chapter 5 game theory and the tools of strategic business analysis

Chapter 5

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 players0 Order of play0 strategies0 Chance 0 Information 0 Payoffs

3

Example 1: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.

Example 1: Prisoners’ Dilemma

5

Prisoner 2

confess silent

Prisoner 1Confess 8, 8 0, 20

Silent 20, 0 1, 1

0 Numbers represent years in jail0 Each has a dominant strategy to confess0 Silent is a dominated strategy0 Nash equilibrium: Confess Confess

Example 2: Matching Pennies

6

Player 2

Heads Tails

Player 1Heads - 1, +1 +1 - 1

Tails +1 - 1 - 1, +1

Example 3: Oligopoly Game

7

General Motors

High price Low price

FordHigh price 500, 500 100, 700

Low price 700, 100 300, 300

0 Similarly for GM0 The Nash equilibrium is Price low, Price low

Game Types

0 Game of perfect information0 Player – knows prior choices

0All other players

0 Game of imperfect information0 Player – doesn’t know prior choices

8

Representing Games

0 The previous examples are of0 Simultaneous games0 Games of imperfect information

Games can be represented visually in0 Bi- matrix form

0 Table0 Dimensions depend on the number of strategies

0 Game tree0 Extensive form game

9

Matching Pennies

10

Player 2

Heads Tails

Player 1Heads - 1, +1 +1 - 1

Tails +1 - 1 - 1, +1

Game of imperfect informationRepresented in bi-matrix form

Extensive form of the game of matching pennies

11

Child 2 does not know whether child 1 chose heads or tails. Therefore, child 2’s information set contains two nodes.

Child 1

Child 2Child 2

Tails Heads

TailsHeads TailsHeads

- 1+1

+1- 1

+1- 1

-1+1

Strategy

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

Game of perfect information

13Player 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

1

2 3

IBM

ToshibaToshiba

UNIXDOS

UNIXDOS UNIXDOS

600200

100100

100100

200600

In extensive form

Strategies

0 IBM: 0 DOS or UNIX

0 Toshiba0 DOS if DOS and UNIX if UNIX0 UNIX if DOS and DOS if UNIX0 DOS if DOS and DOS if UNIX0 UNIX if DOS and UNIX if UNIX

Game of imperfect information

0 Assume instead Toshiba doesn’t know what IBM chooses0 The two firms move at the same time

0 Imperfect information0 Need to modify the game accordingly

Game of imperfect information

17

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.

1

2 3

IBM

ToshibaToshiba

UNIXDOS

UNIXDOS UNIXDOS

600200

100100

100100

200600

In extensive form

Information set

Toshiba’s strategies:• DOS• UNIX

18

Toshiba

DOS UNIX

IBMDOS 600, 200 100, 100

UNIX 100, 100 200, 600

Game of imperfect informationIn normal form

Equilibrium for GamesNash Equilibrium

0 Equilibrium 0 state/ outcome0 Set of strategies0 Players – don’t want to change behavior 0 Given - behavior of other players

0 Noncooperative games0 No possibility of communication or binding

commitments

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Nash Equilibria

chosen is *s when i player to payoff

i player of choicestrategy

choicesstrategy ofarray -

i

),...,(

),...,(*

**1

**1

n

*i

n

ss

s

sss

20

ii

nini

n

Ss

ssssss

sss

in all for

If

mequilibriu Nash a is -

ii

ˆ

),...,ˆ,...,(),...,,...,(

),...,(***

1***

1

**1

21

Toshiba

DOS UNIX

IBMDOS 600, 200 100, 100

UNIX 100, 100 200, 600

Nash Equilibrium: Toshiba-IBMimperfect 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 if0 A gives a higher payoff than B 0 No matter what opposing players do

0 Dominant strategy0 Best for a player0 No matter what opposing players do

0 Dominant-strategy equilibrium0 All players - dominant strategies

22

Oligopoly Game

23

General Motors



Low price 700, 100 300, 300

0 Ford has a dominant strategy to price low 0 If GM prices high, Ford is better of pricing low0 If GM prices low, Ford is better of pricing low

Oligopoly Game

24

General Motors



Low price 700, 100 300, 300

0 Similarly for GM0 The Nash equilibrium is Price low, Price low

Prisoners’ Dilemma

25

Prisoner 2

confess silent

Prisoner 1Confess 8, 8 0, 20

Silent 20, 0 1, 1

0 Numbers represent years in jail0 Each has a dominant strategy to confess0 Silent is a dominated strategy0 Nash equilibrium: Confess Confess

