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# Game Theory - PowerPoint PPT Presentation

Game Theory. “Loretta’s Driving Because I’m Drinking and I’m Drinking Because She’s Driving” - The Lockhorns Cartoon Mike Shor Lecture 3. Review. Understand the game you are in Note if the rules are flexible Anticipate your opponents’ reactions Understand the assumptions

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### Game Theory

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

- The Lockhorns Cartoon

Mike Shor

Lecture 3

• Understand the game you are in

• Note if the rules are flexible

• Understand the assumptions

• Recognize that not everyone else understands them

Game Theory - Mike Shor

• 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

• Game Theory - Mike Shor

• Pure strategy equilibrium

• Consider mixed later

• Dominance

• Dominance solvable

• Only one dominant strategy

• Successive elimination of dominated strategies

• Cell-by-cell inspection

• Game Theory - Mike Shor

• 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

Game Theory - Mike Shor

• Players: Reynolds and Philip Morris

• 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?

Game Theory - Mike Shor

PLAYERS

STRATEGIES

PAYOFFS

Game Theory - Mike Shor

• Regardless of what you think Philip Morris will do

• Game Theory - Mike Shor

A 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:

• P(C,X)>P(A,X) P(C,X)>P(B,X)

• P(C,Y)>P(A,Y) P(C,Y)>P(B,Y)

• P(C,Z)>P(A,Z) P(C,Z)>P(B,Z)

• C is weakly dominant for Firm 1 if:

• Some inequalities are weak (), at least one is strong(>)

Game Theory - Mike Shor

• 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.

Game Theory - Mike Shor

• 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?

Game Theory - Mike Shor

Two 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

• Game Theory - Mike Shor

• Who has a dominant strategy?

• Assume it will be played!

• Other player can plan accordingly.

Game Theory - Mike Shor

• For The Economist: G dominant = S dominated

• Dominated Strategy:

• There exists another strategy which always does better regardless of opponents’ actions

Game Theory - Mike Shor

• 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

Game Theory - Mike Shor

• 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

Bar 2

Game Theory - Mike Shor

• 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?

• Game Theory - Mike Shor

Successive Elimination of Dominated Strategies

Bar 2

\$2

\$4

\$5

\$2

10

,

,

10

14

,

,

12

14

,

,

15

Bar 1

Bar 1

\$4

20

,

,

20

28

,

,

15

12

,

,

14

\$5

15

,

,

28

25

,

,

25

15

,

,

14

Game Theory - Mike Shor

• Often there are no dominated strategies

• Or: reducing the game is not sufficient

• There may be multiple equilibria

• Method:

Cell-by-cell inspection

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

• Game Theory - Mike Shor

• Games of Assurance

• Games of Coordination

• Games of Chicken

Game Theory - Mike Shor

• 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

Firm 2

Game Theory - Mike Shor

• Consider { Invest , Don’t }

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

Firm 2

Game Theory - Mike Shor

• 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

• Game Theory - Mike Shor

• 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 2

Game Theory - Mike Shor

• 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

Game Theory - Mike Shor

• You must put yourself in your rival’s shoes

• Recognize dominant and dominated strategies

• Anticipate that your opponent will recognize them as well

Game Theory - Mike Shor