The Collective Intelligence of Diverse Agents: Micro Foundations of Uncertainty

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The Collective Intelligence of Diverse Agents:

Micro 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

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Agent Based Models

Logical Informal |_____ABM___________________________________|Mathematical Appreciative

Brittle Flexible

|____________________________________ABM____|Mathematical Appreciative

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Model Benchmarking

Real World

Math ABM

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Model Validation

Real World

Math ABM

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Methodological Translation

Real World

Math ABM

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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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The Wisdom of Crowds:Galton’s 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 Crowds:Galton’s 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

a) Corn Flakesb) Rice Krispiesc) Cheeriosd) 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 answer10% Narrowed down to two answers36% Narrowed down to three

answers44% 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 Crowd’s 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 Flakes30: Cheerios or Frosted Flakes30: Cheerios or Rice Krispies

Cheerios gets 50 votes!

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Model 2: Correlated Signal

Suppose that we’re 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 we’re 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 we’ll get the signal b.

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Three PeopleTrue 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 PeopleTrue 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 > 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

Outcome Signal

noise

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Generated Signals

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

f(s|x)

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Generated Signals

True state generates something that is correlated with the state’s value

- luminosity of stars S = L+e

- 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 chef’s quality. Batches temporally separate enough to be considered independent revelations of quality.

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Predictive Model: Lu Hong

Attributes Prediction

model

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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 he’ll raise taxes.

The stock’s 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 NovelModern ArchitecturePrice Earnings RatioModern ArtSKA

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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 F:X 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 SortPlace the following food items in piles with

at least two items per pile:

Broccoli Canned Ham CarrotsFresh Salmon Bananas ApplesSpam Ahi Tuna NY Strip SteakRib Roast Sea Bass Canned

Salmon

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Veggies Fish & Meat Canned StuffBroccoli Fresh Salmon Canned SalmonCarrots Ahi Tuna SpamArugula Niman Pork Canned BeetsFennel Sea Bass Canned Posole

BOBO Sort

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Veggies Fish & Meat Weird StuffBroccoli Fresh Salmon Ahi Tuna Canned Beets Canned Salmon ArugulaCarrots Spam Fennel Niman Pork Canned Posole

Sea Bass

Airstream Sort

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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

CharismaH MH ML L

H Experience

MH ML

L

G G G

G G G G

G

G

B

B

B

B

B

B

B

B

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Experience Interpretation

HExperience MH

ML

L

G G G

G G G G

G

G

B

B

B

B

B

B

B

B

B

B

B

B

B G

G

75 % Correct

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Charisma Interpretation

H MH ML L

75% Correct

G B

G BBBG

G G

BG G

BG G

BG G

G

G

B

B

B

B B

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Balanced Interpretation

H MH ML L

75% Correct H

Extreme on MHone measure.Moderate on MLthe other

L

G G G

G B G G

G

G

B

B

B

B

B

B

B

B

B

G

G

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Voting Outcome

H MH ML L

H

MH ML

L

GGB GGG GBG

GGG GGB G GBG

BGG

BGG

BGB

GBB

BBG

BBB

BBB

BBG

BBG

BGB

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Reality

G G G

G G G G

G

G

B

B

B

B

B

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

G GBG

BGG

BGG

BGB

GBB

BBG

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 |X|=N• Set of outcomes S = {G,B}

• Interpretation: Ij = {mj,1,mj,2…mj,nj} is a partition of X

• P(mj,i) = probability mj,i arises

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Four Types of Independence

1. Independent Interpretations2. Independent Interpreted Signals3. Independently Correct Interpreted

Signals4. 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 “el”equals the product of the probability that j says “i” times the probability k says “el”.

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Why Independent Interpretations

We’re interested in independent interpretations because that’s the best people or agents could do in the binary setting. It’s 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,B3:b}Person 2: {G1,G2, G3 ,B3: g} {B1,B2:b}

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 BG G BG G B

B B G

GGGB

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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Proofg b

g

b

G = XB = 1-X

G = YB = 1-Y

G = ZB = 1-Z

G = WB = 1-W

Prob row correct: (X+Y+2-(W+Z))/4Prob column correct: (X+Z+2-(W+Y))/4Prob both correct: (X+1-W)/4

(X+Y+2-(W+Z))(X+Z+2-(W+Y)) - 4(X+1-W) = (X-W)2 - (Y-Z)2 > 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, K>1, 2r>m>r

A state is a vector of K attributes, (x1,...,xK); takes a value from {0,1}; each xi takes a value from {1,...,m}; each state is equally likely

The outcome function F(x1,...,xK)= if an even number of xi’s have values greater than r; 1- otherwise

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Resurrecting Independence

Interpretation i considers every attribute except attribute xi

Interpreted signal si based on interpretation i equals if an even number of x attributes other than xi have values greater than r; 1- 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 doesn’t 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 Signals:Markets

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 it’s 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