Transcript Eng & Tech Mgmt - Chinese University of Hong Kong

Game Theory
Games of strategy
 Sequential games
 Simultaneous decisions
 Dominated strategies
 Nash equilibrium
 Prisoners’ dilemma

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Sequential decisions







Previously ….
Sequential decisions with uncertainty
Decision trees … with “chance” nodes
but …
“God does not play dice” – Albert Einstein
“Subtle is the Lord, but malicious He is not.”
What about your competitors?
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A sequential “game”



Decisions made in sequence.
Your decision depends on decision made previously
by others, and others’ decisions follow and depend
on yours, etc.
Outcome/payoff depends on all decisions made by
all.
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Lucy Van Pelt vs. Charlie Brown

Lucy Van Pelt holds a football on the ground
and invites Charlie Brown to run up and kick
it. At the last moment, Lucy pulls the ball
away. Charles Brown, kicking air, lands on
his back, and this gives Lucy great perverse
pleasure.
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This time …
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Representing Decisions in a
Game Tree
Pull Ball Away
Accept
Lucy
Charlie
Let Charlie kick
Reject
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,
,
,
6
And then …
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Games of Strategy
Vijay Krishna (Harvard Business School):
Any situation where the choices of two or more
rational decision makers together leads to
gains and losses for them is called a game.
A game may simultaneously involve elements
of both conflict and co-operation among the
decision makers.
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Market Competition - HDTV
Vizio considers entering a market now
monopolised by Samsung. Samsung can
decide to respond by being
accommodating or aggressively fight a
price war. Profit outcomes for both firms
depends on the strategies of both firms.
As Vizio, you can analyse this problem
using Decision Analysis by estimating
probabilities of Samsung’s responses.
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Market Entry –
Decision Tree for Vizio
Accommodate
$100,000 to Vizio
p
Samsung
Enter
1-p
-$200,000 to Vizio
Vizio
Fight price war
Keep out
$0 to Vizio
How to estimate the probabilities? What does p depend on?
If no information, p=0.5? Then Vizio will not enter market.
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-7, 2
Samsung
0, 10
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Vizio
Probability of Samsung’s
response will depend
on Samsung’s payoff
in the different
scenarios
5, 8
Game Tree
Representation
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Market Entry – Game Tree Model
Accommodate
$100,000 to
Samsung
Samsung
-$200,000 to
Vizio
Enter
Vizio
Keep out
$100,000 to
Vizio
Fight price war -$100,000 to
$0 to Vizio
Samsung
$300,000 to Samsung
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Analysing Game Trees
Rule 1: Look Ahead and Reason Back!

For this market alone, Vizio should choose
enter because Samsung (rationally) will
accommodate.

If Samsung worries that Vizio may enter other markets in the
region after this, Samsung may take a tough stand. Vizio
should not enter.
The “payoff” should include all “benefits”.

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Look Ahead & Reason Back
1.
Formulate the game tree of the situation.

2.
Evaluate payoffs at the “leaves” of the tree.

3.
Identify your own and opponent’s strategy at each stage.
This assumes:
 Your opponent’s strategy can be observable.
 Strategy must be irreversible.
Think about what will happen at the end.
Reason backward through the tree.

Identify the best strategy for each player at each stage,
starting at the end.
Note the essence of a game of strategy is interdependence. Your
decision affects your opponent’s decision and your opponent’s
affects yours.
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More complex games
Black-1
P-Q4
Black-1
P-QB4
White-2
P-K4
White-1
Black-1
Theoretically, can map out all possible chess
moves and then select the best sequence of
moves
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White-2
White-2
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Chess - Human vs. Computers



Good chess players can “see”
14 moves ahead!
(1968) David Levy: “No
computer can beat him in 10
years”
Deep Blue




Chess playing machine built by
IBM in the 1990’s
2 to 2.5 million moves per second.
(1996) Deep Blue 1 lost to
world chess champion Gary
Kasparov.
(1997) Deep Blue 2 defeated
Kasparov.
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Deep Blue vs. Kasparov 1996, game 1.
The final position.
a
b
c
d
e
f
g
h
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
a
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b
c
d
e
f
g
h
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Homework?


Draw the game tree for TIC-TAC-TOE.
Sure-win strategies?
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The Election of the Chief
Executive for Hong Kong




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The next Chief Executive of Hong Kong SAR
Government will be “elected”.
Mr. B is Beijing’s favourite candidate.
Ms. C (the potential challenger) considers entering
the race.
Mr. B must determine whether to launch a
preemptive advertising campaign against Ms. C
(expensive) or not (cost-saving).
Ms. C must determine whether to enter the race.
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Election Game Tree
B’s, C’s payoff
In
1, 1
C
Advertise
The larger
the better
Out
B
3, 2
In
2, 4
No Ad
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C
Out
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4, 2
Game Tree
B’s, C’s payoff
In
1, 1
C
Ads
Out
B
3, 2
In
2, 4
No Ads
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C
Out
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4, 2
Game Tree
B’s, C’s payoff
In
1, 1
C
Ads
Out
B
3, 2
In
2, 4
No Ads
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C
Out
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4, 2
Advantage due to Order of
Decisions?



