Friday, October 15, 2010

Is there a prisoner's dilemma in vaccination?

Recently, I have been thinking about vaccination strategies as I was confronted with opinions which I consider at least very risk inviting. Not to spill oil in the fire I will thus anonymize illnesses and use made up probabilities. But let me assure you that for the illness I have in mind the probabilities are such that the story is similar.

Let's consider illness X. For simplicity assume that if you meet somebody with that illness you'll have it yourself a bit later with 100% probability. If you have X then in 1 in 2000 cases you will develop complication C which is lethal. But C itself is not contagious.

Luckily, there exists a vaccination against X that is 100% effective, i.e. if vaccinated you are immune to X. But unfortunately, the veccination itself causes the deadly C in 1 in a million cases.

So, the question is: Should you get vaccinated?

Unfortunately, the answer is not clear: It depends on the probability that if not vaccinated you will run into somebody spreading X. If X is essentially eradicated there is no point in taking the vaccination risk but if X is common it is much safer to vaccinate.

The break even is obviously when that probability is 1 in 500. If it's less likely to meet somebody then it would be to your advantage not to vaccinate.

Unfortunately, the probability of meeting an X infected person depends on how well people are vaxinated: As X is so contagious, if the vaccination rate drops the probability of meeting somebody with X dramatically increases. That is, not vaccinating yourself might be profitable for you but if everybody follows the same strategy the vaccination rate might drop and the society as a whole will see many more cases of C.

If you assume in addition that your information is not perfect and you might be wrong in estimating the probabilities involved it is not clear to me to which fixed point this system evolves.

But it seems likely to me that there are situations where for the society as a whole it is much better if you get vaccinated even if this increases you personal risk of encountering C.