Game Theory
• Game theory models strategic behavior by agents
who understand that their actions affect the actions
of other agents.
• Useful to study
– Company behavior in imperfectly competitive markets
(such as Coke vs. Pepsi).
– Military strategies.
– Bargaining/Negotiations.
– Biology
• A game consists of players, strategies, and
payoffs.
Battle of Bismarck Sea
• We want to model the Battle of the Bismarck Sea.
• Two Admirals: Imamura (Japan) and Kenny (US).
• Japan is in retreat.
• Imamura wants to transport troops in a convoy
from Rabaul to Lae
• Kenny wants to bomb Japanese troops. .
• North route is two days, Southern route is three days.
• It takes one day for Kenny to switch searching routes.
Imamura wants to run convoy from Rabaul to Lae
Battle of Bismarck Sea
Imamura
North
South
-2
-2
North
2
2
Kenny
-3
-1
South 1
3
This representation is called a Normal form Game.
•Imamura wants to transport troops.
• Kenny wants to bomb Japanese troops. .
• North route is two days, Southern route is three days.
• It takes one day for Kenny to switch routes.
Battle of Bismarck Sea
Imamura
North
South
-2
-2
North
2
2
Kenny
-3
-1
South 1
3
Players
• Imamura wants to transport troops.
• Kenny wants to bomb Japanese troops. .
• North route is two days, Southern route is three days.
• It takes one day for Kenny to switch routes.
Battle of Bismarck Sea
Imamura
North
South
-2
-2
North
2
2
Kenny
-3
-1
South 1
3
Players
Imamura’s
Strategies
• Imamura wants to transport troops.
• Kenny wants to bomb Japanese troops. .
• North route is two days, Southern route is three days.
• It takes one day for Kenny to switch routes.
Battle of Bismarck Sea
Kenny’s
Strategies
Imamura
North
South
-2
-2
North
2
2
Kenny
-3
-1
South 1
3
Players
Imamura’s
Strategies
• Imamura wants to transport troops.
• Kenny wants to bomb Japanese troops. .
• North route is two days, Southern route is three days.
• It takes one day for Kenny to switch routes.
Battle of Bismarck Sea
Imamura
North
South
-2
-2
North
2
2
Kenny
-3
-1
South 1
3
Imamura’s
Payoffs:
Each day of
Bombing =
-1 in payoff
• Imamura wants to transport troops.
• Kenny wants to bomb Japanese troops. .
• North route is two days, Southern route is three days.
• It takes one day for Kenny to switch routes.
Battle of Bismarck Sea
Kenny’s
Payoffs:
Imamura
Each day of
Imamura’s
North
South
Bombing =
Payoffs:
1 in payoff North
-2
Each day of
-2
2
2
Bombing =
Kenny
-1 in payoff
-3
-1
South 1
3
• Imamura wants to transport troops.
• Kenny wants to bomb Japanese troops. .
• North route is two days, Southern route is three days.
• It takes one day for Kenny to switch routes.
Battle of Bismarck Sea
Imamura
North
South
-2
-2
North
2
2
Kenny
-3
-1
South 1
3
Notice that Kenny guessing wrong costs him a day.
• Imamura wants to transport troops.
• Kenny wants to bomb Japanese troops. .
• North route is two days, Southern route is three days.
• It takes one day for Kenny to switch routes.
The Prisoners’ Dilemma
Clyde
S
S
Bonnie
C
C
-5
-5
-1
-15
-15
-1
-10
-10
Bonnie and Clyde are caught.
They can confess or be silent.
Nash equilibrium
• A Nash equilibrium is a set of strategies (s1,s2)
– Where each player has no incentive to deviate given
the other players’ strategies.
– (or) Given other equilibrium strategies a player would
choose his equilibrium strategy.
– (or) A best response to a best response.
– u1(s1,s2)>=u1(z,s2) for all possible z and
u2(s1,s2)>=u2(s1,z) for all possible z.
• A pure strategy equilibrium is where each player only
chooses a particular strategy with certainty.
• What are the pure-strategy equilibria in Prisoners’
dilemma?
Trench Warfare: a prisoners’ dilemma
Mars
Not Shoot Shoot
-5
Not Shoot
-1
-5
-15
Venus
-10
-15
Shoot
-1
-10
Are we doomed to the bad outcome?
