The Collective Intelligence of Diverse Agents: Micro Foundations of Uncertainty

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The Collective Intelligence of Diverse
AgentsMicro Foundations of Uncertainty

• Lu Hong
• Scott E Page

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Outline

• Aside on Theoretical Foundations
• The Wisdom of Crowds
• Standard Models
• Interpretation Framework
• Mathematical Results
• Diversity, Democracy, and Markets

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Methodological Tradeoff

• Logical Informal
• ____________________________________________
  
• Mathematical Appreciative
• Brittle Flexible
• ____________________________________________
  
• Mathematical Appreciative

4
Agent Based Models

• Logical Informal
• _____ABM___________________________________
  
• Mathematical Appreciative
• Brittle Flexible
• ____________________________________ABM____
  
• Mathematical Appreciative

5
Model Benchmarking

• Real World
• Math ABM

6
Model Validation

• Real World
• Math ABM

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Methodological Translation

• Real World
• Math ABM

8
Models of Collective Wisdom
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Von Hayek
...it is largely because civilization enables us
constantly to profit from knowledge which we
individually do not possess and because each
individual's use of his particular knowledge may
serve to assist others unknown to him in
achieving their ends that men as members of
civilized society can pursue their individual
ends so much more successfully than they could
alone.
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Aristotle

• For each individual among the many has a share
  of excellence and practical wisdom, and when they
  meet together, just as they become in a manner
  one man, who has many feet, and hands, and
  senses, so too with regard to their character and
  thought.

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Aristotle

• Hence, the many are better judges than a single
  man of music and poetry, for some understand one
  part and some another, and among them they
  understand the whole.
• Politics book 3 chapter 11

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(No Transcript)
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The Wisdom of CrowdsGaltons Steer

• 1906 Fat Stock and Poultry Exhibition, 787 people
  guessed the weight of a steer. Their average
  guess 1,197 lbs.

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The Wisdom of CrowdsGaltons Steer

• 1906 Fat Stock and Poultry Exhibition, 787 people
  guessed the weight of a steer. Their average
  guess 1,197 lbs.
• Actual Weight 1,198 lbs.

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Who Wants to Be a Millionaire
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Experts or Crowds?

• Experts Correct 2/3 of the time
• Audience Correct over 90 of the time

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Three Mathematical Models
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Model 1 Known Information

• Best Selling Cereal of All Time
• Corn Flakes
• Rice Krispies
• Cheerios
• Frosted Flakes

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• Answer
• c) Cheerios

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How Errors Cancel

• Consider a crowd of 100 people
• 10 Know the correct answer
• 10 Narrowed down to two answers
• 36 Narrowed down to three answers
• 44 No clue

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Votes for Correct Answer

• 10 10 Know the correct answer
• 5 10 Narrowed down to two answers
• 12 36 Narrowed down to three answers
• 11 44 No clue
• 38 TOTAL

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Why The Crowds Correct

• The correct answer gets 38 votes.
• Assume that the other 62 votes are spread across
  the other three. Each of those three receives
  around 20 votes.

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N.B.

• The crowd can be correct with very high
  probability even if no one in the crowd knows the
  correct answer.

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The Math

• 40 Cheerios or Corn Flakes
• 30 Cheerios or Frosted Flakes
• 30 Cheerios or Rice Krispies
• Cheerios gets 50 votes!

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Model 2 Correlated Signal

• Suppose that were trying to discover whether or
  not a truck full of sour cream has gone bad due
  to a faulty refrigerator.

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Model 2 Correlated Signal

• Suppose that were trying to discover whether or
  not a truck full of sour cream has gone bad due
  to a faulty refrigerator.
• True State G (good) or B (bad)

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Signals

• Suppose that we can test pints of sour cream and
  get signals (g and b) and that with probability
  3/4, these signals are correct.

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Signals

• Suppose that we can test pints of sour cream and
  get signals (g and b) and that with probability
  3/4, these signals are correct.
• If the sour cream is bad, 3/4 of the time well
  get the signal b.

