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SEEM 35301 Game Theory Games of strategy Sequential games Simultaneous decisions Dominated...
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Transcript of SEEM 35301 Game Theory Games of strategy Sequential games Simultaneous decisions Dominated...
SEEM 3530 1
Game Theory
Games of strategy Sequential games Simultaneous decisions Dominated strategies Nash equilibrium Prisoners’ dilemma
SEEM 3530 2
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?
SEEM 3530 3
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 allall decisions made by all.all.
SEEM 3530 4
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.
SEEM 3530 5
This time …
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Representing Decisions in a Game Tree
Charlie
Reject
Accept Lucy
Pull Ball Away
Let Charlie kick
,
,
,
SEEM 3530 7
And then …
SEEM 3530 8
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.
SEEM 3530 9
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 both firms.
As Vizio, you can analyse this problem using Decision Analysis by estimating probabilities of Samsung’s responses.
SEEM 3530 10
Market Entry – Decision Tree for Vizio
Vizio
Keep out
Enter
Samsung
Accommodate
Fight price war
$0 to Vizio
$100,000 to Vizio
-$200,000 to Vizio
p
1-p
How to estimate the probabilities? What does p depend on?
If no information, p=0.5? Then Vizio will not enter market.
SEEM 3530 11
Game Tree Representation
Probability of Samsung’s response will depend on Samsung’s payoff in the different scenarios
Viz
io
Sam
sung
0, 1
0-7
, 25,
8
Do not enter
Ente
r Mar
ket
Agg
ress
ive
Acc
omm
odat
e
SEEM 3530 12
Market Entry – Game Tree Model
Vizio
Keep out
Enter
Samsung
Accommodate
Fight price war
$0 to Vizio
$300,000 to Samsung
$100,000 to Vizio
$100,000 to Samsung
-$200,000 to Vizio
-$100,000 to Samsung
SEEM 3530 13
Analysing Game Trees
Rule 1: Look Ahead and Reason Back!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”.
SEEM 3530 14
Look Ahead & Reason Back1. Formulate the game tree of the situation.
Identify your own and opponent’s strategy at each stage. This assumes: Your opponent’s strategy can be observable. Strategy must be irreversible.
2. Evaluate payoffs at the “leaves” of the tree. Think about what will happen at the end.
3. 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 decision affects yours.
SEEM 3530 15
More complex games
White-1P-K4
P-Q4
Black-1
Black-1
Black-1
P-QB4 White-2
White-2
White-2
Theoretically, can map out all possible chess moves and then select the best sequence of moves to win the game!
SEEM 3530 16
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.
SEEM 3530 17
Deep Blue vs. Kasparov 1996, game 1.The final position.
a b c d e f g h
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
a b c d e f g h
SEEM 3530 18
Homework?
Draw the game tree for TIC-TAC-TOE. Sure-win strategies?
SEEM 3530 19
The Election of the Chief Executive for Hong Kong
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.
SEEM 3530 20
Election Game Tree
B
No Ad
AdvertiseOut
C
In1, 1
Out
C
In
3, 2
2, 4
4, 2
B’s, C’s payoff
The larger the better
SEEM 3530 21
Game Tree
B
No Ads
AdsOut
C
In1, 1
Out
C
In
3, 2
2, 4
4, 2
B’s, C’s payoff
SEEM 3530 22
Game Tree
B
No Ads
AdsOut
C
In1, 1
Out
C
In
3, 2
2, 4
4, 2
B’s, C’s payoff
SEEM 3530 23
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?
SEEM 3530 24
Game Tree
C
Out
InNo Adv
B
Adv1, 1
No Adv
B
Adv
4, 2
2, 3
2, 4
C’s, B’s payoff
SEEM 3530 25
Better off being first?
Is there a first-mover advantage? What about adoption of new technology?
Better off as a technology leader? Better off as a technology follower?
SEEM 3530 26
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.
SEEM 3530 27
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).
Purchases evenly split the if both magazines have the same story.
SEEM 3530 28
Matrix Representation of Game
Payoff for Time Newsweek
A B
TimeA 35 = 70/2 70
B 30 15 = 30/2
What should Time do?
SEEM 3530 29
Matrix Representation of Game for Newsweek
Payoff to Newsweek
Newsweek A B
Time
A 35 30
B 70 15
No matter what Time does, Newsweek is better off putting (A) as cover story.
SEEM 3530 30
Dominant Strategies
Payoffs Newsweek
Time, Newsweek A B
TimeA 35, 35 70, 30
B 30, 70 15, 15
Choosing (A) is a dominant strategy for both Time and Newsweek!
SEEM 3530 31
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.
SEEM 3530 32
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.
SEEM 3530 33
Pricing example
Time’s Newsweek’s Price
Sales $2 $3
Time’s Price
$2 4 million 8 million
$3 0 million 5 million
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.
