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Would the affected communities voluntarily obtain herd immunity if a cure for COVID-19 becomes available? The conventional wisdom says “no” because of the fact that an individual's decisions does not take into account its positive externality to others, i.e., if you get a vaccine, you are improving the chance that other will get sick. That is why people argue that pandemic is a Prisoners’ Dilemma (PD) game. However, the current paper shows that a more careful investigation of the dynamic transmission of a disease may lead to a different conclusion.

We first study how an infectious disease transmitted via the SIR model, a well-established mathematical epidemic transmission model, and show that the marginal positive externality created by one more person being vaccinated is strictly increasing and becomes 0 at and after the society obtains herd immunity. Based on this finding, we formulate the vaccination game as a public goods game (PD game with more than two individuals) with non-linear externality. The unique mixed-strategy Nash equilibrium predicts that whether or not society obtains herd immunity via voluntary vaccination depends on 1) how infectious the disease is and 2) the relative size of the damage from the disease to the cost of receiving vaccines where both damage and cost covers not only monetary and physical components but also mental psychological ones. When an epidemic is not very infectious and the damage is not too large relative to the cost of vaccination, people have weak incentives to take a vaccine so that it is hard to obtain herd immunity. However, either when the epidemic is sufficiently infectious or when the damage is large enough relative to the cost of vaccination, there is a substantial chance for the society to obtain herd immunity via voluntary vaccination. The data from the controlled laboratory experiments confirms these theoretical findings.

COVID-19 is not very infectious relative to other epidemic (the basic reproduction ratio R0 is only 2-2.5). Moreover, the damage from COVID-19 varies largely depending on the age group. Our analysis predicts that age groups above 65 are likely to receive COVID-19 vaccines voluntarily. However, younger generations seem to have different incentives and they are unlikely to receive vaccines without having some incentives provided by the government. Various policies to decrease psychological cost of vaccines and to increase benefit from vaccines, such as tightly monitoring fake news and providing travel quarantine reliefs for those who receive vaccines, are recommended.