Prisoners’ Dilemma

0 Each player has a dominant strategy0 Equilibrium is Pareto dominated

26

Elimination of Dominated Strategies

0 Dominated strategy0 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

27

28

(a) Eliminating dominated strategies

Player 2

1 2 3

Player 11 2, 0 2, 4 0, 2

2 0, 6 0, 2 4, 0

(b) One step of elimination

Player 2

1 2

Player 11 2, 0 2, 4

2 0, 6 0, 2

(c ) Two steps of elimination

Player 2

1 2

Player 1 1 2, 0 2, 4

29

(a) Eliminated dominated strategies

Player 2

1 2 3

Player 11 20, 0 10, 1 4, -4

2 20, 2 10, 0 2, -2

(b) Reduced game eliminating column 3 first

Player 2

1 2

Player 11 20, 0 10, 1

2 20, 2 10, 0

Games with Many Equilibria

0 Coordination game0 Players - common interest: equilibrium0 For multiple equilibria

0Preferences - differ 0 At equilibrium: players - no change

30

31

Toshiba

DOS UNIX

IBMDOS 600, 200 100, 100

UNIX 100, 100 200, 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

32

Player 2

1 2 3 4 5 6 7 8 9 10

Player 1

1 1, 1 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0

2 0, 0 2, 2 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0

3 0, 0 0, 0 3, 3 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0

4 0, 0 0, 0 0, 0 4, 4 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0

5 0, 0 0, 0 0, 0 0, 0 5, 5 0, 0 0, 0 0, 0 0, 0 0, 0

6 0, 0 0, 0 0, 0 0, 0 0, 0 6, 6 0, 0 0, 0 0, 0 0, 0

7 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 7, 7 0, 0 0, 0 0, 0

8 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 8, 8 0, 0 0, 0

9 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 9, 9 0, 0

10 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 10, 10

Table 11.12

A game with no equilibria in pure strategies

33

General 2

Retreat Attack

General 1 Retreat 5, 8 6, 6

Attack 8, 0 2, 3

The “I Want to Be Like Mike” Game

34

Dave

Wear red Wear blue

Michael Wear 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 satisfactory0 These equilibria are supported by a non credible threat0 They can be eliminated by solving the game in extensive

form using backward induction0 This approach gives us an equilibrium that involve a

credible threat0 We refer to this equilibrium as a sub-game perfect Nash

equilibrium.

35

36

Toshiba

(DOS if DOS,DOS if UNIX)

(DOS | DOS,UNIX | UNIX)

(UNIX | DOS,UNIX | UNIX)

(UNIX | DOS,DOS | UNIX)

IBMDOS 600, 200 600, 200 100, 100 100, 100

UNIX 100, 100 200, 600 200, 600 100, 100

Non credible threats: IBM-ToshibaIn normal form

0 Three Nash equilibria0 Some involve non credible threats.0 Example IBM playing UNIX and Toshiba playing UNIX

regardless:0 Toshiba’s threat is non credible

Backward induction

37

1

2 3

IBM

ToshibaToshiba

UNIXDOS

UNIXDOS UNIXDOS

600200

100100

100100

200600

Subgame perfect Nash Equilibrium

0 Subgame perfect Nash equilibrium is0 IBM: DOS0 Toshiba: if DOS play DOS and if UNIX play UNIX

0 Toshiba’s threat is credible0 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

39

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 1, 1

Right(refuse to go to Aunt Sophie’s House)

-1, -1 2, 0

Rotten kid game in extensive form

40

• 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

RefuseGo to Aunt Sophie’s House

Relent if refuse

Punish if refuse

-1-1

20

11

1

2

Application 1: Collusive Duopoly

0Example: The European voluntary agreement for washing machines in 1998

0The agreement requires firms to eliminate from the market inefficient models

0Ahmed and Segerson (2011) show that the agreement can raise firm profit, however, it is not stable Firm 2

eliminate Keep

Firm 1eliminate $1,000 $1,000 $200 $1,200

keep $1,200 $200 $500 $500

Application 2: Wal-Mart and CFL bulbs market

0 In 2006 Wal-Mart committed itself to selling 1 million CFL bulbs every year

0 This was part of Wal-Mart’s plan to become more socially responsible

0 Ahmed(2012) shows that this commitment can be an attempt to raise profit.

43

1

2 3

Wal-Mart

Small firmSmall firm

Do not commitCommit to output target

Do notCommit Do notCommit

9045

50040

8060

10050

When the target is small

The outcome is similar to a prisoners dilemma


44

1

2 3

Wal-Mart

Small firmSmall firm

Do not commitCommit to output target

Do notCommit Do notCommit

8030

50035

90100

10050

When the target is large

When the target is large enough, we have a game of chicken


chapter 5 game theory and the tools of strategic business analysis

Documents

dos unixdos dos

unix unixunix dos

unix15toshibados dos

heads tails player

dos unixibm dos600

toshiba doesnt

dominant strategy

strategya players strategy