Transcript 1 1
Slide 1
1
Slide 2
problem set 5
from Binmore’s
Fun and Games
p.271 Ex. 34,35
p.273 Ex. 40,41
from Osborne’s
Introd. To G.T.
p.132 Ex. 132.2
p.141 Ex. 141.2
Slide 3
Battle of the sexes
An additional mixed equilibrium??
woman
Ballet
When player 2 mixes (1-β ,β)
player 1’s payoffs are:
Ballet
2,1 0,0
Boxing
0,0 1,2
man
Ballet :
2 1 - β + 0β = 2 - 2β
Boxing :
0 1 - β + 1β = β
Ballet
Boxing
2
Boxing 2 - 2β > β > β
3
3
Slide 4
Battle of the sexes
An additional mixed equilibrium??
When player 1 mixes (1-α , α)
player 2’s payoffs are:
Ballet
Boxing :
Ballet
Boxing
Ballet
2,1 0,0
Boxing
0,0 1,2
man
Ballet :
woman
1 1 - α + 0α = 1 - α
0 1 - α + 2α = 2α
1
Boxing 1 - α > 2α > α
3
4
Slide 5
2
Player 1 : Ballet Boxing 2 - 2β > β > β
3
1
Player 2 : Ballet Boxing 1 - α > 2α > α
3
player 2’s mix
Boxing, Boxing
Player 1’s
Best Response
1
β
Player 2’s
Best Response
Mixd Nash equilibium
α = 1/3, β= 2/3
Ballet, Ballet
α
1
player 1’s mix 5
Slide 6
1/3
Ballet
2/3
Boxing
2/3
2/3
2,1 0,0
2/3
1/3
0,0 1,2
2/3
2/3
6
Slide 7
Mixed Strategies and Domination
x
1,0
6,4
0,9
4.5
3,7
2,3
4,0
3.5
1/2
1/2
A mix of these strategies dominates the middle one
7
Slide 8
A general notation
Let G1 be the matrix of the game describing
p1
player 1’s payoffs
p2
Let player 1 use the mixed strategy
p= •
•
p
n
q1
q2
and let player 2 use the mixed strategy
q •
•
q
then player 1’s payoff is given by:
m
q1
q2
p t G1 q = p1 , p2 , • • pn G1 •
•
q
8
m
Slide 9
Let G( , ) be a game, and let p,q
be mixed strategies
of the two players
Define:
G p,q p G1q, p G2 q
t
t
9
Slide 10
A pair of mixed strategies p* ,q*
is a Nash equilibrium of a game G if
for all mixed strategies p :
p*t G1 q* p t G1 q*
and for all mixed strategies q :
p*t G2 q* p*t G2 q
10
Slide 11
The Inspection Game
• An agency inspects pollution in one of n days.
• A firm will pollute in one of these n days.
• The agency wins if it catches the firm poluting,
otherwise the firm wins.
• Zero-sum game, agency wins = 1, agency loses = -1
• Agency and firm decide each day whether to act or
wait
• If none has acted until the last day (day n), then on
that day the agency wins.
Let Vk be the value of the game beginning on day k,
Vn = 1.
Player 1’s payoff in an equilibrium
of the game beginning at day
11 k
Slide 12
A
wait
Day 1
act
F
F
wait
wait
-1,1
-1,1
act
1 ,-1
Day 2
Day 3
A
wait
act
F
wait
F
wait
act
12
Slide 13
If until day k none has acted, then the game on day k
can be described by:
??
If Vk+1 > -1, then:
act
wait
act
1
-1
wait
-1
Vk+1
and there is no equilibrium in pure strategies
Vn = 1, by induction, we show that if Vk+1 > -1
then so is Vk
13
Slide 14
To find a mixed equilibrium
1-β
act
β
wait
1-2β
act
1
-1
-1+β(1+Vk+1)
wait
-1
Vk+1
2
3 +Vk+1
Player 1’s
best response
14
Slide 15
To find a mixed equilibrium
Player 2’s
best response
2
3 +Vk+1
2
3 +Vk+1
act
wait
1-α
1
-1
α
-1
Vk+1
1-2α
-1+α(1+Vk+1)
15
Slide 16
What is 1’s payoff in
this equilibrium??
