Game theory
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Game Theory. Mike Shor Topic 1 (continued). “ I don ’ t skate to where the puck is, I skate to where it ’ s going to be. ”. - Wayne Gretzky. Defining the Game. Review Strategic interdependence of players You pursue your goals So does everyone else

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Game Theory

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Game theory

Game Theory

Mike ShorTopic 1 (continued)

“I don’t skate to where the puck is, I skate to where it’s going to be.”

- Wayne Gretzky


Defining the game

Defining the Game

  • Review

    • Strategic interdependence of players

    • You pursue your goals

    • So does everyone else

    • Your actions change the environment for others

  • Outline:

    • Defining the rules of the game

    • Manipulating the rules if they are flexible

    • Questioning game theory’s assumptions


Rules of the game

Rules of the Game

  • The participants

    • Players (who )

    • Strategies(what )

    • Payoffs( why )

  • The rules

    • Timing of moves(when )

    • Information( ? )

  • The assumptions

    • Rationality

    • Common knowledge


The participants

The Participants

  • Who?

  • Players

    • Everyone who impacts your earnings

    • Everyone whose actions you care about

  • Notation:

    • N players indexed i,j or 1,2,3,…

    • Nature (randomness) is player 0


The participants1

The Participants

  • What?

  • Strategies

    • A comprehensive plan of action for every contingency in a game

    • An action specifies a move at a specific stage of a game

    • A strategy specifies an action for every possible decision a player can face


The participants2

The Participants

  • What?

  • Notation:

    • sia strategy for player i

    • s a vector of strategies (s1, s2 , s3 , …)

    • s-i a strategy for all players except i

    • Si the set of all possible strategies for i


The participants3

The Participants

  • Why?

  • Payoffs

    • Numbers associated with each outcome

    • Reflect the interests of the players

    • Sometimes just ordinal (A better than B)

    • Sometimes need cardinal (e.g., A = 2B)

  • Notation:

    • Player i’s payoffs given by ui(s) = ui(si,s-i )


The rules

The Rules

  • Timing of moves

    • Are moves simultaneous or sequential?

  • Information

    • Who knows what and when?


Changing the rules

Changing the Rules

  • Are the rules of the game fixed?


Example agenda setting

Example:Agenda Setting

Graduation speaker

Ron Paul, Sarah Palin, or Al Gore?

  • Four committee members prefer:

    Ronto Sarahto Al( R > S > A )

  • Three committee members prefer:

    Sarahto Alto Ron( S > A > R )

  • Two committee members prefer:

    Alto Ronto Sarah( A > R > S )

  • Voting by Majority Rule


Agenda setting continued

Agenda Setting (continued)

  • No majority in three way race

  • Committee president sets voting rules:

    • First, Ron v. Sarah

      Then, Ron/Sarah winner versus Al

    • First, Ron v. Al

      Then, Ron/Al winner versus Sarah

    • First, Sarah v. Al

      Then, Sarah/Al winner versus Ron


Agenda setting continued1

Agenda Setting (continued)

  • Recall member preferences:

    4 (R>S>A) 3 (S>A>R) 2 (A>R>S)

  • Majority rule results:

    • R beats S ; S beats A ; A beats R

  • Voting results:

    • Rvs.Sthen winner versus A

    • Rvs.Athen winner versus S

    • Svs.Athen winner versus R


  • Changing the rules1

    Changing the Rules

    • Are the rules of the game fixed?

      “The battle is won before it is ever fought”

      — Sun Tzu

    When the rules of the game are flexible manipulate them to your advantage.


    The assumptions

    The Assumptions

    • Rationality

    • Common Knowledge


    The assumptions1

    The Assumptions

    • Rationality

      • Players aim to maximize their payoffs

      • Players are perfect calculators

      • Players know their own payoffs

      • Players never make mistakes


    The assumptions2

    The Assumptions

    • Common Knowledge

      • Each player knows the rules of the game

        • Each player knows that each player knows the rules

        • Each player knows that each player knows that each player knows the rules

        • Each player knows that each player knows that each player knows that each player knows the rules

        • Each player knows that each player knows that each player knows that each player knows that each player knows the rules

        • Etc. Etc. Etc.


    The assumptions3

    The Assumptions

    • Some piece of information X

      • Knowledge: I know X

      • Mutual knowledge: Each of us knows X

      • Common Knowledge: Each of us knows that each of us knows X, …

    • The rules of the game are common knowledge (players, strategies, payoffs, timing, information) are common knowledge


    Common knowledge

    Common Knowledge

    • Do people really behave as the theory predicts?

    • If not, what good is the theory?


    Common knowledge1

    Common Knowledge

    • Behavioral game theory:

      The rule of three

    • Game theory:

      The rule of infinity

    • Example: the stock market


    The stock market rationality

    The Stock Market& Rationality

    “I can calculate the motions of heavenly bodies, but not the madness of people”

    - Sir Isaac Newton

    (upon losing £20,000 in the South Sea Bubble in 1720)

    If you knew the exact date of the next stock market crash, what would you do?


    Which month am i

    Which Month Am I?

    “_________

    “_________

    “_________

    “_________

    “_________

    “_________

    “_________

    “_________

    • often is a good month for stocks”

    • is typically a strong month for stocks”

    • tends to be a good month for stocks”

    • is usually a good month for stocks”

    • is typically a bad month for the stock market”

    • is usually the worst month for stocks”

    • is traditionally a cruel month for stocks”

    • is the worst month for stocks”


    Fun with probabilities

    Fun with Probabilities

    “Almost every generation, it seems, has its October stock-market crash.”

    — Wall Street Journal, October 2008

    • Why is October so bad?


    October

    October

    • October 19, 1987

      • Dow dropped 23%

    • October 1-10, 2008

      • Dow dropped 22%


    Fun with probabilities1

    Fun with Probabilities

    If, in each month, the stock market is equally likely to go up or go down, then there is over an 80% chance that some month will be “bad” three years in a row!


    The worst month for stocks

    The Worst Month for Stocks

    “October. This is one of the peculiarly dangerous months to speculate in stocks.

    – Wall Street Journal, quoting Mark Twain


    Wall street journal may 1988

    Wall Street JournalMay 1988

    Front Page Story:

    “Clear Road Ahead …”

    • “Will record gains through the fall”

    • “October looms on the horizon … the dreaded though inevitable slide.”


    Game theory

    1988

    MAY

    4

    3

    2

    1

    0


    Ipo lock ups

    IPO Lock-Ups

    • A company wishes to “go public”

      • It offers to sell shares to the public

      • Typically only 20% of the total shares are offered

      • The remained are held by “insiders”

    • Lock-up

      • precludes insiders from selling shares, typically for six months

    • What happens when the lock-up expires?


    Ipo lock ups1

    IPO Lock-Ups

    “Increased supply after the lock-up expiration will drive price down. It is simply economics of supply and demand.”

    — Wall Street Journal reporter


    Excess returns prior to ipo lock up expiration

    0.005

    0

    -0.005

    -0.010

    (Cumulative Abnormal)

    Return

    -0.015

    -0.020

    -0.025

    -0.030

    -20

    -16

    -12

    -8

    -4

    0

    4

    8

    12

    16

    Event Day

    Excess Returns prior to IPO Lock-Up Expiration


    Summary

    Summary

    • You’re always in a game

    • The rules are likely to be flexible

    • Think one step ahead

    • The rule of three steps

    • Looking ahead:

      • Equilibrium:

        What are the outcome of games?


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