KEVIN DORST
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Stranger Apologies

How to Polarize Rational People

9/12/2020

22 Comments

 
(1700 words; 8 minute read)

The core claim of this series is that political polarization is caused by individuals responding rationally to ambiguous evidence.

To begin, we need a possibility proof: a demonstration of how ambiguous evidence can drive apart those who are trying to get to the truth. That’s what I’m going to do today.

​I’m going to polarize you, my rational readers.
In my hand I hold a fair coin. I’m going to toss it twice (…done).  From those two tosses, I picked one at random; call it the Random Toss. How confident are you that the Random Toss landed heads?  50-50, no doubt––it’s a fair coin, after all.

​But I’m going to polarize you on this question. What I’ll do is split you into two groups—the Headsers and the Tailsers—and give those groups different evidence. What’s interesting about this evidence is that we all can predict that  it’ll lead Headsers to (on average) end up more than 50% confident that the Random Toss landed heads, while Tailsers will end up (on average) less than 50% confident. That is: everyone (yourselves included) can predict that you’ll polarize.

The trick? I’m going to use ambiguous evidence.

First, to divide you. If you were born on an even day of the month, you’re a Headser; if you were born on an odd day, you’re a Tailser.  Welcome to your team.

You’re going to get different evidence about how the coin-tosses landed. That evidence will come in the form of word-completion tasks. In such a task, you’re shown a string of letters and some blanks, and asked whether there’s an English word that completes the string. For instance, you might see a string like this:

P_A_ET

And the answer is: yes, there is a word that completes that string. (Hint: what is Venus?) Or you might see a string like this:

CO_R_D

And the answer is: no, there is no word that completes that string.

That’s the type of evidence you’ll get. You’ll be given two different word-completion tasks—one for each toss of the coin. However, Headsers and Tailser will be given different tasks. Which one they’ll see will depend on how the coin landed.

The rule:
  • For each toss of the coin, Headsers will see a completable string (like ‘P_A_ET’) if the coin landed heads; they’ll see an uncompletable string (like ‘CO_R_D’) if it landed tails.
  • Conversely: for each toss, Tailsers will see a completable string if the coin landed tails; they’ll see an uncompletable string if it landed heads.

Here’s your job. Click on the appropriate link below to view a widget which will display the tasks for your group. You’ll view the first world-completion task, and then enter how confident you are (between 0–100%) that the string was completable. Enter “100” for “definitely completable”, “50” for “I have no idea”, “0” for “definitely not completable”, and so on.

If you’re a Headser, the number you enter is your confidence that the coin landed heads on the first toss. If you’re a Tailser, it’s your confidence that the coin landed tails—so the widget will subtract it from 100 to yield your confidence that the coin landed heads. (If you’re 60% confident that the coin landed tails, that means you’re 100–60 = 40% confident that it landed heads.)

You’ll do this whole procedure twice—once for each toss of the coin. Then the widget will tell you what your average confidence in heads was, across the two tosses. This is how confident you should be that the Random Toss landed heads, given your confidence in each individual toss. And this average is the number that will polarize across the two groups.

Enough set up; time to do the tasks.
  • If you’re a Headser, click here. 
  • If you’re a Tailser, click here.
Welcome back.  You have now, I predict, been polarized.

This is a statistical process, so your individual experience may differ. But my guess is that if you’re a Headser, your average confidence in heads was greater than 50%; and if you’re  Tailser, your average confidence in heads was less than 50%.

I’ve run this study. Participants were divided into Headsers and Tailsers. They each saw four word-completion tasks. Here’s how the two groups’ average confidence in heads (i.e. their confidence that the Random Toss landed heads) evolved as they saw more tasks:
Picture
The average confidence in heads of each group after seeing 0–4 independent word-completion tasks. Blue is Headsers; orange is Tailsers. Bars represent 95% confidence intervals for the mean of each group’s average confidence at each stage.
Both groups started out 50% confident on average, but the more tasks they saw, the more this diverged. By the end, the average Headser was 58% confident that the Random Toss landed heads, while the average Tailser was 36% confident of it.

