A Logic of Diversity
Scott E Page
Complex Systems, Political Science, Economics
and
Institute for Social Research
University of Michigan
Santa Fe Institute
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The dim boy claps because the others clap.
- Richard Hugo
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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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Co-Authors
Lu Hong: Mathematics of Diversity
Jenna Bednar: Cultural Diversity and
Institutional Path Dependence
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The Diversity Mantra
Identity
Diverse
Diversity
Perspectives
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Extending The Mantra
Identity
Diverse
Better
Diversity
Perspectives
Outcomes
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Enlarging The Mantra
Identity
Diverse
Better
Diversity
Perspectives
Outcomes
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Identity
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Training
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Experiential
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Enlarging The Mantra
Identity,
Training,
Diverse
Better
Experiential
Perspectives
Outcomes
Diversity
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Today’s Talk:
Unpacking The First Box
Diverse
Perspectives
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Wednesday’s Talk:
Demonstrating Causality
Better
Diverse
Perspectives
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Outcomes
Today’s Talk
Describing the differences inside of our
heads - cognitive differences.
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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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A Most Important Question
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A Most Important Question
Where do you keep your ketchup?
Fridge?
Cupboard?
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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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The Diversity Mantra
Identity
Diverse
Diversity
Perspectives
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Diverse Perspectives?
Perspectives
Heuristics
Interpretations
Mental Models
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Perspectives
A perspective is a representation of the
set of possible solutions.
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The Value of Perspectives
Most great breakthroughs in science
result from new perspectives.
Newton: Planetary Motion
Mendeleyev: Periodic Table
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Diverse Perspectives
(x,y)
(r,)
Cartesian
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Polar
Ben and Jerry
y
chunk
size
x
z
number of chunks
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Consultant 1
x
z
y
caloric rank
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Consultant 2
z
x
masticity
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y
Rugged Landscapes
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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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Mt Fuji Landscape
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Caloric Landscape
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Chew Time Landscape
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Value of Consultants
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Perspectives in Strategic
Contexts
A perspective can also simplify a strategic
context. What was hard can become
easy.
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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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4/29/98 Page-De Marchi Match
D: 7
P: 6
D: 5
P: 3
D: 1
P: 9
(12)
(9)
(6,8,12)
(12,9,15)
De Marchi offers Draw!!
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7th Grade Algebra
A Magic Square
834
159
672
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Page-De Marchi Again
_ __
___
_X_
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Page-De Marchi Again
_ __
___
0X_
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Page-De Marchi Again
_ __
_X_
0X_
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Page-De Marchi Again
_ 0_
_X_
0X_
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Page-De Marchi Again
_ 0_
XX_
0X_
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Page-De Marchi Again
_ 0_
XX0
0X_
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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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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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Water Flow Problem
Three valves (x,y,z): open = 1, closed = 0
Flow: x + y + z - 2xy - 2yz - 2xz + 4xyz
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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
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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.
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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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Heuristic Example
Fill in the blank
• 1
2
3
5
_
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13
Answer
1
2
3
5
8
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13
Heuristic
1 2 3
5
8
13
– xi+2- xi+1 =xi
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Next Question
1
4
_
16
25
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36
Answer
1
4
9
16
25
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36
Heuristic
1
4
9
16
25
– x i2
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36
Last One!
1 2
6
_
1806
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Answer
1
2
6
42
1806
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Heuristic
1
2 6
42
1806
– xi+1 – xi = xi 2
– 6 – 2 = 22
– 42 – 6 = 62
– 1806 – 42 = 422
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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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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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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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“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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Real Life Examples
“Kerry is a liberal”
Soccer moms and NASCAR Dads
Price Earnings Ratios
Autism
Modern Art
SKA
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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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The Outcome Function
student
advisor
O
C
A
O
G
G
G
B
C
G
G
B
G
A
B
G
B
B
R
G
B
B
B
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R
Advisor Type Interpretation
prediction
advisor
O
G
G
G
B
G
C
G
G
B
G
G
A
B
G
B
B
B
R
G
B
B
B
B
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Student Type Interpretation
O
student
C
A
G
G
G
B
G
G
B
G
B
G
B
B
G
B
B
B
G
G
B
B
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R
prediction
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.
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Miles Davis
Experts parse the world more finely than
the rest of.
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Chicago El 1992
red line max 70K riders
blue line max 80K riders
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Chicago El 1992
red line max 70K riders
blue line max 80K riders
40K
70K
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80K
30K
Chicago El 1992
red line max 70K riders
blue line max 80K riders
40K
80K
70K
30K
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Chicago El 1992
yellow line max 40K riders
purple line max 80K riders
40K
70K
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80K
30K
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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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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Toolboxes and IQs
Suppose 50 possible tools
Sarah knows 20
Frank knows 12
What are odds that Sarah knows all that
Frank knows?
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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
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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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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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Many Ladders Model
# of Ladders
1
2
3
4
5
Probability
Sarah > Frank
100%
29%
9%
1%
0.005%
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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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We Believe What We Know
Discipline
Math
Physics
Economics
Political Science
Literature
# of Ladders
Very Few
Very Few
Few
Several
Many
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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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What’s Next
- Individual diversity influences collective
performance.
- Explain``Wisdom of Crowds”
- See that diversity and ability merit equal
standing
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