1

n-Person Games

in

Normal Form

Chapter 5

2

Fundamental Differences with 3 Players: the SpoilersCounterexamplesThe theorem for games like Chess does not generalizeThe solution theorem for 0-sum, 2-player games does not generalizeA player playing the spoiler

3

Indeterminate three-person game

w, l, l

l, w, l

Top

Bottom

3

4

Multiple solutions, 3-person, zero-sum game

1, -1/2, -1/2 -1/2, 1, -1/2

3-Top 3-Bottom

5

Competitive Advantage and Market Niche with 3 PlayersThe row-column matrix representation for 3 playersGames where no player plays the spoilerPure and mixed strategy equilibria for 3-player games

6

Competitive Advantage, three firms

New Technology Stay Put

0, 0, 0

-a, a/2, a/2

New Technology

Stay Put

Firm 3 - New Technology

a/2, -a, a/2

-a/2, -a/2, a

Firm 2

Firm 1New

Technology Stay Put

a/2, a/2, -a

-a/2, a, -a/2

New Technology

Firm 3 - Stay Put

Firm 2

Firm 1

Stay Put

a, -a/2, -a/2

0, 0, 0

7

Competitive Advantage, three firms: Strategy for Firm 1

New Technology Stay Put

0, 0, 0

-a, a/2, a/2

New Technology

Stay Put

Firm 3 - New Technology

a/2, -a, a/2

-a/2, -a/2, a

Firm 2

Firm 1New

Technology Stay Put

a/2, a/2, -a

-a/2, a, -a/2

New Technology

Firm 3 - Stay Put

Firm 2

Firm 1

Stay Put

a, -a/2, -a/2

0, 0, 0

8

Competitive Advantage, three firms: Strategy for Firm 2

New Technology Stay Put

0, 0, 0

-a, a/2, a/2

New Technology

Stay Put

Firm 3 - New Technology

a/2, -a, a/2

-a/2, -a/2, a

Firm 2

Firm 1New

Technology Stay Put

a/2, a/2, -a

-a/2, a, -a/2

New Technology

Firm 3 - Stay Put

Firm 2

Firm 1

Stay Put

a, -a/2, -a/2

0, 0, 0

9

Competitive Advantage, three firms: Strategy for Firm 3

New Technology Stay Put

0, 0, 0

-a, a/2, a/2

New Technology

Stay Put

Firm 3 - New Technology

a/2, -a, a/2

-a/2, -a/2, a

Firm 2

Firm 1New

Technology Stay Put

a/2, a/2, -a

-a/2, a, -a/2

New Technology

Firm 3 - Stay Put

Firm 2

Firm 1

Stay Put

a, -a/2, -a/2

0, 0, 0

10

Competitive Advantage, three firms:The Nash equilibrium

New Technology Stay Put

0, 0, 0

-a, a/2, a/2

New Technology

Stay Put

Firm 3 - New Technology

a/2, -a, a/2

-a/2, -a/2, a

Firm 2

Firm 1New

Technology Stay Put

a/2, a/2, -a

-a/2, a, -a/2

New Technology

Firm 3 - Stay Put

Firm 2

Firm 1

Stay Put

a, -a/2, -a/2

0, 0, 0

11

Market Niche for three firms

Enter Stay Out

-50, -50, -50

0, -50, -50Stay Out

Firm 3 - Enter

-50, 0, -50

Firm 2

Firm 1 Stay Out

-50, -50, 0

0, 100, 0

Firm 3 - Stay Out

Firm 2

Firm 1

Stay Out

100, 0, 0

0, 0, 0

Enter

Enter Enter

0, 0, 100

12

Market Niche, three firms: Strategy for Firm 1

Enter Stay Out

-50, -50, -50

0, -50, -50Stay Out

Firm 3 - Enter

-50, 0, -50

Firm 2

Firm 1 Stay Out

-50, -50, 0

0, 100, 0

Firm 3 - Stay Out

Firm 2

Firm 1

Stay Out

100, 0, 0

0, 0, 0

Enter

Enter Enter

0, 0, 100

13

Market Niche, three firms: Strategy for Firm 2

Enter Stay Out

-50, -50, -50

0, -50, -50Stay Out

Firm 3 - Enter

-50, 0, -50

Firm 2

Firm 1 Stay Out

-50, -50, 0

0, 100, 0

Firm 3 - Stay Out

Firm 2

Firm 1

Stay Out

100, 0, 0

0, 0, 0

Enter

Enter Enter

0, 0, 100

14

Market Niche, three firms: Strategy for Firm 3

Enter Stay Out

-50, -50, -50

0, -50, -50Stay Out

Firm 3 - Enter

-50, 0, -50

Firm 2

Firm 1 Stay Out

-50, -50, 0

0, 100, 0

Firm 3 - Stay Out

Firm 2

Firm 1

Stay Out

100, 0, 0

0, 0, 0

Enter

Enter Enter

0, 0, 100

15

Market Niche, three firms: Three pure strategy equilibria

Enter Stay Out

-50, -50, -50

0, -50, -50Stay Out

Firm 3 - Enter

-50, 0, -50

Firm 2

Firm 1 Stay Out

-50, -50, 0

0, 100, 0

Firm 3 - Stay Out

Firm 2

Firm 1

Stay Out

100, 0, 0

0, 0, 0

Enter

Enter Enter

0, 0, 100

16

Mixed Strategy equilibrium in Market Niche with 3 players

From the standpoint of the market, the distribution of number of firms in the market niche, according to mixed strategy equilibria is as follows:

p(3 firms enter) = .08p(2 firms enter) = .31p(1 firm enters) = .42p(No firm enters)= .19

17

3-Player Versions of Coordination, Deal-Making, and Advertising

Video System Coordination with 3 firms Let’s Make a Deal with 3 firms Cigarette Advertising on Television with 3 firms

