Problems from Ch 8 The OS2 story Introduced by IBM in 1987 to compete with MS Windows Faster and more reliable than Windows but not many applications available A Model of Attracting Developers ID: 474191
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Slide1
Playing in the DarkProblems from Ch 8Slide2
The OS2 story
Introduced by IBM in 1987 to compete
with MS Windows.
Faster and more reliable than Windows
but not many applications available.Slide3
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 -3Slide4
OS2 Developers Game TreeSlide5
How many regular, proper subgames does this game have?
0
1
2
3
7Slide6
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
.Slide7
What actually happenedIBM developed OS2Not 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
. Slide8
Mixed strategy equilibrium?There is also a mixed strategy Nash equilibriumPrediction of this model is consistent with history of OS2
Developers do not know what each other will do and choose mixed strategiesSlide9
Finding equilibriumSuppose 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
?Slide10
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/pSlide11
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.Slide12
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>Slide13
Prediction of Mixed strategy equilibriumIBM 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=27/32Slide14
IBM and Software Developers: Problem 2, p 282Slide15
How many regular, proper subgames does this game have?
0
1
2
3
7Slide16
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.Slide17
Subgame between 2 and 3 ifIBM and Company 1 develop
D
DND
D
2,2
1,0
DND
0,1
0,0
Company 3
Company 2Slide18
Subgame between 2 and 3 ifIBM develops and company 1 does not
D
DND
D
1,1
-1,0
DND
0,-1
0,0
Company 3
Company 2Slide19
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 0Slide20
Problem 1 Slide21
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.Slide22
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 1Slide23
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 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
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/cSlide25
Solving for SPNEFind the pure strategy Nash for this reduced game. (use stars) What is the
subgame
perfect Nash equilibrium for the entire game?Slide26
Problem 8, p 286Slide27
How many strategies does Player 1 have?3
6
8
9
12Slide28
How many regular, proper subgames are there?
A) 1
B) 2
C) 3D) 4
E) 5Slide29
Subgames after b1 and c1b1 reduces to payoff 1,1,1
c1 reduces to payoff 2,1,0
What about a1?Slide30
Game between 2 and 3 if 1 goes aSlide31
There are two SPNEsa1/d1/d1, b2/a2/a2, a3 And c1/d1/d1, a2/a2/a2, b3
One N.E. that is not SPNE is
a1/d1/e1,b2/a2/a2,a3
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