Imagine that there are just two cafés in your town at which you can dine. At Café Safe the menu is limited and rather bland, your parent’s music is playing over the PA system, and people are friendly but stand-offish. However, the place is clean, and they follow all health protocols. Business is good at Café Safe and it is expanding its space for more customers, but it is increasingly easy to get a table.

Across the street is Café Risky, where the food is much more varied, the entertainment is better, and the patrons are just having a good old time, despite dining elbow-to-elbow. In fact, the food “is to die for,” because they are lax on their food safety protocols and a small number of customers actually die every week from eating the food. If you eat there often enough, you will likely get at least mild food poisoning. But despite the risks, business is booming at Café Risky!

Welcome to post-Covid-vaccination Florida (and a lot of other places in the U.S.). Where are you eating these days? As a loyal Café Safe customer, I am asking, “How in the world does Café Risky stay in business?” It turns out Russian mathematician Andrey Markov figured out why around the turn of the 20th century, and understanding his “Markov chain” concept will be key to shutting Café Risky down. The short answer is, however, that unless the “health department” aggressively steps in, Café Risky will likely be with us a long time.

Markov chains

I first ran into the math of Markov chains while studying computer science and working in the auto industry a very long time ago. It works pretty well as a technique to model brand loyalty for consumer choices like cars and restaurants, because it accounts for why some declining brands hang on well past the point when you would think they would die off. And it can also demonstrate why even promising new products have a hard time getting off the ground and capturing new customer loyalty.

The math for modeling two brands or restaurants in a town is pretty simple and we will look at that here. In the real world, you often need to model multiple brands, and the math gets more complex. But thanks to computers, and the simplification of the Markov chain concept, it still works well enough to focus talk about marketing strategy on some key factors.

Here is the simple diagram for our two “Covid cafés”:

The concept behind Markov chains is that people tend move, say from week to week in this case, from one brand or restaurant to another not in absolute numbers, but rather as a percentage of the current number of customers. In other words, when trying to predict how many Ford customers Kia can pick off, the more reliable number to examine is the percentage of current Ford customers that are vulnerable to switching, not the absolute count of those people.

And so, in our “Covid cafés” depicted in the circles above, there are two loyalty-based percentages that we critically need to determine, shown as the arrows at top and bottom:

1. What percentage of people who currently practice safe Covid protocols are likely to engage in new risky behavior this week? These folks include the unvaccinated frontline workers who normally take care, the “backsliders” tempted by the better food across the street, and the “cheaters,” who don’t tell their “safe” friends who they have been dining with on the sly. We will call this the “safe-to-risky defection rate,” which is indicated by the top blue arrow above.
2. What percentage of people who currently engage in risky Covid behavior are getting newly vaccinated this week, or are newly convinced to start practicing safe protocols? This is the “risky-to-safe-defection rate,” which is indicated by the bottom orange arrow.

Basically, these two defection rates work against each other. The remaining two arrows are the “café loyalty percentages.” Mathematically they are just what is left over after the first two percentages are subtracted. Keeping the “safe loyalty” percentage as high as possible will be critical to the success of our vaccination programs.

We are not sure yet what happens when vaccinated people prematurely “backslide” into risky Covid behavior, but the continuing high infection and death counts in many places, including Florida where I live, indicate that Café Risky is still doing very good business. That is disturbing.

The simulation results

This simple case can be modeled with a Google Sheets spreadsheet, which you can access at this link and download to modify on your own. What my example numbers show is that, even if our new vaccinations and the “safe convert rate” exceeds the “backsliders/cheater rate,” there comes a point where the populations in both cafés will stabilize. Changing the rates in the spreadsheet will determine where that new equilibrium point is and how long it takes to get there, but both cafés will find their natural equilibrium. That is the lesson of Markov chains.

In this simulation, I started with two cafes of equal size, with a conservative “defection to risky” rate of 5% per week and an aggressive “defection to safe” rate of 10% per week. Even with that optimistic growth in Café Safe numbers, the numbers in both cafés plateau by week 28 with Café Risky holding at 50% the size of Café Safe. There is an inevitable point where Café Risky’s loyal customers will sustain it. Defections out are offset by more defections in because Café Safe has become proportionally much bigger.

What those two defection percentages really are is anybody’s guess right now. With a more aggressive “defection to safe” percentage I could push the mix to perhaps 80-20 in favor of Café Safe, but it gets increasingly hard to push the Café Risky numbers lower than 20% (and I don’t think we are close). You can download the simulation spreadsheet and try some numbers yourself. Technical notes are included at the end of this post.

So, what does this mean in the real world?

In the restaurant or automobile business, sometimes that lower equilibrium point is sustainable at the low plateau level and sometimes it is not, in which case the brand goes out of business. You can likely think of many long-faded brands that are still on the grocery store shelves, however, sustained by a small, but sufficient level of loyal buyers. My point here is that if you think that Café Risky is “going out of business,” you have another think coming.

The “obstinate and dissident loyalty” factor that keeps people dining at Café Risky has a powerful and vocal political base, one that is driven by aversion to science and susceptibility to conspiracy theories like QAnon. The continued presence of Café Risky will continue to put all of us at risk, preventing the vaccines from getting herd immunity to the required levels that let the rest of us resume our pre-Covid normal lives.

I still have on my arm the large scar from a childhood smallpox vaccination. I received, along with virtually every kid in the country at the time, both the shot for the Salk polio vaccine and the later sugar cube containing the Sabin polio vaccine. Science and political will have defeated both of these horrible diseases. But imagine what would have happened if any of these vaccines had been introduced in today’s political environment. We would still be seeing iron lungs and pock-marked kids as common parts of our daily life.

Technical notes:

• You can think of each week in this case as the current “state” of our two cafés, designated in the spreadsheet as “A” (Café Safe) and “B” (Café Risky).
• For the next period, the probability of “A” customers switching to “B” is called “P(B|A)” (the probability of B, given A). Likewise, the probability of “B” customers switching to “A” is called “P(A|B)” (the probability of A, given B).
• From those two it is easy to calculate the percentage of loyal customers for each café, “P(A|A)” and “P(B|B)”.
• Apply those four values to the current state and you get next week’s state. As the stronger café gets more customers, its weekly defections will also get larger if the rate remains the same. At some point, the absolute numbers of defections from each café will balance each other.
• You can apply the same logic to three or more “brands” but the number of “defection rates” you need among the multiple brands grows quickly. Still, it is possible to monitor “real world” brand defections via sampling polls to get a handle on how your brand is doing.
• You can copy lines at the bottom of the spreadsheet to extend the number of periods. It turns out that the initial state of each café only impacts how long it takes to reach equilibrium. The end result ratio of “A” to “B” will be the same for any given mix of defection rates.

Related posts:

• The math of changing your mind