18

Video System Coordination, three firms: The payoff matrices

Beta VHS

1, 1, 1

Firm 3 - Beta

Firm 2

Firm 1

Firm 3 - VHS

Firm 2

Firm 1

0, 0, 0

Beta VHS

Beta

VHS

Beta

VHS 1, 1, 1

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

19

Video System Coordination, three firms: Strategy for Firm 1

Beta VHS

1, 1, 1

Firm 3 - Beta

Firm 2

Firm 1

Firm 3 - VHS

Firm 2

Firm 1

0, 0, 0

Beta VHS

Beta

VHS

Beta

VHS 1, 1, 1

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

20

Video System Coordination, three firms: Strategy for Firm 2

Beta VHS

1, 1, 1

Firm 3 - Beta

Firm 2

Firm 1

Firm 3 - VHS

Firm 2

Firm 1

0, 0, 0

Beta VHS

Beta

VHS

Beta

VHS 1, 1, 1

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

21

Video System Coordination, three firms: Strategy for Firm 3

Beta VHS

1, 1, 1

Firm 3 - Beta

Firm 2

Firm 1

Firm 3 - VHS

Firm 2

Firm 1

0, 0, 0

Beta VHS

Beta

VHS

Beta

VHS 1, 1, 1

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

22

Video System Coordination, three firms: Two pure strategy equilibria

Beta VHS

1, 1, 1

Firm 3 - Beta

Firm 2

Firm 1

Firm 3 - VHS

Firm 2

Firm 1

0, 0, 0

Beta VHS

Beta

VHS

Beta

VHS 1, 1, 1

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

23

Let’s make a deal, three players: Payoffs in millions of dollars

Yes No

5, 5, 5

Player 3 - Yes

Player 2

Player 1

Player 2

Player 1

0, 0, 0

Yes No

Yes

No

Yes

No

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

Player 3 - No

0, 0, 0

24

Let’s make a deal, three players: Strategy for player 1

Yes No

5, 5, 5

Player 3 - Yes

Player 2

Player 1

Player 2

Player 1

0, 0, 0

Yes No

Yes

No

Yes

No

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

Player 3 - No

0, 0, 0

25

Let’s make a deal, three players: Strategy for player 2

Yes No

5, 5, 5

Player 3 - Yes

Player 2

Player 1

Player 2

Player 1

0, 0, 0

Yes No

Yes

No

Yes

No

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

Player 3 - No

0, 0, 0

26

Let’s make a deal, three players: Strategy for player 3

Yes No

5, 5, 5

Player 3 - Yes

Player 2

Player 1

Player 2

Player 1

0, 0, 0

Yes No

Yes

No

Yes

No

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

Player 3 - No

0, 0, 0

27

Let’s make a deal, three players: Five pure strategy equilibria

Yes No

5, 5, 5

Player 3 - Yes

Player 2

Player 1

Player 2

Player 1

0, 0, 0

Yes No

Yes

No

Yes

No

0, 0, 0

0, 0, 00, 0, 0 0, 0, 0

0, 0, 0

Player 3 - No

