed) Nash
  equilibrium for any finite game

17

• Sequential games.
• Games in extensive form
• versus
• in normal form.

18

• Battle of Sexes again

Ballet
( 1, 2)
Girl
( -1, -1)
Ballet
Soccer
Boy
Ballet
( 0, 0)
Soccer
( 2, 1)
Soccer
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• Strategies as Plans of Actions.
• Boys strategies Ballet, and Soccer.
• Girls Strategies
• Ballet strategy
• Soccer strategy
• Strategy to follow and
• Strategy to oppose.