Game Theoryand Electronic Communications Markets

Bernhard von Stengel

London School of Economics

1

Non-cooperative game theory

• Players motivated by individual incentives

• Interactions resulting in payoffs

Explains:

• Selfish but collectively damaging behaviour

• How to think strategically

• More than one possible equilibrium (stable outcome)

• Rules of the game matter

• Selfish behaviour in networks

2

Explain selfish behaviorPrice-setting game

high

high low

price

price

price price

low

1

2

1

3

2

0

1

3

02

3

Explain selfish behaviorPrice-setting game

high

high low

price

price

price price

low

1

2

1

3

2

0

1

3

02

3

Explain selfish behaviorPrice-setting game

high

high low

price

price

price price

low

1

2

1

3

2

0

1

3

02

3

Explain selfish behaviorFishing game

over−

fishing

sustainable

overfishing

fishing

fishing

sustainable

1

2

1

3

2

0

1

3

02

price competition positive for consumers, but same game betweenfishing nations detrimental for all!

3

Thinking strategically: Epson vs. HP

stay out enter (laser printer market)

lower prices

fight

HP

Epson

Epson

14

−2 −1

3

2

5

0keep prices

back out

4

Thinking strategically: Epson vs. HP

stay out enter (laser printer market)

lower prices

fight

HP

Epson

Epson

14

−2 −1

3

2

5

0keep prices

back out

4

Thinking strategically: Epson vs. HP

stay out enter (laser printer market)

HP

Epson

3

2

5

0keep prices

4

Thinking strategically: Epson vs. HP

stay out enter (laser printer market)

lower prices

fight

HP

Epson

Epson

14

−2 −1

3

2

5

0keep prices

back out

4

Thinking strategically: Epson vs. HP

lower prices

fight

Epson

14

−2 −1

back out

stay out enter (laser printer market)

HP

Epson

3

2

5

0keep prices

4

Thinking strategically: Epson vs. HP

lower prices

fight

Epson

14

−2 −1

back out

stay out enter (laser printer market)

HP

Epson

3

2

5

0keep prices

4

Thinking strategically: Epson vs. HP

lower prices

fight

Epson

14

−2 −1

back out

stay out enter (laser printer market)

HP

Epson

3

2

5

0keep prices

4

Thinking strategically: Epson vs. HP

lower prices

fight

Epson

14

−2 −1

back out

stay out enter (laser printer market)

HP

Epson

3

2

5

0keep prices

4

More than one possible equilibrium

Equilibrium = one strategy for each player, which is optimalwhen the other players’ strategies stay fixed.

A game may have more than one equilibrium:

For example, in the Epson vs. HP game:

Equilibrium 1: (stay out , lower prices)

Equilibrium 2: (enter / fight , keep prices)

not an equilibrium: (enter / fight , lower prices)

5

More than one possible equilibrium

Equilibrium = one strategy for each player, which is optimalwhen the other players’ strategies stay fixed.

A game may have more than one equilibrium:

For example, in the Epson vs. HP game:

Equilibrium 1: (stay out , lower prices)

Equilibrium 2: (enter / fight , keep prices)

not an equilibrium: (enter / fight , lower prices)

5

More than one possible equilibrium

Equilibrium = one strategy for each player, which is optimalwhen the other players’ strategies stay fixed.

A game may have more than one equilibrium:

For example, in the Epson vs. HP game:

Equilibrium 1: (stay out , lower prices)

Equilibrium 2: (enter / fight , keep prices)

not an equilibrium: (enter / fight , lower prices)

5

The bandwidth choice game

Which equilibrium?

Evolved from starting distribution:

bandwidth

bandwidthlow

high

high low

width

band−

width

band−

1

2

1

1

5

0

1

1

5 0

6

The bandwidth choice game

Which equilibrium?

Evolved from starting distribution:

bandwidth

bandwidthlow

high

high low

width

band−

width

band−

1

2

1

1

5

0

1

1

5 0

6

The bandwidth choice gameWhich equilibrium?

