Nash-Equilibrium for Two-Person Games · Nash-Equilibrium for Two-Person Games Chapter 3 2 Zero-sum Games and Constant-sum Games ... solutions: Strategy for Player 1 1, -1-1, 1 0,

1

Nash-Equilibrium for Two-Person

Games

Chapter 3

2

Zero-sum Games and Constant-sum Games

Definition of zero-sum games Examples: Poker, Battle of the Networks

The arrow diagram for a 2×2 game in normal form

The arrows point towards a Nash equilibrium

Transforming a constant-sum game into a zero-sum game

3

Battle of the Networks, normal form: The payoff matrix

55%, 45%

50%, 50%

52%, 48%

45%, 55%

Sitcom Sports

Sitcom

Sports

Network 2

Network 1

4

Battle of the Networks, normal form: Strategy for Network 1

55%, 45%

50%, 50%

52%, 48%

45%, 55%

Sitcom Sports

Sitcom

Sports

Network 2

Network 1

Network 1’s Best Responses are underlined.

5

Battle of the Networks, normal form: Strategy for Network 2

55%, 45%

50%, 50%

52%, 48%

45%, 55%

Sitcom Sports

Sitcom

Sports

Network 2

Network 1


6

Battle of the Networks, normal form: The Equilibrium

55%, 45%

50%, 50%

52%, 48%

45%, 55%

Sitcom Sports

Sitcom

Sports

Network 2

Network 1

All Best Responses are underlined.

7

Battle of the Networks, zero-sum form: The payoff matrix

10% , -10%

0, 0

4%, -4%

-10%, 10%

Sitcom Sports

Sitcom

Sports

Network 2

Network 1

8

Battle of the Networks, zero-sum form: Strategy for Network 1

10%, -10%

0, 0

4%, -4%

-10%, 10%

Sitcom Sports

Sitcom

Sports

Network 2

Network 1


9

Battle of the Networks, zero-sum form: Strategy for Network 2

10% , -10%

0, 0

4%, -4%

-10%, 10%

Sitcom Sports

Sitcom

Sports

Network 2

Network 1


10

Battle of the Networks, zero-sum form: The equilibrium

10%, -10%

0, 0

4%, -4%

-10%, 10%

Sitcom Sports

Sitcom

Sports

Network 2

Network 1

All Best Responses are underlined.

11

Why look for Nash equilibrium?

Equilibrium conceptNo player has incentive to change strategy

Analogy to dominance solvabilityPlay of strictly dominant strategies leads to Nash equilibriumWeak dominance solution is also Nash equilibrium

12

Nash equilibrium of a game like chess

1

L

R

2

(l, w)L

R (w, l)

(d, d)

a) Extensive Form

(w, l)(l, w)

(d, d) (d, d)

L

L

R

R1

2

b) Normal Form

13

Competitive Advantage

The game Competitive Advantage and the economic realities it reflectsThe solution of Competitive Advantage, showing why firms are driven to adopt new technologies

14

Competitive Advantage: The payoff matrix

0, 0

-a, a

a, -a

0 , 0

Stay Put

New Technology

Stay Put

Firm 2

Firm 1New Technology

15

Competitive Advantage: Strategy for Firm 1

0, 0

-a, a

a, -a

0 , 0

Stay Put

New Technology

Stay Put

Firm 2


16

Competitive Advantage: Strategy for Firm 2

0, 0

-a, a

a, -a

0 , 0

Stay Put

New Technology

Stay Put

Firm 2


17

Competitive Advantage: The equilibrium

0, 0

-a, a

a, -a

0 , 0

Stay Put

New Technology

Stay Put

Firm 2


18

1-Card Stud Poker

A model of a more complicated game, made simpler by fewer kinds of cards and smaller handsHow imperfect information is created by the deal of the cardsPrinciples of Poker reflected by the solution of 1-card StudThe concept of dominated strategy

19

Stud Poker, the rules

1. Each player places 1 chip into the pot2. Each player is dealt 1 card, face-down

1. Deck has 50% Aces and 50% Kings2. No one looks at their card

3. Each player decides whether to make a further bet of one more chip or not. A player does this by placing 1 chip in hand behind his or her back (or not), out of sight of the other player.

