Game theory
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Game theory






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Game theory Game theory Presentation Transcript

  • Outline
    • What is Game Theory?
    • History of Game Theory
    • Key Elements of a game
    • Types of games
    • Pure Strategies & Mixed Strategies
    • 2 players Zero-Sum games
    Nash equilibrium
    Coordination games
    Applications of game theory
  • What is game theory?
    Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology (most notably evolutionary biology and ecology), engineering, political science, international relations, computer science, and philosophy.
    • In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose.
    • A design tool.
    • The mathematics of human interactions.
    • A promise for the unification of social sciences.
  • History of game theory
    • von Neumann wrote a key paper in 1928
    • 1944: “Theory of Games and Economic Behavior” by von Neumann and Morgenstern
    • 1950: Nash invents concept of Nash equilibrium
    • Game theory booms after this…
    • 1994: Harsanyi, Nash, and Selten win Nobel Prize in economics for game theory work
  • Key elements of a game
    • Players: Who is interacting?
    • Strategies: What are their options?
    • Payoffs: What are their incentives?
    • Information: What do they know?
    • Rationality: How do they think?
  • Types of game
    • Cooperative or non-cooperative
    • Zero sum and non-zero sum
    • Simultaneous and sequential
    • Perfect information and imperfect information
    • Finite & Infinite Strategies
  • Pure Strategies
    • The upper value of the game is equal to the minimum of the maximum values in the columns.
    • The lower value of the game is equal to the maximum of the minimum values in the rows.
  • An Example
  • Mixed strategy
    A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy.
  • 2-players Zero-Sum games
    Penny Matching:
    • Each of the two players has a penny.
    • Two players must simultaneously choose whether
    to show the Head or the Tail.
    • Both players know the following rules:
    -If two pennies match (both heads or both
    tails) then player 2 wins player 1’s penny.
    -Otherwise, player 1 wins player 2’s penny.
    Player 2
    Player 1
  • Prisoner’s dilemma
    • No communication:
    - Strategies must be undertaken without
    the full knowledge of what the other
    players (prisoners) will do.
    • Players (prisoners) develop dominant strategies but are not necessarily the best one.
  • Payoff matrix for prisoner’s dilemma
  • Equilibrium
    Strategic equilibrium refers to the “solution” of a game: A state which a game will tend towards
    The Prisoner’s Dilemma has one (unique Nash) equilibrium (rat out-rat out)
    No one player can unilaterally change his strategy for a better outcome: ”I can do no better, given that the other player keeps doing what he is doing.”
  • Nash’s Equilibrium
    A Nash equilibrium is a situation in which none of them have dominant Strategy and each player makes his or her best response
    (S, T) is Nash equilibrium if S is the best strategy to T and T is the best strategy to S
    John Nash shared the 1994 Nobel prize in Economics for developing this idea!
    This equilibrium occurs when each player’s strategy is optimal, knowing the strategy's of the other players.
  • Coordination games
    coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies.
    situations in which all parties can realize mutual gains, but only by making mutually consistent decisions
  • Examples
    Models the strategic conflict when two players
    have to choose their priorities
    Pure coordination game
  • Exp2 –Stag hunt
    A situation in which both players (hunters) can benefit if they cooperate
    (hunting a stag). However, cooperation might fail, because each hunter has an alternative which is safer because it
    does not require cooperation to succeed (hunting a hare).
  • Applications of Game Theory
    • Psychology
    • Law
    • Military Strategy
    • Management
    • Sports
    • Game Playing
    • Mathematics
    • Computer Science
    • Biology
    • Economics
    • Political Science
    • International Relations
    • Philosophy