14.12 Oyun Teorisi

14.12 Oyun Teorisi
Muhamet Yıldız
Güz 2005
Sınav 1 Çözümleri
1. 1. sorunun çözümü
(a) 1. oyuncu her iki oyunda da aynı kazanç fonksiyonuna sahip, dolayısıyla
1. oyuncu açıkça aynı tercihlere sahip.
Ya 2. oyuncu? Başka deyişle, 2. oyuncunun sağdaki oyundaki kazançları
soldaki oyundaki kazançlarının, negatif olmayan afin bir transformasyonu
mudur? Yine başka bir deyişle, öyle a ≥ 0 ve b varmıdır ki, 0a + b =
0, 1a+b = 1, 4a+b = 3 ve 2a+b = 2 olsun? Ilk ikisini sağlamak için, a = 1
ve b = 0 olmalı, ancak bunlar 3. eşitliği sağlamazlar. Demekki, böyle bir
transformasyon yoktur ve 2. oyuncu aynı tercihlere sahip değildir.
(b) 1. oyuncunun sağdaki oyundaki kazançları soldaki oyundaki kazançlarının,
negatif olmayan afin bir transformasyonu mudur? Başka bir deyişle, öyle
a ≥ 0 ve b varmıdır ki, 0a + b = 1, 6a + b = 4, 2a + b = 2 ve 4a + b = 3
olsun? Evet, bu eşitlikleri çözerek görebilirsiniz ki, a = 1/2 ve b = 1 öyle
sayılardır.
2. oyuncunun sağdaki oyundaki kazançları soldaki oyundaki kazançlarının,
negatif olmayan afin bir transformasyonu mudur? Başka bir deyişle, öyle
a ≥ 0 ve b varmıdır ki, 1a + b = 0, 4a + b = 1, 7a + b = 2 ve −2a + b = −1
olsun? Evet, bu eşitlikleri çözerek görebilirsiniz ki, a = 1/3 ve b = −1/3
öyle sayılardır.
Evet her iki oyuncu da aynı tercihlere sahip.
2. 2. sorunun çözümü
1
transformation of the payoffs in the game on the left? In other words, do there
exist a ! 0 and b with 1a+b = 0, 4a+b = 1, 4a+b = 1, 7a+b = 2, −2a+b = −1,
and 1a + b = 0? Yes, you can solve the equations and see that a = 1/3 and
b = −1/3 are such an a and b.
So yes, both players have the same preference relation on lotteries with
2. oyuncu
için,
L’yi
kesin domine eder. Rasyonel bir oyuncu olarak, 2.
strategy
profiles
as M
their
outcomes.
oyuncu asla L oynamaz. 2’nin rasyonel olduğunu bilen 1. oyuncu asla B oyAnswer to Problem 2
namaz, çünkü indirgenmiş oyunda A B’yi kesin domine eder. Elimizde alttaki
For player 2 playing M strictly dominates playing L. As a rational player,
oyun
player 2kalır:
will never play L. Knowing that player 2 is rational, player 1 will never
play B since A strictly dominates B in the remaining game. Then we are left
with the following game:
M
R
A 4,1 1,0
C 2,0 2,2
In this reduced game, the pure Nash equilibria are obvious: (A, M ) and
(C, R) .
Indirgenmiş oyunda, saf Nash dengeleri açıktır: (A,M) ve (C,R).
Şimdi de karma stratejilere bakalım.1 1. oyuncu için P (A) = p ve P (C) = 1−p,
2. oyuncu içinse P (M ) = q ve P (R) = 1 − q olsun. O zaman bu olasılıkların
Now, let’s look at the mixed equlibrium. Let P (A) = p and P (C) = 1 − p
sağlaması
gerekenler
for
player 1 and
P (M ) = q and P (R) = 1 − q for player 2. Then the conditions
that these probabilities have to satisfy are
∗ qthe
+ 1mixed
∗ (1 − equlibrium.
q) = 2 Let P (A) = p and P (C) = 1 − p
Now, let’s look4 at
∗ p) +
(1 −Pp)
+ 2player
∗ (1 −2.p)Then the conditions
for player 1 and P 1(M
=0q∗and
(R)== 10−∗qpfor
that these probabilities have to satisfy are
As a result we get
4 ∗ q + 1 ∗ (1 − q) = 2
q = 1/3
1 ∗olarak
p + 0 ∗ (1 − p) = 0 ∗ p + 2 ∗ (1 − p)
şeklindedir. Sonuç
p = 2/3
As a result we get
Then the mixed strategy Nash equlibrium is
!
