Game Theory on vaccination
The article: https://www.pnas.org/content/101/36/13391
The source of the article:
Chris T. Bauch and David J. D. Earn
PNAS September 7, 2004 101 (36) 13391-13394; https://doi.org/10.1073/pnas.0403823101
Edited by Maurice R. Hilleman, Merck Institute for Vaccinology, West Point, PA, and approved July 21, 2004 (received for review May 28, 2004)
For the parents with newly born babies, they face a prison dilemma because of the voluntary vaccination policies. Let’s suppose Family A with a new-born baby. While most of the vaccines are proved safe, there is still a chance to cause deaths and they can cause negative effects to the babies and families, such as potential short-term or long-term side effects, financial costs, time costs and pain from injection, etc. When most of the population around Family A have themselves vaccinated, it will have little probability for the baby in Family A to get infected even the baby hasn’t taken vaccination. In this situation, Family A will have a higher payoff by choosing not to vaccinate. But when a less surrounding population are vaccinated, the risk of getting infected with no vaccination will increase by a large extent. In this situation, Family A will have a higher payoff by choosing to vaccinate. So, there exists a prison dilemma and each family is a player in the game: Players have to choose between two strategies with different payoffs on whether to take vaccination or not, depended on the choices made by other players in the population.
This article particularly considers the game theory in the vaccination situation and incorporate the epidemic model into the theory of game. This study can be particularly useful under the pandemic age, when people also need to choose between vaccinating or not vaccinating based on their payoffs and decisions made by others.
This article uses the risk of morbidity of babies, either from vaccination or infection, to measure the payoff of the whole population. The article uses this concept to construct the following equation, where r v and r i to denote the morbidity risks from vaccination and infection, respectively, and πp to denote the probability that an unvaccinated baby will eventually be infected if the vaccine coverage level in the population is p. And E is the probability of mobility for the whole population because of either vaccination or infection. Then the article uses the concept of Nash Equilibria to solve for P, which denotes the probability of a single family to choose vaccination. Worth notably, from the whole population level, P can also denote the probability of the whole population to choose to vaccinate.
Since there are two variables in this equation, we can scale on one variable into constant. Let r = r v/r I,, then the equation can be written as the following.
This article introduces a new concept that is not mentioned in the class: convergently stable Nash equilibria. It is used in the context of the population level and can be used to estimate the probability of a collective behavior. Incorporating the Game Theory and the epidemic model, this article manages to solve for a convergently stable Nash equilibria (CSNE), which is a unique strategy that P=P*. With strategy P=P* convergently stable, families will choose to start playing strategies near P* and ultimately adopt P*. Then we can use P* to represent the percentage of population who choose to vaccinate.
With a series of mathematical induction, the article has the result as shown in the following graph. At each (lifelong immunity) level, there exists a relationship between r and P*. Then we can use the P* to estimate the number of families in the whole population that will choose to take vaccination, which is socially significant and can help to determine many public health policies.
