michigania 02005 a logic of diversity scott e page complex systems, political science, economics and...
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Michigania 02005
A Logic of Diversity
Scott E Page
Complex Systems, Political Science, Economics and
Institute for Social Research
University of MichiganSanta Fe Institute
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Michigania 02005
The dim boy claps because the others clap.
- Richard Hugo
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Michigania 02005
A Logic of Diversity
I am going to replace abstract concepts, metaphors, and mantras with formal frameworks to produce a logic of individual diversity and its aggregative implications.
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Michigania 02005
Lu Hong: Mathematics of Diversity
Jenna Bednar: Cultural Diversity and Institutional Path Dependence
Co-Authors
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Michigania 02005
The Diversity Mantra
Identity
Diversity
Diverse
Perspectives
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Michigania 02005
Extending The Mantra
Identity
Diversity
Diverse
Perspectives
Better
Outcomes
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Michigania 02005
Enlarging The Mantra
Identity
Diversity
Diverse
Perspectives
Better
Outcomes
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Michigania 02005
QuickTime™ and aTIFF (Uncompressed) decompressor
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IdentityQuickTime™ and a
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QuickTime™ and aTIFF (LZW) decompressor
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Michigania 02005
QuickTime™ and aTIFF (Uncompressed) decompressor
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TrainingQuickTime™ and a
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QuickTime™ and aTIFF (Uncompressed) decompressor
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QuickTime™ and aTIFF (Uncompressed) decompressor
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QuickTime™ and aTIFF (Uncompressed) decompressor
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Michigania 02005
QuickTime™ and aTIFF (Uncompressed) decompressor
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ExperientialQuickTime™ and a
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QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
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QuickTime™ and aTIFF (Uncompressed) decompressor
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QuickTime™ and aTIFF (Uncompressed) decompressor
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Michigania 02005
Enlarging The Mantra
Identity,
Training,
Experiential
Diversity
Diverse
Perspectives
Better
Outcomes
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Michigania 02005
Today’s Talk: Unpacking The First Box
Diverse
Perspectives
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Michigania 02005
Wednesday’s Talk: Demonstrating Causality
Diverse
Perspectives
Better
Outcomes
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Michigania 02005
Today’s Talk
Describing the differences inside of our heads - cognitive differences.
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Michigania 02005
Brief Intermission
Link to training (calculus, physics, etc..) obvious.
Link to experience (we reason based on past cases) also clear
But what of identity and culture?
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Michigania 02005
A Most Important Question
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Michigania 02005
A Most Important Question
Where do you keep your ketchup?
Fridge?Cupboard?
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Michigania 02005
The Follow-up Questions
Shoes on or off in your house?Cross street when the red hand is
flashing but no cars are present?Read newspaper at breakfast table?When you greet friends do you hug?
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Michigania 02005
The Diversity Mantra
Identity
Diversity
Diverse
Perspectives
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Michigania 02005
Diverse Perspectives?
PerspectivesHeuristics
InterpretationsMental Models
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Michigania 02005
Perspectives
A perspective is a representation of the set of possible solutions.
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Michigania 02005
The Value of PerspectivesMost great breakthroughs in science
result from new perspectives.
Newton: Planetary MotionMendeleyev: Periodic Table
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Michigania 02005
Diverse Perspectives
Cartesian Polar
(x,y)
(r
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Michigania 02005
Ben and Jerry
chunk size
number of chunks
y
x
z
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Michigania 02005
Consultant 1
caloric rank
zx y
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Michigania 02005
Consultant 2
masticity
z x y
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Michigania 02005
Rugged Landscapes
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Michigania 02005
Perspectives and Difficulty
A perspective creates a landscape where the elevation of each solution equals its value. The better the perspective, the less rugged the landscape.
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Michigania 02005
Mt Fuji Landscape
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Michigania 02005
Caloric Landscape
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Michigania 02005
Chew Time Landscape
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Michigania 02005
Value of Consultants
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Michigania 02005
Perspectives in Strategic Contexts
A perspective can also simplify a strategic context. What was hard can become easy.
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Michigania 02005
Sum to Fifteen – Herb Simon
Setup: Cards numbered 1-9 face up on table
Play: Players alternate selecting cards
Object: To hold exactly three cards that add up to fifteen
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Michigania 02005
4/29/98 Page-De Marchi Match
D: 7P: 6D: 5 (12)P: 3 (9)D: 1 (6,8,12)P: 9 (12,9,15)
De Marchi offers Draw!!
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Michigania 02005
7th Grade Algebra
A Magic Square
8 3 4 1 5 9 6 7 2
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Michigania 02005
Page-De Marchi Again
_ _ _ _ _ _
_ X _
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Michigania 02005
Page-De Marchi Again
_ _ _ _ _ _
0 X _
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Michigania 02005
Page-De Marchi Again
_ _ _ _ X _
0 X _
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Michigania 02005
Page-De Marchi Again
_ 0 _ _ X _
0 X _
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Michigania 02005
Page-De Marchi Again
_ 0 _ X X _
0 X _
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Michigania 02005
Page-De Marchi Again
_ 0 _ X X 0
0 X _
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Michigania 02005
An Equivalence
It can be shown that tic tac toe on the magic square is equivalent to sum to fifteen.
In one perspective the game is hard. In the other perspective, the game is easy.
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Michigania 02005
What Is Hard Can Be Easy
Theorem: For any problem there exists a representation such that the problem of finding an optimal solution is easy.
