Wstęp do Teorii Gier
Zeus and Athena
• Zeus Music is a market leader in producing a modern audio
equipment.
• Athena Acoustics is a smaller but very innovative firm
• Both firms have invented a new hexaphonic sound system
of a total audio surrounding. (You hang a guy at some
height and put 6 speakers around him)
• There is uncertainty about the size of a market.
– There is 50-50 chance of small ($24 mln profits) and big ($40
mln profits) market
• The two firms have to decide whether to launch a highest
quality audio system designed for a very demanding
customers or a cheaper system designed for less
demanding customers.
Zeus and Athena
• If the market is small, it is better to sell highest quality
audio system
• If the market is big, there will be more demand for lower
quality system
• Athena is prepared better than Zeus to produce the more
innovative system
• But Zeus is a widely known company and has more
marketing potential in selling the less innovative system
• Taking into account all these factors, analysts from Zeus
estimated market shares of both firms
• Since estimation was based on publicly available
information Zeus believes that analysts from Athena have
similar predictions.
Information sets to be added later
Different Assumptions about
information A1
•
Both firms do not reveal their decisions about production (and
they do not know what nature did)
Different Assumptions about
information A1
ATHENA
LQ
ZEUS
HQ
LQ
(23,9)
(18,14)
HQ
(18,14)
(20,12)
ATHENA
LQ
ZEUS
HQ
LQ
(23,9)
(18,14)
HQ
(18,14)
(20,12)
Different Assumptions about
information A1
ATHENA
LQ
ZEUS
HQ
LQ
(23,9)
(18,14)
HQ
(18,14)
(20,12)
Mixed strategy Nash Equilibrium: for both firms the same:
2/7 LQ, 5/7 HQ
Value of a game: (19 and 3/7 for Zeus; 12 and 4/7 for Athena)
Different Assumptions about
information A2
• Zeus has to make production decision earlier.
• Athena – a smaller and more flexible firm – may decide after
observing decision made by Zeus
Different Assumptions about
information A2
ATHENA
ZEUS
LQ
HQ
LQ/LQ
(23,9)
(18,14)
LQ/HQ
(23,9)
(20,12)
HQ/LQ
(18,14)
(18,14)
HQ/HQ
(18,14)
(20,12)
ATHENA
ZEUS
LQ
HQ
LQ/LQ
(23,9)
(18,14)
LQ/HQ
(23,9)
(20,12)
HQ/LQ
(18,14)
(18,14)
HQ/HQ
(18,14)
(20,12)
Different Assumptions about
information A2
ATHENA
ZEUS
LQ
HQ
LQ/LQ
(23,9)
(18,14)
LQ/HQ
(23,9)
(20,12)
HQ/LQ
(18,14)
(18,14)
HQ/HQ
(18,14)
(20,12)
2 Nash equilibria: Athena: Always choose different than Zeus
(LQ, HQ/LQ), (HQ,HQ/LQ)
Value for Zeus 18, value for Athena 14
Different Assumptions about
information A3
• Zeus has to make production decision earlier.
• But before it conducts a very thorough and costly market research,
which will allow to determine whether the market is small or big
• Athena will not know the outcome of this research but will know
that it took place
Different Assumptions about
information A3
ATHENA
LQ/LQ
LQ/HQ
ZEUS
HQ/LQ
HQ/HQ
LQ/LQ
(23,9)
(16,16)
(25,7)
(18,14)
LQ/HQ
(23,9)
(20,12)
(23,9)
(20,12)
HQ/LQ
(18,14)
(12,20)
(24,8)
(18,14)
HQ/HQ
(18,14)
(16,16)
(22,10)
(20,12)
ATHENA
LQ/LQ
LQ/HQ
ZEUS
HQ/LQ
HQ/HQ
LQ/LQ
(23,9)
(16,16)
(25,7)
(18,14)
LQ/HQ
(23,9)
(20,12)
(23,9)
(20,12)
HQ/LQ
(18,14)
(12,20)
(24,8)
(18,14)
HQ/HQ
(18,14)
(16,16)
(22,10)
(20,12)
Different Assumptions about
information A3
ATHENA
LQ/LQ
LQ/HQ
ZEUS
HQ/LQ
HQ/HQ
LQ/LQ
(23,9)
(16,16)
(25,7)
(18,14)
LQ/HQ
(23,9)
(20,12)
(23,9)
(20,12)
HQ/LQ
(18,14)
(12,20)
(24,8)
(18,14)
HQ/HQ
(18,14)
(16,16)
(22,10)
(20,12)
Nash equilibrium: Zeus (dominant strategy): produce HQ
systems if market is small; produce LQ systems if market is big.