Vk+1 - 1
3 +Vk+1
2
3 +Vk+1
2
3 +Vk+1
1 +Vk+1
3 +Vk+1
2
3 +Vk+1
act
wait
1
-1
-1
Vk+1
EQUILIBRIUM
16
Slide 17
Vk+1 - 1
3 +Vk+1
1 +Vk+1
3 +Vk+1
2
3 +Vk+1
act
wait
1-α
1
-1
α
-1
Vk+1
Vk+1 - 1
2 Vk+1 + 1
Vk =
= -1+
> -1
3 +Vk+1
3 +Vk+1
So if Vk+1 > -1 then Vk > -1, hence for all j’s Vj > -1.
17
Slide 18
To solve the difference equation
Vk+1 - 1
1 + Vk =
+1
3 +Vk+1
1+Vk =
2 Vk+1 + 1
3 +Vk+1
1
1
1
+ =
1 + Vk+1 2 1 + Vk
1
wk+1 + = wk
2
Change the variable
1
wk =
1 + Vk
1
1
wn =
=
1 + Vn 2
18
Slide 19
0.5n
1
0.5(n-k+1)
k
1.5
n -2
1
n -1
0.5
n
wk
n-k+1 = remaining days on the morning of day k
n
1-k+1
ww
+ = wk
k+1
k =
2 2
1
11 - n - k
wn =
Vk ==
1 + Vn 2n - k + 1
19
Slide 20
On day k each player acts with prob.
1 +Vk+1
3 +Vk+1
2
3 +Vk+1
act
wait
1 +Vk+1
1 + Vk+1
1/wk+1
11
-1
3 +Vk+1 2 + 1 + Vk+1 2 + 1/wk+1 2wk+1 + 1
-1
Vk+1
1
n-k+1
When there are s days left, each acts with probability 1/s.
A simpler way to solve this game
20
Slide 21
Each of the players has n strategies.
Strategy k: to act on day k
●
●
●
1
2
1
1
-1
-1
2
-1
1
-1
●
-1
1
1
●
1
●
n
n
-1
-1
There is a unique mixed strategy equilibrium in
which each strategy has equal probability 1/n
1
21
Slide 22
2
-1
n
1
n
1
n
1
2
1
1
-1
-1
2
-1
1
-1
●
-1
●
●
n
1
1
●
n
●
1
●
2
-1
n
1
n
-1
-1
1
If a player has not acted until period k, then he has equal
probabilities of acting in any of the remaining days. 22
Slide 23
Reporting a Crime
A group of n people observes a crime.
Each may report the crime or refrain from
reporting.
The utility of having the crime reported is v, the
cost of reporting is c, (v > c)
If no one reports the utility is 0.
There are Nash equilibria in which exactly
one person reports the crime.
Is there a symmetric equilibrium?
(an equilibrium in which all play the same strategy)
23
Slide 24
Is there a symmetric equilibrium?
(an equilibrium in which all play the same strategy)
There are no symmetric pure strategy equilibria
All report, or all not report are no equilibria !!!
Look for a symmetric mixed strategy equilibrium in
which each one reports with probability p
If a player reports he gets
v -c
If a player does not report he gets
0 1 - p
n-1
+ v 1 - 1 - p
n-1
24
Slide 25
report
not report
v -c
0 1 - p
n-1
+ v 1 - 1 - p
If this is an equilibrium, then:
n-1
v - c = 0 1 - p + v 1 - 1 - p
c
n-1
= 1 - p
v
c
p = 1-
v
n-1
n-1
1
n-1
25
Slide 26
n
1
n-1
n
p
c
v
1
n-1
c
p = 1-
v
c
1
v
1
n-1
1
n-1
26
Slide 27
Minimax & Maximin
Strategies
Given a game G( , ) and a strategy s of player 1:
min G1 s,t
t
is the worst that can happen to player 1 when he
plays strategy s.
He can now choose a strategy s for which this
‘worst scenario’ is the best
max min G1 s,t
s
t
27
Slide 28
A strategy s is called a minimax (security) strategy if
min
minGG
s,t max
maxmin
minG
G1s,t
σ,t .
1 s,t
σs
t t
max
s
min G1 s,t s
t
min G1 s',t s'
t
{
{
tt
min G1 s,t
t
min G1 s',t
t
28
Slide 29
A strategy s is called a maximin (security) strategy if
minG
G11 s,t
s,t max
maxmin
minG
G11σ,t
s,t .
min
t
t
s
σ
t
t
These can be defined for mixed strategies as well.
max = sup , min = inf
Similarly, one may define
min max G1 s,t
t
s
If the game is strictly competitive then this is the
best of the ‘worst case scenarios’ of player 2.
