Vaccinating your children and the Prisoner’s Dilemma

3 February 2015

Oh those crazy anti-vaxxers and their selfish agendas! Why are they hell-bent on turning their children into disease factories? Etc, etc.

Of course I’m in the pro-vaccination camp, because of science and reason and all of those wonderful things, but this post isn’t about why the anti-vaccination camp is so lame and ignorant and dangerous. It’s about why so many parents are making the choices that they make. Of course, I’ll try get in all the cheap shots that I can, because people who willfully and recklessly endanger children deserve all the pillorying that’s coming to them.

What is herd immunity?

Immunisation works through something called herd immunity. The spread of diseases works on something called the chain of infection, which describes how you got the flu from your girlfriend because she was kissing your best friend, and he had the flu. If enough people have had this season’s strain of flu, and therefore can’t be infected again, then the spread of flu slows down.

If enough of the population has had a disease, or has been immunised against the disease, then that indirectly provides protection to the rest of the population who can still be infected, and this is known as herd immunity. Herd immunity, for most infectious diseases, needs quite a high percentage of the population to be vaccinated.

If you look at the Wikipedia link there’s a table in the middle that shows what the required percentages are for different diseases. The higher the R0 (the number of people that can be infected by one disease carrier if the entire population is unprotected) the larger the percentage of the population that needs immunisation for herd immunity to take effect.

A mumps carrier infects 4 to 7 people in an unprotected community, and so at least 75% immunisation is needed for herd immunity. A measles carrier infects between 12 and 18 unprotected people, and so at least 83% of the population needs to be innoculated for herd immunity to happen.

So, yeah, herd immunity. Some people can’t be immunised for whatever reason (like a weak immune system or an egg allergy) and they benefit from the herd immunity. In economic terms, there is a positive externality created by the immunised, from which the unimmunised benefit.

The Prisoner’s Dilemma (PD)

The Prisoner’s dilemma is a staple of game theory, which is the branch of economics dealing with strategic behaviour. It describes a situation involving two or more people who have the choice to cooperate with each other or to ‘defect’. The outcomes for each person depend both on their own actions and the actions of every other person involved.

So here’s the backstory for the PD. Two suspects are arrested for a big crime. The police only have circumstantial evidence and can only successfully prosecute the two for a lesser crime, carrying a lesser sentence – unless they can get one or both of the suspects to confess to the crime.

The two men (sexist, I know, but statistically most crime is committed by men) are separated and put into solitary confinement. The police detective visits the first man (known as Mr A) and gives him a pep talk. He gives the same pep talk to the second man (known as Mr B):

“Hi Mr A. As things stand, you’re going to jail for two years, if you refuse to confess to your crimes. You and Mr B will have a nice stretch of time to keep each other company. If you do confess to your crimes and implicate Mr B then we’ll chop two years off your sentence and you can go free. Mr B, however, will spend the next ten years in jail.”

“If you keep quiet and Mr B rats you out then he’ll go free and you’ll spend ten years in jail. And if you both confess then you’ll each get eight years in jail: ten years’ sentence minus two years for your confession.”

Bear in mind that ‘cooperation’ and ‘defection’ refer to Mr A’s actions in relation to Mr B, not to the police: he ‘cooperates’ by remaining silent and he ‘defects’ by blabbing and implicating Mr B. What is Mr A to do?

There are four possible scenarios with four payoffs for Mr A:

– I cooperate and Mr B cooperates. I receive 2 years in jail

– I defect and Mr B cooperates. I receive 0 years in jail

– I cooperate and Mr B defects. I receive 10 years in jail

– I defect and Mr B defects. I receive 8 years in jail

(The payoffs are the same for Mr B if you substitute ‘B’ for ‘A’ in the above four scenarios)

For Mr A, it’s always in his interests to defect, no matter what Mr B does. If Mr B cooperates, Mr A can cut two years off his sentence by defecting. Mr A also cuts two years off his sentence if he defects when Mr B defects.

The PD shows how actions which are individually rational can lead to a general outcome which is the worst possible outcome for the group. Examples of real-world PDs include price wars between cellphone companies / airlines, overfishing/exploitation of common resources, and the decision to immunise or not immunise.

When is it rational to not vaccinate?

Let’s imagine a society where the immunisation rate is 100%. Everyone immunises their kids and nobody listens to Jenny McCarthy (although they might watch her on TV with the sound off). You’re a new parent and you have to decide whether or not to immunise your kid.

If the whole society has been immunised then there’s no downside in not immunising your kid, because herd immunity has already been achieved. You can’t reduce the risk to your child by immunising her, so why do it?

On the other hand, there is a downside in immunising your kid. It’s not very high but it’s there. Maybe your kid will have an adverse reaction, or maybe your kid will be that unlucky 1 in 4 000 that develops bruising / bleeding from the jab.

This link shows that the risks of complications from immunisation are far lower than the risks of complications from contracting measles. But if there’s no risk of contracting measles, thanks to herd immunity, then it’s rational to not immunise.

This ‘rationality’ is highly qualified and related to this hypothetical case of universal immunisation. It assumes that there are informed parents who are able to understand and compare statistical probabilities and assess risks. In reality, many parents are underestimating the risks associated with disease and grossly overestimating the risks associated with immunisation.

Also, we are close to losing herd immunity in places where the anti-vaxxer nonsense has taken root. We’ve never had 100% immunisation rates, and if we fall below 80% (and then 70%) immunisation rates then we’ll lose herd immunity and I won’t even be able to defend the defective parents with hypothetical scenarios.

The good news is that if immunisation rates fall just a bit more, the payoffs from the four different scenarios will change enough so that, technically, we won’t have a PD anymore. Then we can chalk the increase in preventable diseases to the fear and ignorance of anti-vaxxer parents and we won’t be able to argue that their decisions are rational.


One thought on “Vaccinating your children and the Prisoner’s Dilemma

  1. Russell Lamberti

    Hey Pauly,

    Two quibbles with your link. 1) it’s hard to see the link on your page unless you happen to mouse over it (blog text style change can fix this). 2) The risks are not apples with apples because the odds of contracting measles is not 100%, but the odds of having a vaccine if you decide to have a vaccine are 100%. If the odds in a modern, clean environment of unvaccinated children contracting measles is say 20% (erring on the high estimate), then the left hand infographic gets squashed into the final two lines of people – 3 people would be red, 5 people would be yellow, and 92 people would be green. Also, the right hand infographic seems to assume that vaccinated children cannot contract measles, which is of course false. If vaccine failure rate is 5% when exposed outright to measles (plausible), then the right hand infographic would have 1 red person, 17 yellow people, and 82 green people. This makes the decision a lot more marginal, no?


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