0, 0, 0

28

Stonewalling Watergate

Watergate as a 3-person Prisoner’s DilemmaStrictly dominant strategies and uniqueness of equilibriumEquilibria which are bad for the players

29

Stonewalling Watergate: D = Dean, E = Ehrlichman, H = Halderman

Stonewall Talk

-3, -3, -3

H - Stonewall

E

D

E

D

-5, -2, -2

Talk

Talk

Stonewall

Talk

-5, -5, -2

-2, -5, -2-2, -5, -5 -2, -2, -5

-5, -2, -2

-4, -4, -4

H - Talk

Stonewall

Stonewall

30

Stonewalling Watergate: Strategy for Dean

Stonewall Talk

-3, -3, -3

H - Stonewall

E

D

E

D

-5, -2, -2

Talk

Talk

Stonewall

Talk

-5, -5, -2

-2, -5, -2-2, -5, -5 -2, -2, -5

-5, -2, -5

-4, -4, -4

H - Talk

Stonewall

Stonewall

31

Stonewalling Watergate: Strategy for Ehrlichman

Stonewall Talk

-3, -3, -3

H - Stonewall

E

D

E

D

-5, -2, -2

Talk

Talk

Stonewall

Talk

-5, -5, -2

-2, -5, -2-2, -5, -5 -2, -2, -5

-5, -2, -5

-4, -4, -4

H - Talk

Stonewall

Stonewall

32

Stonewalling Watergate: Strategy for Halderman

Stonewall Talk

-3, -3, -3

H - Stonewall

E

D

E

D

-5, -2, -2

Talk

Talk

Stonewall

Talk

-5, -5, -2

-2, -5, -2-2, -5, -5 -2, -2, -5

-5, -2, -5

-4, -4, -4

H - Talk

Stonewall

Stonewall

33

Stonewalling Watergate: The Nash equilibrium

Stonewall Talk

-3, -3, -3

H - Stonewall

E

D

E

D

-5, -2, -2

Talk

Talk

Stonewall

Talk

-5, -5, -2

-2, -5, -2-2, -5, -5 -2, -2, -5

-5, -2, -5

-4, -4, -4

H - Talk

Stonewall

Stonewall

34

Symmetry and Games with Many Players

A compact notation for utility functionsA generalized symmetry sufficient conditionA symmetric game may have asymmetric equilibria

35

Solving Symmetric Games with Many Strategies

A test for when a game is symmetricSymmetry makes games easier to solveSolving a game of common interest by exploiting the symmetry of the game

The Nash Demand Game

36

The Nash demand game:the payoff matrix

0,0

0, 0

0, 0

0, 0

$0

$2

$0

Player 2Player 1 $2

1, 1

2, 0

0, 2 0, 1

1, 0$1

$1

37

The Nash demand game:player 1’s strategy

0,0

0, 0

0, 0

0, 0

$0

$2

$0

Player 2Player 1 $2

1, 1

2, 0

0, 2 0, 1

1, 0$1

$1

38

The Nash demand game:player 2’s strategy

0,0

0, 0

0, 0

0, 0

$0

$2

$0

Player 2Player 1 $2

1, 1

2, 0

0, 2 0, 1

1, 0$1

$1

39

The Nash demand game:Nash equilibrium

0,0

0, 0

0, 0

0, 0

$0

$2

$0

Player 2Player 1 $2

1, 1

2, 0

0, 2 0, 1

1, 0$1

$1

40

Stag Hunt

Game requiring cooperation for efficient outcomeAdding third player leads to qualitatively different outcome