4. Each player simultaneously reveals further bet.5. If only one player has made a further bet, then that player wins

the pot and no cards need be shown. 6. If both players have made a further bet of either 1 or 0 chips,

then there is a showdown. High card wins the pot; in case of a tie, the players split the pot.

20

One-Card Stud Poker, extensive form

0 1 2

1/4

1/4

1/4

1/4

(A, A)

(A, K)

(K, A)

(K, K)

Bet

Pass

Bet

Bet

Bet

Pass

Pass

Pass

(0, 0)(a, -a)(-a, a)(0, 0)(a+b, -a-b)

(a, -a)(-a, a)(a, -a)(-a-b, a+b)(a, -a)(-a, a)(-a, a)(0, 0)(a, -a)(-a, a)(0, 0)

21

One-Card Stud Poker, normal form: The payoff matrix

0, 0

-a, a

a, -a

0, 0

Pass

Bet

Pass

Bet

22

One-Card Stud Poker, normal form: Strategy for player 1

0, 0

-a, a

a, -a

0, 0

Pass

Bet

Pass

Bet

23

One-Card Stud Poker, normal form: Strategy for player 2

0, 0

-a, a

a, -a

0, 0

Pass

Bet

Pass

Bet

24

One-Card Stud Poker, normal form: The equilibrium

0, 0

-a, a

a, -a

0, 0

Pass

Bet

Pass

Bet

25

Nash Equilibria of 2-Player, 0-Sum Games

Two examples of 2-player, 0-sum games with multiple Nash equilibriaEvery equilibrium of a 2-player, 0-sum game has the same payoffs (proved by contradiction for the 2×2 case)

26

Two-person, zero-sum game with two solutions: The payoff matrix

1, -1

1, -1

0, 0

0, 0

Right

Left

Right

Player 2

Player 1 Left

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Two-person, zero-sum game with two solutions: Strategy for player 1

1, -1

1, -1

0, 0

0, 0

Right

Left

Right

Player 2

Player 1 Left

28

Two-person, zero-sum game with two solutions: Strategy for player 2

1, -1

1, -1

0, 0

0, 0

Right

Left

Right

Player 2

Player 1 Left

29

Two-person, zero-sum game with two solutions: Two equilibria

1, -1

1, -1

0, 0

0, 0

Right

Left

Right

Player 2

Player 1 Left

30

Two-person, zero-sum game with four solutions: The payoff matrix

1, -1

-1, 1

0, 0

0, 0

Right

Left

Right

Player 2Player 1 Left

0, 0

0, 0

0, 0 1, -1

-1, 1Center

Center

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Two-person, zero-sum game with four solutions: Strategy for Player 1

1, -1

-1, 1

0, 0

0, 0

Right

Left

Right


0, 0

0, 0

0, 0 1, -1

-1, 1Center

Center

32

Two-person, zero-sum game with four solutions: Strategy for Player 2

1, -1

-1, 1

0, 0

0, 0

Right

Left

Right


0, 0

0, 0

0, 0 1, -1

-1, 1Center

Center

33

Two-person, zero-sum game with four solutions: Four equilibria

1, -1

-1, 1

0, 0

0, 0

Right

Left

Right


0, 0

0, 0

0, 0 1, -1

-1, 1Center

Center

34

Von Neumann’s Theorem for2-Player Zero-Sum Games

Every equilibrium of a 2-player zero-sum game has the same payoffsNo benefit to cooperation

35

Proof of van Neumann’s theorem: By contradiction, suppose a > b

a, -a

d, -d

c, -c

b, -b

21

36

2-Player Variable Sum Games

The concept of a variable sum gameMost games in business are variable sumA variable sum game may have equilibria with different payoffsNecessary and sufficient conditions for a solutionThe sufficient condition of undominated strategiesGames as parables

