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Game Theory. Topic 2 Simultaneous Games. “ Loretta ’ s driving because I ’ m drinking and I ’ m drinking because she ’ s driving. ”. - The Lockhorns. Review. Understanding the game Noting if the rules are flexible Anticipating our opponents ’ reactions Thinking one step ahead

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Game theory

Game Theory

Topic 2

Simultaneous Games

“Loretta’s driving because I’m drinking and I’m drinking because she’s driving.”

- The Lockhorns


Review
Review

  • Understanding the game

  • Noting if the rules are flexible

  • Anticipating our opponents’ reactions

  • Thinking one step ahead

  • Where does this lead us?

    • We’ve defined the “game” but not the outcome


Equilibrium
Equilibrium

  • The likely outcome of a game when rational, strategic agents interact

    • Each player is playing his or her best strategy given the strategy choices of all other players

    • No player has incentive to change his or her action unilaterally

  • Outline:

    • Model interactions as games

    • Identify the equilibria

    • Decide when they are likely to occur


Cigarette advertising on tv

1964

1970

Cigarette Advertising on TV

  • All US tobacco companies advertised heavily on TV

  • Surgeon General issues official warning

    • Cigarette smoking may be hazardous

  • Cigarette companies fear lawsuits

    • Government may recover healthcare costs

  • Companies strike agreement

    • Carry the warning label and cease TV advertising in exchange for immunity from federal lawsuits.


  • Strategic interaction
    Strategic Interaction

    • Players: Reynolds and Philip Morris

    • Strategies: Advertise or Not Advertise

    • Payoffs: Companies’ Profits

    • Environment:

      • 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?


    Strategic form of a game

    PAYOFFS

    Strategic Form of a Game

    PLAYERS

    STRATEGIES


    What to do
    What to Do?

    If you are advising Reynolds, what strategy do you recommend?


    Best replies
    Best Replies

    • A strategy is a best reply to some opponents’ strategy if it does at least as well as any other strategy

    • si is a best replyto s-i if

      for every si’


    Solving the game
    Solving the Game

    • Best reply for Reynolds:

      • If Philip Morris advertises:

      • If Philip Morris does not advertise:


    Dominance
    Dominance

    • A strategy is dominantif it outperforms all other strategies no matter what opposing players do

    • Games with dominant strategies are easy to solve

      • No need for “what if …” thinking


    Dominance1
    Dominance

    • sistrictly dominates si’ if

      for every s-i

      (the payoff is strictly higher for any strategies of the other players)

    • siweakly dominates si’ if

      for every s-i, and

      for some s-i


    Dominance2
    Dominance

    • A strategy si is strictly dominatedif some strategy si’ strictly dominates it

    • A strategy si is the dominant strategy if it strictly dominates all other strategies

    • A strategy si is weakly dominated if some strategy si’ weakly dominates it

    • A strategy si is the weakly dominant strategy if it weakly dominates all others


    Dominance3
    Dominance

    If you have a dominant strategy, and no ability to agree on an alternate course of action, use it

    If your opponent has a dominant strategy, and no ability to agree on an alternate course of action, then expect her to play it


    Prisoner s dilemma
    Prisoner’s Dilemma

    Optimal

    • Both players have a dominant strategy

    • The equilibrium results in lower payoffs for each player

    Equilibrium


    Prisoner s dilemma1
    Prisoner’s Dilemma

    • Both players have a dominant strategy (s1,s1)

      u11 > u21u12 > u22

    • The equilibrium results in lower payoffs for each player

      u22 > u11

    • The above two statements imply:

      u12 > u22 >u11 > u21


    Cigarette advertising
    Cigarette Advertising

    • After the 1970 agreement:

      • Cigarette advertising decreased by $63 million

      • Industry Profits rose by $91 million


    Prisoner s dilemma2
    Prisoner’s Dilemma

    • The dominant strategy will be played


    How to win a bidding war by bidding less
    How to Win a Bidding War by Bidding Less?