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Michigania 02005
Water Flow Problem
Three valves (x,y,z): open = 1, closed = 0
Flow: x + y + z - 2xy - 2yz - 2xz + 4xyz
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Michigania 02005
Walsh Functions
Walsh Function #3 W(x,y,z) = 0 if #1’s is evenW(x,y,z) = 1 if #1’s is odd
W(x,y,z) = x + y + z - 2xy - 2yz - 2xz + 4xyz
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Michigania 02005
CautionDiverse perspectives create more adjacencies,
and therefore more solutions. Those additional solutions include better solutions only if the perspectives are appropriate to the problem.
More need not imply more better.
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Michigania 02005
Heuristics
Heuristics are techniques that we use for finding solutions. They can take many forms
- simulated annealing algorithms
- rule of 72
- do the opposite
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Michigania 02005
Heuristic Example
Fill in the blank
• 1 2 3 5 _ 13
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Michigania 02005
Answer
1 2 3 5 8 13
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Michigania 02005
Heuristic
1 2 3 5 8 13– xi+2- xi+1 =xi
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Michigania 02005
Next Question
1 4 _ 16 25 36
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Michigania 02005
Answer
1 4 9 16 25 36
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Michigania 02005
Heuristic
1 4 9 16 25 36– xi
2
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Michigania 02005
Last One!
1 2 6 _ 1806
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Michigania 02005
Answer
1 2 6 42 1806
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Michigania 02005
Heuristic
1 2 6 42 1806– xi+1 – xi = xi
2
– 6 – 2 = 22
– 42 – 6 = 62
– 1806 – 42 = 422
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Michigania 02005
Combining Heuristics
1 2 6 42 1806
– xi+1 – xi = xi 2
This is a combination of the first two heuristics
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Michigania 02005
One plus one equals THREE
By knowing two heuristics, you know three heuristics: The two individual heuristic plus the combined heuristic.
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Michigania 02005
Interpretations
Reality consists of many variables or attributes. People cannot include them all. Therefore, we either- consider only some attributes - lump things together
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Michigania 02005
“Lump to Live”
If we did not lump various experiences, situations, and events into categories, we could not draw inferences, make generalities, or construct mental models.
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Michigania 02005
Real Life Examples
“Kerry is a liberal”Soccer moms and NASCAR DadsPrice Earnings RatiosAutismModern ArtSKA
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Michigania 02005
An Example
• Students and advisors can have one of four personality types:– Obsessive– Curious– Ambitious– Rule Following
• Outcome function F maps each pair into an outcome which is either good or bad.
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Michigania 02005
The Outcome Function
G G
G
G
G
G
G
B
B
B
BB
B B B
O C A R
O
C
A
R
student
advisorG
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Michigania 02005
Advisor Type Interpretation
G G
G
G
G
G
G
B
B
B
BB
B B B
O
C
A
R
advisor G
prediction
G
G
B
B
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Michigania 02005
Student Type Interpretation
G G
G
G
G
G
G
B
B
B
BB
B B B
G
O C A Rstudent
predictionG G B B
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Michigania 02005
Making Horse RacesThis is why we differ on our
predictions of what will happen with stock prices, who will win sporting events, and who is a likely terrorist -- we look at the world differently.
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Michigania 02005
Miles DavisExperts parse the world more finely
than the rest of.
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Michigania 02005
Chicago El 1992
red line max 70K riders blue line max 80K riders
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Michigania 02005
Chicago El 1992
red line max 70K riders blue line max 80K riders
40K
70K
80K
30K
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Michigania 02005
Chicago El 1992
red line max 70K riders blue line max 80K riders
40K
70K
80K
30K
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Michigania 02005
Chicago El 1992
yellow line max 40K riders purple line max 80K riders
70K
80K
30K
40K
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Michigania 02005
Toolboxes vs Measuring Sticks
We can think of a person’s ability as her collection of tools -- her perspectives, her heuristics, and her mental models -- and not as an IQ score.
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Michigania 02005
More Toolbox Combinatorics
With one hundred tools, the number of collections of ten tools equals
17,310,309,456,440
Compare this to the number of I.Q.s!
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Michigania 02005
Toolboxes and IQs
Suppose 50 possible toolsSarah knows 20 Frank knows 12
What are odds that Sarah knows all that Frank knows?
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Michigania 02005
Toolboxes and IQs
Suppose 50 possible toolsSarah knows 20 Frank knows 12
What are odds that Sarah knows all that Frank knows? About 4 in a billion
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Michigania 02005
An Implicit Cheat
I assumed that any tool can be acquired. (“I think I’ll learn string theory.”) That may not be true. It could be that tools have an ordering - to learn one tool you must first learn another.
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Michigania 02005
Ladder Model
Suppose the tools are arranged in a ladder, so that to learn tool nine you must first learn tools 1-8.
Sarah now knows tools 1-20 Frank now knows tools 1-15
Sarah is “smarter” than Frank.
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Michigania 02005
Many Ladders Model
Probability# of Ladders Sarah > Frank
1 100%2 29%3 9%4 1%5 0.005%
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Michigania 02005
A Puzzle
Why do people in the humanities and the arts believe in the value of diversity and why do people in the sciences not?
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Michigania 02005
We Believe What We Know
Discipline # of Ladders Math Very Few Physics Very Few Economics Few
Political Science Several Literature Many
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Michigania 02005
Summary
- We don’t apply our IQ directly- We apply tools
- Perspectives, heuristics, interpretations, mental models
- Tools are superadditive (42)- Cannot universally compare
intelligences- Can compare domain specific
intelligence
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Michigania 02005
What’s Next
- Individual diversity influences collective performance.
- Explain``Wisdom of Crowds”- See that diversity and ability merit
equal standing