Free response question

IEO past paper: 2021 International Economics Olympiad, open questions

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: $$\begin{array}{lcc} \text{Person P...} & \text{...and meets someone vaccinated} & \text{...and meets someone non-vaccinated} \\ \hline \text{...is vaccinated} & 0 & 0.05 \\ \text{...is not vaccinated} & 0.15 & 0.4 \end{array}$$ 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.