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Polarization: theory and evidence Jean-Pierre Benoît Juan Dubra.

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1 Polarization: theory and evidence Jean-Pierre Benoît Juan Dubra

2 Outline Two groups come into a lab and report beliefs Typeθ1θ1 < θ2< θ2 …<θn<θn Higher in FOSD SenseTypeθ1θ1 < θ 2 …< θ n Probability.2.1….4Probability.3.2….3 The two groups observe a common signal S and report beliefs again Larger belief increases in FOSDLower Belief decreases in FOSD Typeθ1θ1 θ2θ2 …θnθn θ1θ1 < θ 2 …< θ n Probability.1 ….5Probability.4.3….2 Can the results be obtained with Bayesian model? Obtained in a “reasonable” Bayesian model? Can Bayesian model explain “agreed upon” patterns? Can Bayesian model explain a particular experiment?

3 Lord, Ross, and Lepper (1979) Darley and Gross (1983) Plous (1991) -- nuclear Nyhan and Reifler (2010) -- WMD Gerber and Green (1999) – Bayesian implications Miller, McHoskey, Bane, and Dowd (1993) – CP, direct response Kuhn and Lao (1996) Munro and Ditto (1997) -- homosexuality Kunda (1987) – motivated reasoning Liberman and Chaiken (1992) – defensive processing Lord, Lepper, and Preston (1984) -- corrective measures

4 Acemoglu, Chernozhukov, and Yildiz (2009) Andreoni and Mylovanov (2012) Baliga, Hanany, and Klibanoff (2012) Dixit and Weibull (2007) Glaeser and Sunstein (2013) Kondor (2012) Rabin and Schrag (1999)

5 A Hypothesis-Confirming Bias in Labeling Effects Darley and Gross (1983) Hannah: Nine years old, fourth grade, Caucasian 70 Princeton undergraduates. Question: What is Hannah’s grade level? Scale: Kindergarten -- sixth grade, in 3-month intervals.

6 Demographic Information Negative Expectancy Condition: Father: Meat-packer Mother: Seamstress Both: High school education Positive Expectancy Condition Father: Attorney Mother: Free-lance writer Both: College graduates Hannah’s neighbourhood

7 Performance Information Subjects viewed a video of Hannah answering problems that ranged from “easy” to “difficult”.

8 Indicate Hannah’s Ability Liberal ArtsNo-PerformancePerformance Lower Class Upper Class ReadingNo-PerformancePerformance Lower Class Upper Class MathematicsNo-PerformancePerformance Lower Class Upper Class

9 What should happen? Initial responses: A – 4.25, B – A and B receive common information: A and B harmonize if, following the information, A and B's responses both rise or both fall. A and B moderate if, following the information, A's response falls and B's response rises. A and B polarize if, following the information, A's response rises and B's response falls.

10 Mixed Performance Hannah correctly answers some “difficult” questions but misses some “easy” questions. Hannah sometimes concentrates, sometimes is distracted. The two group’s reports move further apart (polarize) Prior literature has said “we have to explain how could two people’s beliefs move further apart”: – Impossible with Bayes let’s do ambiguity (Baliga et al.) – Impossible with Bayes let’s assume that people differ on their beliefs of how types map to signals (Acemoglu et al) – Andreoni&Mylovanov and Kondor do it with 2 dimensions We say that is the “wrong” question.

11 Mixed Performance 1.Hannah's mixed performance shows her to be an average student. 2.Hannah manages to answer some difficult questions and to concentrate when she wants to. Hannah may not be the best student, but she is well above average (say, 4.5 or above). 3.Hannah misses some easy questions and cannot maintain her concentration. Hannah may not be the worst student, but she is well below average (say, 3.5 or below).

12 Let’s take exactly that story, and let – B be “distractions = Boredom” – I be “distractions = I nability to focus” State Space {B, I } x {3.5, 4, 4.5}. Each state has p=1/6 Signal 1: A = { b,i } w P( b | B,j) = 2/3, P( b | I,j) = 1/3 Signal 2: S = {sometimes distracted, focused} = {d,f} Probability of signal d in each state B½½1 I 1½½ Posterior after b and common signal d B1/6 1/3 I 1/61/12 Marginal1/31/45/12 Posterior after i and common signal d B1/12 1/6 I 1/31/6 Marginal5/121/41/3

13 What needs to be explained? Given next to last slide “certainly not” that people could polarize. We had two very simple and plausible interpretations of the data that implied in one case that Hannah was better than the prior, and in the other that she was worse. What do the results actually mean? – Apart from the simple example, there are many other plausible interpretations which are not “it’s a bias”. – “Big point”: be careful when reading these papers (the alternative hypothesis is never stated). Maybe it’s a cheap point, but economists have embraced the idea that it is a bias.

