As of the time of writing this post, I have a model that can roughly explain the phenomenon of polarization, the role of confirmation bias in the extreme magnetization (tendency toward a single extremist viewpoint) in a single community with only slight initial magnetization, and how the role of a moderating authority (such as our own moderating team) can prevent extremist opinions from taking root.

The oft-cited claim:

"There is no need for outside interference in the marketplace of ideas. The better ideas will consistently win out over the weaker ideas."

There is a model for opinion dynamics that can be used to justify that above statement, though it does not seem to model real-world opinion dynamics well at all.

The weak model is this:

•  Assume that the opinions can range from some low number to a high number.  (In the analysis in this thread, I chose to model the the opinions ranging from 1 to 5).  Each person has some (not necessarily random) opinion between 1 and 5.

•  Low numbers represent one extreme on one issue and high numbers represent the opposite extreme of the same issue.

•  Over time, conversations happen.

•  Each conversation consists of a speaker, a listener, and a specific question on which side of a given border between two positions one should take.  For example, suppose the question were "Is the best opinion somewhere in position 3 or less, or 4 or more?"  (We'll call this "question 3.5" as it asks about the border between positions 3 and 4.)  Anyone with an opinion of 1, 2, or 3 would say "the best opinion is 3 or less" and will be described as taking the negative side on that question.  Anyone with an opinion of 4 or 5 would say "the best opinion is 4 or more" and will be described as taking the positive side on that question.

•  In the event that the speaker favors the positive side, if the listener is of the position on the immediate negative side of the question (i.e, position 3 if the question is 3.5, position 1 if the question is 1.5, etc,) then there is some probability (10% in these simulations) that the listener's position will move one step to the positive side.

•  If the listener's position is already on the positive side of the question, then the listener does not change xyr opinion, as xe already agrees with the speaker.

•  If the listener's position was more than one step to the negative side of the question (i.e, the listener had position 2 and the question was 3.5) then the listener's opinion does not change, due to being too far removed.

•  Perfectly analogous dynamics happen if the speaker's position is on the negative side of the question.

If one runs a computer program starting with a slightly slanted population of size 100, 30 of whom have position 1, 25 of whom have position 2, 20 of whom have position 3, 15 of whom have position 4, and 10 of whom have position 5, and allows 1,000,000 conversations to happen, following the rules set above.  Note that the initial magnetization (average opinion value) is 2.50, slightly lower than the middle opinion of 3, for a slight negative slant.

For a typical simulation, the 100 people will form a consensus, either at opinion 2 or opinion 3.  The histograms for twelve such runs are as follows:

The code used to run the simulations is attached as a .txt file.  It is written for MATLAB, and will have to be renamed to have the file extension .m to run.

Mathematically, why does this happen?  Why is there consensus, and why is it usually at a position so close to the initial magnetization of the population?

We'll actually answer the second question first.  Let's first ask the question "Under what conditions is the magnetization likely to increase, likely to decrease, or remain roughly the same?"  As it turns out, under any conditions, for this model, the magnetization is likely to remain roughly constant.

Let's think about the probability of the magnetization (a function m(t) of time) increasing by one "step" (that is, 1/100, 1 divided by the population size).  We'll calculate this probability in terms of the speaker's opinion S, the listener's opinion L. and the question Q.  An increase will happen under the following conditions:

• S > L

• Q = L + 0.5

• The 10% chance of an increase actually happens.

Thus, we get the probability P(m(t) = m(t-1) + 1/100) = P(S > L) × P(Q = L + 0.5) × 0.10.

A decrease will happen under the following conditions:  

• S < L

• Q = L - 0.5

• The 10% chance of an increase actually happens.

Thus, we get the probability P(m(t) = m(t-1) - 1/100) = P(S < L) × P(Q = L - 0.5) × 0.10.

Because each person is just as likely to be the speaker as the listener, the probabilities  P(S > L) and P(L < S) are equal.  Because each question is as likely to be asked in this model as any other, the probabilities P(Q = L + 0.5) and P(Q = L - 0.5) are equal.  Thus, the probability of an increase in magnetization and the probability of a decrease in magnetization are equal, and the magnetization will have no expected long-term increase or decrease.  We call this the martingale property, and say that the magnetization is a martingale.

This addresses the question of why convergence happens near the original magnetization most of the time.  (Not always, as evidenced by the histogram in the right column of the middle row.  Statistical noise and randomness do happen.)

For that, we consider the same martingale question about the extreme positions.  More formally, suppose that there are people with at least three positions.  There is a highest occupied position and a lowest occupied position.  Is the number of people in each of those two extreme positions likely to increase, likely to decrease, or remain the same?

