Playing in the Dark - PowerPoint Presentation

Slide1
Slide2

Playing in the Dark

Problems from

Ch 8Slide3

The OS2 story

Introduced by IBM in 1987 to compete

with MS Windows.

Faster and more reliable than Windows

but not many applications available.Slide4

A Model of Attracting Developers

(Platform Externalities)

Assume 3 companies who could develop applications. Not

cevelop

gives payoff 0

If all 3 develop, each gets payoff of 3 and

IBM gets 20

If 2 develop, each gets payoff of 2 and IBM gets 15

If 1 develops, each gets -1 and IBM gets -2

If 0 develop, each gets 0 and IBM gets -3Slide5

OS2 Developers Game TreeSlide6

How many regular, proper

subgames

does this game have?

0

1

2

3

7Slide7

Pure strategy Nash

equilibria

In the

subgame

where IBM develops OS2 there are 2 pure strategy Nash

equilibria

. All develop. None develop.

IBM would develop OS2 if all 3 companies develop apps and would not if none do.

So there are 2

subgame

perfect pure strategy Nash

equilibria

.Slide8

What actually happened

IBM developed OS2

Not enough software firms developed software for OS2 to make it viable.

IBM gave up on OS2 and lost a bundle.

This is not either of the two predicted

subgame

perfect Nash

equilibria

. Slide9

Mixed strategy equilibrium?

There is also a mixed strategy Nash equilibrium

Prediction of this model is consistent with history of OS2

Developers do not know what each other will do and choose mixed strategiesSlide10

Finding equilibrium

Suppose each developer develops with probability p.

In equilibrium all are indifferent between developing and not developing.

If you develop, probability that 2 others develop is p

2

What is probability that 1 other develops? Probability that nobody else

deveops

?Slide11

If 3 players each develop with independent probability p, what

is the probability that if you develop, exactly one of the other two players will develop?

A) p

B) 1-p

C) p(1-p)

D) 2p(1-p)

E) 1/pSlide12

Expected payoff to Develop if all develop with probability p

Expected payoff from developing is

3 p

2

+ 1x2p(1-p)-1(1-p)

2

Simplfies

to 4p-1

The payoff from not developing is 0.

Players will use mixed strategy if

4p-1=0, so p=1/4.Slide13

IBM’s expected profit from developing OS2 if developers use

misced

strategies

Probability 3 develop is (1/4)(1/4)(1/4)=1/64

Probability 2 develop is 3(1/4)(1/4)(3/4)=9/64

Probability 1 develop is 3(3/4)(3/4)=27/64

Probability none develop is (3/4)(3/4(3/4)=27/64

Expected Profit for IBM

with OS2 is

20(1/64)+15(9/64)-2(27/64)-3(27/64)= 20/64>Slide14

Prediction of Mixed strategy equilibrium

IBM would develop OS2, realizing that it might not succeed, but that there enough chance that it would succeed and the winnings if it does are large enough so that it is worth trying.

Probability that fewer than 2 companies adopt

and OS2 fails is 27/64+27/64=27/32Slide15

IBM and Software Developers: Problem 2, p 282Slide16

How many regular, proper

subgames

does this game have?

0

1

2

3

7Slide17

Nash

equiibria

of regular subgames

Only Nash equilibrium in

subgame

where IBM and Company 1 develop has 2 develop and 3 develop. Payoff to Company 1 is then 2

In

subgame

where IBM develops and Company 1 does not develop there are 2 Nash

equilibria

for 2 and 3. Both develop or neither develops. In either case, payoff to company 1 is 0.Slide18

Subgame

between 2 and 3 if

IBM and Company 1 develop

D

DND

D

2,2

1,0

DND

0,1

0,0

Company 3

Company 2Slide19

Subgame

between 2 and 3 if

IBM develops and company 1 does not

D

DND

D

1,1

-1,0

DND

0,-1

0,0

Company 3

Company 2Slide20

What will company 1 do and what will IBM do?

In

subgame

where company 2 chooses, only

subgame

perfect equilibrium is Company 1 develops and so do companies 2 and 3.

Only

subgame

perfect equilibrium has all 3 companies developing if IBM develops.

What will IBM do in a

subgame

perfect N.E?

Devlop

gives payoff of 5. Don’t gives 0Slide21

Problem 1 Slide22

Find regular proper

subgames

and trim the tree

In a

subgame

perfect equilibrium, what will 3 do at node on left?

What will 3 do at node on right?

Draw trimmed tree on blackboard.Slide23

x/x

y/x

x/y

y/y

a

1,1,1

2,3,1

1,1,1

2,3,1

b

0,3,2

0,3,2

3,0,3

3,0,3

Player 3 goes d/c/c

Player 2

Player 1Slide24

x/x

y/x

x/y

y/y

a

1,1,1

1,2,0

1,1,1

1,2,1

b

0,4,1

0,4,1

3,0,3

3,0,3

Player 3 goes d/d/c

Player 2

Player 1Slide25

x/x

y/x

x/y

y/y

a

1,1,1

1,2,0

1,1,1

1,2,1

b

0,4,1

0,4,1

3,0,3

3,0,3

Player 3 goes d/d/c

x/x

y/x

x/y

y/y

a

1,1,1

2,3,1

1,1,1

2,3,1

b

0,3,2

0,3,2

3,0,3

3,0,3

Player 2

Player 3 goes d/c/cSlide26

Solving for SPNE

Find the pure strategy Nash for this reduced game. (use stars)

What is the

subgame

perfect Nash equilibrium for the entire game?