"
q = 1/3
2
2
1
1
A + p C,= M
+ R
3
3
3 2/3 3
thestrategy
mixed strategy
and Then
the pure
NE are Nash equlibrium is
!
"
2 M )1 and1(C, R)2.
(A,
A + C, M + R
3
3strateji
3 Nash
3 dengesi
elde ederiz. Bu durumda, karma
and
the puretostrategy
NE 3are
Answer
Problem
M ) and
(C,below.
R) . We first eliminate action y
(a) The backwards induction(A,
outcome
is as
for player 2, by assuming that player 2 is sequentially rational and hence
will
not to
play
y, which 3is conditionally dominated by x. We also eliminate
Answer
Problem
action r for player 1, assuming that player 1 is sequentially rational. This is
r is conditionally
Second,
that player
(a) because
The backwards
inductiondominated
outcome isby
as l.below.
Weassuming
first eliminate
action2y
isfor
sequentially
rational
and
that
player
2
knows
that
player
1
is
sequentially
player 2, by assuming that player 2 is sequentially rational and hence
rational,
c. This is dominated
because, knowing
that
player
1 is
will not we
playeliminate
y, which bisand
conditionally
by x. We
also
eliminate
sequentially
rational,
player
2
would
know
that
1
will
not
play
r,
and
hence
action r for player 1, assuming that player 1 is sequentially rational. This is
bbecause
would lead
to payoff of 0,dominated
and that
playing
c would
leadthat
to aplayer
payoff2
2 by
r is conditionally
by l.
Second,
assuming
ofis 1.
Being
sequentially
rational
she
must
play
a.
Finally,
assuming
that
sequentially rational and that player 2 knows that player 1 is sequentially
(i)
player
1
is
sequentially
rational,
(ii)
player
1
knows
that
player
rational, we eliminate b and c. This is because, knowing that player21isis
sequentially
(iii)
playerknow
1 knows
2 knows
that
sequentiallyrational,
rational, and
player
2 would
that that
1 willplayer
not play
r, and hence
player
1
is
sequentially
rational,
we
eliminate
L.
This
is
because
(ii)
and
b would lead to payoff of 0, and that by playing c would lead to a payoff
(iii)
lead
player
1
to
conclude
that
2
will
play
a
and
x,
and
thus
by
(i)
he
of 1. Being sequentially rational she must play a. Finally, assuming that
plays
R.
(i) player 1 is sequentially rational, (ii) player 1 knows that player 2 is
sequentially rational, and (iii) player 1 knows that player 2 knows that
1 ∗ p + 0 ∗ (1 − p)
q
As a result we get p
=
=
1/3
2/3
=
0 ∗ p + 2 ∗ (1 − p)
Then the mixed strategy Nash equlibriumq is = 1/3
p = " 2/3
!
2
2
1
1
A+
C, Nash
M +equlibrium
R
Then the mixed
strategy
is
3
3
3
3
!
"
2
2
1
1
and the pure strategy NE are
A + ve
C, M + R
idir ve saf strateji Nash dengeleri de (A,M)
3
3 (C,R)’dirler.
3
3
(A, M ) and (C, R) .
andçözümü
the pure strategy NE are
3. 3. sorunun
(A, M ) and (C, R) .
(a) Geriye
doğru tümevarım
çözümü şöyledir. Önce, 2. oyuncunun sıralı
Answer
to Problem
3
rasyonel olduğunu varsayarak 2. oyuncu için x tarafından koşullu domine
(a) The backwards
induction
outcome
Answer
to Problem
3 is as below. We first eliminate action y
edilen y’yi eliyoruz. 1. oyuncunun da sıralı rasyonel olduğunu varsafor player 2, by assuming that player 2 is sequentially rational and hence
(a) eliyoruz.
The
backwards
induction
outcome
is as by
below.