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• Three People
• True State B
• Correct Outcomes
• P1 P2 P3 Probability
• b b b (3/4)(3/4)(3/4) 27/64
• b b g (3/4)(3/4)(1/4) 9/64
• b g b (3/4)(1/4)(3/4) 9/64
• g b b (1/4)(3/4)(3/4) 9/64
• Total 54/64

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• Three People
• True State B
• Incorrect Outcomes
• P1 P2 P3 Probability
• g g g (1/4)(1/4)(1/4) 1/64
• b g g (3/4)(1/4)(1/4) 3/64
• g b g (1/4)(3/4)(1/4) 3/64
• g g b (1/4)(1/4)(3/4) 3/64
• Total 10/64

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General Model

• With probability p gt 0.5, people get the correct
  signal. Therefore, if N people get signals, pN
  get the correct signal.
• As N gets large, the expected probability of a
  correct vote goes to one.

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Model 3 Averaging of Noise

• Suppose that we want to predict the luminosity of
  a star. Each of 100 people stationed around the
  globe takes out a light meter and takes a
  reading.

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Model 3 Averaging of Noise

• Suppose that we want to predict the luminosity of
  a star. Each of 100 people stationed around the
  globe takes out a light meter and takes a
  reading.
• Call the reading for person k, r(k)

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Noise/Interference

• The signal that a person gets equals the true
  luminosity, L, plus or minus an error term, due
  to ambient light, humidity or who knows what.
• r(k) L e(k)
• e(k) is the error term

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Noises Off

• The average of the signals equals L plus the
  average of the error terms
• r(1) r(2) r(N)/N L e(1) e(2)
  ..e(N)/N
• If the error terms are, on average, zero, then
  they all cancel, and the prediction is accurate.

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Important Questions

• Why should we assume that these error terms are,
  on average, equal to zero?
• Why should we assume the signals are independent?
• Is this how an ABM would capture collective
  wisdom?

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Markets and Democracy

• Model 1 Some people know the answer
• Model 2 People get signals that are
  probabilistically correct
• Model 3 People see the true state plus an error

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• NONE do.

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Signal
noise
Outcome
Signal
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Generated Signals

• True state of the world x
• Signal s
• Joint probability distribution f(s,x)
• Conditional probability distribution f(sx)

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Generated Signals

• True state generates something that is
  correlated with the states value
• - luminosity of stars
• S Le
• - quality of a product good, bad
• s True quality with prob p

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A Generated Signal

• A chef of unknown quality produces batches of
  risotto. Each batch is a signal of the chefs
  quality. Batches temporally separate enough to be
  considered independent revelations of quality.

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Predictive Model Lu Hong
model
Attributes
Prediction
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Interpretations
Reality consists of many variables or attributes.
People cannot include them all. Therefore, we
consider only some attributes or lump things
together into categories. (Reed 1972, Rosch
1978)
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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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Predictive Models

• Edwards is a liberal therefore hell raise
  taxes.
• The stocks price earnings ratio is high
  therefore, the stock is a bad investment.

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How Do We Predict?

• We parse the world into categories and make
  predictions based on those interpretations.

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Interpretations

• Victorian Novel
• Modern Architecture
• Price Earnings Ratio
• Modern Art
• SKA

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Predictive Models

• I love SKA music!!

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Model Interpreted Signals

• Situations/objects in the world have many
  attributes (x1, x2, x3 . xn)
• Outcome function maps situations to
  outcomes/states FX S
• Agents have predictive models based on subsets of
  attributes.

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People
We differ in how we categorize. Thus, we
differ in our predictions.
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Pile Sort

• Place the following food items in piles with at
  least two items per pile
• Broccoli Canned Ham Carrots
• Fresh Salmon Bananas Apples
• Spam Ahi Tuna NY Strip Steak
• Rib Roast Sea Bass Canned Salmon

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BOBO Sort
Veggies Fish Meat Canned Stuff Broccoli
Fresh Salmon Canned Salmon Carrots Ahi
Tuna Spam Arugula Niman Pork Canned
Beets Fennel Sea Bass Canned Posole
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Airstream Sort
Veggies Fish Meat Weird Stuff Broccoli
Fresh Salmon Ahi Tuna Canned Beets Canned
Salmon Arugula Carrots Spam Fennel
Niman Pork Canned Posole Sea Bass
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Agents
Differ in location in space or on network Differ
in type Therefore, differ in pieces of
information that they use
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An Example

• What follows is an example in which a crowd of
  three people make a collective prediction.

