> the individual risk of infection is low precisely because everyone else vaccinates their childrenYes, this is true. However, this is easy for individuals to take into account as the percentage of vaccinated individuals changes: if fewer people get vaccinated, the incentive for a particular individual to do so increases. So I don't see this in itself as a collective action problem.
> A Nash equilibrium generates a suboptimal outcome, where disease prevalence increases to the point where the perceived risks are equal.
Not quite until perceived risks are equal; until perceived costs are equal. In other words, the equilibrium (which can change over time as the percentage of vaccinated individuals changes--see above) is where the expected cost of getting vaccinated exactly equals the expected cost of not getting vaccinated.
This in itself does not mean the outcome is suboptimal. However...
> this has negative externalities for others, such as those too young to take a vaccine or who are allergic
...this does, since the externalities are not considered by individuals when making their risk-cost calculations.
> Collective action problems like this are usually resolved with some kind of coordinating authority. In some environments that authority can be normal social suasion, but in others it needs to be explicitly regulatory.
The main coordinating authority in our society seems to me to be schools requiring kids to be vaccinated. That sets a hard ceiling on the percentage of non-vaccinated children, since parents who do not want to vaccinate have to either home school or pay for some form of private tutoring.
The question would be whether this needs to be a hard requirement, vs. simply an assessment of how many kids are actually vaccinated in the population and whether that percentage is low enough to have a significant negative impact on herd immunity.
But even that's not really the question, since, as I noted above, individuals can factor in how much herd immunity there is when making their own vaccination decisions. The only real coordination issue is due to the negative externalities: some percentage of people can't get vaccinated and so herd immunity is the only thing protecting them. So the real question is whether the percentage of people vaccinated because they voluntarily choose to get vaccinated after considering their individual costs vs. benefits is low enough that the risk to the people who can't get vaccinated is unacceptably high because of decreasing herd immunity.
I don't know if anyone has actually tried to analyze the issue this way.