Game Theory “Loretta’s Driving Because I’m Drinking and I’m Drinking Because She’s Driving” - The Lockhorns Cartoon Mike Shor Lecture 3

Game Theory

“Loretta’s Driving Because I’m Drinking and I’m Drinking

Because She’s Driving”- The Lockhorns Cartoon

Mike ShorLecture 3

Game Theory - Mike Shor 2

Review Understand the game you are in

Note if the rules are flexible

Anticipate your opponents’ reactions

Understand the assumptions• Recognize that not everyone else understands them



Equilibrium Nash Equilibrium:

• A set of strategies, one for each player, such that each player’s strategy is best for her given that all other players are playing their equilibrium strategies

Best Response:• The best strategy I can play given the strategy

choices of all other players

Everybody is playing a best response• No incentive to unilaterally change my strategy


Identifying the Equilibrium Pure strategy equilibrium

• Consider mixed later

Dominance• Dominance solvable• Only one dominant strategy

Successive elimination of dominated strategies

Cell-by-cell inspection


Cigarette Advertising on TV All US tobacco companies advertised

heavily on TV

Surgeon General issues official warning• Cigarette smoking may be hazardous

Cigarette companies’ reaction• Fear of potential liability lawsuits

Companies strike agreement• Carry the warning label and cease TV

advertising in exchange for immunity from federal lawsuits.

1964

1970


Strategic Interactions Players: Reynolds and Philip Morris Strategies: { Advertise , Do Not Advertise } Payoffs: Companies’ Profits

Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor

How to represent this game?


Normal (Strategic) Form

PLAYERS

STRATEGIESPAYOFFS

Philip Morris

No Ad Ad

Reynolds No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30


Normal Form

Best reply for Reynolds:• If Philip Morris advertises: advertise• If Philip Morris does not advertise: advertise

Regardless of what you think Philip Morris will do

Advertise!

Philip Morris

No Ad Ad

ReynoldsNo Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30


Dominant StrategyA strategy that outperforms all other choices

no matter what opposing players do Firm 1’s strategies: { A, B, C } Firm 2’s strategies: { X, Y, Z } C is strictly dominant for Firm 1 if:

(C,X)>(A,X) (C,X)>(B,X) (C,Y)>(A,Y) (C,Y)>(B,Y) (C,Z)>(A,Z) (C,Z)>(B,Z)

C is weakly dominant for Firm 1 if: Some inequalities are weak (), at least one is strong(>)


Dominance Solvable

If each player has a dominant strategy, the game is dominance solvable

What is the equilibrium of the cigarette advertising game?

COMMANDMENT

If you have a dominant strategy, use it.

Expect your opponent to use her dominant strategy if she has one.


Cigarette Advertising After the 1970 agreement, cigarette

advertising decreased by $63 million Profits rose by $91 million Prisoner’s Dilemma An equilibrium is NOT necessarily efficient

What if the game is not dominance solvable?


A Strategic SituationTwo firms competing over sales

Time and The Economist must decide upon the cover story to run some week.

The big stories of the week are:• A presidential scandal (labeled S), and• A proposal to deploy US forces to Grenada (G)

Neither knows which story the other magazine will choose to run


One Dominant Strategy

Who has a dominant strategy? Assume it will be played! Other player can plan accordingly.

The Economist

G S

TimeS 100 , 100 0 , 90

G 95 , 100 95 , 90


Dominated Strategies

For The Economist: G dominant = S dominated

Dominated Strategy:• There exists another strategy which always does

better regardless of opponents’ actions

The Economist

G S

TimeS 100 , 100 0 , 90

G 95 , 100 95 , 90


Successive Elimination of Dominated Strategies If a strategy is dominated,

eliminate it The size and complexity of the game

is reduced Eliminate any dominant strategies

from the reduced game Continue doing so successively


Example: Tourists & Natives• Two bars (bar 1, bar 2) compete• Can charge price of $2, $4, or $5• 6000 tourists pick a bar randomly• 4000 natives select the lowest price bar

$2 $4 $5

Bar 1

$2 10 , 10 14 , 12 14 , 15

$4 12 , 14 20 , 20 28 , 15

$5 15 , 14 15 , 28 25 , 25

Bar 2


Successive Elimination of Dominated Strategies Does any player have a dominant

strategy? Does any player have a

dominated strategy?• Eliminate the dominated strategies• Reduce the normal-form game• Iterate the above procedure

What is the equilibrium?


Successive Elimination of Dominated Strategies

$4 $5

Bar 1$4 20 , 20 28 , 15

$5 15 , 28 25 , 25

25 , 25

28 , 15

14 , 15

$5$4

15 , 2815 , 14$520 , 2012 , 14$4Bar 114 , 1210 , 10$2

$2

,

,

,

, ,

, , Bar 1, ,

Bar 2

Bar 2


No Dominated Strategies Often there are no dominated strategies

• Or: reducing the game is not sufficient

There may be multiple equilibria Method:

Cell-by-cell inspection Ask:

Is each player playing the best response to the other player?


Types of Games Games of Assurance Games of Coordination Games of Chicken


Games of Assurance Two firms each earning $45,000 Both can invest the $45,000 into R&D R&D successful only if both invest If R&D successful, each earns $95,000

Invest Don’t

Firm 1Invest 50 , 50 0 , 45

Don’t 45 , 0 45 , 45

Firm 2


Consider { Invest , Don’t }

Both players have an incentive to change their strategy: NOT an equilibrium

Cell-by-cell Inspection

Invest Don’t

Firm 1Invest 50 , 50 0 , 45

Don’t 45 , 0 45 , 45

Firm 2


Assurance Outcomes Two equilibria exist Both firms prefer (I ,I) to (D,D)

• Payoffs of 50 to each firm instead of 45 However, investing is risky

• Must have assurances How to achieve assurance?

• Strategic moves: commit to choosing I• Sequential moves: leader chooses

the equilibrium


Games of Coordination Joint ventures and the choice of supplier Two firms engaged in joint venture Must use the same supplier, but

each firm has a preferred supplier

Firm 2A B

Firm 1A 100 , 50 0 , 0

B 0 , 0 50 , 100


Coordination Outcomes Two equilibria exist Firms prefer different equilibria How to achieve the most

desirable outcome for you?• Strategic moves: commit to choosing A• Sequential moves: leader chooses

the equilibrium


Summary You must put yourself in

your rival’s shoes

Recognize dominant and dominated strategies

Anticipate that your opponent will recognize them as well

Documents

Game Theory “Loretta’s Driving Because I’m Drinking and I’m Drinking Because She’s Driving” - The Lockhorns Cartoon Mike Shor Lecture 3