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Reality

• Charisma
• H MH ML L
• H
• Experience
• MH
• ML
• L

G
G
G
B
G
G
G
G
B
G
B
B
B
G
B
B
B
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Experience Interpretation

• H
• Experience
• MH
• ML
• L

75 Correct
G
G
G
B
B
B
G
G
G
G
G
B
B
G
G
B
B
B
B
G
B
B
B
B
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Charisma Interpretation

• H MH ML L
• 75
• Correct

G
B
G
G
B
G
G
B
G
G
G
B
G
G
B
B
G
B
B
B
G
B
B
B
G
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Balanced Interpretation

• H MH ML L
• 75 Correct H
• Extreme on MH
• one measure.
• Moderate on ML
• the other
• L

G
G
G
B
B
G
B
G
G
B
G
B
B
B
G
G
B
B
B
G
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Voting Outcome

• H MH ML
  L
• H
• MH
• ML
• L

GGB
GGG
GBG
BGB
GGG
GGB
G
GBG
GBB
BGG
BBG
BBB
BBG
BGG
BBB
BBG
BGB
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Reality

G
G
G
B
G
G
G
G
B
G
B
B
B
G
B
B
B
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Row and Column Correct

GGB
GGG
GGG
GGB
G
BBG
BBB
BBB
BBG
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Row and Column Split

GBG
BGB
G
GBG
GBB
BGG
BBG
BGG
BGB
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Key Idea

• Think of these predictions as signals. To
  differentiate them from our standard, generated
  signals, call them interpreted signals.

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Independence of Interpreted Signals

• Consider the interpreted signals based on
  charisma and on experience.
• Each was correct with probability 0.75

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Both Row and Column Correct

GGB
GGG
GGG
GGB
G
BBG
BBB
BBB
BBG
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Negative Correlation

• Probability Correct Prediction 0.75
• Probability Both Correct 0.5
• If Independent, Probablility Both Correct 0.56

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Conditional Independence?

• Probability each is correct conditional on the
  outcome G equals 0.75
• Probability both correct conditional on the
  outcome G equals 0.5
• Correctness of the predictions is negatively
  correlated conditional on the outcome being good.

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Binary Interpreted Signals

• Set of objects XN
• Set of outcomes S G,B
• Interpretation Ij mj,1,mj,2mj,nj is a
  partition of X
• P(mj,i) probability mj,i arises

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Four Types of Independence

• Independent Interpretations
• Independent Interpreted Signals
• Independently Correct Interpreted Signals
• Conditionally Independent Interpreted Signals

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Independent Interpretations

• P(mji and mkl) P(mji)P(mkl)
• Probability j says i and k say elequals the
  product of the probability that j says i times
  the probability k says el.

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Why Independent Interpretations
Were interested in independent interpretations
because thats the best people or agents could do
in the binary setting. Its the most diverse
two predictions could be. Captures a world in
which agents or people look at distinct pieces of
information.
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Independent Interpretations
Claim If two interpretations are independent,
then X can be represented by a K dimensional
rectangle with the two interpretations looking at
non overlapping subsets of variables.
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Independent Not Different
Independent interpretations must rely on the same
fundamental representation and look at different
parts of it. Thus, to say that two people have
independent perspectives is to say that they look
at the world the same way but look at different
parts of the same representation.
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Independent Interpreted Signals

• Interpreted signal sj (mji) prediction by j
  given in set I
• Interpreted signals are independent iff sj (mji)
  and sk (mkl) are independent random variables.

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• Claim Independent interpretations imply
  independent interpreted signals
• pf if what we see is independent, what we
  predict has to be independent.

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• Claim Independent interpreted signals need not
  imply independent interpretations.
• pf Outcomes G1,G2,G3,B1,B2,B3
• Person 1 G1,G2 B1 g G3,B2,B3b
• Person 2 G1,G2, G3 ,B3 g B1,B2b
• Independent interpreted signals
• P(g,b) P(g,.)P(.,b)

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Independently Correct Interpreted Signals

• C(sj (mji)) 1 if prediction is correct, 0 else
• Predictions are independently correct iff
• C(sj (mji)) and C(Sk(mkl)) are independent random
  variables.

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• Claim Independent predictions need not be
  independently correct predictions.
• Pf recall our example. The predictions were
  independent but they were not independently
  correct.

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• A prediction is reasonable if it is correct at
  least half of the time.
• A prediction is informative if it is correct more
  than half of the time.

