1. Game Theory
Lecture 4
Game Theory
Lecture 4

2. An Introduction to Game theory
problem set 4
from M. Osborne’s
An Introduction to Game theory
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3. 3 , 3 0 , 6 6 , 0 1 , 1 C Prisoners’ Dilemma C cooperate
Some (classical) examples of simultaneous games
Prisoners’ Dilemma C
Cooperate D
defect
C cooperate
3 , 3
0 , 6
6 , 0
1 , 1
The ‘D strategy strictly dominates the C strategy

4. G1(s1,t) > G1(s2,t) G1(s1,t) ≥ G1(s2,t) G1(s1,t) > G1(s2,t)
Strategy s1 strictly dominates strategy s2
if for all strategies t of the other player
G1(s1,t) > G1(s2,t)
Strategy s1 weakly dominates strategy s2
if for all strategies t of the other player
G1(s1,t) ≥ G1(s2,t)
and for some t
G1(s1,t) > G1(s2,t)
weakly

5. Successive deletion of
dominated strategies
example of strict dominance X
1 , 5
2 , 3
7 , 4
3 , 3
4 , 7
5 , 2 X
Nash Equilibrium

6. ???
weak dominance
an example
1 , 0
1 , 4
1 , 2
0 , 3
2 , 3 X ? ?

7. Successive deletion of dominated strategies And Sub-game Perfectness
Example 2 1
0 , 3
1 , 4
1 , 2
1 , 0
2 , 3 1 l r 2 1 2

8. rr lr rl ll
1 , 0
1 , 4
1 , 2
0 , 3
2 , 3 rr lr rl ll 2 1
0 , 3
1 , 4
1 , 2
1 , 0
2 , 3 r l

9. rr lr rl ll rr lr rl ll ( r,l ) ( l,l ) 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3
Sub-game perfect equiibrium
0 , 3
1 , 2
1 , 4 1 2
( r,l )
( l,l )
2 , 3
1 , 0

10. rr lr rl ll rr lr rl ll rr lr rl ll rr lr rl ll rr lr rl ll X X X X X
1 , 0
1 , 4
1 , 2
0 , 3
2 , 3 rr lr rl ll
1 , 0
1 , 4
1 , 2
0 , 3
2 , 3 rr lr rl ll
1 , 0
1 , 4
1 , 2
0 , 3
2 , 3 rr lr rl ll
1 , 0
1 , 4
1 , 2
0 , 3
2 , 3 rr lr rl ll
1 , 0
1 , 4
1 , 2
0 , 3
2 , 3 X X
weakly dominating 1 1 l r 2 2 2 1 X X X
0 , 3
1 , 2
1 , 4 1 2
( x , l )
( x , l )
2 , 3
1 , 0
delete ( x , r )
delete ( x , r )

11. Another example of a simultaneous game
The Stag Hunt
Stag
Hare
2 , 2
0 , 1
1 , 0
1 , 1
A generalization to n person game:
There are n types of stocks. Stock of type k yields
payoff k if at least k individuals chose it,
otherwise it yields 0.

12. Another example of a simultaneous game
The Stag Hunt
Stag
Hare
2 , 2
0 , 1
1 , 0
1 , 1
Stag
Hare
2 , 2
0 , 1
1 , 0
1 , 1
Equilibria
payoff dominant equilibrium
risk dominant equilibrium

13. Change of payoffs
Stag
Hare
2 , 2
0 , 1.99
1.99 , 0
1.99 ,1.99

14. Bach or Stravinsky (BOS)
Yet another example of a simultaneous game
Battle of the sexes
Bach or Stravinsky (BOS)
Ballet
Boxing
2 , 1
0 , 0
1 , 2
Ballet
Boxing
2 , 1
0 , 0
1 , 2
woman
man
Equilibria

15. Battle of the sexes woman man 2 , 1 0 , 0 1 , 2
Yet another example of a simultaneous game
Battle of the sexes
Ballet
Boxing
2 , 1
0 , 0
1 , 2
woman
man
A generalization to a bargaining situation

16. Nash Demand Game Two players divide a Dollar.
Each demands an amount ≥ 0.
Each receives his demand if the
total amount demanded is ≤ 1.
Otherwise they both get 0.
Demand of player 2
Demand of player 1 1

17. Nash Demand Game equilibria a continuum of equilibria
Demand of player 2
Demand of player 1 1

18. Matching Pennies 1 , -1 -1 , 1 last example of a simultaneous game
head
tails
1 , -1
-1 , 1
no pure strategies equilibrium exists

19. Matching Pennies 1 , -1 -1 , 1 last example of a simultaneous game
head
tails
1 , -1
-1 , 1
no pure strategies equilibrium exists

20. Mixed strategies Matching Pennies 1 , -1 -1 , 1
last example of a simultaneous game
Matching Pennies
Mixed strategies
head
tails
1 , -1
-1 , 1
A player may choose
head with probability 
and tails with probability 1- 
no pure strategies equilibrium exists

21. his payoff is the lottery:
last example of a simultaneous game
Mixed strategies
Matching Pennies
1-
head 
tails
1 , -1
-1 , 1
player 2 mixes:
head
tails
1 , -1
-1 , 1
if player 1 plays ‘head’
his payoff is the lottery:
if the payoffs are in terms of his vN-M utility
then his utility from the lottery is

22.  1-  Mixed strategies Matching Pennies 1 , -1 -1 , 1 1 , -1 -1 , 1
last example of a simultaneous game
Mixed strategies
Matching Pennies
1- 
head 
tails
1 , -1
-1 , 1
player 2 mixes:
head
tails
1 , -1
-1 , 1
Similarly, if player 1 plays ‘tails’ his payoff is …….