First-mover advantage?
Mr. B (first) sets the stage for Ms. C
(second). Mr. B can look ahead to Ms. C’s
optimal response and make the move to his
advantage.
Can Ms. C improve her situation by acting
first?
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Game Tree
C’s, B’s payoff
Adv
1, 1
B
In
No Adv
C
4, 2
Adv
2, 3
Out
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B
No Adv
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2, 4
Better off being first?


Is there a first-mover advantage?
What about adoption of new technology?


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Better off as a technology leader?
Better off as a technology follower?
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Simultaneous Decisions
In the chess example, the sequence of decisions
alternate between the players.
In other situations, the decision may not be
sequential but simultaneous.
Tic-tac-toe (sequential)
Stone-paper-scissors (simultaneous)
In simultaneous games, the payoffs to the players
are still interdependent on chosen strategies of
ALL players.
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Time vs. Newsweek



Each week, these magazines decide on what story to
put on the cover.
They do not know the other’s decision until
publication.
Suppose there are two “hot” stories:




(A): Anna Chapman, the Russian Spy,
(B): British Petroleum Oil Spill damage
Newsstand buyers only purchase if story is on cover.
70% interested in (A) and 30% in (B).

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Purchases evenly split the if both magazines have the same
story.
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Matrix Representation of Game
Payoff for Time
Newsweek
A
B
A
35 = 70/2
70
B
30
15 = 30/2
Time
What should Time do?
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Matrix Representation of Game
for Newsweek
Payoff to
Newsweek
Newsweek
A
B
A
35
30
B
70
15
Time
No matter what Time does, Newsweek is better off putting
(A) as cover story.
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Dominant Strategies
Payoffs
Newsweek
Time, Newsweek
A
B
A
35, 35
70, 30
B
30, 70
15, 15
Time
Choosing (A) is a dominant strategy for
both Time and Newsweek!
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Dominant Strategy



A dominant strategy is one that makes a
player better off than he would be if he used
any other strategy, no matter what strategy
his opponent uses.
A strategy is dominated if there is another
strategy that under no circumstances leads to
a lower payoff, and sometimes yields a better
payoff.
Note: For some games, there may be no
dominant strategy for some players.
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Properties of a dominant strategy
1: A dominant strategy dominates your other
strategies, NOT your opponent!
Even with your dominant strategy, your payoff could
be smaller than your opponents.
2: A dominant strategy does not requires that
the worst possible outcome of the dominant
strategy is better than the best outcome of an
alternative strategy.
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Pricing example
Suppose there are just two possible pricing choices: $3 (a
profit margin of $2 per copy) and $2 ($1 per copy).
Customers will always buy the lower-priced magazine.
Profits are split equally between the two. The total
readership is 5 million if the price is $3, and rises to 8
million if the price is only $2.
Time’s
Sales
Time’s
Price
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Newsweek’s Price
$2
$3
$2
4 million
8 million
$3
0 million
5 million
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Analysing Games
Rule 2: If you have a dominant
strategy, use it !
Rule 3: Eliminate any dominated
strategies from consideration,
and do so successively!
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Eliminating Dominated
Strategies - Example




American ship at A,
Iraqi ship at I.
Iraqi plans to fire a missile at
American ship; American ship
plans to fire a defense missile
to neutralize the attack
(simultaneously).
Missiles programmed to
A
(possibly) turn every 20
seconds.
If missile not neutralised in 60
seconds, American ship sinks!
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B
C
F
E
D
G
I
H
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Possible strategies (paths)
I1I2I3I4I5I6I7I8IFCB IFEB IFED IFEH IHGD IHED IHEB IHEF

For American,
A1ABCF
H
O
O
O
O
O
O
H

A2ABEF
O
H
H
H
O
H
H
H

A3ABEH
O
H
H
H
O
H
H
H
A4ABED
O
H
H
H
H
H
H
H
A5ADGH
O
O
O
H
H
O
O
O
A6ADEH
O
H
H
H
O
H
H
H
A7ADEF
O
H
H
H
O
H
H
H
A8ADEB
H
H
H
H
O
H
H
H
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



A2, A3 dominated by A4,
A6, A7 dominated by A8,
A1 is dominated by A8,
A5 is dominated by A4,
Only A4 and A8 not
dominated.
Similarly for Iraqi.

