Chapter 11 Game Theory and the Tools of Strategic Business Analysis

Game Theory 0 Game theory applied to economics by John Von Neuman and Oskar Morgenstern 0 Game theory allows us to analyze different social and economic situations

Games of Strategy Defined 0 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 0 A game is comprised of 0 Number of players 0 Order to play 0 Choices 0 Chance 0 Information 0 Utility 3

Representing Games 0 Game tree 0 Visual depiction 0 Extensive form game 0 Rules 0 Payoffs 4

Game Types 0 Game of perfect information 0 Player – knows prior choices 0 All other players 0 Game of imperfect information 0 Player – doesn’t know prior choices 5

Strategy 0 A player’s strategy is a plan of action for each of the other player’s possible actions

Game of perfect information 7 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 IBM Toshiba UNIX DOS UNIX DOS UNIX DOS In extensive form

Strategies 0 IBM: 0 DOS or UNIX 0 Toshiba 0 DOS if DOS and UNIX if UNIX 0 UNIX if DOS and DOS if UNIX 0 DOS if DOS and DOS if UNIX 0 UNIX if DOS and UNIX if UNIX

9 Toshiba (DOS | DOS, DOS | UNIX) (DOS | DOS, UNIX | UNIX) (UNIX | DOS, UNIX | UNIX) (UNIX | DOS, DOS | UNIX) IBM DOS600, , 100 UNIX100, , , 100 Game of perfect information In normal form

Game of imperfect information 0 Assume instead Toshiba doesn’t know what IBM chooses 0 The two firms move at the same time 0 Imperfect information 0 Need to modify the game accordingly

Game of imperfect information 11 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 IBM Toshiba UNIX DOS UNIX DOS UNIX DOS In extensive form Information set Toshiba’s strategies: DOS UNIX

12 Toshiba DOSUNIX IBM DOS600, , 100 UNIX100, , 600 Game of imperfect information In normal form

Another example: Matching Pennies 13 Player 2 HeadsTails Player 1 Heads- 1, Tails , +1

Extensive form of the game of matching pennies 14 Child 2 does not know whether child 1 chose heads or tails. Therefore, child 2’s information set contains two nodes. Child 1 Child 2 Tails Heads Tails Heads Tails Heads

Equilibrium for Games Nash Equilibrium 0 Equilibrium 0 state/ outcome 0 Set of strategies 0 Players – don’t want to change behavior 0 Given - behavior of other players 0 Noncooperative games 0 No possibility of communication or binding commitments 15

Nash Equilibria 16

17 Toshiba DOSUNIX IBM DOS600, , 100 UNIX100, , 600 Nash Equilibrium: Toshiba-IBM imperfect Info game The strategy pair DOS DOS is a Nash equilibrium. Are there any other equilibria?

Dominant Strategy Equilibria 0 Strategy A dominates strategy B if 0 A gives a higher payoff than B 0 No matter what opposing players do 0 Dominant strategy 0 Best for a player 0 No matter what opposing players do 0 Dominant-strategy equilibrium 0 All players - dominant strategies 18

Oligopoly Game 19 General Motors High priceLow price Ford High price500, , 700 Low price700, , Ford has a dominant strategy to price low 0 If GM prices high, Ford is better of pricing low 0 If GM prices low, Ford is better of pricing low

Oligopoly Game 20 General Motors High priceLow price Ford High price500, , 700 Low price700, , Similarly for GM 0 The Nash equilibrium is Price low, Price low

The Prisoners’ Dilemma 0 Two people committed a crime and are being interrogated separately. 0 The are offered the following terms: 0 If both confessed, each spends 8 years in jail. 0 If both remained silent, each spends 1 year in jail. 0 If only one confessed, he will be set free while the other spends 20 years in jail.

Prisoners’ Dilemma 22 Prisoner 2 confesssilent Prisoner 1 Confess8, 80, 20 Silent20, 01, 1 0 Numbers represent years in jail 0 Each has a dominant strategy to confess 0 Silent is a dominated strategy 0 Nash equilibrium: Confess Confess

Prisoners’ Dilemma 0 Each player has a dominant strategy 0 Equilibrium is Pareto dominated 23

Elimination of Dominated Strategies 0 Dominated strategy 0 Strategy dominated by another strategy 0 We can solve games by eliminating dominated strategies 0 If elimination of dominated strategies results in a unique outcome, the game is said to be dominance solvable 24

25 (a) Eliminating dominated strategies Player Player 1 12, 02, 40, 2 20, 60, 24, 0 (b) One step of elimination Player 2 12 Player 1 12, 02, 4 20, 60, 2 (c ) Two steps of elimination Player 2 12 Player 112, 02, 4

26 (a) Eliminated dominated strategies Player Player 1 120, 010, 14, , 210, 02, -2 (b) Reduced game eliminating column 3 first Player 2 12 Player 1 120, 010, 1 220, 210, 0

Games with Many Equilibria 0 Coordination game 0 Players - common interest: equilibrium 0 For multiple equilibria 0 Preferences - differ 0 At equilibrium: players - no change 27

28 Toshiba DOSUNIX IBM DOS600, , 100 UNIX100, , 600 Games with Many Equilibria The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX

Normal Form of Matching Numbers: coordination game with ten Nash equilibria 29 Player Player 1 11, 10, 0 2 2, 20, 0 3 3, 30, 0 4 4, 40, 0 5 5, 50, 0 6 6, 60, 0 7 7, 70, 0 8 8, 80, 0 9 9, 90, 0 100, 0 10, 10

Table A game with no equilibria in pure strategies 30 General 2 RetreatAttack General 1Retreat5, 86, 6 Attack8, 02, 3

The “I Want to Be Like Mike” Game 31 Dave Wear redWear blue MichaelWear red(-1, 2)(2, -2) Wear blue(1, -1)(-2, 1)

Credible Threats 0 An equilibrium refinement: 0 Analyzing games in normal form may result in equilibria that are less satisfactory 0 These equilibria are supported by a non credible threat 0 They can be eliminated by solving the game in extensive form using backward induction 0 This approach gives us an equilibrium that involve a credible threat 0 We refer to this equilibrium as a sub-game perfect Nash equilibrium. 32

33 Toshiba (DOS | DOS, DOS | UNIX) (DOS | DOS, UNIX | UNIX) (UNIX | DOS, UNIX | UNIX) (UNIX | DOS, DOS | UNIX) IBM DOS600, , 100 UNIX100, , , 100 Non credible threats: IBM-Toshiba In normal form 0 Three Nash equilibria 0 Some involve non credible threats. 0 Example IBM playing UNIX and Toshiba playing UNIX regardless: 0 Toshiba’s threat is non credible

Backward induction IBM Toshiba UNIX DOS UNIX DOS UNIX DOS

Subgame perfect Nash Equilibrium 0 Subgame perfect Nash equilibrium is 0 IBM: DOS 0 Toshiba: if DOS play DOS and if UNIX play UNIX 0 Toshiba’s threat is credible 0 In the interest of Toshiba to execute its threat

Rotten kid game 0 The kid either goes to Aunt Sophie’s house or refuses to go 0 If the kid refuses, the parent has to decide whether to punish him or relent 36 Player 2 (a parent) (punish if the kid refuses) (relent if the kid refuses) Player 1 (a difficult child) Left (go to Aunt Sophie’s House) 1, 1 Right (refuse to go to Aunt Sophie’s House) -1, -12, 0

Rotten kid game in extensive form 37 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 Kid Parent Refuse Go to Aunt Sophie’s House Relent if refuse Punish if refuse