(That difference is statistically significant; the 95%-confidence interval for the mean difference between the groups’ final average confidence in heads is [16.02, 26.82]; the Cohen’s d effect size is 1.58—usually 0.8 is considered “large”. For a full statistical report, including comparison to a control group with unambiguous evidence, see the Technical Appendix.)

Upshot: the more word-completion tasks Headsers and Tailsers see, the more they polarize.
​
The crucial question: Why?
(800 words left)
Getting a complete answer to this—and to why such polarization should be considered rational—will take us a couple more weeks.  But the basic idea is simple enough.

​A word-completion task presents you with evidence that is asymmetrically ambiguous. It’s easier to know what to think if there is a completion than if there’s no completion. If there is a completion, all you have to do is find one, and you know what to think.  But if there’s no completion, then you can’t find one; but nor can you be certain there is none—for you can’t rule out the possibility that there’s one you’ve missed.

Staring at `_E_RT’, you may be struck by a moment of epiphany—`HEART!’—and thereby get unambiguous evidence that the string is completable.

But staring at `ST_ _RE’, no such epiphany is forthcoming; the best you’ll get is a sense that it’s probably not completable, since you haven’t yet found one.  Nevertheless, you should remain unsure whether you’ve made a mistake: “Maybe it does have a completion and I should know it; maybe in a second, I’ll think to myself, `It is completable—duh!’”  This self-doubt is the sign of ambiguous evidence, and it prevents you from being too confident that it’s not completable.

The result? When you’re presented with a string that’s completable, you often get strong, unambiguous evidence that it’s completable; when you’re presented with a string that’s not completable, you can only get weak, ambiguous evidence that it’s not. Thus when the string is completable, you should often be quite confident that it is; when it’s not, you should never be very confident that it’s not.

I polarized you by exploiting this asymmetry. Headsers saw completable strings when the coin landed heads; Tailsers saw them when it landed tails. That means that Headsers were good at recognizing heads-cases and bad at recognizing tails-cases, while Tailsers were good at recognizing tails-cases and bad at recognizing heads-cases.

As a result, if you ask Headsers, they’ll say, “It’s landed heads a lot!”; and if you ask Tailsers, they’ll say, “It’s landed tails a lot!”. They polarize.

Here it’s worth emphasizing a subtle point but important point—one that we’ll return to. The ambiguous/unambiguous-evidence distinction is not the weak/strong-evidence distinction. Ambiguous evidence is evidence that you should be unsure how to react to; unambiguous evidence is evidence that you should be sure how to react to.

Ambiguous evidence is necessarily weak, but unambiguous evidence can be weak too. Example: if I tell you I’m about to toss a coin that’s 60% biased towards heads, that is weak but unambiguous evidence--weak because you shouldn’t be very confident it’ll land heads, but unambiguous because you know exactly how confident to be (namely, 60%).

The claim is that it is asymmetries in ambiguity—not asymmetries in strength—which drive polarization. We can test this by comparing our ambiguous-evidence Headsers and Tailsers to a control group that received evidence that was sometimes strong and sometimes weak, but always (relatively) unambiguous. (It came in the form of draws from an urn; see the Technical Appendix for details.)

Here is the evolution of the Headsers and Tailsers who got unambiguous evidence:
Picture
The average confidence in heads of each group after seeing 0–4 independent draws from an urn. Blue is Headsers; orange is Tailsers. Bars represent 95% confidence intervals for the mean of each group’s average confidence at each stage.

As you can see, there is some divergence (most liked a “response bias” because of the phrasing of the questions), but significantly less divergence than in our ambiguous-evidence case. (Again, see the technical appendix for a statistical comparison.)

Upshot: ambiguous evidence can be used to drive polarization. 

That concludes my possibility proof. The rest of this project will try to figure out what it means.  To do that, we need to look further into both the theoretical foundations and the real-world applications.

To preview the foundations: there is a clear sense which both Headsers and Tailsers are succeeding at getting to the truth of the matter—for each coin flip, they each tend to get more accurate about how it landed. The trouble is that this accuracy is asymmetric, and as a result they end up with very different overall pictures of the outcomes of the series of coin flips.