Only I1 and I5 are not
dominated.
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Simplified Game
American
C
B
Iraqi
A
I1IFCB
I5IHGD
A4ABED
O
H
A8ADEB
H
O
vs. Iraqi
American
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F
E
D
G
I
H
No dominant
strategy for
either player!
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Nash Equilibrium
A set of strategies constitute
a Nash Equilibrium if:
no player can benefit by
changing her strategy while
the other players keep their
strategies unchanged.

Each player’s strategy is the
“best-response” to the other
players’ set of strategies.
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Dominant Strategy Equilibrium



Higher viewership
means more
advertising revenues
for both TV stations.
Each TV station has a
dominant strategy.
In this case, the
equilibrium for this
game is obvious.
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TVB payoff
ATV
ATV payoff
SoapOpera
News &
Analysis
SoapOpera
55%,
45%
52%,
48%
News &
Analysis
50%,
50%
45%,
55%
TVB
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Dominant Strategy Equilibrium

If, in a game, each player has a dominant
strategy, and each player plays the dominant
strategy, then that combination of (dominant)
strategies and the corresponding payoffs are
said to constitute the dominant strategy
equilibrium for that game.
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Nash Equilibrium
Payoffs
Newsweek
Time, Newsweek
A
B
A
42, 28
70, 30
B
30, 70
18, 12
Time
No dominant strategy for Newsweek.
Unique Nash equilibrium.
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Example - “Chicken”
James Dean
Swerve
Don’t
swerve
Mad
Swerve
C, C
C, H
Max
Don’t
swerve
H, C
D, D
H>C>D
No
dominant strategy
Two Nash equilibria
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Choosing among Multiple
Equilibria


Some games have
multiple equilibria.
“Rule of the road”



Brit
Drive on Drive on
left
right
Hong Kong, Britain,
Australia, Japan (left)
China, Europe,
Mexico, USA (right)
The social convention
of the locale
determines which
equilibrium to choose.
Drive on
left
, D, D
Drive on
right
D, D ,
Yank
Sweden switch from left to
right in 1967.
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In-class exercise (Texas A&M)
Each of you owns a
production plant and
can choose to
produce 1 or 2 units
of a product.
 More total production
will lower price and
hence profit.
What would you do?

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# of
“1”
Payoff to Payoffs to
“1” firms “2” firms
0
$0.50
1
$0.04
$0.54
2
$0.08
$0.58
:
:
:
29
$1.16
$1.66
30
$1.20
$1.70
:
:
:
59
$2.36
$2.86
60
$2.40
$2.90
:
:
:
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Is a Nash equilibrium “good”
for the players?
Just because a game has an equilibrium does not mean
that those strategies are “best” for the players.
 Prisoners’ dilemma:



Two burglars, Bob and Al, are captured at the scene of a
burglary and interrogated separately by the police.
Each has to choose whether or not to confess.
Outcomes:



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If neither man confesses, then both will serve only one year.
If both confesses, both will go to prison for 10 years.
However, if one burglar confesses and implicates the other, and
the other burglar does not confess, the one who has collaborated
with the police will go free, while the other burglar will go to prison
for 20 years.
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1950 – Dresher & Flood (Rand)
A. W. Tucker
Prisoners’ dilemma
Punishment
Al
confess
deny
confess
10,10
0,20
deny
20,0
1,1
Bob


For each player, the dominant strategy is to confess!
Unique Nash equilibrium!
• Both play the dominant strategy but create mutually
disastrous outcome! Both would be better off by denying!
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Cartels



Companies or countries form an alliance to jointly
make price and production decisions.
World Trade Organisation (WTO) /
General Agreement on Tariffs and Trade (GATT)
The Organization of Petroleum Exporting Countries
(OPEC) is a cartel.

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the mission of OPEC is to coordinate and unify the policies
of its Member Countries and ensure the stabilization of oil
markets in order to secure a regular supply of petroleum to
consumers, a steady income to producers …
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OPEC – Maintaining a Cartel
Iraq’s output
Profits
Iran



2 mb
4 mb
2 mb
46 , 42
26 , 44
4 mb
52 , 22
32 , 24
Total output:
4mb 6mb 8mb
Price per barrel:
$25 $15 $10
Extraction costs: Iran: $2/barrel;
Iraq: $4/barrel
Dominant strategy: produce at higher level !
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Ensuring Co-operation




The dominant strategy equilibrium results
in each producing 4 million barrels and
achieving 56 million in total joint profit.
Suppose OPEC countries have agreed to
maintain production at 2 mb per day.
If members produces 2 million barrels each
(as agreed), they will make 88 million in
total joint profit.
Is it possible to achieve cooperation, when
the dominant strategy is to cheat?
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Detection of Cheating




Co-operation is difficult when the reward for
cheating is high.
How to tell if some member cheated and
produced more?
The price is US$25 per barrel only if
members maintained low production. If price
drops below $25, then someone has cheated!
What if demand actually decreased?
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Identifying cheaters

In a two-player game, an honest party
knows who cheated.