29
1
Slide 2
problem set 5
from Binmore’s
Fun and Games
p.271 Ex. 34,35
p.273 Ex. 40,41
from Osborne’s
Introd. To G.T.
p.132 Ex. 132.2
p.141 Ex. 141.2
Slide 3
Battle of the sexes
An additional mixed equilibrium??
woman
Ballet
When player 2 mixes (1-β ,β)
player 1’s payoffs are:
Ballet
2,1 0,0
Boxing
0,0 1,2
man
Ballet :
2 1 - β + 0β = 2 - 2β
Boxing :
0 1 - β + 1β = β
Ballet
Boxing
2
Boxing 2 - 2β > β > β
3
3
Slide 4
Battle of the sexes
An additional mixed equilibrium??
When player 1 mixes (1-α , α)
player 2’s payoffs are:
Ballet
Boxing :
Ballet
Boxing
Ballet
2,1 0,0
Boxing
0,0 1,2
man
Ballet :
woman
1 1 - α + 0α = 1 - α
0 1 - α + 2α = 2α
1
Boxing 1 - α > 2α > α
3
4
Slide 5
2
Player 1 : Ballet Boxing 2 - 2β > β > β
3
1
Player 2 : Ballet Boxing 1 - α > 2α > α
3
player 2’s mix
Boxing, Boxing
Player 1’s
Best Response
1
β
Player 2’s
Best Response
Mixd Nash equilibium
α = 1/3, β= 2/3
Ballet, Ballet
α
1
player 1’s mix 5
Slide 6
1/3
Ballet
2/3
Boxing
2/3
2/3
2,1 0,0
2/3
1/3
0,0 1,2
2/3
2/3
6
Slide 7
Mixed Strategies and Domination
x
1,0
6,4
0,9
4.5
3,7
2,3
4,0
3.5
1/2
1/2
A mix of these strategies dominates the middle one
7
Slide 8
A general notation
Let G1 be the matrix of the game describing
p1
player 1’s payoffs
p2
Let player 1 use the mixed strategy
p= •
•
p
n
q1
q2
and let player 2 use the mixed strategy
q •
•
q
then player 1’s payoff is given by:
m
q1
q2
p t G1 q = p1 , p2 , • • pn G1 •
•
q
8
m
Slide 9
Let G( , ) be a game, and let p,q
be mixed strategies
of the two players
Define:
G p,q p G1q, p G2 q
t
t
9
Slide 10
A pair of mixed strategies p* ,q*
is a Nash equilibrium of a game G if
for all mixed strategies p :
p*t G1 q* p t G1 q*
and for all mixed strategies q :
p*t G2 q* p*t G2 q
10
Slide 11
The Inspection Game
• An agency inspects pollution in one of n days.
• A firm will pollute in one of these n days.
• The agency wins if it catches the firm poluting,
otherwise the firm wins.
• Zero-sum game, agency wins = 1, agency loses = -1
• Agency and firm decide each day whether to act or
wait
• If none has acted until the last day (day n), then on
that day the agency wins.
Let Vk be the value of the game beginning on day k,
Vn = 1.
Player 1’s payoff in an equilibrium
of the game beginning at day
11 k
Slide 12
A
wait
Day 1
act
F
F
wait
wait
-1,1
-1,1
act
1 ,-1
Day 2
Day 3
A
wait
act
F
wait
F
wait
act
12
Slide 13
If until day k none has acted, then the game on day k
can be described by:
??
If Vk+1 > -1, then:
act
wait
act
1
-1
wait
-1
Vk+1
and there is no equilibrium in pure strategies
Vn = 1, by induction, we show that if Vk+1 > -1
then so is Vk
13
Slide 14
To find a mixed equilibrium
1-β
act
β
wait
1-2β
act
1
-1
-1+β(1+Vk+1)
wait
-1
Vk+1
2
3 +Vk+1
Player 1’s
best response
14
Slide 15
To find a mixed equilibrium
Player 2’s
best response
2
3 +Vk+1
2
3 +Vk+1
act
wait
1-α
1
-1
α
-1
Vk+1
1-2α
-1+α(1+Vk+1)
15
Slide 16
What is 1’s payoff in
this equilibrium??