37

Let’s Make a Deal: The payoff matrix

$15M, $15M

0, 0

0, 0

0, 0

No

Yes

No

Director

Movie Star Yes

38

Let’s Make a Deal: Strategy for the Movie Star

$15M, $15M

0, 0

0, 0

0, 0

No

Yes

No

Director

Movie Star Yes

39

Let’s Make a Deal: Strategy for the Director

$15M, $15M

0, 0

0, 0

0, 0

No

Yes

No

Director

Movie Star Yes

40

Let’s Make a Deal: Two equilibria

$15M, $15M

0, 0

0, 0

0, 0

No

Yes

No

Director

Movie Star Yes

41

Video System Coordination: The payoff matrix

1, 1

0, 0

0, 0

1, 1

VHS

Beta

VHS

Firm 2

Firm 1 Beta

42

Video System Coordination: Strategy for Firm 1

1, 1

0, 0

0, 0

1, 1

VHS

Beta

VHS

Firm 2

Firm 1 Beta

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Video System Coordination: Strategy for Firm 2

1, 1

0, 0

0, 0

1, 1

VHS

Beta

VHS

Firm 2

Firm 1 Beta

44

Video System Coordination: The two equilibria

1, 1

0, 0

0, 0

1, 1

VHS

Beta

VHS

Firm 2

Firm 1 Beta

45

-1, -1

2, -2

-2, 2

-1, -1

Confess

Cooperate

Confess

Player 2

Player 1Cooperate

(remain silent)

(remain silent)

All numbers are years of time served

Prisoner’s Dilemma: The payoff matrix

46

-1, -1

2, -2

-2, 2

-1, -1

Confess

Cooperate

Confess

Player 2

Player 1Cooperate

(remain silent)

(remain silent)


Prisoner’s Dilemma: Strategy for Player 1

47

-1, -1

2, -2

-2, 2

-1, -1

Confess

Cooperate

Confess

Player 2

Player 1Cooperate

(remain silent)

(remain silent)


Prisoner’s Dilemma: Strategy for Player 2

48

-1, -1

2, -2

-2, 2

-1, -1

Confess

Cooperate

Confess

Player 2

Player 1Cooperate

(remain silent)

(remain silent)


Prisoner’s Dilemma: The equilibrium

49

Cigarette Advertising on Television

Advertising as a strategic variableHeavy advertising by each firm as a dominant strategy and a game equilibriumHow the ban on cigarette advertising on television in 1971 raised industry profitsThe prisoner’s Dilemma

50

Cigarette Television Advertising: The payoff matrix

$60M, $20M

Don’t advertise on television

Company 2

Company 1

$50M, $50M

$27M, $27M

Advertise on television


Don’t advertise on television $20M, $60M

51

Cigarette Television Advertising: Strategy for Company 1

$60M, $20M


Company 2

Company 1

$50M, $50M

$27M, $27M




52

Cigarette Television Advertising: Strategy for Company 2

$60M, $20M


Company 2

Company 1

$50M, $50M

$27M, $27M




53

Cigarette Television Advertising: The equilibrium

$60M, $20M


Company 2

Company 1

$50M, $50M

$27M, $27M




54

2-Player Games with Many Strategies

Using calculus to maximize utility and solve gamesEquilibrium as the solution of a pair of first order conditionsAdvertising budgets as continuous strategic variables

55

Game with many strategies,the corners

x2 = 1

Player2

Player 1

1, 1x1 = 1

x1 = 0

x2 = 0

0, 0

0, 00, 0

56

Existence of Equilibrium

The best response functionA fixed point theoremThe Existence of Equilibrium

57

Strictly dominance solvable game, best response function

10

0 x1

x2

1

f1(x2)

f2(x1) Nash Equilibrium

58

Strictly dominance solvable game, best response mapping

100

x2

1

x1

100000

x2

1000

f(x*)

(500 , 250)x1

59

Advertising, many strategies, best response functions

100010000

0 x1

x2

1000f1(x2)

f2(x1)

X* = (500 , 250)

60

Advertising, many strategies, best response functions

100000

x2

1000

x*

(500 , 250)x1

100000

x2

1000

f(x*)

(500 , 250)x1

f

Documents

Nash-Equilibrium for Two-Person Games · Nash-Equilibrium for Two-Person Games Chapter 3 2 Zero-sum Games and Constant-sum Games ... solutions: Strategy for Player 1 1, -1-1, 1 0,