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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. 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

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
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

Game Theory - Mike Shor

equilibrium
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

Game Theory - Mike Shor

identifying the equilibrium
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

Game Theory - Mike Shor

cigarette advertising on tv
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

Game Theory - Mike Shor

strategic interactions
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?

Game Theory - Mike Shor

normal strategic form
Normal (Strategic) Form

PLAYERS

STRATEGIES

PAYOFFS

Game Theory - Mike Shor

normal form
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!

Game Theory - Mike Shor

dominant strategy
Dominant Strategy

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

dominance solvable
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.

Game Theory - Mike Shor

cigarette advertising
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?

Game Theory - Mike Shor

a strategic situation
A Strategic Situation

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

one dominant strategy
One Dominant Strategy
  • Who has a dominant strategy?
  • Assume it will be played!
  • Other player can plan accordingly.

Game Theory - Mike Shor

dominated strategies
Dominated Strategies
  • 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

successive elimination of dominated strategies
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

Game Theory - Mike Shor

example tourists natives
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

Bar 2

Game Theory - Mike Shor

successive elimination of dominated strategies1
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?

Game Theory - Mike Shor

successive elimination of dominated strategies2
Bar 2Successive 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

no dominated strategies
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?

Game Theory - Mike Shor

types of games
Types of Games
  • Games of Assurance
  • Games of Coordination
  • Games of Chicken

Game Theory - Mike Shor

games of assurance
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

Firm 2

Game Theory - Mike Shor

cell by cell inspection
Cell-by-cell Inspection
  • Consider { Invest , Don’t }
  • Both players have an incentive to change their strategy: NOT an equilibrium

Firm 2

Game Theory - Mike Shor

assurance outcomes
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

Game Theory - Mike Shor

games of coordination
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 2

Game Theory - Mike Shor

coordination outcomes
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

Game Theory - Mike Shor

summary
Summary
  • 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

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