Evolved from starting distribution:

bandwidth

bandwidthlow

high

high low

width

band−

width

band−

1

2

1

1

5

0

1

1

5 0

6

The bandwidth choice gameWhich equilibrium? Evolved from starting distribution:

1

70 %30 %

1.5bandwidth

bandwidthlow

high

high low

width

band−

width

band−

1

2

1

1

5

0

1

1

5 0

6

The bandwidth choice gameWhich equilibrium? Evolved from starting distribution:

50 % 50 %

2.5

1

bandwidth

bandwidthlow

high

high low

width

band−

width

band−

1

2

1

1

5

0

1

1

5 0

6

The bandwidth choice gameWhich equilibrium? Evolved from starting distribution:

4

20 %80 %

1

bandwidth

bandwidthlow

high

high low

width

band−

width

band−

1

2

1

1

5

0

1

1

5 0

6

The bandwidth choice gameWhich equilibrium? Evolved from starting distribution:

100 % 0 %

5

1

bandwidth

bandwidthlow

high

high low

width

band−

width

band−

1

2

1

1

5

0

1

1

5 0

6

Rules of the game matter

The Quality Game

changed to a game with commitment

high

buy don’t

low

quality

quality

buy1

2

1

3

2

0

1

1

02

⇒ Commitment power can help both players!

7

Rules of the game matter

The Quality Game

changed to a game with commitment

high

buy don’t

low

quality

quality

buy1

2

1

3

2

0

1

1

02

⇒ Commitment power can help both players!

7

Rules of the game matter

The Quality Game

changed to a game with commitment

high

buy don’t

low

quality

quality

buy1

2

1

3

2

0

1

1

02

⇒ Commitment power can help both players!

7

Rules of the game matter

The Quality Game changed to a game with commitment

12 0

0 132

1

buybuy

2

1

buy buy

2

don’t don’t

low qualityhigh quality

high

buy don’t

low

quality

quality

buy1

2

1

3

2

0

1

1

02

⇒ Commitment power can help both players!

7

Rules of the game matter

The Quality Game changed to a game with commitment

12 0

0 132

1

buybuy

2

1

buy buy

2

don’t don’t

low qualityhigh quality

high

buy don’t

low

quality

quality

buy1

2

1

3

2

0

1

1

02

⇒ Commitment power can help both players!

7

Rules of the game matter

The Quality Game changed to a game with commitment

12 0

0 132

1

buybuy

2

1

buy buy

2

don’t don’t

low qualityhigh quality

high

buy don’t

low

quality

quality

buy1

2

1

3

2

0

1

1

02

⇒ Commitment power can help both players!

7

Rules of the game matter

The Quality Game changed to a game with commitment

12 0

0 132

1

buybuy

2

1

buy buy

2

don’t don’t

low qualityhigh quality

high

buy don’t

low

quality

quality

buy1

2

1

3

2

0

1

1

02

⇒ Commitment power can help both players!

7

Selfish routing in networks

A B

x delayed by x

1−x delayed by 1

flow 1

Selfish routing 33% worse than optimal, centrally planned routing

(33% longer delay is worst possible if delay functions are linear,e.g. flow x delayed by x)

8

Selfish routing in networks

optimal: 1−x = 0.5

optimal: x = 0.5

average delay: 0.25 + 0.5 = 0.75

A B

x delayed by x

1−x delayed by 1

flow 1

Selfish routing 33% worse than optimal, centrally planned routing

(33% longer delay is worst possible if delay functions are linear,e.g. flow x delayed by x)

8

Selfish routing in networks

selfish: x = 1.0

average delay: 1.0 + 0 = 1.0

selfish: 1−x = 0

A B

x delayed by x

1−x delayed by 1

flow 1

Selfish routing 33% worse than optimal, centrally planned routing

(33% longer delay is worst possible if delay functions are linear,e.g. flow x delayed by x)

8

Selfish routing in networks

selfish: x = 1.0

average delay: 1.0 + 0 = 1.0

selfish: 1−x = 0

A B

x delayed by x

1−x delayed by 1

flow 1

Selfish routing 33% worse than optimal, centrally planned routing

(33% longer delay is worst possible if delay functions are linear,e.g. flow x delayed by x)

8

Selfish routing can be optimal

D

C

BA

x delayed by x

1−x delayed by 1

flow 1

1−x delayed by 1−x

x delayed by 1

Increasing network capacity can worsen equilibrium congestion!

9

Selfish routing can be optimal

optimal: x = 0.5

optimal: 1−x = 0.5

optimal average delay: 0.75 + 0.75 = 1.5

D

C

BA

x delayed by x

1−x delayed by 1

flow 1

1−x delayed by 1−x

x delayed by 1

Increasing network capacity can worsen equilibrium congestion!