14 What do the answers mean? “Subjects who were given only demographic information about the child demonstrated a resistance to making expectancy-consistent attributions on the ability indexes.” “Their estimations of the child’s ability level tended to cluster closely around the one concrete fact they had at their disposal: the child’s grade in school.”

15 What do the answers mean? Base-rate information represents probabilistic statements about a class of individuals, which may not be applicable to every member of the class. Thus, regardless of what an individual perceives the actual base rates to be, rating any one member of the class requires a higher standard of evidence.

16 Don’t Prejudge: Say different from 4 only if 75% sure. Lower class child: – Most likely 3.5; 35% chance she is exceptional, and thus 4. Upper class child: – Most likely 4.5; 35% chance she is exceptional, and thus 4. No performance video – Subjects answer “4” Performance video: Hannah is not exceptional – Subjects answer “3.5” or “4.5”

17 “A teacher, for example, would be extremely hesitant to conclude that a black child had low ability unless that child supplied direct behavioral evidence validating the application of the label.” Darley & Gross (…): – Don’t say bad unless she performs badly – Don’t say good unless she performs well This other (rational) example: – Don’t say bad/good unless sufficiently confident.

18 So far, I have said that we can explain certain results with a Bayesian model (low bar, and Andreoni&Mylovanov and Kondor have done it). Next, we’ll raise the bar about what it is that one should try to do. But before that I will discuss what are the “deliverables”: – Polarization in Expected value? – Polarization in FOSD?

19 Expected Value Lower ClassExceptionalAverage (-)Average (+)ExceptionalMean Probability1/81/21/41/8 Grade Video signal: Hannah is pretty average Lower class: 3.33 Higher class: 3.67 Upper ClassExceptionalAverage (-)Average (+)ExceptionalMean Probability1/81/41/21/8 Grade

20 FOSD Polarization (Baliga, Hanany, and Klibanoff) (Dixit and Weibull) Rich Poor

21 Baliga, Hanany, Klibanoff Definition Fix two individuals with beliefs η and η’ over Φ (in R) and with common support such that η’ stochastically dominates η. After they both observe a signal x whose likelihood given θ ∈ Φ is π θ (x), we say that polarization occurs if and only if the resulting posterior beliefs lie further apart in the sense of fosd. Theorem (fosd) Polarization cannot occur if the two individuals use Bayesian updating. Precludes moderation. Unrelated to experiment (e.g. in Hannah people report one summary statistic for whole beliefs, not the whole distribution).

22 Many “dimensions” to get polarization Andreoni and Mylovanov: The essential feature of our model is simply that the optimal action depends on relative values of different dimensions of the information space and, as such, contains at least one fewer degree of freedom. Kondor: The main result is based on a simple observation. Agents’ opinions about the opinions of others (higher-order expectations) respond differently to public information than agents’ opinions about the fundamentals of an economic object.

23 One dimension – only the type matters Θ = {2, 3, 4, 5} uniform beliefs S = {s 2, s 3, s 4, s 5 } 2345 S2S2 ¾¼00 S3S3 1/8½¼ S4S4 ¼½ S5S5 00¼¾ 2345 S2S2 ¾¼00 S3S3 ½¼ S4S4 ¼½ S5S5 00¼¾ LikelihoodsPosteriors 

24 Type (grade)2345 Subject I (s 3 )1/81/21/41/8 Subject II (s 4 )1/81/41/21/8 M Θ = {2, 3, 4, 5} S = {s 2, s 3, s 4, s 5 }} 2345 L1000 M0110 H0001 Likelihoods T= {L, M, H}

25 Lumping Type (grade)2345 Subject I (s 3 )1/81/21/41/8 Subject II (s 4 )1/81/41/21/8 Bad: t ∈ {2,3} Good: t ∈ {4,5} Θ = {2, 3, 4, 5}S = {s 2, s 3, s 4, s 5 }} If not knowing the real type space you collect beliefs about B or G you would find fosd polarization. And conclude that there’s a bias.

26 All the previous discussion was about “if we are trying to explain experiments, what is it that experiments tell us?” – They don’t tell us anything about FOSD – Despite Baliga et al, you could obtain FOSD in an experiment, if you don’t know the exact type space. – From Baliga also: FOSD seems too strong (precludes moderation). Still, we can derive our results in FOSD, so we will proceed with this “higher bar” for polarization.

