## The Game Theory of (Anti) Vaccination

I recently read an article about the prevalence of parents not vaccinating their children in certain (read: affluent) LA communities. While I vehemently disagree with the anti-vaccination movement, there are certain game-theoretic incentives that might compel people not to vaccinate. The idea is very closely related to the prisoner’s dilemma–one of the most studied scenarios in game theory.

Imagine that you do believe that vaccines had the potential to cause harm. Maybe not everyone vaccinated is harmed by the vaccine, but you believe there is a chance that the vaccine itself will cause some malady. Let’s quantify the (perceived) harm caused by vaccination to be say, $$4$$ harm units. If an otherwise rational person chooses not vaccinate, it is probably because the perceived harm done by the vaccine is greater than the perceived harm from the disease the vaccine is meant to prevent. So let’s quantify the (anticipated) harm caused by not vaccinating to be, say, $$1$$ harm unit. In this (overly simplified and probably inaccurate) world, any reasonable person would choose no vaccine ($$1$$ harm unit) over vaccination ($$4$$ harm units).

Here is the problem: by not vaccinating your own children, you put the entire rest of the population at a greater risk of the disease. We can formalize this as follows. Assume there is a population of $$n$$ people, each of whom chooses either a vaccine or no vaccine. A person who gets vaccinated incurs $$4$$ harm units, but if a person chooses not be vaccinated, the entire population of $$n$$ people incur $$1$$ harm unit.

Bob is deciding whether or not to vaccinate. Regardless of Bob’s choice, some people will vaccinate, while others will not. Suppose $$k$$ people choose not to vaccinate. Then every person incurs $$k$$ harm units from those people. Thus, if Bob chooses to vaccinate, he will incur $$k + 4$$ harm units (while each non-vaccinator incurs $$k$$), but if he chooses not to, he will only incur $$k + 1$$ harm units (as will the other non-vaccinators). Thus, Bob has an incentive to not vaccinate. Bob is no martyr (in fact, he is downright selfish) so he decides not to vaccinate.

What is interesting about this scenario, is that any individual can improve their own situation (i.e., incur less harm) by selfishly choosing not to vaccinate. But if everyone (or even more than $$4$$ people) choose not to vaccinate, everyone is worse off than if they had all chosen to vaccinate. In fact, if everyone acts in their own best interest, everyone achieves the worst possible outcome!