Playing in the DarkProblems from Ch 8

The OS2 story

Introduced by IBM in 1987 to competewith MS Windows.

Faster and more reliable than Windowsbut not many applications available.

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 andIBM 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 -3

OS2 Developers Game Tree

How many regular, proper subgames does this game have?

A) 0B) 1C) 2D) 3E) 7

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.

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.

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 strategies

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 p2 What is probability that 1 other develops? Probability that nobody else deveops?

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/p

Expected payoff to Develop if all develop with probability p

• Expected payoff from developing is 3 p2+ 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.

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 is20(1/64)+15(9/64)-2(27/64)-3(27/64)= 20/64>

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=27/32

IBM and Software Developers: Problem 2, p 282

How many regular, proper subgames does this game have?

A) 0B) 1C) 2D) 3E) 7

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.

Subgame between 2 and 3 ifIBM and Company 1 develop

D DNDD 2,2 1,0DND 0,1 0,0

Company 3

Company 2

Subgame between 2 and 3 ifIBM develops and company 1 does not

D DNDD 1,1 -1,0DND 0,-1 0,0

Company 3

Company 2

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 0

Problem 1

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.

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 1

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 1

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 3 goes d/c/c

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?

Problem 8, p 286

How many strategies does Player 1 have?

A) 3B) 6C) 8D) 9E) 12

How many regular, proper subgames are there?

• A) 1• B) 2• C) 3• D) 4• E) 5

Subgames after b1 and c1

• b1 reduces to payoff 1,1,1• c1 reduces to payoff 2,1,0• What about a1?

Game between 2 and 3 if 1 goes a

There are two SPNEs

• a1/d1/d1, b2/a2/a2, a3 And c1/d1/d1, a2/a2/a2, b3

One N.E. that is not SPNE isa1/d1/e1,b2/a2/a2,a3

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