To preview the applications: these ambiguity-asymmetries can be exploited. Fox News can spin its coverage so that information that favors Trump is unambiguous, while that which disfavors him is ambiguous. MSNBC can do the opposite.  So when we divide into those-who-watch-Fox and those-who-watch-MSNBC, we are, in effect, dividing ourselves into Headsers and Tailsers.  As a result, although we are getting locally more informed whenever we tune into these programs, our global pictures of Trump are getting pulled further and further apart.

In fact, the same sort of ambiguity-asymmetry plays out in many different settings—helping explain why group discussions push people to extremes, why individuals favor news sources that agree with their views, and why partisans interpret shared information in radically different ways.

That’s where we’re headed. To get there, we need to get a few more basics on the table. Empirically: what mechanisms have led to the recent rise in polarization? And theoretically: what would it mean for this polarization—in our Headser/Tailser game, and in real life—to be “rational”?

We’ll tackle those two questions in the coming weeks. Then we’ll put all the pieces together, and examine the role that ambiguity-asymmetries in evidence play in the mechanisms that drive polarization.


What next?
If you liked this post, consider signing up for the newsletter, following me on Twitter, or spreading the word.
For a full statistical analysis of the experiment, see the technical appendix.  No doubt those with more experimental expertise can find ways it could be improved––suggestions most welcome!
Next post: I’ll sketch a story of the empirical mechanisms that drive polarization; with that on the table, we'll move on to evaluating them normatively.


PS. Thanks to Branden Fitelson and especially Joshua Knobe for much help with the experimental design and analysis.  
22 Comments
Peter Gerdes
9/12/2020 02:24:14 pm

Ok, sorry to nitpick a bit but the OED has a truly scary amount of words:

Conred:

Provision or allowance for maintenance, aliment; pension.

J. Grant Hist. Burgh Schools Scotl. i. i. 4 Pope Innocent IV. subsequently confirmed to Kelso the churches and schools of Roxburgh, free from all synodal rent and conreds.



Reply
Kevin
9/15/2020 08:02:13 am

Ha, fair enough! Good to know. That one didn't some up in the word-search algorithm I used, but I suppose I should be more careful in delineating what counts.

Reply
Peter Gerdes
9/17/2020 11:27:27 am

The OED lets you search with ? as wildcards which is only way I figured this out.

I know it's a personality flaw :-).

Kevin
10/3/2020 10:17:13 am

Ha, amazing, good to know! Will have to use that one from now on :).

Peter Gerdes
9/12/2020 03:05:42 pm

Maybe I'm missing something but I don't see the sense in which this suggests any deviation from the standard Bayesian model.

I mean how is this experiment any different (on a subjective probability type understanding) than an experiment which said ok if the coin lands and you are in group 1 if the coin lands heads we tell you with 95% probability and say nothing with 5% probability while if the coin lands tails we say nothing with 95% probability and tell you how it landed with 5% probability.

What I'd hypothesize is happening is that either asking only about the heads result rather than asking about tails plus the fact that people are better at reasoning about certainty than uncertainty together bias people to only consider the information gained via the heads route and not to consider the information gained via the fact that they didn't get a highly confident result in tails.

Specifically, I predict that you could run the test without ambiguous evidence using the setup I just described and you would get similar results because it's just much easier for people to think 'Ohh, I have high confidence info it is heads' than to think 'I didn't get high confidence info that it was tails so I should increase my probability of heads accordingly'.

Reply
Peter Gerdes
9/12/2020 03:08:01 pm

Ohh yah, and vice verss for the Tailsers.

Specifically, is it not true that you can use this to win money from people if they really are willing to makes bets in accordance with that data and isn't that pretty sold grounds to describe it as irrational?

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Peter Gerdes
9/12/2020 05:05:11 pm

Ohh, and sorry for forgetting to add that while I'm not yet convinced this is all super interesting and exactly the kind of thing that I think philosophers should be doing more of rather than merely trying to spin out more supposedly a priori features of rationality so even if my hypothesis turns out to be correct this is very much great work.

