Does Nash Equilibrium predict the existence of vaccine reluctance? I was listening to a lecture on Nash Equilibrium, which stated that a Nash Equilibrium by definition occurs at a point that no players in the game have an incentive to change their strategy - everyone is playing their best response to everyone else. It occurred to me, though: it seems to me that 100% vaccination rate is not a Nash Equilibrium because herd immunity for a good vaccine would likely occur far earlier than that - once herd immunity is achieved, unvaccinated people would have far less incentive to change their strategy and get vaccinated.
Am I missing something, or does Nash Equilibrium occur at exactly the point where herd immunity is achieved - no one regrets their decision or wishes that they chose a different strategy, and no one has an incentive to change their strategy? (For any actual biologists reading, yes, I do realize that I'm being very simplistic about how vaccines and herd immunity actually works).
That being said, if I'm right about this, does game theory predict that some people would be reluctant to get the vaccine simply because they're &quot;freeloading&quot; (for lack of a better term) on people who already got the vaccine?
It depends on how you build the game. You may have games with several Nash equilibria, with a dominant strategy equilibrium (which is a special case of Nash equilibrium) or no equilibria at all.
Let us consider a game with 3 players, where herd immunity is achieved at 40% and where players incur a small cost for taking the vaccination and make a big gain when herd immunity is achieved. In this case you have 3 Nash equilibria (when 2 players decide to get the vaccine, nobody has an incentive to deviate).
In another game you may have 3 players who don't care about herd immunity, but that want to become immune (say the disease can come back at any time). In this case you have dominant strategy equilibrium: everybody gets the vaccine.
Finally let's construct a game with 2 players, where herd immunity is achieved at 40%. In this scenario players incur a small cost for taking the vaccination, and a bigger cost if they are the only player who takes the vaccination [1]. Again, herd immunity gives a gain. This can be modelled with a payout matrix such as:





PlayerA Vax
PlayerA No Vax




Player B Vax
1,1
2,-1


Player B No Vax
-1,2
0,0




then the game doesn't have a Nash equilibrium (you may have noticed that this is the same payout matrix of the prisoner's dilemma).
To answer your question, a game that includes vaccine reluctance may have several equilibria or none at all. Vaccine reluctance is an a-priori to your model, which can be factored in when building the payout matrix. Then you can analyze the game an try to predict possible outcomes.
[1] For instance if players are members of an anti-vax community/group, they may incur a reputation cost if other members don't get the vaccine.