    • The battle for Federated (1988)

      • Parent of Bloomingdales

  • Current share price ≈ $60

  • Expected post-takeover share price ≈ $60

  • Macy’s offers $70/share

    • contingent on receiving 50% of the shares

  • Do you tender your shares to Macy’s?


  • How to win a bidding war continued
    How to Win a Bidding War (continued)

    • Robert Campeau bids $74 per share not contingent on amount acquired

    • Campeau’s Mixed Scheme:

      • If less than 50% tender their shares, each receives:

        $74 per share

      • If more than 50% tender their shares, (if X% tender), each receives:


    The federated game
    The Federated Game

    • To whom do you tender your shares?


    How to win a bidding war
    How to Win a Bidding War

    • Each player has a dominant strategy: Tender shares to Campeau

    • Resulting Price:

      (½ x 74) + (½ x 60) = $67

    • BUT: Macy’s offered $70 !


    Dominant strategies
    Dominant Strategies

    “The biggest, looniest deal ever. ”

    – Fortune Magazine, July 1988

    on Campeau’s acquisition of Federated Stores


    Prisoner s dilemma examples
    Prisoner’s Dilemma Examples

    • Pricing by Firms

      • High or low prices?

      • Value menus and loyalty programs

    • Divorce

      • Hire attorneys or proceed amicably?

    • Nuclear Weapons

      • Build or don’t build weapons?

    • State governments

      • Inducements to attract business to a state


    Dominated strategies
    Dominated Strategies

    • Two restaurants compete

      • Can charge price of $30, $50, or $60

    • Customer base consists of tourists and natives

      • 600 tourists pick randomly

      • 400 natives select the lowest price

    • Marginal costs are $10


    Tourists natives
    Tourists & Natives

    • Example scenario:

      • Restaurant 1: $50, Restaurant 2: $60

      • Restaurant 1 gets:

        300 tourists + 400 natives

        = 700 customers x ($50-$10) = $28K

      • Restaurant 2 gets:

        300 tourists + 0 natives

        = 300 customers x ($60-$10) = $15K


    Tourists natives1
    Tourists & Natives

    in thousands of dollars

    R. 2


    Iterated deletion of dominated strategies
    Iterated Deletion of Dominated Strategies

    • Does any player have a dominated strategy?

      • Eliminate the strictly dominated strategies

      • Reduce the size of the game

      • Repeat: Iterate the above procedure



    No dominated strategies
    No Dominated Strategies

    • Often there are no dominated strategies

    • Some games may have multiple equilibria

    • Equilibrium selection becomes an issue

    • Method:

      For each player, find the best response to every strategy of the other player


    Equilibrium1
    Equilibrium

    • An outcome in which every player is playing a best response to the strategies of all other players.

    • An equilibriumis a strategy profile s such that si is a best reply to s-i for all i.



    Games of coordination
    Games of Coordination

    • Complements & technology adoption

      • Two complementing firms

      • Must use same technology, but each firm has a preferred technology

      • Equilibrium does not offer a unique prediction

      • Commit (or go first) to win!

    Firm 2


    Games of assurance
    Games of Assurance

    • Joint research ventures

      • Each firm may invest $50,000 into an R&D project

      • Project succeeds only if both invest

      • If successful, each nets $75,000

    Firm 2


    Games of chicken
    Games of Chicken

    • Entry into small markets

    Firm 2


    The right game to play
    The Right Game to Play

    • Why do we “solve” games?

    • To know which one to play!

      • How do internal corporate changes impact the outcome of strategic interaction?

    • Some games are better than others


    Capacity constraints
    Capacity Constraints

    • Can decreasing others’ added value increase our profits?

    • Can decreasing total industry value increase our profits?


    Multiple equilibria
    Multiple Equilibria

    • What is the predictive power of game theory when there are multiple equilibria?

      • Sometimes nothing ?

      • Refinements

        • Focal points

        • Efficiency

        • Evolutionary stability

        • Fairness

        • Risk dominance


    Summary
    Summary

    • Games have predictable outcomes

      • Notice dominant & dominated strategies

    • Select the right game to play

    • Looking ahead:

      • Sequential Games:

        How do games unfold over time?