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 Prisoner’s DilemmaB
Confess Deny
Confess
A Deny
-3*, -3*
0, -5
-5, 0
-1, -1

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

7. The Prisoner’s 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 A’s choice
is optimal given B’s 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
2*, 1*
0, 0
Soccer
Boy Ballet
-1, -1
1*, 2*

10. 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 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 p - q + pq
= 4 pq - 2 p - 2 q + 1
= 2 p (2 q - 1) + (1 - 2 q ) ,
we have
if q > 1/2 ,
p = [0, 1] if q = 1/2 ,
if q < 1/2 .

15. Similarly,
if p > 1/2 ,
q = [0, 1] if p = 1/2 ,
if p < 1/2 . q 1 N p

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
( 0, 0)
Soccer
( 2, 1)
Soccer

19. Strategies as Plans of Actions.
Boy’s strategies: Ballet, and Soccer.
Girl’s Strategies:
Ballet strategy;
Soccer strategy;
Strategy to follow; and
Strategy to oppose.