Game theory


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Game theory

  1. 1. BY: AMAR KUMAR<br />USN:1PI07ME019<br />GAME THEORY <br />
  2. 2. Outline<br /><ul><li>What is Game Theory?
  3. 3. History of Game Theory
  4. 4. Key Elements of a game
  5. 5. Types of games
  6. 6. Pure Strategies & Mixed Strategies
  7. 7. 2 players Zero-Sum games</li></ul>Nash equilibrium<br />Coordination games<br />Applications of game theory<br />
  8. 8. What is game theory?<br />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.<br /><ul><li>In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose.
  9. 9. A design tool.
  10. 10. The mathematics of human interactions.
  11. 11. A promise for the unification of social sciences.</li></li></ul><li>History of game theory<br /><ul><li>von Neumann wrote a key paper in 1928
  12. 12. 1944: “Theory of Games and Economic Behavior” by von Neumann and Morgenstern
  13. 13. 1950: Nash invents concept of Nash equilibrium
  14. 14. Game theory booms after this…
  15. 15. 1994: Harsanyi, Nash, and Selten win Nobel Prize in economics for game theory work</li></li></ul><li>Key elements of a game<br /><ul><li>Players: Who is interacting?
  16. 16. Strategies: What are their options?
  17. 17. Payoffs: What are their incentives?
  18. 18. Information: What do they know?
  19. 19. Rationality: How do they think?</li></li></ul><li>Types of game<br /><ul><li>Cooperative or non-cooperative
  20. 20. Zero sum and non-zero sum
  21. 21. Simultaneous and sequential
  22. 22. Perfect information and imperfect information
  23. 23. Finite & Infinite Strategies</li></li></ul><li>Pure Strategies<br /><ul><li>The upper value of the game is equal to the minimum of the maximum values in the columns.
  24. 24. The lower value of the game is equal to the maximum of the minimum values in the rows. </li></li></ul><li>An Example<br />
  25. 25. Mixed strategy<br />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.<br />
  26. 26. 2-players Zero-Sum games<br />Penny Matching: <br /><ul><li> Each of the two players has a penny.
  27. 27. Two players must simultaneously choose whether </li></ul> to show the Head or the Tail. <br /><ul><li> Both players know the following rules:</li></ul> -If two pennies match (both heads or both <br /> tails) then player 2 wins player 1’s penny. <br /> -Otherwise, player 1 wins player 2’s penny.<br />Player 2<br />Player 1<br />
  28. 28. Prisoner’s dilemma<br /><ul><li>No communication:</li></ul> - Strategies must be undertaken without <br /> the full knowledge of what the other <br /> players (prisoners) will do.<br /><ul><li> Players (prisoners) develop dominant strategies but are not necessarily the best one.</li></li></ul><li>Payoff matrix for prisoner’s dilemma<br />Jane <br />Bob<br />
  29. 29. Equilibrium <br />Strategic equilibrium refers to the “solution” of a game: A state which a game will tend towards<br />The Prisoner’s Dilemma has one (unique Nash) equilibrium (rat out-rat out)<br />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.”<br />
  30. 30. Nash’s Equilibrium<br />A Nash equilibrium is a situation in which none of them have dominant Strategy and each player makes his or her best response<br />(S, T) is Nash equilibrium if S is the best strategy to T and T is the best strategy to S<br />John Nash shared the 1994 Nobel prize in Economics for developing this idea!<br />This equilibrium occurs when each player’s strategy is optimal, knowing the strategy's of the other players.<br />
  31. 31. Coordination games<br />coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies.<br />situations in which all parties can realize mutual gains, but only by making mutually consistent decisions<br />
  32. 32. Examples <br />Models the strategic conflict when two players <br />have to choose their priorities<br />Pure coordination game<br />
  33. 33. Exp2 –Stag hunt<br />A situation in which both players (hunters) can benefit if they cooperate<br />(hunting a stag). However, cooperation might fail, because each hunter has an alternative which is safer because it<br />does not require cooperation to succeed (hunting a hare).<br />
  34. 34. Applications of Game Theory<br /><ul><li>Psychology
  35. 35. Law
  36. 36. Military Strategy
  37. 37. Management
  38. 38. Sports
  39. 39. Game Playing
  40. 40. Mathematics
  41. 41. Computer Science
  42. 42. Biology
  43. 43. Economics
  44. 44. Political Science
  45. 45. International Relations
  46. 46. Philosophy</li>