A Logic of Diversity

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A Logic of Diversity
Scott E Page Complex Systems, Political Science,
Economics and Institute for Social
Research University of Michigan Santa Fe
Institute
2
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.

4

Co-Authors

• Lu Hong Mathematics of Diversity
• Jenna Bednar Cultural Diversity and
  Institutional Path Dependence

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The Diversity Mantra

Identity Diversity
Diverse Perspectives
6
Extending The Mantra

Identity Diversity
Diverse Perspectives
Better Outcomes
7
Enlarging The Mantra

Identity Diversity
Diverse Perspectives
Better Outcomes
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Identity
9
Training
10
Experiential
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Enlarging The Mantra

Identity, Training, Experiential Diversity
Diverse Perspectives
Better Outcomes
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Todays Talk Unpacking The First Box

Diverse Perspectives
13
Wednesdays Talk Demonstrating Causality

Better Outcomes
Diverse Perspectives
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Todays 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?

18
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 Diversity
Diverse 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 Polar

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Ben and Jerry

• chunk
• size
• number of chunks

y

x
z
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Consultant 1

• caloric rank

z
x
y

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Consultant 2

• masticity

z
x
y

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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 (12)
• P 3 (9)
• D 1 (6,8,12)
• P 9 (12,9,15)
• De Marchi offers Draw!!

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7th Grade Algebra

• A Magic Square
• 8 3 4
• 1 5 9
• 6 7 2

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Page-De Marchi Again

• _ _ _
• _ _ _
• _ X _

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Page-De Marchi Again

• _ _ _
• _ _ _
• 0 X _

39
Page-De Marchi Again

• _ _ _
• _ X _
• 0 X _

40
Page-De Marchi Again

• _ 0 _
• _ X _
• 0 X _

41
Page-De Marchi Again

• _ 0 _
• X X _
• 0 X _

42
Page-De Marchi Again

• _ 0 _
• X X 0
• 0 X _

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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 1s is even
• W(x,y,z) 1 if 1s 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 _ 13

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Answer

• 1 2 3 5 8 13

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Heuristic

• 1 2 3 5 8 13
• xi2- xi1 xi

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Next Question

• 1 4 _ 16 25 36

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Answer

• 1 4 9 16 25 36

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Heuristic

• 1 4 9 16 25 36
• xi2

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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
• xi1 xi xi 2
• 6 2 22
• 42 6 62
• 1806 42 422

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Combining Heuristics

• 1 2 6 42 1806
• xi1 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
O C A R
G
G
G
B
O
G
G
G
B
C
advisor
G
B
B
B
A
G
B
B
B
R
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Advisor Type Interpretation
prediction
G
G
G
B
O
G
C
G
G
G
B
G
advisor
B
G
B
B
B
A
R
B
G
B
B
B
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Student Type Interpretation
student
O C A R
G
G
G
B
G
G
B
G
G
B
B
B
G
B
B
B
prediction
G
G
B
B
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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
80K
70K
30K
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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
80K
70K
30K
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Toolboxes vs Measuring Sticks

• We can think of a persons 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 Ill 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

• Probability
• of Ladders Sarah gt Frank
• 1 100
• 2 29
• 3 9
• 4 1
• 5 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 of Ladders
• Math Very Few
• Physics Very Few
• Economics Few
• Political Science Several
• Literature Many

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Summary

• We dont 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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Whats Next

• Individual diversity influences collective
  performance.
• ExplainWisdom of Crowds
• See that diversity and ability merit equal
  standing