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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

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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

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Enlarging The Mantra

Identity,

Training,

Experiential

Diversity

Diverse

Perspectives

Better

Outcomes

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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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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 Value of Perspectives

- Most great breakthroughs in science result from new perspectives.
- Newton: Planetary Motion
- Mendeleyev: Periodic Table

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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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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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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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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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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

Probability

# of LaddersSarah > 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

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