additional Nash equilibriaasymmetric outcomespossibility of free riding

41

Stag Hunt, two hunters:The payoff matrix

3, 3

1, 0

0, 1

1, 1

hunt big

Hunter 2

Hunter 1 hunt small

hunt big

hunt small

42

Stag Hunt, two hunters:Strategy for hunter 1

3, 3

1, 0

0, 1

1, 1

Hunter 2

Hunter 1 hunt big hunt small

hunt big

hunt small

43

Stag Hunt, two hunters:Strategy for hunter 2

3, 3

1, 0

0, 1

1, 1

Hunter 2

Hunter 1 hunt big hunt small

hunt big

hunt small

44

Stag Hunt, two hunters:The equilibrium

3, 3

1, 0

0, 1

1, 1

Hunter 2

Hunter 1 hunt big hunt small

hunt big

hunt small

45

Stag Hunt, three hunters:The payoff matrix

hunt big hunt big

hunt small hunt small

hunter 1 hunter 1

hunter 2hunt big hunt small hunt big hunt small

hunter 2

hunter 3: hunt big hunter 3: hunt small

3, 3, 3

5, 3, 3 1, 1, 0

3, 5, 3 3, 3, 5 0, 1, 1

1, 0, 1 1, 1, 1

46

Stag Hunt, three hunters:Strategy for hunter 1

hunt small

hunt big hunt big

hunt small

hunter 1 hunter 1

hunter 2hunt big hunt small hunt big hunt small

hunter 2

hunter 3: hunt big hunter 3: hunt small

3, 3, 3

5, 3, 3 1, 1, 0

3, 5, 3 3, 3, 5 0, 1, 1

1, 0, 1 1, 1, 1

47

Stag Hunt, three hunters:Strategy for hunter 2

hunt small

hunt big hunt big

hunt small

hunter 1 hunter 1

hunter 2hunt big hunt small hunt big hunt small

hunter 2

hunter 3: hunt big hunter 3: hunt small

3, 3, 3

5, 3, 3 1, 1, 0

3, 5, 3 3, 3, 5 0, 1, 1

1, 0, 1 1, 1, 1

48

Stag Hunt, three hunters:Strategy for hunter 3

hunt small

hunt big hunt big

hunt small

hunter 1 hunter 1

hunter 2hunt big hunt small hunt big hunt small

hunter 2

hunter 3: hunt big hunter 3: hunt small

3, 3, 3

5, 3, 3 1, 1, 0

3, 5, 3 3, 3, 5 0, 1, 1

1, 0, 1 1, 1, 1

49

Stag Hunt, three hunters:Nash equilibria

hunt small

hunt big hunt big

hunt small

hunter 1 hunter 1

hunter 2hunt big hunt small hunt big hunt small

hunter 2

hunter 3: hunt big hunter 3: hunt small

3, 3, 3

5, 3, 3 1, 1, 0

3, 5, 3 3, 3, 5 0, 1, 1

1, 0, 1 1, 1, 1

50

The Tragedy of the Commons

Games played on a commonsThe equilibrium of such a game has a tragic outcomeExternalitiesFirst Welfare TheoremThe case of the Geysers of Northern California

51

Tragedy of the Commons:score sheet

Payoff = 5(10 - xi) + xi (23 – 0.25 Σxi)

4

3

2

1

payoffstrategy (xi )

52

Tragedy of the Commons:Commons production function

0 6 12X

F(X)

3.6

53

Tragedy of the Commons

0 6 12X

F(X)

5 10

Nash, 5 players

Nash, 10 players

Ultimate Tragic NEF(X)/X = 0.1

Economic Zone Uneconomic Zone

3.6

54

Tragedy of the Commons

0 6 12X

F(X)

0.1

average product

marginal product

efficient outcome

tragic outcome

55

Appendix. Tragedy of the Commons in the Laboratory

Playing a game in a behavior laboratoryTragic outcomes of a game played on a commons in a laboratoryUnexplained phenomena