23. When β = 0.5 player 1 is indifferent between the two strategies
last example of a simultaneous game
Mixed strategies
Matching Pennies
1- 
head
 tails
1 , -1
-1 , 1
tails
player 2 mixes:
head
tails
1 , -1
-1 , 1
He prefers to play ‘head’ if:
He prefers to play ‘tails’ if:
When β = 0.5 player 1 is indifferent
between the two strategies
and any mix of the two

24. β (1-α , α) α Player 1 prefers to play ‘head’ if: Player 2’s
Best Response
function ??
Player 1 prefers to play ‘tails’ if:
Player 1 is indifferent when
player 2’s mix 1
Player 1’s
Best Response
function β
(1-α , α)
player 1’s mix
tails
head α 1

25. β α When player 1 plays ‘head’ often Player 2 prefers to play ‘tails’
Player 2’s
Best Response
function ??
Player 2’s
Best Response
function
Nash equilibium
α = β= 0
player 2’s mix 1
Player 1’s
Best Response
function β
player 1’s mix α 1

26. 1/2
head
1/2 tails
1 , -1
-1 , 1
tails

27. Exercises from M. Osborne’s An Introduction to Game Theory
EXERCISE 30.1 (Variants of the Stag Hunt) Consider variants of the n-hunter Stag Hunt in which only m hunters, with 2 ≤ m < n, need to pursue the stag in order to catch it. (Continue to assume that there is a single stag.) Assume that a captured stag is shared only by the hunters who catch it. Under each of the following assumptions on the hunters’ preferences, find the Nash equilibria of the strategic game that models the situation.
As before, each hunter prefers the fraction 1 / n of the stag to a hare
Each hunter prefers the fraction 1 / k of the stag to a hare, but prefers a hare to any smaller fraction of the stag, where k is an integer with m ≤ k ≤ n.
The following more difficult exercise enriches the hunters’ choices in the Stag Hunt. This extended game has been proposed as a model that captures Keynes’ basic insight about the possibility of multiple economic equilibria, some of which are undesirable (Bryant 1983, 1994).
Next exercise

28. EXERCISE 31.1 (Extension of the Stag Hunt) Extend the n-hunter Stag Hunt by giving each hunter K (a positive integer) units of effort, which she can allocate between pursuing the stag and catching hares. Denote the effort hunter i devotes to pursuing the stag by ei , a nonnegative integer equal to at most K. The chance that the stag is caught depends on the smallest of all the hunters’ efforts, denoted minj ej. (“A chain is as strong as its weakest link.”) Hunter i’s payoff to the action profile (e , en ) is 2minjej -ei . (She is better off the more likely the stag is caught, and worse off the more effort she devotes to pursuing the stag, which means the catches fewer hares.) Is the action profile (e, e), in which every hunter devotes the same effort to pursuing the stag, a Nash equilibrium for any value of e? (What is a player’s payoff to this profile? What is her payoff if she deviates to a lower or higher effort level?) Is any action profile in which not all the players’ effort levels are the same a Nash equilibrium? (Consider a player whose effort exceeds the minimum effort level of all players. What happens to her payoff if the reduces her effort level to the minimum?)
Next exercise

29. Next exercise 2.7.5 Hawk-Dove
The Game in the next exercise captures a basic feature of animal conflict.
EXERCISE 31.2 (Hawk-Dove) Two animals are fighting over some prey. Each can be passive or aggressive. Each prefers to be aggressive if its opponent is passive, and passive if its opponent is aggressive; given its own stance, it prefers the outcome in which its opponent is passive to that in which its opponent is aggressive. Formulate this situation as a strategic game and find its Nash equilibria.
Next exercise

30. EXERCISE 34.1 (Guessing two-thirds of the average) Each of three people announces an integer from 1 to K. If the three integers are different, the person whose integer is closest to 2/3 of the average of the three integers wins $1. If two or more integers are the same , $1 is split equally between the people whose integer is closest to 2/3 of the average integer. Is there any integer k such that the action profile (k,k,k), in which every person announces the same integer k, is a Nash equilibrium? (If k ≥ 2, what happens if a person announces a smaller number?) Is any other action profile a Nash equilibrium? (What is the payoff of a person whose number is the highest of the three? Can she increase this payoff by announcing a different number?)
Last excercise

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Game theory is used widely in political science, especially in the study of elections. The game in the following exercise explores citizens’ costly decisions to vote.
EXERCISE 34.2 (Voter participation) Two candidates , A and B, compete in an election. Of the n citizen, k support candidate A and m (= n - k) support candidate B. Each citizen decides whether to vote, at a cost, for the candidate she supports, or to abstain. A citizen who abstains receives the payoff of 2 if the candidate she supports wins, 1 if this candidate ties for first place , and 0 if this candidate loses. A citizen who votes receives the payoffs 2 - c, 1 - c, and -c in these three cases, where 0 < c < 1.
For k = m = 1, is the game the same (except for the names of the actions) as any considered so far in this chapter?
For k = m, find the set of Nash equilibria. (Is the action profile in which everyone votes a Nash equilibrium? Is there any Nash equilibrium in which one of the candidates wins by one vote? Is there any Nash equilibrium in which one of the candidates wins by two or more votes?)
What is the set of Nash equilibria for k < m?
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