Game theory
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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

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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


Game theory

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 2

    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


    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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