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Chapter 5

Chapter 5. Game Theory and the Tools of Strategic Business Analysis. Game Theory. Game theory applied to economics by John Von Neuman and Oskar Morgenstern Game theory allows us to analyze different social and economic situations. Games of Strategy Defined.

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Chapter 5

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  1. Chapter 5 Game Theory and the Tools of Strategic Business Analysis

  2. Game Theory • Game theory applied to economics by John Von Neuman and Oskar Morgenstern • Game theory allows us to analyze different social and economic situations

  3. Games of Strategy Defined • Interaction between agents can be represented by a game, when the rewards to each depends on his actions as well as those of the other player • A game is comprised of • Number of players • Order of play • strategies • Chance • Information • Payoffs

  4. Example 1:Prisoners’ Dilemma • Two people committed a crime and are being interrogated separately. • The are offered the following terms: • If both confessed, each spends 8 years in jail. • If both remained silent, each spends 1 year in jail. • If only one confessed, he will be set free while the other spends 20 years in jail.

  5. Example 1: Prisoners’ Dilemma • Numbers represent years in jail • Each has a dominant strategy to confess • Silent is a dominated strategy • Nash equilibrium: Confess Confess

  6. Example 2: Matching Pennies

  7. Example 3: Oligopoly Game • Similarly for GM • The Nash equilibrium is Price low, Price low

  8. Game Types • Game of perfect information • Player – knows prior choices • All other players • Game of imperfect information • Player – doesn’t know prior choices

  9. Representing Games • The previous examples are of • Simultaneous games • Games of imperfect information Games can be represented visually in • Bi- matrix form • Table • Dimensions depend on the number of strategies • Game tree • Extensive form game

  10. Matching Pennies Game of imperfect information Represented in bi-matrix form

  11. Extensive form of the game of matching pennies Child 1 Heads Tails Child 2 Child 2 Heads Tails Heads Tails +1 - 1 +1 - 1 - 1 +1 • 1 +1 Child 2 does not know whether child 1 chose heads or tails. Therefore, child 2’s information set contains two nodes.

  12. Strategy • A player’s strategy is a plan of action for each of the other player’s possible actions

  13. Game of perfect information In extensive form IBM DOS UNIX Toshiba Toshiba 1 2 DOS UNIX DOS UNIX 3 200 600 600 200 100 100 100 100 Player 2 (Toshiba) knows whether player 1 (IBM) moved to the left or to the right. Therefore, player 2 knows at which of two nodes it is located

  14. Strategies • IBM: • DOS or UNIX • Toshiba • DOS if DOS and UNIX if UNIX • UNIX if DOS and DOS if UNIX • DOS if DOS and DOS if UNIX • UNIX if DOS and UNIX if UNIX

  15. Game of perfect information In normal form

  16. Game of imperfect information • Assume instead Toshiba doesn’t know what IBM chooses • The two firms move at the same time • Imperfect information • Need to modify the game accordingly

  17. Game of imperfect information In extensive form IBM Information set DOS UNIX • Toshiba’s strategies: • DOS • UNIX Toshiba Toshiba 1 2 DOS UNIX DOS UNIX 3 600 200 100 100 100 100 200 600 Toshiba does not know whether IBM moved to the left or to the right, i.e., whether it is located at node 2 or node 3.

  18. Game of imperfect information In normal form

  19. Equilibrium for GamesNash Equilibrium • Equilibrium • state/ outcome • Set of strategies • Players – don’t want to change behavior • Given - behavior of other players • Noncooperative games • No possibility of communication or binding commitments

  20. Nash Equilibria

  21. Nash Equilibrium: Toshiba-IBM imperfect Info game The strategy pair DOS DOS is a Nash equilibrium. Are there any other equilibria?

  22. Dominant Strategy Equilibria • Strategy A dominates strategy B if • A gives a higher payoff than B • No matter what opposing players do • Dominant strategy • Best for a player • No matter what opposing players do • Dominant-strategy equilibrium • All players - dominant strategies

  23. Oligopoly Game • Ford has a dominant strategy to price low • If GM prices high, Ford is better of pricing low • If GM prices low, Ford is better of pricing low

  24. Oligopoly Game • Similarly for GM • The Nash equilibrium is Price low, Price low

  25. Prisoners’ Dilemma • Numbers represent years in jail • Each has a dominant strategy to confess • Silent is a dominated strategy • Nash equilibrium: Confess Confess

  26. Prisoners’ Dilemma • Each player has a dominant strategy • Equilibrium is Pareto dominated

  27. Elimination of Dominated Strategies • Dominated strategy • Strategy dominated by another strategy • We can solve games by eliminating dominated strategies • If elimination of dominated strategies results in a unique outcome, the game is said to be dominance solvable

  28. Games with Many Equilibria • Coordination game • Players - common interest: equilibrium • For multiple equilibria • Preferences - differ • At equilibrium: players - no change

  29. Games with Many Equilibria The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX

  30. Normal Form of Matching Numbers: coordination game with ten Nash equilibria

  31. Table 11.12 A game with no equilibria in pure strategies

  32. The “I Want to Be Like Mike” Game

  33. Credible Threats • An equilibrium refinement: • Analyzing games in normal form may result in equilibria that are less satisfactory • These equilibria are supported by a non credible threat • They can be eliminated by solving the game in extensive form using backward induction • This approach gives us an equilibrium that involve a credible threat • We refer to this equilibrium as a sub-game perfect Nash equilibrium.

  34. Non credible threats: IBM-Toshiba In normal form • Three Nash equilibria • Some involve non credible threats. • Example IBM playing UNIX and Toshiba playing UNIX regardless: • Toshiba’s threat is non credible

  35. Backward induction IBM DOS UNIX Toshiba Toshiba 1 2 DOS UNIX DOS UNIX 3 100 100 600 200 100 100 200 600

  36. Subgame perfect Nash Equilibrium • Subgame perfect Nash equilibrium is • IBM: DOS • Toshiba: if DOS play DOS and if UNIX play UNIX • Toshiba’s threat is credible • In the interest of Toshiba to execute its threat

  37. Rotten kid game • The kid either goes to Aunt Sophie’s house or refuses to go • If the kid refuses, the parent has to decide whether to punish him or relent

  38. Rotten kid game in extensive form Kid Go to Aunt Sophie’s House Refuse Parent 1 2 Punish if refuse Relent if refuse -1 -1 2 0 1 1 • The sub game perfect Nash equilibrium is: Refuse and Relent if refuse • The other Nash equilibrium, Go and Punish if refuse, relies on a non credible threat by the parent

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