27 Lord, Ross, and Lepper 151 subjects complete a questionnaire 48 are selected two weeks later. – 24 Proponents: Favour capital punishment, believe it has a deterrent effect, think most relevant research supports their views. – 24 Opponents: Oppose capital punishment, doubt it has a deterrent effect, believe most relevant research supports their views.

28 Two Studies Kroner and Phillips (1977) compared murder rates for the year before and the year after adoption of capital punishment in 14 states. In 11 of 14 states, murder rates were lower after adoption of capital the death penalty. This research supports the deterrent effect of the death penalty. Palmer and Crandall (1977) compared murder rates in 10 pairs of neighbouring states with different capital punishment laws. In 8 of the 10 pairs, murder rates were higher in the state with capital punishment. This research opposes the deterrent effect of the death penalty. Two-page description of methodology. Studies also flipped Studies are fictional, but characteristic of research found in the literature cited in judicial decisions.

29 Attitudes Polarize “It is an impressive demonstration of assimilation biases that contending factions both believe the same data to justify their position ‘objectively’.”

30 Updating Model T|G F|B T|GF|B T: Capital punishment deters F: Does not deter

31 Ancillary State Matters Ambiguous Signals: different meanings in different ancillary states. T|G F|B T|BF|G T: Capital punishment deters F: Does not deter

32 Recall from Hannah “ambiguous”: distracted meant different things in different “ancillary” states – B “distracted=boredom”; I “distracted=inability to focus” State Space {B, I } x {3.5, 4, 4.5}. Each state has p=1/6 Signal 1: A = { b,i } w P( b | B,j) = 2/3, P( b | I,j) = 1/3 Signal 2: S = {sometimes distracted, focused} = {d,f} Probability of signal d in each state B½½1 I 1½½ Posterior after b and common signal d B1/6 1/3 I 1/61/12 Marginal1/31/45/12 Posterior after i and common signal d B1/12 1/6 I 1/31/6 Marginal5/121/41/3

33 Plausible? T|50% F|75% T|25%F|50% T: Capital punishment deters F: Does not deter Selection No Selection Info: % of states in which crime rises.

34 State Space {S,N} x {T,F}. Each state has p=1/4 Signal 1: A = { s,n } w P( s | S,j) = 2/3, P( s | N,j) = 1/3 Signal 2: % states with increase in crime rate = {25%,50%,75%} Probability of signal 50% in each state TF S3/41/2 N 3/4 Posterior after s & common signal 50%. TF S2/54/15 N2/151/5 Marginal8/157/15 Posterior after s & common signal 50%. TF S1/52/15 N4/152/5 Marginal7/158/15

35 Can we do it? yes (others too). With a plausible model? Yes (previous slides). The focus on plausibility is new. Next: – Should you expect the result? Yes (new) Theorem Empirical “proof” – Predict patterns? Yes (new) When would polarization occur Experts polarize more (Theorem). Unexpected! People who are more certain polarize more. Unexpected!

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44 Theorem: Bayesian model says Polarization “should” occur in setting of experiments Selection No Selection Believe trueBelieve false False: CP does not deter True: CP deters % % % % Prior Evidence 50% New Evidence 50%

45 “Proof” that result is expected: Affirmative Action Miller, McHoskey, Bane, and Dowd (1993) No net polarization. “Why did relatively more subjects in this study report a depolarization of their attitudes? We have no convincing answer. Subjects may have been less familiar with detailed arguments about affirmative action relative to capital punishment.” Information orthogonal to how they were selected

46 Who Polarizes? Experts and people with strong opinions. Capital Punishment (Miller…): – Polarization: Extreme pro ≈ 52%, moderate pro ≈ 26%, moderate anti ≈ 39%, extreme anti ≈ 47%. – Depolarization: Extreme pro ≈ 7%, moderate pro ≈ 18%, moderate anti ≈ 17%, extreme anti ≈ 11%. – No change ≈ 50%. Nuclear: – Attitude polarization highest among subjects who reported high issue involvement and strong convictions. Page 46

47 Who polarizes? Experts On experts: If experts are people who know “all” the available information (say: number and nature of nuclear incidents), they will have observed the same prior information S. For the general population, those who have seen ambiguous signals (of the same kind as C) will polarize, while the rest won’t.

48 More polarization for people with strong opinions.


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