Game Theory and the Nash Equilibrium

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Game Theoryand the Nash Equilibrium

• Eponine Lupo

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What is Game Theory?

• Game Theory is a mathematical theory that deals
  with models of conflict and cooperation.
• It is a precise and logical description of a
  strategic setting
• It can be applied to many social sciences,
  evolutionary biology, and has many applications
  in economics.
• Game Theory is often used in more complex
  situations where chance and a players choice are
  not the only factors that are contributing to the
  outcome.
• Ex. Oil deposits

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Definitions

• Gamessituations where the outcome is determined
  by the strategy of each player
• Strategya complete contingent plan outlining all
  the actions a player will do under all possible
  circumstances
• Key assumption players are rational with
  complete information and want to maximize their
  payoffs

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Examples

• Classic Games
• Matching Pennies
• Coordination
• Battle of the Sexes
• Prisoners Dilemma
• Normal Form
• Extensive Form
• Strategiespure strategy set
• Solution
• Nash Equilibrium (D,D)

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Mixed Strategy

• A probability distribution over the pure
  strategies for a player
• Must add up to 1 or 100
• Infinite number of mixed strategies
• Choose a mixed strategy to keep opponents
  guessing
• Use a mixed strategy if the game is not solvable
  using pure strategies (no cominant or efficient
  strategies)

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Solution Concepts

• DominancePrisoners Dilemma
• S1 is dominated by S11 if S11 gives Player 1
  better payoffs than S1, no matter what the other
  players do.
• Compares 1 strategy to another of a single player
• Iterated DominancePigs
• EfficiencyPareto Coordination
• S is more efficient than S1 if everyone prefers S
  to S1
• Compares 2 strategy combinations involving all
  players
• S is efficient if there is nothing thats more
  efficient than S.
• Best Response
• S1 is a Best Response to S2 if S1 gives player 1
  the highest payoff given player 2 is playing S2

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Nash Equilibrium

• Named after John Nash
• American mathematician
• Subject of A Beautiful Mind
• Definition A strategy profile is a Nash
  equilibrium if and only if each players
  prescribed strategy is a best response to the
  strategies of the others.
• No player can do better by unilaterally changing
  his or her strategy
• Equilibrium that is reached even if it is not the
  best joint outcome

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• Pure and Mixed Strategy N.E.
• Some games do not have a pure strategy N.E.
• One always exists in a mixed form
• All finite games have at least one N.E.
• A N.E. will/must be played in the last stage
• In a Mixed N.E., each player chooses his
  probability mixture to maximize his value
  conditional on the other players selected
  probability mixture.

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N.E. in Classical Games

• Matching Penniesmixed strategy only
• (.5,.5)X(.5,.5)
• Coordination
• Prisoners Dilemma

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N.E. in R

• Find the Dominant strategies
• Find the Best Responses for each player
• Find the pure strategy N.E.

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Whats Next?

• Find the mixed strategy N.E. for 2X2 games
• Find more than 1 mixed strategy NE
• 2 player games with more than 2 strategies
• 3 player games

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Questions?Comments?Suggestions?