Athena: always produce HQ systems
(HQ/LQ,HQ/HQ)
Value for Zeus: 22, value for Athena: 10
Different Assumptions about
information A3
• Interesting: Athena’s strategy changed from “always choose
different than Zeus” to “always choose HQ systems”,
despite the fact that the only new information for her was
that Zeus has conducted market research.
– The reasoning is as follows: Since Zeus chose HQ, Athena wins
from choosing LQ only in case the market is big. Since Athena
knows about market research, it is clear for her that since Zeus
chose HQ, it cannot be that the market is big – and hence in this
situation it is better for Athena to produce HQ
• It requires quite a subtle reasoning.
• Zeus increased the profits from 18 to 22. So market
research is profitable if its cost does not exceed 4.
Different Assumptions about information A4
• What if Athena also knew the results of the market research – the
same as at the beginning
Different Assumptions about
information A5
• What if Zeus hides from Athena that it conducted
a market research?
• Athena will have wrong idea about the game
being played.
– Athena will think that the game is as in A2
– Zeus will know that the right game is as in A3
• Athena plays HQ/LQ (optimal in A2)
– Zeus will exploit that an play HQ/LQ and will get 24
instead of 22
• Making market research secret brings Zeus another $2mln.
Duopol Stackelberga (1934)
• Tak samo jak w przypadku
konkurencji ilościowej 2 firm
wg Cournot:
– Tylko zamiast jednocześnie,
jedna z firm może zdecydować
bądź zobowiązać się wcześniej
niż inna co do produkcji
• Model nazywa się wówczas
oligopolem Stackelberga
Model Cournot - przypomnienie
• Zysk ze sprzedaży i-tej firmy
• Najlepsze odpowiedzi graczy:
• Równowaga i zyski w równowadze:
Exercise
• Two firms produce identical good.
• Each firm decides upon its production levels.
• Inverse demand is p(x1,x2)=20-2x1-2x2 (or 0 if
x1+x2>10)
• Cost function is the same for both players
c(xi)=8xi, i=1,2
1. Cournot duopoly – determine Nash
equilibrium
Model Stackelberga
1
• Gra ma dwa etapy:
– Firma 1 wybiera poziom
produkcji (lub zdolności
produkcyjnych) q1 ≥ 0
– Firma 2 widzi wybór pierwszej
i również wybiera poziom
produkcji q2 ≥ 0
• Zyski firm są takie same
q1
2
q2
Π1,Π2
Równowaga indukcji wstecznej
• Racjonalność sekwencyjna firmy II wymaga, że
będzie najlepiej odpowiadać na jakikolwiek
wybór q1:
• Firma I natomiast wybierze q1 tak, aby
maksymalizować:
Równowaga indukcji wstecznej
• Równowaga indukcji wstecznej
• Wynik gry w równowadze
• Zyski w równowadze
– korzyść lidera
First-mover advantage
Exercise continued
2. Player 1 decides first, Player 2 after observing
what player 1 has done
– The structure of the game is common knowledge
a) Find SPNE
b) Consider the following pair of strategies: x1=4,
x2=2 if x1<4; x2=1 if x1=4; x2=1.5 if x1>4. Is this
a (imperfect) Nash Equilibrium?
c) Show that a pair of strategies: x1=3; x2=1.5
irrespective of an observed x1 is not a NE.
PRODUCT
INNOVATOR
FOLLOWER
WINNER
Jet Airliners
De Havilland (Comet)
Boeing (707)
Follower
Float glass
Pilkington
Corning
Leader
X - Ray Scanner
EMI
General Electric
Follower
Office P.C.
Xerox
IBM
Follower
VCRs
Ampex/Sony
Matsushita
Follower
Diet Cola
R.C. Cola
Coca Cola
Follower
Instant Cameras
Polaroid
Kodak
Leader
Pocket Calculator
Bowmar
Texas Instruments
Follower
Microwave Oven
Raytheon
Samsung
Follower
Plain Paper Copiers
Xerox
Canon
Not clear
Fiber Optic Cable
Corning
many companies
Leader
Video Games Players
Atari
Nintendo/Sony
Followers
Disposable Diapers
Proctor & Gamble
Kimberly-Clark
Leader
Web browser
Netscape
Microsoft
Follower
Cholesterol lowering
margarine
Raisio
Unilever
Follower
MP3 players
Diamond Multimedia
Apple
Follower