Let a be the minimum occupied position and let b be the maximum occupied position.  Let N(a) and N(b) be the numbers of people with each of extreme positions a and b, respectively.

The probability that N(a) will increase is P(S = a) × P(L = a + 1) × P(Q = a + 0.5) × 0.10.

The probability that N(a) will decrease is P(L = a) × P(S ≥ a + 1) × P(Q = a + 0.5) × 0.10.

Because the speaker and listener have equal probability of having a given opinion, and because the probability of having an opinion at least a + 1 is more than the probability of having specifically opinion a + 1, N(a) is more likely to decrease than to increase.  We call this having the supermartingale property (super rather than sub because the current observable value is an overestimate of any future value.)

An analogous argument can be used to show that N(b) is also a supermartingale.

In the case that there are only two occupied opinions and b = a + 1, then both N(a) and N(b) are martingales and the eventual movements between the two states will follow a symmetric lazy random walk, and will eventually converge to all settling on a or b.

Thus far, we have not entertained the possibility of there being various different "echo chambers" with different slants.  Suppose that we have ten such echo chambers and that five have the aforementioned slight negative slant (30 for opinion 1, 25 for opinion 2, 20 for opinion 3, 15 for opinion 4, and 10 for opinion 5) and that the others have the equal and opposite positive slant.

Suppose that, when a conversation happens, 90% of the time, the speaker and listener are forced to be from the same chamber.  Because there are 10 times as many people, we use 10,000,000 conversations rather than 1,000,000.

The results of twelve such simulations are shown.  Note that the consensus hasn't had time to run to completion yet, at there are ten times as many people.

The results are shown below:

The results shouldn't be surprising.  The magnetization of the entire society itself is a martingale, as are the magnetizations of each of the individual echo chambers.

If individual chambers converge to consensus at opinion 3, they can be expected to stay there, while if they converge to another opinion, they can be expected to slowly move toward the dominant opinion 3 from other chambers.  Note that this motion would be slower, as only 1% of the conversations would be interchamber conversations with a listener in that chamber.

The code is attached.

The above pictures show the main problem with this model:  Initially slanted echo chambers unslanted themselves and each converged to the same neutral magnetization.  Our observations on political polarization contradict those results.  When reality contradicts the results of a model, that model is flawed.

The question now arises:  If the above model (whose setup initially seemed reasonable) isn't qualitatively accurate, what modifications need to be made?

I hypothesize that the missing piece pertains to confirmation bias.  That is, each person has some strength of allegiance to either the positive or negative side (or possibly neither) and the strength of that allegiance makes it easier for that person to shift xyr opinion toward that side and harder to shift toward the opposite side.

We assign each person both a position on a scale of 1-5 and an allegiance (on a scale of 0-100).  High allegiances indicate allegiance toward the positive side, low allegiances indicate allegiances toward the negative side, and an allegiance of 50 indicates no confirmation bias in either direction.  We assume that, upon hearing the speaker's opinion, the listener may change xyr opinion, and then will shift xyr allegiance slightly (by 10% of the distance to that extreme).

And the distribution of allegiances is here:

The assumption, from looking at the graphs, is that the echo chambers themselves are magnetized, and that the end result is strong polarization.  An analysis of the individual echo chambers confirms this.  (The image below shows the distributions of opinions and allegiances for each echo chamber for a single simulation.)

And here's the text file with the code used to run the above simulations:

An attempt to limit extremism and polarization:

Having observed the problems of political extremism and polarization, we're left with the question:  how do we fix the problem?

If a laissez-faire approach to free speech will consistently lead to the rise of extremist positions, heavily magnetized communities, and a polarized populace, the answer seems to be that some regulations should be placed on speech.  Ideally, the speech regulations shouldn't be too heavy.

In this next simulation, we assume that the following regulatory measures are implemented:

• 10% of the "conversations" are the regulators stating "opinions 1 and 5 are too extreme".

•  This serves to shift the listener's allegiance one "10% step" in each direction.  Note that this will have the net effect of slightly demagnetizing the listener, as a 10% step in the direction that the listener is already magnetized will have a milder effect than a 10% step toward the further extreme.

•  While the ordering of those two subsequent shifts is relevant, we randomize that ordering.

•  The regulators' heavy messaging of "the extreme opinions are wrong" also serves to suppress 90% of the speakers who would say "take the positive side of question 4.5" or "take the negative side of question 1.5".

The results of twelve simulations are plotted here:

And the chamber-by-chamber plots for the first simulation are plotted here:

We see that this serves to do the following:

•  It eliminates the dangerous extreme opinions.

•  It prevents the allegiances of the members from staying too extreme, instead hovering around the 33 or 67 values.

•  It DOES NOT prevent a stark divide between the negative-slanted and positive-slanted communities.