Still, the cheating party may deny they cheated.
When there are many players, even when
cheating has been detected, it may be
difficult to identify who cheated !
If voluntary cooperation is not possible, how
about making use of punishment?

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In a one-period game, there is no solution to
achieve reciprocal co-operation.
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Punishment – Credible Threat?





If the game repeats, cooperation may be enforced.
Suppose Iran begins to cheat and produces 4 million
barrel per day secretly.
 Iran’s profit goes up from 46 to 52 million per day.
When Iraq finds out, Iraq also produces 4 million
barrels.
 Iran’s profit goes down to 46 to 32 million per day.
Assume it takes a month for Iraq to know.
 Iran’s total profit through cheating: 6x30= 180 million
Iraq retaliates by increasing production.
 Iran’s cheating gain will be wiped out in 13 days
(i.e., 180 million / 14 million)
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Competition or Collusion?

DVD player vendors: Fortress
Broadway
wholesale: $1500, retail: $3000
Broadway: lowest price guarantee:
“refund double the price difference”
Should Fortress cut its price to $2750?

What will the consumer do?

How will Broadway respond?



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“Implicit” Cartel




If Fortress tries to increase its market share by
lowering its price to $2750.
Customers will buy from Broadway and claim
from a $500 rebate. The “selling price” for
Broadway is effectively $2500; lower than
Fortress’ price of $2750.
In response, Broadway will not give away
rebates but lower its price to $2750.
Fortress becomes worse off … so why bother?
Collusion is enforced by “announcing” the
punishment!
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Sustaining Co-operation



Mechanism must Detect cheating and Deter
cheating.
Which Punishment?
Simplicity and clarity


Certainty


Easy for potential cheaters to evaluate consequences.
Players have confidence that defection will be punished
and co-operation rewarded.
Severity

not to “fit the crime” but for deterrence!
Risk of mistakes?
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Tit-for-tat
Exodus 21:22-25
 If men who are fighting hit a pregnant woman
and she gives birth prematurely but there is
no serious injury, the offender must be fined
whatever the woman’s husband demands.
 But if there is a serious injury, you are to take
life for a life, eye for eye, tooth for tooth, hand
for hand, burn for burn, wound for wound,
bruise for bruise.
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Tit-for-tat Strategy
Co-operates in the first period, thereafter
mimics the rival’s action from previous rounds





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Clarity (simple to implement)
Niceness (does not initiates cheating)
Provocability (it never lets cheating go unpunished)
Forgiveness (does not hold a grudge, willing to restore
cooperation)
“Chain reaction” of mistakes?
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Hatfields and McCoys



This is one of best-documented stories on
inter-family feud (1878 – 1891).
Early settlers in the Tug Valley on the
Kentucky and West Virginia border.
Feud started over the disputed ownership of
a pig!
Kentucky
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West
Virginia
58
Tit-for-tat with misperception
Round Hatfield McCoys Round Hatfield McCoys
1
P
P
6
P
A
2
P
P
7
A
P
3
P
P
8
P
A
4
P
P*
9
A
P*
5
A
P
10
A
A
6
P
A
* Misperceived as an “A”
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A
A
12
A
A
Mis-perception leads
to perpetual
retaliation!
Nuclear conflict?
Cuban missile crisis
(1962).
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Tit-for-tat Strategy

When misperceptions are possible, in the long run
tit-for-tat will spend half the time cooperating and
half of it defecting.


When the probability of a misperception is small, it will take
a lot longer for this phenomenon to occur.
When the probability is 50%, whatever you do will not have
any affect on your opponent.



Opponent will perceive aggression with 0.5 probability.
When the probability is 50%, there is no hope of
achieving co-operation. One should always attack!
Feud never ends …
Should one be more forgiving?
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中庸之道 (The Moderate
Chinese Way)




Tit-for-tat is quick to punish opponent who has a
long history of cooperation.
Other responses:
(Matthew 5:38): “But I tell you, do not resist an evil
person. If someone strikes you on the right cheek,
turn to him the other also.”
A more forgiving tit-for-tat:



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Begin cooperating
Continue cooperating, but keep count of how many times
the other side appears to be have defected while you have
cooperated.
If the this percentage becomes unacceptable, revert to titfor-tat.
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Summary






Games of strategy
Sequential games
Simultaneous decisions
Dominated strategies
Nash equilibrium
Prisoners’ dilemma
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