Michigania 02005 A Logic of Diversity Scott E Page Complex Systems, Political Science, Economics and Institute for Social Research University of Michigan

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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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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Lu Hong: Mathematics of Diversity

Jenna Bednar: Cultural Diversity and Institutional Path Dependence

Co-Authors

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The Diversity Mantra

Identity

Diversity

Diverse

Perspectives

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Extending The Mantra

Identity

Diversity

Diverse

Perspectives

Better

Outcomes

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Enlarging The Mantra

Identity

Diversity

Diverse

Perspectives

Better

Outcomes

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TrainingQuickTime™ and a










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Enlarging The Mantra

Identity,

Training,

Experiential

Diversity

Diverse

Perspectives

Better

Outcomes

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Today’s Talk: Unpacking The First Box

Diverse

Perspectives

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Wednesday’s Talk: Demonstrating Causality

Diverse

Perspectives

Better

Outcomes

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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

Diversity

Diverse

Perspectives

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Diverse Perspectives?

PerspectivesHeuristics

InterpretationsMental Models

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Perspectives

A perspective is a representation of the set of possible solutions.

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The Value of PerspectivesMost great breakthroughs in science

result from new perspectives.

Newton: Planetary MotionMendeleyev: Periodic Table

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Diverse Perspectives

Cartesian Polar

(x,y)

(r

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Ben and Jerry

chunk size

number of chunks

y

x

z

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Consultant 1

caloric rank

zx 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: 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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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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_ _ _ _ _ _

0 X _

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_ _ _ _ X _

0 X _

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_ 0 _ _ X _

0 X _

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_ 0 _ X X _

0 X _

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_ 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 #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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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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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– xi+2- xi+1 =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– xi

2

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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– 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 DadsPrice Earnings RatiosAutismModern ArtSKA

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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

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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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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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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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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Miles DavisExperts 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


40K

70K

80K

30K

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Chicago El 1992


40K

70K

80K

30K

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Chicago El 1992

yellow line max 40K riders purple line max 80K riders

70K

80K

30K

40K

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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 toolsSarah 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 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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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

Probability# of Ladders Sarah > 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 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

Documents

Michigania 02005 A Logic of Diversity Scott E Page Complex Systems, Political Science, Economics and Institute for Social Research University of Michigan