michigania 02005 a logic of diversity scott e page complex systems, political science, economics and...
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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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Page-De Marchi Again
_ _ _ _ _ _
0 X _
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Page-De Marchi Again
_ _ _ _ X _
0 X _
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Page-De Marchi Again
_ 0 _ _ X _
0 X _
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Page-De Marchi Again
_ 0 _ X X _
0 X _
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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 #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
red line max 70K riders blue line max 80K riders
40K
70K
80K
30K
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Chicago El 1992
red line max 70K riders blue line max 80K riders
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