The math of changing your mind – part 1

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In a recent post I explored the idea of collective delusion, where our self-concept of things like our susceptibility to advertising or the under-estimates of our gambling losses are betrayed by the total size of the advertising and gambling markets. In this post, I will look at how collective delusion can extend to our most fervently-held political convictions and the process by which to model how minds are changed during a political campaign.

In short, you have likely changed your mind politically in significant ways, and yet also likely can’t admit your own movement. How did that happen? Well, it happened one day at a time.

Markov chains

Many years ago now, when I was writing computer-based simulations for new manufacturing processes in the automobile industry, we sometimes used a mathematical technique called the Markov chain, named after the Russian mathematician Andrey Markov (1856–1922). Markov chains look at how the “states” of dynamic systems change over time. For instance, you can take a “snapshot” of a manufacturing process every hour, say an assembly line conveyor, track its past changes, and then predict its future performance if certain parameters are modified. [1]

A simple example, to be explored in more mathematical detail the second part of this post, would be to project what happens when a new restaurant (“B” in the diagram below) opens up near an existing one (restaurant “A”), and starts aggressively advertising its presence. Each day is a new “state,” with some customers “defecting” over to the new restaurant every day, some of them permanently, but then some customers of the new restaurant are also “defecting back” to the original one. If we can determine the “defection rates” in each direction, then we can project how each restaurant will perform over time.

Markov Defections

Alternatively, think Barnes & Noble bookstores after Amazon came into the market. Where would the Amazon “hit” level off?

Simple Markov chains can be solved with a calculator, while more complex ones will require a spreadsheet to track the “states,” because the result can be counter-intuitive. Even more complex chains might require a customized computer simulation. You can download the spreadsheet for this model in the upcoming Part Two.

If our original restaurant had 1000 customers a day (a busy place!) then a 10% “defection rate” would mean that it would lose 100 customers the first day, and 90 the second day (10% of the 900 remaining customers). However, also assume that soon 5% of the new restaurant’s customers begin “defecting back” to the original restaurant, and that’s where it gets messy.

It turns out that if the aforementioned pair of restaurants follows the classic Markov scenario, these “defection rates” will trade-off against one another over time, until at some point a new “steady state” emerges. At that point, the number of customer defections in either direction will balance out, even if at different rates. And then, at that point, each restaurant in our example will stabilize to a new number of loyal customers.

The original restaurant stops losing its total number of customers (stabilizing at a lower overall number) and the new restaurant, at some point, stops gaining new total customers, at least until the defection rates change due to some intervention. Defections continue in either direction, but at this new steady state, the numbers balance each other out.

So who wins the restaurant wars?

What would happen if the upstart restaurant in our example was able to pull 10% of the existing restaurant’s customers away each new business day, however 5% of the new restaurant’s customers also “defect back” to the original place every day? Our first guess might be that these rates would drive the first restaurant completely out of business over time, but our guess would be wrong. If the defection rates held steady in both directions, it turns out that the original restaurant would stop bleeding customers after about 36 days. The bad news is that this new “steady state” would be at about one-third the number of customers for that first restaurant than they originally had, the other two-thirds going to the new place.

If you want to see the Markov simulation of this scenario, stay tuned to this blog for the Part Two post ,in the queue, where I present the math visually and let you tweak the numbers yourself.

Of course, this result assumes that the original restaurant doesn’t “fight back” with deals and advertising, or that no new customers come into the mix. But we could simulate these factors as well, testing out by how much we would need to change the relative mix of “defection rates” or attract new customers to keep our business at its current level.

Indeed, this is where this type of simulation gets very valuable, as we can keep “tweaking” the variables to see the extent of the changes that we need to make. The results are not always intuitive. We could even add more restaurants to the mix, and the Markov chain math will handle as complex a set of restaurants as our estimates of defection data might allow.

How about political decision-making?

To illustrate a similar example in politics, what happens when a new challenger candidate enters a political race against an entrenched incumbent? Markov chain simulations can also be useful in predicting the outcome of a primary or general election. The math gets much more complicated when a third or fourth candidate enters the race, but if we can get a handle on “defection rates” between each pair of candidates, through polling or observation, then we can write a simple computer simulation to “watch” the movement of supporters between candidates for each day of the campaign as it progresses. And as we go along, we can test and tweak our assumptions to improve the accuracy of our predictions.

Just like the restaurants, “defections” in either direction start to balance out over time, and a fairly stable number of loyalists for each side typically remains. However, if the election happens before the “steady state” returns (which is a frequent occurrence), then the race may be a toss-up.

A fantastical example that would never happen in real life

We’ll save the messy math for Part Two, but assume that you wanted to completely flip the priorities of an established political party, say in these areas:

Markov Change

We could create a simulation to test each of these propositions separately, but let’s start with an “all-or-nothing” acceptance of the new “platform.” Like our restaurant example, the key is to get the “defection rate” in one direction (from existing “favored” candidates to the challenger) to be greater than the “reverse defection rate,” the people who drift back to other candidates. In other words, our political advertising needs to bring people into our camp, and then keep them long enough through loyalty-building so that the “reverse defection rate” is smaller than the rate of new people coming in.

We can tackle this conversion one person at a time, because that is how minds are changed. But add up those singles and a quantifiable “defection rate” emerges.

If we can do this, then the size of the net defection rate that we achieve will determine how long it will take to reach our “steady state” of loyal followers. We need to reach a plurality of voters by the day of the election.

As demonstrated in the prior post of this series, advertising works, even if we are under the “collective delusion” that it does not work on us. And this obviously includes political advertising. We know this because the above four “classic Republican positions” (and more) have reversed in an amazingly short period of time. Which types of messages work best is a subject for another time, but clearly fear and bigotry are “best sellers.”

My take on this math behind the changing of political minds had me tracking friends and relatives during the 2016 presidential campaign, as well as since, to watch their social media postings turn 180 degrees one by one on the above issues. New, substitute “memes,” first start appearing, like the “King David” meme from pastors justifying Donald Trump’s questionable moral behavior, which I wrote about in an earlier post. Then the criticism on that issue stops on the social media feed, followed by the embrace of the new position as if the old one was never held.

A Markov chain “defection” in real life.

If you want to hang around to see the Markov Chain math in action, subscribe to this blog using the box on the left side of your screen, or click on the Facebook or Twitter icons to be notified when Part Two gets posted.

Part Two of this series has now been posted.


  1. For a visual introduction to Markov chains, I recommend Powell, Victor. “Markov Chains Explained Visually.” Explained Visually.

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