We
first eliminate
eliminate
r’yi
Bunun
nedeni l’in
r’ı koşullu
domine
etmesidir.
Ikinciaction y
willyarak
not play
y,
which
is
conditionally
dominated
x. We
also
for player 2, by assuming that player 2 is sequentially rational and hence
olarak,
sıralı
rasyonel olduğunu
ve12’nin
1’in sıralı rasyonel
action
r for2’nin
player
1, assuming
that player
is sequentially
rational.olduğunu
This is
will not play y, which is conditionally dominated by x. We also eliminate
because
r is varsayarak,
conditionally
dominated
by l.that
Second,
assuming
2
bildiğini
ve
c’yi
eliyoruz.
Bunun
nedeni,
1’in that
sıralıplayer
rasyonel
action r for bplayer
1, assuming
player
1 is sequentially
rational. This is
is sequentiallybecause
rationalr and
that
player
2
knows
that
player
1
is
sequentially
is conditionally
dominated by l.bilir
Second,
that
olduğunu bilen 2. oyuncu
1’in r oynamayacağını
ve buassuming
durumda
b 0player 2
rational, we eliminate
b and
c. This
because,
that
player
is
is sequentially
rational
andisthat
player 2knowing
knows that
player
1 is 1sequentially
getirirken,
c
oynamak
1
getirir.
Sıralı
rasyonel
olduğundan,
a
oynamalı.
sequentially rational,
2 would
not play
r, andthat
hence
rational,player
we eliminate
b know
and c. that
This1iswill
because,
knowing
player 1 is
Son olarak,
1. oyuncunun
olduğunu,
b would
lead sequentially
to(i)payoff
ofrational,
0, and sıralı
that
by2 would
playing
c would
lead
tooyuncunun
aplay
payoff
playerrasyonel
know
that 1(ii)
will1.not
r, and hence
b would
to payoff
of must
0, and
that a.
by
playing
would1’in
lead
to a payoff
of 1.2. Being
sequentially
rational
she
play
that
oyuncunun
sıralılead
rasyonel
olduğunu
bildğini
veFinally,
(iii)1’incassuming
2’nin
sıralı
1. Being sequentially
she must
play that
a. Finally,
that
(i) player 1 isofsequentially
rational,rational
(ii) player
1 knows
playerassuming
2 is
rasyonel olduğunu
bildiğini
bildiğini rational,
varsayarak
L’yi
eliyoruz.
Bunun
ne- 2 is
(i)
player
1
is
sequentially
(ii)
player
1
knows
that
player
sequentially rational, and (iii) player 1 knows that player 2 knows that
deni,1 (ii)
vesequentially
(iii)’nin 1’in
2’nin we
aand
veeliminate
x(iii)
oynayacağına
kanaat
ve that
rational,
player
knows
thatgetirmesini
player
knows
player
is sequentially
rational,
L. 1This
is because
(ii)2and
player
is sequentially
we
L. This
da1 (i)
R oynamasını
sağlamasıdır.
(iii)dolayısıyla
lead player
to1sayesinde
conclude
that
2rational,
will play
aeliminate
and x, and
thusisbybecause
(i) he (ii) and
(iii) lead player 1 to conclude that 2 will play a and x, and thus by (i) he
plays R.
(b) 1. oyuncunun
stratejisi vardır, 2. oyuncununsa 6 (seçilecek eylemlerle
plays4 R.
(b) Player
1 has
strategies
playerwhile
2 hasplayer
6 (named
the actions
be to be
adlandırılan).
(b) 4Player
1 has while
4 strategies
2 has 6by(named
by theto
actions
chosen).
chosen).
ax ay
bx
cy cx cy
ax ayby bxcx by
Ll 0, 1 0, 1Ll 0,0,11 0,0,1 1 0, 0,
1
0,
1
0, 11 0, 1 0, 1
Lr 0, 1 0, 1Lr 0,0,11 0,0,1 1 0, 0,
1 1 0,0,11 0, 1 0, 1
Rl
1,
2
1,
2
2,
0 1 2,1,00 1, 1 1, 0
Rl 1, 2 1, 2 2, 0
2, 0 1,
1, 24 1,
2 4−1,1,4 1 −1,
Rr 1, 2 1, 2Rr −1,
−1,
1, 04 1, 1 1, 0
(c) Rasyonelleştirilebilir stratejileri2 hesaplayın.