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• Claim Informative predictions need not be
  reasonable conditional on every state

G
G G B
G
G G B
G G B
G
B B G
B
Conditional on state B, the prediction is correct
2/5 of the time
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Claim Independent, informative interpreted
signals that predict good and bad outcomes with
equal likelihood must be negatively correlated in
their correctness.
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Proof
g b
G X B 1-X
G Y B 1-Y

g b
G W B 1-W
G Z B 1-Z
Prob row correct (XY2-(WZ))/4 Prob column
correct (XZ2-(WY))/4 Prob both correct
(X1-W)/4 (XY2-(WZ))(XZ2-(WY)) -
4(X1-W) (X-W)2 - (Y-Z)2 gt 0
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Negative Result
We cannot assume independent signals and be
consistent with independent interpretations.
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What Does This All Mean?
The following assumptions which are common in in
literature are inconsistent with independent
interpreted signals States G,B equally
likely Signals g,b independent conditional
on the state across agents.
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However, much of the time, mathematical models do
not assume unconditional independence, but
independence conditional on the true outcome.
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Negative Conditional Correlation
Claim If interpreted signals are informative and
independent, then they must be negatively
correlated conditional on at least one outcome.
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Negative Correlation
Claim If interpreted signals are informative and
independent, then they must be negatively
correlated conditional on at least one
outcome. Independence conditional on the state
is impossible
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Positive Result
Claim Independent, informative interpreted
signals that predict good and bad outcomes with
equal likelihood that are correct with
probability p exhibit negative correlation equal
to 1 - (1/4(p-p2))

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Amazing Result
Claim The complexity of the outcome function
does not alter correlation other than through the
accuracy of the interpreted signals

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Resurrecting Independence
We can obtain independence if we relax the
assumption that people use independent
interpretations and if we make some incredibly
heroic assumptions about the topology over states
and how people construct categories.
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Resurrecting Independence

• K, r, m are positive integers, Kgt1, 2rgtmgtr
• A state is a vector of K attributes, (q,
  x1,...,xK) q takes a value from 0,1 each xi
  takes a value from 1,...,m each state is
  equally likely
• The outcome function F(q,x1,...,xK)q if an even
  number of xis have values greater than r 1-q
  otherwise

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Resurrecting Independence

• Interpretation i considers every attribute except
  attribute xi
• Interpreted signal si based on interpretation i
  equals q if an even number of x attributes other
  than xi have values greater than r 1-q otherwise

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• Claim Any outcome function that produces
  conditionally independent interpreted signals is
  isomorphic to this example.

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One Left Out
The only way to align conditionally independent
signals with interpreted signals is to assume
each person leaves out a different
attribute. This doesnt make sense if seen from
an incentive standpoint.

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Diversity in Democracy Markets
Diverse interpretations -- interpretations that
use distinct attributes create negatively
correlated signals.

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Collective Accuracy
If we take the collective prediction to be equal
to the average of individuals predictions, then
the following holds. Collective Error Average
Error - Variance

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Efficient Individual Signals
Suppose that agents evolve predictive models
(interpreted signals) and that each new category
has a cost. Then, there exist efficient (but
not accurate) interpreted signals. See Fryer
and Jackson

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Efficient Collective Signals
Suppose that we take the distribution of the
accuracy of signals as given, then it is possible
to determine which signals to include. See N.
Johnson

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Evolved Interpreted Signals Democracy
Suppose that we allow agents to evolve
interpretations. Over time, the agents become
more accurate, but the collection becomes less
accurate due to the reduction in diversity
(variance). Kollman and Page

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Evolved Interpreted SignalsMarkets
Markets create incentives for people to look at
different attributes. In an auction setting, it
may be incentive compatible to look at distinct
attributes -- providing micro foundations for
both Aristotle and Hayek.

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Small Groups
With endogenous information acquisition members
of small groups should be able to look at
different attributes and do better than
independence would predict.

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Large Groups
Even with endogenous information acquisition and
the incentives to think differently, people may
not be able to generate enough encodings to avoid
positive correlation. Thus, those limiting
results as N gets large may not hold.

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Summary

• Conceptual Contribution
• Shown difference between ABM and Mathematical
  models of signals
• Linked to psychology and shown how diversity
  might explain signals
• Shown chink in armor of independence assumption
  (maybe its too convenient)

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Summary

• Contributions
• Collective Wisdom depends on either
• Smart people or
• Diversity
• Can expect large groups to find the best barbeque
  in NC (generated signals) but not to make the
  correct choice on a proxy vote

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Summary

• Extensions
• Explore complexity
• How does the mapping from attributes to outcomes
  effect signal correlation and accuracy for more
  than two people?
• Explore endogenous information