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Michigania 02005
A Logic of Diversity
Scott E Page
Complex Systems, Political Science, Economics
and
Institutefor Social Research
University of Michigan
SantaFeInstitute
Michigania 02005
The dim boy claps because the others clap.
- Richard Hugo
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.
Michigania 02005
Lu Hong: Mathematics of Diversity
Jenna Bednar: Cultural Diversity and
Institutional Path Dependence
Co-Authors
Michigania 02005
The Diversity Mantra
Identity
Diversity
Diverse
Perspectives
Michigania 02005
Extending The Mantra
Identity
Diversity
Diverse
Perspectives
Better
Outcomes
Michigania 02005
Enlarging The Mantra
Identity
Diversity
Diverse
Perspectives
Better
Outcomes
Michigania 02005
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Identity
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Michigania 02005
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Training
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Michigania 02005
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Experiential
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Michigania 02005
Enlarging The Mantra
Identity,
Training,
Experiential
Diversity
Diverse
Perspectives
Better
Outcomes
Michigania 02005
Today’s Talk:
Unpacking The First Box
Diverse
Perspectives
Michigania 02005
Wednesday’s Talk:
Demonstrating Causality
Diverse
Perspectives
Better
Outcomes
Michigania 02005
Today’s Talk
Describing the differences inside of our
heads - cognitive differences.
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?
Michigania 02005
A Most Important Question
Michigania 02005
A Most Important Question
Where do you keep your ketchup?
Fridge?
Cupboard?
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?
Michigania 02005
The Diversity Mantra
Identity
Diversity
Diverse
Perspectives
Michigania 02005
Diverse Perspectives?
Perspectives
Heuristics
Interpretations
Mental Models
Michigania 02005
Perspectives
A perspective is a representation of the
set of possible solutions.
Michigania 02005
The Value of Perspectives
Most great breakthroughs in science
result from new perspectives.
Newton: Planetary Motion
Mendeleyev: Periodic Table
Michigania 02005
Diverse Perspectives
Cartesian Polar
(x,y)
(r,θ)
Michigania 02005
Ben and Jerry
chunk
size
number of chunks
y
x
z
Michigania 02005
Consultant 1
caloric rank
zx y
Michigania 02005
Consultant 2
masticity
z x y
Michigania 02005
Rugged Landscapes
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.
Michigania 02005
Mt Fuji Landscape
Michigania 02005
Caloric Landscape
Michigania 02005
Chew Time Landscape
Michigania 02005
Value of Consultants
Michigania 02005
Perspectives in Strategic
Contexts
A perspective can also simplify a strategic
context. What was hard can become
easy.
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
Michigania 02005
4/29/98 Page-De Marchi Match
D: 7
P: 6
D: 5 (12)
P: 3 (9)
D: 1 (6,8,12)
P: 9 (12,9,15)
De Marchi offers Draw!!
Michigania 02005
7th
Grade Algebra
A Magic Square
8 3 4
1 5 9
6 7 2
Michigania 02005
Page-De Marchi Again
_ _ _
_ _ _
_ X_
Michigania 02005
Page-De Marchi Again
_ _ _
_ _ _
0 X_
Michigania 02005
Page-De Marchi Again
_ _ _
_ X_
0 X_
Michigania 02005
Page-De Marchi Again
_ 0 _
_ X_
0 X_
Michigania 02005
Page-De Marchi Again
_ 0 _
XX_
0 X_
Michigania 02005
Page-De Marchi Again
_ 0 _
XX0
0 X_
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.
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.
Michigania 02005
Water Flow Problem
Three valves (x,y,z): open = 1, closed = 0
Flow: x + y + z - 2xy - 2yz - 2xz + 4xyz
Michigania 02005
Walsh Functions
Walsh Function #3
W(x,y,z) = 0 if #1’s is even
W(x,y,z) = 1 if #1’s is odd
W(x,y,z) = x + y + z - 2xy - 2yz - 2xz + 4xyz
Michigania 02005
Caution
Diverse 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.
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
Michigania 02005
Heuristic Example
Fill in the blank
• 1 2 3 5 _ 13
Michigania 02005
Answer
1 2 3 5 8 13
Michigania 02005
Heuristic
1 2 3 5 8 13
– xi+2- xi+1 =xi
Michigania 02005
Next Question
1 4 _ 16 25 36
Michigania 02005
Answer
1 4 9 16 25 36
Michigania 02005
Heuristic
1 4 9 16 25 36
– xi
2
Michigania 02005
Last One!
1 2 6 _ 1806
Michigania 02005
Answer
1 2 6 42 1806
Michigania 02005
Heuristic
1 2 6 42 1806
– xi+1 – xi = xi
2
– 6 – 2 = 22
– 42 – 6 =62
– 1806 – 42 =422
Michigania 02005
Combining Heuristics
1 2 6 42 1806
– xi+1 – xi = xi
2
This is a combination of the first two heuristics
Michigania 02005
One plus one equals THREE
By knowing two heuristics, you know three
heuristics: The two individual heuristic plus
the combined heuristic.
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
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.
Michigania 02005
Real Life Examples
“Kerry is a liberal”
Soccer moms and NASCAR Dads
Price Earnings Ratios
Autism
Modern Art
SKA
Michigania 02005
An Example
• Studentsand advisorscan haveoneof four
personality types:
– Obsessive
– Curious
– Ambitious
– RuleFollowing
• Outcomefunction F mapseach pair into an outcome
which iseither good or bad.
Michigania 02005
TheOutcomeFunction
G G
G
G
G
G
G
B
B
B
BB
B B B
O C A R
O
C
A
R
student
advisor
G
Michigania 02005
Advisor TypeInterpretation
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
Michigania 02005
Student TypeInterpretation
G G
G
G
G
G
G
B
B
B
BB
B B B
G
O C A R
student
predictionG G B B
Michigania 02005
Making Horse Races
This 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.
Michigania 02005
Miles Davis
Experts parse the world more finely than
the rest of.
Michigania 02005
Chicago El 1992
red line max 70K riders blue line max 80K riders
Michigania 02005
Chicago El 1992
red line max 70K riders blue line max 80K riders
40K
70K
80K
30K
Michigania 02005
Chicago El 1992
red line max 70K riders blue line max 80K riders
40K
70K
80K
30K
Michigania 02005
Chicago El 1992
yellow line max 40K riders purple line max 80K riders
70K
80K
30K
40K
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.
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!
Michigania 02005
Toolboxes and IQs
Suppose 50 possible tools
Sarah knows 20
Frank knows 12
What are odds that Sarah knows all that
Frank knows?
Michigania 02005
Toolboxes and IQs
Suppose 50 possible tools
Sarah knows 20
Frank knows 12
What are odds that Sarah knows all that
Frank knows? About 4 in a billion
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.
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.
Michigania 02005
Many Ladders Model
Probability
# of Ladders Sarah > Frank
1 100%
2 29%
3 9%
4 1%
5 0.005%
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?
Michigania 02005
We Believe What We Know
Discipline # of Ladders
Math Very Few
Physics Very Few
Economics Few
Political Science Several
Literature Many
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
Michigania 02005
What’s Next
- Individual diversity influences collective
performance.
- Explain``Wisdom of Crowds”
- See that diversity and ability merit equal
standing

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