9

Selfish routing can be optimal

selfish =

selfish = optimal: x = 0.5

optimal: 1−x = 0.5

optimal average delay: 0.75 + 0.75 = 1.5

D

C

BA

x delayed by x

1−x delayed by 1

flow 1

1−x delayed by 1−x

x delayed by 1

Increasing network capacity can worsen equilibrium congestion!

9

Add a shortcut

1−x+z delayed by 1−x+z

x−z delayed by 1

z delayed by 0

D

C

BA

x delayed by x

1−x delayed by 1

flow 1

Increasing network capacity can worsen equilibrium congestion!

9

Add a shortcut

1−x+z delayed by 1−x+z

x−z delayed by 1

z delayed by 0

optimal average delay: 0.75 + 0.75 = 1.5

optimal: x = 0.5, z=0

optimal: 1−x = 0.5

D

C

BA

x delayed by x

1−x delayed by 1

flow 1

Increasing network capacity can worsen equilibrium congestion!

9

Add a shortcut

selfish: x = z = 1.0

average delay: 1.0 + 0 + 1.0 = 2.01−x+z delayed by 1−x+z

x−z delayed by 1

z delayed by 0

D

C

BA

x delayed by x

1−x delayed by 1

flow 1

Increasing network capacity can worsen equilibrium congestion!

9

Add a shortcut

selfish: x = z = 1.0

average delay: 1.0 + 0 + 1.0 = 2.01−x+z delayed by 1−x+z

x−z delayed by 1

z delayed by 0

optimal average delay: 0.75 + 0.75 = 1.5

D

C

BA

x delayed by x

1−x delayed by 1

flow 1

Increasing network capacity can worsen equilibrium congestion!

9

Braess’s paradox

selfish: x = z = 1.0

average delay: 1.0 + 0 + 1.0 = 2.01−x+z delayed by 1−x+z

x−z delayed by 1

z delayed by 0

optimal average delay: 0.75 + 0.75 = 1.5

D

C

BA

x delayed by x

1−x delayed by 1

flow 1

Increasing network capacity can worsen equilibrium congestion!

9

Summary: Game theory

• provides tools for analysing interactive decisions

• very suitable for: few players, or large number of similarplayers; competitive dynamics of markets; “thinking ahead”

• important are: incentives; rules of the game;information of the players

• central concept: equilibrium (not always unique)

• explain: selfish behaviour in competition, networks androuting

• many more applications: design of auctions, optimal bidding,. . .

10

Summary: Game theory

• provides tools for analysing interactive decisions

• very suitable for: few players, or large number of similarplayers; competitive dynamics of markets; “thinking ahead”

• important are: incentives; rules of the game;information of the players

• central concept: equilibrium (not always unique)

• explain: selfish behaviour in competition, networks androuting

• many more applications: design of auctions, optimal bidding,. . .

10

Summary: Game theory

• provides tools for analysing interactive decisions

• very suitable for: few players, or large number of similarplayers; competitive dynamics of markets; “thinking ahead”

• important are: incentives; rules of the game;information of the players

• central concept: equilibrium (not always unique)

• explain: selfish behaviour in competition, networks androuting

• many more applications: design of auctions, optimal bidding,. . .

10

Summary: Game theory

• provides tools for analysing interactive decisions

• very suitable for: few players, or large number of similarplayers; competitive dynamics of markets; “thinking ahead”

• important are: incentives; rules of the game;information of the players

• central concept: equilibrium (not always unique)

• explain: selfish behaviour in competition, networks androuting

• many more applications: design of auctions, optimal bidding,. . .

10

Summary: Game theory

• provides tools for analysing interactive decisions

• very suitable for: few players, or large number of similarplayers; competitive dynamics of markets; “thinking ahead”

• important are: incentives; rules of the game;information of the players

• central concept: equilibrium (not always unique)

• explain: selfish behaviour in competition, networks androuting

• many more applications: design of auctions, optimal bidding,. . .

10

Summary: Game theory

• provides tools for analysing interactive decisions

• very suitable for: few players, or large number of similarplayers; competitive dynamics of markets; “thinking ahead”

• important are: incentives; rules of the game;information of the players

• central concept: equilibrium (not always unique)

• explain: selfish behaviour in competition, networks androuting

• many more applications: design of auctions, optimal bidding,. . .

10