Not in trench warfare of WWI.
This happens since the game is repeated –
that is, played several times.
Repeated games: forever punish
• Forever punish strategy: if someone cheats, punish
forever.
–
–
–
–
Cheating gives short term gain but long term loss.
Gain of 4 for the time of cheating.
Loss of 5 from the next period on.
Whether this stops cheating depends upon how you
value today compared to tomorrow (discount rate).
• Forever punish is not so great if there is uncertainty.
Why?
• Many tribal religions do not have explicit ethical code as
part of the religion: no stealing, no murder, etc.
Coordination Problem
Jim
VHS
VHS
Sean
Beta
Beta
1
1
0.5
0
0
0.5
2
2
• Jim and Sean want to have the same VCR.
• Beta is a better technology than VHS.
Information Technology
• Phones, Faxes, e-mail, etc. all have the following
property:
– Network externalities: The more people using it the
more benefit it is to each user.
• Computers, VCRs, PS2s, also have this property
in that both software can be traded among users
and the larger the user market, the larger number
of software titles are made.
• How do markets operate with such externalities?
Discussion points
• Competitors: VHS vs. Beta, Qwerty vs. Dvorak,
Windows vs. Mac, Playstation vs. Xbox.
• Does the best always win?
• Standardization helps with network externalities.
– Drive on left side vs. right side. Out of 206 countries
144 (70%) are rhs.
– Left is more nature for an army: swords in right hand,
mounting horses. (Napolean liked the other way.)
– Sweden switched from left to right in 1967.
• Lots of networks: Religions and Languages.
Penalty Kick
Kick L
Kicker
Kick R
Goalie
Dive L
Dive R
1
-1
-1
1
1
-1
1
-1
• A Kicker can kick a ball left or right.
• A Goalie can dive left or right.
Mixed Strategy equilibrium
• Happens in the Penalty kick game.
• Notice that if the Kicker kicks 50-50 (.5L+.5R),
the Goalie is indifferent to diving left or right.
• If the Goalie dives 50-50 (.5L+.5R), the Kicker is
indifferent to kicking left or right.
• Thus, (.5L+.5R,.5L+.5R) is a mixed-strategy N.E.
• Nash showed that there always exists a Nash
equilibrium (which includes mixed strategies).
Do you believe it?
• Can we empirically test this theory?
• Yes!
– Study was done with the Italian football league.
– Step 1: See if the strategies are really left or right.
– Step 2: Calculate payoffs. How? If when the goalie
guesses correctly, there is no goal 100% of the time the
payoffs are 0 for the kicker and 100 for the goalie. If
there no goal 80% of the time, then the payoffs are 20
for the kicker and 80 for the goalie, etc…
– Step 3: Calculate the Nash equilibrium.
– Step 4: Compare.
Do you believe it?
• Do they really choose only L or R? Yes. Kickers 93.8% and
Goalie 98.9%.
• Kickers are either left or right footed. Assume R means kick
in “easier” direction. Below is percentage of scoring.
Kick L
Kick R
Dive L
Dive R
58.3
94.97
92.91
69.92
• Nash prediction for
(Kicker, Goalie)=(41.99L+58.01R, 38.54L+61.46R)
• Actual Data
=(42.31L+57.69R, 39.98L+60.02R)
Parking Enforcement Game
Student Driver
Park OK Park in Staff
Check
University
Don’t
-5
-5
5
5
-5
-95
5
0
• Student can decide to park in staff parking.
• University can check cars in staff parking lot.
What happens?
•
•
•
•
If the University checks, what do the students do?
If the students park ok, what does the Uni do?
If the uni doesn’t check, what do the students do?
If the students park in the staff parking, what does
the uni do?
• What is the equilibrium of the game?
• What happens if the university makes it less harsh
a punishment to only –10. Who benefits from this?
Who is hurt by this?
Answer
• Student parks legally 1/3 of the time and the
uni checks 1/10 of the time.
• With lower penalty, student parks legally
1/3 of the time and the uni checks 2/3 of the
time.
• Who’s expected payoff changes? No one.
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Game Theory - אוניברסיטת חיפה