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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 allall decisions made by all.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

Charlie

Reject

Accept Lucy

Pull Ball Away

Let Charlie kick

,

,

,

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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 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

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.

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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

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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

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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”.

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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.

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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!

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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                                                         8

7                                                      7

6                                                   6

5                                                   5

4                                                      4

3                                          3

2                                                   2

1                                                      1

   a 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

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

No Ad

AdvertiseOut

C

In1, 1

Out

C

In

3, 2

2, 4

4, 2

B’s, C’s payoff

The larger the better

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Game Tree

B

No Ads

AdsOut

C

In1, 1

Out

C

In

3, 2

2, 4

4, 2

B’s, C’s payoff

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Game Tree

B

No Ads

AdsOut

C

In1, 1

Out

C

In

3, 2

2, 4

4, 2

B’s, C’s payoff

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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

Out

InNo Adv

B

Adv1, 1

No Adv

B

Adv

4, 2

2, 3

2, 4

C’s, B’s payoff

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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?

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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).

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

TimeA 35 = 70/2 70

B 30 15 = 30/2

What should Time do?

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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.

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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!

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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

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.

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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!

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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 (possibly) turn every 20 seconds.

If missile not neutralised in 60 seconds, American ship sinks!

E IA

FCB

D HG

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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

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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!

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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.

TVB payoff ATV

ATV payoffSoap-Opera

News & Analysis

TVB

Soap-Opera

55%, 45%

52%, 48%

News & Analysis

50%, 50%

45%, 55%

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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

TimeA 42, 28 70, 30

B 30, 70 18, 12

No dominant strategy for Newsweek.

Unique Nash equilibrium.

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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

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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.

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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

: : :

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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:

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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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

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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. 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

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

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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? 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 Collusion is enforced by “announcing” the punishment!punishment!

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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?

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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 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!

KentuckyWest Virginia

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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).

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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?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: 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.

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Summary

Games of strategy Sequential games Simultaneous decisions Dominated strategies Nash equilibrium Prisoners’ dilemma