2
Ilk olarak,(c)LlCompute
ve Lr Rlthetarafından
kesin dominestrategies.
edilirler. 1’in rasyonel
set of all rationalizable
olduğunu varsayarak, Ll ve Lr oynamayacağı sonucuna varıyoruz. Ll ve
First, Ll and Lr are strictly dominated by strategy Rl. Assuming that player
Lr’yi eliyoruz
ve indirgenmiş
oyun
alttaki
halininot
alıyor
1 is rational,
we conclude
that
he would
play Ll and Lr. We eliminate Ll
and Lr, so the game is reduced to
ax ay
bx
by
cx cy
Rl 1, 2 1, 2 2, 0
2, 0 1, 1 1, 0
Rr 1, 2 1, 2 −1, 4 −1, 4 1, 1 1, 0
Now for player 2 cx and cy are strictly dominated by ax. Hence, assuming
that (i) player 2 is rational,
3 and that (ii) player 2 knows that player 1 is rational,
we eliminate cx and cy. This is because, by (ii), 2 knows that 1 will not play Ll
and Lr, and hence by (i) she would not play cx and cy. The game is reduced to
ax ay
bx
by
Rl 1, 2 1, 2 2, 0
2, 0
Rr 1, 2 1, 2 −1, 4 −1, 4
There is no strictly dominated strategy in the remaining game. Therefore,
all the remaining strategies are rationalizable.
(c) Compute the set of all rationalizable strategies.
First, Ll and Lr are strictly dominated by strategy Rl. Assuming that player
1 is rational, we conclude that he would not play Ll and Lr. We eliminate Ll
and
Lr, için
so the
is reduced
to
Şimdi, 2. oyuncu
cxgame
ve cy
ax tarafından
kesin domine edilirler.
ax ay
bx
by
cx cy
(i) 2’nin rasyonelRl
olduğunu
1, 2 1,ve
2 (ii)2, 2’nin
0
2,1’in
0 rasyonel
1, 1 1, 0olduğunu bildiğini
2 1, 2 −1,
4 −1,
4 1, 1(ii)1,sayesinde,
0
varsayarak cx ve Rr
cy’yi1,eliyoruz.
Bunun
nedeni,
2 1’in Ll
Now for player 2 cx and cy are strictly dominated by ax. Hence, assuming
ve Lr oynamayacağını biliyoruz, dolayısıyla, (i) sayesinde de cx ve cy
that (i) player 2 is rational, and that (ii) player 2 knows that player 1 is rational,
oynamaycaktır.
Oyun alttaki
haline
indirgenir
we eliminate
cx and
cy. This
is because, by (ii), 2 knows that 1 will not play Ll
and Lr, and hence by (i) she would not play cx and cy. The game is reduced to
ax ay
bx
by
Rl 1, 2 1, 2 2, 0
2, 0
Rr 1, 2 1, 2 −1, 4 −1, 4
There is no strictly dominated strategy in the remaining game. Therefore,
the remaining
strategies
rationalizable.
Kalan oyundaallbaşka
kesin domine
edilenare
strateji
yoktur. Dolayısıyla, tüm
kalan stratejiler Answer
rasyonelleştirilebilirdir.
to Problem 4
Denote by y the winner of the first round of voting (either the bill or the
4. 4. sorunun çözümü
amendment). In the second round of voting between 0.6 and y, the Moderates
will vote for whichever is closer to 0.5; the Democrats will vote for the higher tax
Ilk tur oylamanınrate
galibine
y diyelim
(kanun
değişiklik).
and the
Republicans
willtasarısı
vote forya
theda
lower
tax rate. Ikinci
Since the Democrats
Republicans
will always
the winner will be
tur oylamada, 0.6and
ile the
y arasında,
Ilımlılar
hangisiback
0.5’edifferent
yakınsaproposals,
ona oy verewhichever
is
also
backed
by
the
moderates.