Also, looking at the code I notice that "FR_ _ L" is in the non-completion list but isn't that completed by "FRAIL"?

Reply
Peter Gerdes
9/12/2020 05:06:16 pm

Ignore that last remark I confused the lists. Damn lack of edit!

Kevin
9/15/2020 08:14:49 am

Thanks! Yeah good for you not to be convinced yet---will keep me on my toes :).

I'm definitely live to the worry that this isn't anything special to do with ambiguous evidence, and more of the argument for why it is will come in a couple weeks when I get the models on the table (sorry to keep coming back to the "just wait..." refrain, but I'm trying not to pack too much into these posts at once!).

The basic empirical argument, though, is that in the unambiguous version of the study I ran (where people were seeing draws from an urn which either contained a black marble or didn't, and definitely contained some non-black marbles---sometimes they got strong evidence (black marble), other times they got weak evidence (non-black marble)), the amount of polarization was significantly less. (Did you have a different set-up in mind? I was thinking that the control condition was basically the version you were thinking of, but maybe I'm missing something.)

That suggests to me that while part of what's going on may have to do with reasoning with uncertainty vs. reasoning with confidence, there's something else going on as well. In particular, when you see a non-black marble, it's relatively striaghtforward how your confidence should evolve. (What are the different possible contents of the urn; Bayes formula; presto.) But when you see a non-completable string, it's much less clear---should you just condition on the fact that you didn't find one? Well, you know more than that---namely how it looks, and what inklings you have about how word-like it is, etc. Trouble is those details of your evidence aren't cleanly introspectible (hard to distinguish the "pretty word-like" feeling from the "kinda word-like" feeling, for instance), so don't make for a partition of possibilities that you can update on. In particular, if we consider cases of people who didn't find a word (credence < 1 there is one), then of those who were looking at a non-word, their average credence was 44%, but of those looking at word, their average credence was 52% (statistically sig difference---more details coming soon). That suggests that there's a level of discrimination between cases that their evidence gives them access to, but which they can't clearly introspect and so they can't "correct" for it as a Bayesian with unambiguous evidence would (if that makes any sense... perhaps too cryptically put at this point.)

Peter Gerdes
9/17/2020 11:23:11 am

So yah I had something a bit different in mind. I got about as far as figuring out how to get my webserver to record answers returned in a survey in going to setup a test but if you've got the setup already done it might be quicker.

Basically take the same Headser and Tailster division as before but change the instructions as follows:

If you are a headster and the coin lands heads we will inform you of how the coin landed 80% of the time and give you no info in the other 20% of times. If it lands tails we tell you how the coin landed 20% of the time an say nothing the other 80%. (Reverse heads and tails for tailsters). Note they just get either "Sorry no info" or "The coin landed X" no other info other than a description of setup.

Basically, my hypo is that you have one outcome where you don't need to do any complex processing and react to the presence of some clear signal to be sure something is heads up against an outcome where you have to infer based on your failure to see some event in a less intuitive way.

Basically my hypo is that people are less able to quickly infer that it must have been heads because I didn't get clear evidence it was tails than they are to infer it must have been heads because they got heads evidence.

Indeed, I'd be shocked if this wasn't the case since in an evo context we rarely face cases with a known constrained list of outcomes. As such in some sense we should be quicker to see something as heads evidence when it says heads occured than when it is presented as saying tails doesn't.

Also vaguely reminds me of what is going on in the wasson selection cases.

I’m curious enough that if you aren't going to test it Ill eventually get to it but it seems like it might be worth testing something as close as possible to your headster case but with explicit probabilistic mechanisms rather than ambiguity.

Peter Gerdes
9/17/2020 11:26:08 am

Let me add that I do think ambiguity will enhance the effect but if we see even a less strong effect on my alternate proposed experiment then I think it suggests that the mechanism is via some kind of mental complexity and that having to evaluate how likely it is that this puzzle is completed by a word you aren't thinking of is part of that complexity.

Reply
Kevin
10/3/2020 10:16:37 am

Interesting! Sorry to be slow on the reply here.