Vk+1 - 1
3 +Vk+1
2
3 +Vk+1
2
3 +Vk+1
1 +Vk+1
3 +Vk+1
2
3 +Vk+1
act
wait
1
-1
-1
Vk+1
EQUILIBRIUM
16
Slide 17
Vk+1 - 1
3 +Vk+1
1 +Vk+1
3 +Vk+1
2
3 +Vk+1
act
wait
1-α
1
-1
α
-1
Vk+1
Vk+1 - 1
2 Vk+1 + 1
Vk =
= -1+
> -1
3 +Vk+1
3 +Vk+1
So if Vk+1 > -1 then Vk > -1, hence for all j’s Vj > -1.
17
Slide 18
To solve the difference equation
Vk+1 - 1
1 + Vk =
+1
3 +Vk+1
1+Vk =
2 Vk+1 + 1
3 +Vk+1
1
1
1
+ =
1 + Vk+1 2 1 + Vk
1
wk+1 + = wk
2
Change the variable
1
wk =
1 + Vk
1
1
wn =
=
1 + Vn 2
18
Slide 19
0.5n
1
0.5(n-k+1)
k
1.5
n -2
1
n -1
0.5
n
wk
n-k+1 = remaining days on the morning of day k
n
1-k+1
ww
+ = wk
k+1
k =
2 2
1
11 - n - k
wn =
Vk ==
1 + Vn 2n - k + 1
19
Slide 20
On day k each player acts with prob.
1 +Vk+1
3 +Vk+1
2
3 +Vk+1
act
wait
1 +Vk+1
1 + Vk+1
1/wk+1
11
-1
3 +Vk+1 2 + 1 + Vk+1 2 + 1/wk+1 2wk+1 + 1
-1
Vk+1
1
n-k+1
When there are s days left, each acts with probability 1/s.
A simpler way to solve this game
20
Slide 21
Each of the players has n strategies.
Strategy k: to act on day k
●
●
●
1
2
1
1
-1
-1
2
-1
1
-1
●
-1
1
1
●
1
●
n
n
-1
-1
There is a unique mixed strategy equilibrium in
which each strategy has equal probability 1/n
1
21
Slide 22
2
-1
n
1
n
1
n
1
2
1
1
-1
-1
2
-1
1
-1
●
-1
●
●
n
1
1
●
n
●
1
●
2
-1
n
1
n
-1
-1
1
If a player has not acted until period k, then he has equal
probabilities of acting in any of the remaining days. 22
Slide 23
Reporting a Crime
A group of n people observes a crime.
Each may report the crime or refrain from
reporting.
The utility of having the crime reported is v, the
cost of reporting is c, (v > c)
If no one reports the utility is 0.
There are Nash equilibria in which exactly
one person reports the crime.
Is there a symmetric equilibrium?
(an equilibrium in which all play the same strategy)
23
Slide 24
Is there a symmetric equilibrium?
(an equilibrium in which all play the same strategy)
There are no symmetric pure strategy equilibria
All report, or all not report are no equilibria !!!
Look for a symmetric mixed strategy equilibrium in
which each one reports with probability p
If a player reports he gets
v -c
If a player does not report he gets
0 1 - p
n-1
+ v 1 - 1 - p
n-1
24
Slide 25
report
not report
v -c
0 1 - p
n-1
+ v 1 - 1 - p
If this is an equilibrium, then:
n-1
v - c = 0 1 - p + v 1 - 1 - p
c
n-1
= 1 - p
v
c
p = 1-
v
n-1
n-1
1
n-1
25
Slide 26
n
1
n-1
n
p
c
v
1
n-1
c
p = 1-
v
c
1
v
1
n-1
1
n-1
26
Slide 27
Minimax & Maximin
Strategies
Given a game G( , ) and a strategy s of player 1:
min G1 s,t
t
is the worst that can happen to player 1 when he
plays strategy s.
He can now choose a strategy s for which this
‘worst scenario’ is the best
max min G1 s,t
s
t
27
Slide 28
A strategy s is called a minimax (security) strategy if
min
minGG
s,t max
maxmin
minG
G1s,t
σ,t .
1 s,t
σs
t t
max
s
min G1 s,t s
t
min G1 s',t s'
t
{
{
tt
min G1 s,t
t
min G1 s',t
t
28
Slide 29
A strategy s is called a maximin (security) strategy if
minG
G11 s,t
s,t max
maxmin
minG
G11σ,t
s,t .
min
t
t
s
σ
t
t
These can be defined for mixed strategies as well.
max = sup , min = inf
Similarly, one may define
min max G1 s,t
t
s
If the game is strictly competitive then this is the
best of the ‘worst case scenarios’ of player 2.
29