Introduction to Game Theory

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Introduction to Game Theory
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What is Strategic Behavior?

• The hallmark of a game is strategic behavior
• What you choose affects both me and you.
• And vice versa
• It is no longer enough for you to choose the best
  action for yourself.
• You must chose the best action, knowing that what
  I choose affects you and that I am fully aware of
  the way we are interacting.

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Great Book
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Anticipate your rivals response

• Many games occur sequentially
• You move
• Then I move
• Then you react
• Example Charlie Brown and Lucy with a football

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The First Basic Principle

• Rule 1 Look ahead and reason back
• This principle is called backward induction
• You can use it in decision that do not involve
  the strategic behavior of your rival
• But it is especially powerful when you can
  anticipate what your rival will actually do.
• So it is very valuable in game theory

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Application Bargaining

• There is an ice-cream cake. Round 1 its my turn
• I offer you a piece of a certain size
• You say yes or no
• This is the end of round 1
• Round 2 now its your turn
• You offer me a piece of a certain size
• I say yes or no
• This is the end of round 2
• The ice cream cake will completely melt by the
  end of round 1
• What happens?

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Bargaining (part 2)

• The ice cream cake will completely melt after the
  second round
• What happens?
• The ice cream cake will completely melt after the
  third round
• What happens?

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Answers

• In the first case, I get the whole cake
• In the second case, I get half the cake
• In the third case, I get 2/3 of the cake

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Some lessons

• The order of moves matters quite a bit
• The value of your outside options matters quite a
  bit too

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Seeing through your rival

• Often we both need to act at the same time
• This means I need to think about what you are
  doing.
• In the previous examples, I saw what you did and
  then reacted to your choice
• Now I need to think about what you may well do.
• You are in the same quandary
• We both understand these facts

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Times Sales
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Dominant Strategy

• Which choice is best for Time, NO MATTER WHAT
  Newsweek does?
• This is a dominant strategy

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Times Sales
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Newsweeks Sales
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Newsweeks Sales
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Equilibrium for this game

• Both firms run with the cover story about AIDS
• This game is dominance solvable
• It is too bad that they both end up losing money
  compared with some other arrangement such as
  alternating back and forth between hot covers

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Equilibrium for this game

• Both firms run with the cover story about AIDS
• This game is dominance solvable
• It is too bad that they both end up losing money
  compared with some other arrangement such as
  alternating back and forth between hot covers

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Slightly more interesting game

• Slight bias in favor of Time
• Same cover story means 60 of the people go to
  Time

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Times Sales
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Newsweeks Sales
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Solution to this Game

• Newsweek can figure out what time will do
• Time doesnt even care what Newsweek will choose
• Rule 2 If you have a dominant strategy, then use
  it!

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Offense Expected Yards
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Special kind of game

• This is a zero-sum game
• Whatever the offense gets, the defense loses
• And vice versa
• Blitz is a dominated strategy

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Offense Expected Yards
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(Reduced Game has a dominant strategy for Offense)
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The Solution

• The defense knows not to blitz
• The offense knows that it is best to pass
• The defense will chose to counter the pass
• Rule 3 Eliminate dominated strategies and go on
  doing so successively

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Americas Payoffs
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Farawayastans Payoffs
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Nash Equilibrium

• Two mutual (simultaneous) best responses
• If America fights, Farawayastan will fight
• If Farawayastan fights, Amercia will fight
• There is a Peaceful equilibrium too
• There is an equilibrium with randomized responses
  too

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Final Principle

• Rule 4 Having exhausted the simple avenues of
  finding a dominant strategy or eliminating
  dominated strategies, the next thing to do is to
  look for an equilibrium of the game.

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Conclusion

• Game theory is the most exciting branch of
  economics.
• Many sciences are now using it
• Read Dixit and Nalebuff!