If you don't get around to testing it, I might try something like this at some point, but I definitely won't get around to it for awhile—so do let me know if you go ahead with it, and what you find!

I'm still not 100% sure why the mechanism you're describing is different than the urn task I had them do, insofar as I take it "no information" is (despite the label) just weak information, and the urn task was doing was setting it up so that Headsers are more likely to see strong information when it lands heads and weak information when it lands tails. In other words, I'm not sure why telling them "no information" is different than telling them "the marble wasn't black". I may just not be fully following though. Maybe shoot me an email with more thoughts? (kevindorst@pitt.edu). I'm much better at replying to emails.

I'm also inclined to think I'm fine with the conclusion that ambiguity enhances the effect. Honestly, I *am* inclined to think that something like "complexity" of the task will be a good proxy for ambiguity in the technical sense, for when tasks are complex you often should be unsure whether you've reasoned through them properly. So it's possible that the variant you're thinking of might be another way of inducing a sort of ambiguity. ("What do I make of the fact that they haven't told me anything?") For instance, I'd certainly think that as we make it harder and harder to calculate the likelihoods, people will be worse at accounting for the "no info/weak" evidence, in part because I'd think that makes it more ambiguous.

André Martins link
9/13/2020 02:40:03 am

Very interesting to read, but I think there is one important error. It seems like base rate bias but not quite so. Basically, if people can complete the missing letters in possible words at a certain rate, they should estimate the chance of head and tails based on that. Assume, as an example, people can find the word, when there is one, 90% of times. And assume I am a Tailser.

In this case, there is 50% chance the coin will turn a tail. That means I identify the word 45% of times and miss it 5%. In the remaining 50%, the coin landed on heads and I get nothing. Under these circumstances, if I fail at completing the word, I must conclude there is only 1 chance in 11 that it was tails. Not more. If people are giving larger subjective estimates, they are making mistakes.

If I do not get 90% of the times right but other number, figures will change. But people seem to be overestimating the chances of tails. In any case, we get that, with the proper probabilities, my estimate the coin landed on tails, for these numbers should be

0,45.1 +0,55*(1/11) = 0.5

When people do not report 0.5, that is because they are making a mistake. A very subtle one, but a mistake anyway. So, in this case, not an actual rational behavior.

Reply
Kevin
9/15/2020 08:19:14 am

Yes, I agree---if people's evidence were *just* the (unambiguous) evidence about whether or not they completed a word, and they updated on that by conditioning, then they would not polarize---for exactly the reasons you spell out. The argument is (and is going to be made more clear in a couple weeks after the apparatus for ambiguous evidence is clearly on the table) is that that is NOT all the evidence they have. I just wrote some of this in reply to Peter above so copying and pasting the relevant part of the argument:

When you see a non-completable string, it's much less clear---should you just condition on the fact that you didn't find one? Well, you know more than that---namely how it looks, and what inklings you have about how word-like it is, etc. Trouble is those details of your evidence aren't cleanly introspectible (hard to distinguish the "pretty word-like" feeling from the "kinda word-like" feeling, for instance), so don't make for a partition of possibilities that you can update on. In particular, if we consider cases of people who didn't find a word (credence < 1 there is one), then of those who were looking at a non-word, their average credence was 44%, but of those looking at word, their average credence was 52% (statistically sig difference---more details coming soon). That suggests that there's a level of discrimination between cases that their evidence gives them access to, but which they can't clearly introspect and so they can't "correct" for it as a Bayesian with unambiguous evidence would (if that makes any sense... perhaps too cryptically put at this point.)

Sorry if that's uninterpretable at this point. Week 5 will be the post where I give the full ambiguous-evidence model of this task---please do let me know what you think when that comes out, if you get a chance!

Reply
André Martins link
9/15/2020 08:56:53 am

I do agree you have something interesting here. But it does sound far more like a bias than actual rational behavior. It would be a reasonable bias, as the mathematics to actually be perfectly rational is not even simple, I would not expect anyone's brain to do that right. But let me advance it.