We
can
denote
the winner of the
cekler; Demokratlar dah yüksek olan vergi oranına, Cumhuriyetçiler de düşük
second round of voting by f (y) defined as follows:
olan vergi oranına oy verecekler. Demokratlar ve Cumhuriyetçiler hep farklı
f (y) =
0.6 if y >
0.6 oy
or verdikleri
y < 0.4
vergi oranlarına oy verecekleri için, kazananı
Ilımlıların
neye
be= alttaki
y if y ∈
[0.4,
0.6]
lirleyecek. Ikinci turun kazananını f (y) olarak
gibi
tanımlayabiliriz
For the first round of voting, the moderates will choose whichever of x1 and
x2 will cause the outcome of the second round of voting to be closer to 0.5; i.e.
f (y)
= 0.6
y > 0.6 veya
0.4
|0.5 −yf<
(x)
|. Similarly, the Democrats will choose
they
choose
argeger
minx∈{x
1 ,x2 }
arg maxx∈{x1 ,x2 } f (x). And the Republicans will choose arg minx∈{x1 ,x2 } f (x).
y eger
y ∈ [0.4,
0.6]
Again, since=the
Democrats
and
the Republicans will always back different proposals in the first round, the winner will be whichever is also backed by the
Ilk tur için, ılımlılar,
x1 ve therefore
x2 ’den ikinci
sonucu
0.5’e
yakın
ya- given x1 , the
|0.5
− en
f (x)
|. Then,
moderates;
y = turdaki
arg minx∈{x
1 ,x2 }
Democrats is min {0.6, 0.5 + |0.5 − x1 |}; i.e. if x1 is
optimal
of x2 for the
pacak olanı seçerler,
yani, choice
argmin
x∈{x1 ,x2 } |0.5−f (x)|’i seçerler. Benzer şekilde,
larger than is 0.4, they would choose x2 to be as large as possible while being
Demokratlar da argmax
Cumhuriyetçiler de, argminx∈{x1 ,x2they
x∈{x
} f (x)’i
1 ,xthan
2 } f (x)’i
choose
closer to
0.5
is xseçerler.
1 (so that the moderates back x2 ); otherwise
to
be
smaller
than
0.5
in
introducing
0.6.
Then,
if
the
Republicans
choose
x
seçerler. Ilk turda, Demokratlar ve Cumhuriyetçiler hep1farklı tasarıları seçecekleri
the bill, the Democrats will introduce an amendment with a higher tax rate,
için, kazanan ılımlıların
seçtikleri olacaktır, yani y = argminx∈{x ,x2 } |0.5 −
that the moderates back in both rounds. So the best 1that
the Republicans can
f (x)|. O zaman, xdo
optimal x2 seçimi min{0.6, 0.5+
1 veriliyken,
is to chooseDemokratların
x1 = 0.5.
Thus,
full description
strategies of the
three
parties in equilibrium
|0.5−x1 |}’dir, yani, eğer
x1 a0.4’ten
büyükse, of
x2the
’yi olabildiğince
büyük
seçerler,
are as follows: Republicans choose x1 = 0.5; in the first round, they vote
0.5’e x1 ’in olduğundan daha yakın olacak şekilde (böylelikle ılımlılar x2 ’yi
desteklerler); aksi durumda 0.6 seçerler. O zaman, eğer Cumhuriyetçiler x1 ’i
3
4
0.5’ten daha küçük seçerlerse, Demokratlar ılımlıların iki turda da destekleyecekleri daha yüksek bir vergi önerirler. Dolayısıyla, Cumhuriyetçilerin yapabileceklerinin en iyisi x1 = 0.5 seçmektir.
Dolayısıyla, dengede üç oyuncunun stratejileri şöyledir: Cumhuriyetçiler ilk
turda x1 = 0.5 seçerler ve argminx∈{x1 ,x2 } f (x) için oy verirler, ikinci turda
min{0.6, y} için oy verirler. Demokratlar x2 = min{0.6, 0.5 + |0.5 − x1 |}
seçerler, ilk turda, argmaxx∈{x1 ,x2 } f (x) için oy kullanırlar ve ikinci turda
max{0.6, y} için oy kullanırlar. Ilımlılarsa, ilk turda argminx∈{x1 ,x2 } |0.5−f (x)|
için, ikinci turdaysa argminz∈{0.6,y} |0.5 − z| için oy kullanırlar.
5