Chapter 11

1. Chapter 11 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 to play • Choices • Chance • Information • Utility

4. Representing Games • Game tree • Visual depiction • Extensive form game • Rules • Payoffs

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

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

7. 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

8. 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

9. Game of perfect information In normal form

10. 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

11. 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.

12. Game of imperfect information In normal form

13. Another example: Matching Pennies

14. 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.

15. 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

16. Nash Equilibria

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

18. 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

19. 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

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

21. The 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.

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

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

24. 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

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

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

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

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

31. The “I Want to Be Like Mike” Game

32. 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.

33. 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

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

35. 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

36. 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

37. 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