SEEM 3530 34
Analysing Games
Rule 2: Rule 2: If you have a dominant If you have a dominant strategy, use itstrategy, use it !!
Rule 3: Eliminate any dominated Rule 3: Eliminate any dominated strategies from consideration, strategies from consideration, and do so successively!and do so successively!
SEEM 3530 35
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 (possibly) turn every 20 seconds.
If missile not neutralised in 60 seconds, American ship sinks!
E IA
FCB
D HG
SEEM 3530 36
Possible strategies (paths)
For American, 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.
I1-IFCB
I2-IFEB
I3-IFED
I4-IFEH
I5-IHGD
I6-IHED
I7-IHEB
I8-IHEF
A1-ABCF
H O O O O O O H
A2-ABEF
O H H H O H H H
A3-ABEH
O H H H O H H H
A4-ABED
O H H H H H H H
A5-ADGH
O O O H H O O O
A6-ADEH
O H H H O H H H
A7-ADEF
O H H H O H H H
A8-ADEB
H H H H O H H H
SEEM 3530 37
Simplified Game
American Iraqi
vs. IraqiI1-
IFCBI5-
IHGD
American
A4-ABED
O H
A8-ADEB
H O
EI
AF
C
B
D HG
No dominant strategy for either player!
SEEM 3530 38
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.
SEEM 3530 39
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.
TVB payoff ATV
ATV payoffSoap-Opera
News & Analysis
TVB
Soap-Opera
55%, 45%
52%, 48%
News & Analysis
50%, 50%
45%, 55%
SEEM 3530 40
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.
SEEM 3530 41
Nash Equilibrium
Payoffs Newsweek
Time, Newsweek A B
TimeA 42, 28 70, 30
B 30, 70 18, 12
No dominant strategy for Newsweek.
Unique Nash equilibrium.
SEEM 3530 42
Example - “Chicken”
H > C > D
No dominant strategyTwo Nash equilibria
James Dean
Swerve Don’tswerve
Mad Swerve C, C C, H
Max Don’tswerve
H, C D, D
SEEM 3530 43
Choosing among Multiple Equilibria Some games have
multiple equilibria. “Rule of the road”
Hong Kong, Britain, Australia, Japan (left)
China, Europe, Mexico, USA (right)
The social convention of the locale determines which equilibrium to choose.
Brit
Drive on left
Drive on right
Yank
Drive on left , D, D
Drive on right D, D ,
Sweden switch from left to right in 1967.
SEEM 3530 44
In-class exercise (Texas A&M)
Each of you owns a production plant and can choose to produce 11 or 22 units of a product.
More total production will lower price and hence profit.
What would you do?What would you do?
# of “1” Payoff to “1” firms
Payoffs to “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
: : :
SEEM 3530 45
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:
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.
SEEM 3530 46
Prisoners’ dilemma
Punishment Al
confess deny
Bobconfess 10,10 0,20
deny 20,0 1,1
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!
1950 – Dresher & Flood (Rand) A. W. Tucker
SEEM 3530 47
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. 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 …
SEEM 3530 48
OPEC – Maintaining a Cartel
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 !Dominant strategy: produce at higher level !
Profits Iraq’s output
2 mb 4 mb
Iran 2 mb 46 , 42 26 , 44
4 mb 52 , 22 32 , 24
SEEM 3530 49
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?
SEEM 3530 50
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?
SEEM 3530 51
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? In a one-period game, there is no solution to
achieve reciprocal co-operation.
SEEM 3530 52
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)
SEEM 3530 53
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?
SEEM 3530 54
“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 Collusion is enforced by “announcing” the punishment!punishment!
SEEM 3530 55
Sustaining Co-operation
Mechanism must Detect cheating and Deter cheating.
Which Punishment? Simplicity and clarity
Easy for potential cheaters to evaluate consequences. Certainty
Players have confidence that defection will be punished and co-operation rewarded.
Severity not to “fit the crime” but for deterrence!
Risk of mistakes? Risk of mistakes?
SEEM 3530 56
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.
SEEM 3530 57
Tit-for-tat Strategy
Co-operates in the first period, thereafter mimics the rival’s action from previous rounds 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?
SEEM 3530 58
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!
KentuckyWest Virginia
SEEM 3530 59
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 11 A A
* Misperceived as an “A” 12 A A
Mis-perception leads to perpetual retaliation!
Nuclear conflict?
Cuban missile crisis (1962).
SEEM 3530 60
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?Should one be more forgiving?
SEEM 3530 61
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: 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 tit-for-tat.
SEEM 3530 62
Summary
Games of strategy Sequential games Simultaneous decisions Dominated strategies Nash equilibrium Prisoners’ dilemma