Transcript Slide 1

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problem set 4
from M. Osborne’s
An Introduction to Game theory
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Some (classical) examples of simultaneous games
Prisoners’ Dilemma
C
cooperate
D
defect
C
D
Cooperate
defect
3,3
0,6
6,0
1,1
The ‘D strategy strictly dominates the C strategy
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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)
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Successive
deletion
of
example of strict dominance
dominated strategies
X
X
1,5
3,3
2,3
7,4
4,7
5,2
Nash Equilibrium
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weak dominance
???
an example
X
1,0 1,4 1,0 1,4
1,2 1,2 0,3 0,3
2,3 1,4 2,3 1,4
1,2 1,2 0,3 0,3
?
?
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Successive deletion of
dominated strategies
And Sub-game Perfectness
Example
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l
2
0,3
1
r
2
2
1,2
1
1,4
2,3
1
2
1,0
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rr
lr
rl
ll
rr 1 , 0 1 , 4 1 , 0 1 , 4
lr 1 , 2 1 , 2 0 , 3 0 , 3
rl 2 , 3 1 , 4 2 , 3 1 , 4
1
1
ll l , 12 r 1 , 2 0 , 3 0 , 3
2
0,3
2
2
1,2
1
1,4
2,3
1
2
1,0
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rr
lr
rl
ll
rr 1 , 0 1 , 4 1 , 0 1 , 4
lr 1 , 2 1 , 2 0 , 3 0 , 3
rl 2 , 3 1 , 4 2 , 3 1 , 4
1
l
2
0,3
1
ll 1 , 2 1 , 2 0 , 3 0 , 3
r
2
2
1,2
1
1,4
2,3
1
Sub-game perfect equiibrium
2
1,0
( r,l ) ( l,l )
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rr
lr
rl
ll
X rr 1 , 0 1 , 4 1 , 0 1 , 4
X lr 1 , 2 1 , 2 0 , 3 0 , 3
weakly dominating
1
l
2
0,3
1
ll 1 , 2 1 , 2 0 , 3 0 , 3
r
2
2
1,2
rl 2 , 3 1 , 4 2 , 3 1 , 4
X
1
1,4
2,3
1
X
X
2
1,0
(x,l) (x,l)
delete ( x , r ) delete ( x , r10
)
Another example of a simultaneous game
The Stag Hunt
Stag
Hare
Stag
2,2 0,1
Hare
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,
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otherwise it yields 0.
Another example of a simultaneous game
The Stag Hunt
Stag
Hare
Stag
2,2 0,1
Hare
1,0 1,1
Equilibria
payoff dominant equilibrium
risk dominant equilibrium
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Change of payoffs
Stag
Hare
Stag
2,2
0 , 1.99
Hare
1.99 , 0
1.99 ,1.99
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Yet another example of a simultaneous game
Battle of the sexes
Bach or Stravinsky (BOS)
woman
Ballet
man
Boxing
Ballet
2,1 0,0
Boxing
0,0 1,2
Equilibria
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Yet another example of a simultaneous game
Battle of the sexes
woman
Ballet
man
Boxing
Ballet
2,1 0,0
Boxing
0,0 1,2
A generalization to a bargaining situation
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Nash Demand Game
Demand of player 2
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.
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Demand of player 1
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Nash Demand Game
Demand of player 2
equilibria
a continuum of equilibria
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Demand of player 1
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last example of a simultaneous game
Matching Pennies
head
tails
head
1 , -1 -1 , 1
tails
-1 , 1 1 , -1
no pure strategies equilibrium exists
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last example of a simultaneous game
Matching Pennies
head
tails
head
1 , -1 -1 , 1
tails
-1 , 1 1 , -1
no pure strategies equilibrium exists
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last example of a simultaneous game
Matching Pennies
Mixed strategies
head
head
A player may choose
head with probability 
tails
tails
1 , -1 -1 , 1
-1 , 1 1 , -1
and tails with probability 1- 
no pure strategies equilibrium exists
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last example of a simultaneous game
Matching Pennies
Mixed strategies
player 2 mixes:
if player 1 plays ‘head’
his payoff is the lottery:
1- β β 


1
-1


1-

head
tails
tails
head
1 , -1 -1 , 1
tails
-1 , 1 1 , -1
if the payoffs are in terms of his vN-M utility
then his utility from the lottery is
β  -1 +  1 - β 1 = 1 - 2β
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last example of a simultaneous game
Matching Pennies
Mixed strategies
player 2 mixes:
1- β

 -1
β

1
1- 

head
tails
tails
1 - 2β
head
1 , -1 -1 , 1
2β - 1
tails
-1 , 1 1 , -1
Similarly, if player 1 plays ‘tails’ his payoff is …….
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last example of a simultaneous game
Matching Pennies
Mixed strategies
player 2 mixes:
1- 

head
tails
1 - 2β
head
1 , -1 -1 , 1
2β - 1
tails
-1 , 1 1 , -1
He prefers
play
He prefers to play ‘tails’ if:
Whento
β=
0.5 ‘head’
playerif:
1 is indifferent
1 - between
2β > 2β the
- 1 two strategies 2β - 1 > 1 - 2β
and any mix of the two
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0.5 > β
β > 0.5
Player 1 prefers to play ‘head’ if:
Player 1 prefers to play ‘tails’ if:
Player 1 is indifferent when
player 2’s mix
1
0.5 > β
β > 0.5
β = 0.5
Player 2’s
Best Response
function ??
Player 1’s
Best Response
function
β
(1-α , α)
head
α
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tails 1 player 1’s mix
Player 2’s
Best Response
function
function ??
When
player 1 plays ‘head’ often
Nash equilibium
Player
prefers
to play ‘tails’
α =2 β=
0
player 2’s mix
1
Player 1’s
Best Response
function
β
α
1
player 1’s mix
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1/2
1/2
head
tails
0
head
1 , -1 -1 , 1
0
tails
-1 , 1 1 , -1
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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.
a. As before, each hunter prefers the fraction 1 / n of the stag to a hare
b. 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
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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 (e1 . . ., 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
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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
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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.
a. For k = m = 1, is the game the same (except for the names of the actions) as any
considered so far in this chapter?
b. 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?)
c. What is the set of Nash equilibria for k < m?
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