Vaccination Dilemmas
The city of Vaccineville has a lot of residents. During the coronavirus pandemic, the city authorities are not imposing a lockdown but are urging all residents to get vaccinated. A vaccine is available in the city and can be administered to any resident who wants to get it.
The likelihood for a person P to contract the coronavirus depends on whether he or she is vaccinated, as well as what proportion of other residents are vaccinated. For simplicity, assume that every day all residents meet in random pairs, so the probability of meeting a vaccinated person equals the proportion \alpha of residents that are vaccinated (assume that the population is very large, so whether one person is vaccinated does not affect \alpha to any significant extent). The risk for person P to contract the coronavirus in different matches are given by their probability of being infected that are included in the following table:
All residents consider vaccination a costly procedure. Even though the vaccine is offered free of charge, the cost may come in the form of the time needed to be spent visiting a doctor, lack of trust in the effectiveness and safety of the vaccine, side effects, etc. A person's utility equals the probability of not being infected when she meets someone. For those vaccinated, the cost of vaccination that is equal to 0.3 is subtracted from utility.
(a) (10 rp) Suppose people decide individually whether to get vaccinated; each of them maximizes their utility. We say that people's decisions form a Nash equilibrium if no one can benefit by changing their decision with others' decisions fixed. What fraction of Vaccineville residents will get vaccinated in the Nash equilibrium?
The probability of being infected for a vaccinated person is 0 \cdot \alpha + 0.05 \cdot (1 - \alpha) = 0.05 - 0.05\alpha. The probability of being infected for a non-vaccinated person is 0.15\alpha + 0.4(1 - \alpha) = 0.4 - 0.25\alpha. In equilibrium, utilities must be equal, otherwise a few non-vaccinated people would regret and get the vaccination or vise versa.
1 - (0.05 - 0.05\alpha) - 0.3 = 1 - (0.4 - 0.25\alpha).
In equilibrium, a^*=0.25.
(b) (10 rp) Vaccinetown is just like Vaccineville in all aspects, except that its authorities force some of its citizens to get vaccinated. In particular, vaccination is mandatory for doctors and teachers, which together constitute 20% of the town's population. All other residents decide individually whether to get vaccinated or not. What fraction of Vaccinetown residents will get vaccinated in the Nash equilibrium?
Residents who are forced to be vaccinated are included in \alpha, so marginally, the utilities remain the same. So, 25%.
(c) (10 rp) Now suppose that cost of vaccination is different across the population. It is not 0.3 for everyone, but instead, it ranges from zero for some people to very high for others. The authorities want to shift the equilibrium so that more people get vaccinated by making vaccination mandatory for some of them. They know what groups of people have low, medium, and high vaccination costs and can choose for whom to make vaccination mandatory. Knowing that it is politically infeasible to make vaccination mandatory to everyone, what is your recommendation regarding this matter? Explain using the concept of Nash Equilibrium.
People with low costs of vaccination are likely to do it voluntarily. So, medium and high-cost population groups remain. Reasoning in terms of the Nash Equilibrium, forcing high-cost people into vaccination can make vaccination less attractive for some medium-cost residents (not to mention it would require more effort), so the total number of immunized people will not increase. For each of them, the probability of meeting a non-vaccinated person decreases (see part (b) with a similar effect). So, authorities should force into vaccination those with middle cost of vaccination.