I used 90% in the example, to illustrate my point. But you are right people do not know that value and that might change from one case to another. Despite that, I would guess (I haven't done the calculations on paper yet, just in my less reliable than paper head) that

1) If I change 90% for a specific value p, p will cancel out and you always get 50% at the end (easy to check).

2) Now it gets tricky. The actual full Bayesian analysis here requires a probability distribution over p, f(p). And f(p) is not well defined, distinct rational agents could, perhaps should, have different f(p). To get the final estimate (50% or not), we would need to integrate over all values of p, 0<= p <= 1 . And it might look like this could lead to anything. But this integral is calculating the expected value of chance of tails when p changes. Since that chance is always 50% (item 1), the expected value will be 50% no matter the function f(p) a rational,agent would use. And you get 50% regardless of the assumptions.

3) Of course, our brains would not do that. It is reasonable they behave the way you observed. And that can indeed help with the polarization. So, what you found out does contribute to it. But it is a bias, not a perfect rational, mathematical behavior.

Kevin
10/3/2020 10:20:11 am

Thanks Andre! I just posted the argument that this sort of predictable polarization can be rational—it can be a "Bayesian" update in the sense of satisfying the value of evidence, but not being the result of conditioning on a partition, in a way that leads to an expected shift in credence. Curious to hear what you think, if you get a chance to read it!

(https://www.kevindorst.com/stranger_apologies/rational-polarization-can-be-profound-persistent-and-predictable)

Matt Vermaire
9/13/2020 12:19:54 pm

The attention to ambiguous evidence is cool, but I'm wondering if it's really where we should look. Can't you get rational polarization just by presenting *different* evidence to different groups? If you just set up conservatives and liberals with different trusted evidence-providers (Fox and MSNBC, say), doesn't that already explain rational divergence? Or is a goal of the project to explain the polarization even apart from the assumption of differential trust of that sort?

Reply
Kevin
9/15/2020 08:23:19 am

This is a great question (thanks!). The reason will be spelled out more in the week 4 post, but to preview: if people are Bayesians with unambiguous evidence (the standard model of rational belief), then *no matter what evidence they get*, they can't expect their opinions to shift in a particular direction. Intuitively, the thought goes: "if I can now predict that when I get more evidence I'll raise my confidence in P, then I should *now* raise my confidence in P---so I can't predict I'll raise my confidence, after all".

Thus although we can expect that people with different prior beliefs can diverge upon getting different evidence, each individual can't possibly expect *themselves* to be part of that divergence. In particular, they must expect any divergence to be *the other side* getting more extreme, while they will stay put with their opinions. And, I think (and will argue!), this is not what real polarization looks like. Becca and I could each expect that we ourselves would get more extreme in our opinions. And for that sort of predicted self-polarization, it turns out that we'll need ambiguous evidence.

Reply
Matt Vermaire
9/15/2020 03:59:55 pm

Ah, I see that I missed an earlier post emphasizing predictability. Interesting! Looking forward to future episodes.

Reply
Quentin
7/22/2021 06:13:40 pm

Maybe this overlaps with other questions, but couldn't the same effect be explained by the fact that people systematically overestimate the possibility to complete the string when they didn't find a word (or underestimate their capacity to find a word even it's there)?

Reply
Kevin
7/23/2021 02:42:57 pm

Definitely! Super good question. This is in fact the reason why I ended up running the "unambiguous" evidence case, with the urns. The first-pass reply is that it's not *just* so-called "conservatism" in probabilities that is leading them to under-react to weak evidence, because the urn case also has the strong/weak evidence asymmetry but has less polarization.

That said, you might think there's something special about word-completion tasks that makes people be less confident that there's no word when there's not one. I'd want to say that this is exactly what we should expect from the fact that the evidence is ambiguous—and the models of the word-completion task I give later in the series are supposed to illustrate why that need not be a signal of irrationality.

But that's all very compressed in these posts. I'm currently finalizing the draft of the full paper on this, which goes into much more detail on these sorts of worries, so hopefully when I post that, that'll help!

Reply
Joseph Smith link
10/9/2022 10:19:20 pm

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    Kevin Dorst

    Philosopher at MIT, trying to convince people that their opponents are more reasonable than they think

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