Can vaccine distribution and appointment setting up be considered game theory? New to game theory and I started thinking about the difficulty of setting up an appointment for the covid vaccine.
Could this be something to study in game theory?
Would we need to figure out the Nash equilibrium?
The way I see it is we have N players, and each player may or may not want a vaccine, but we also have a government player who wants the most vaccinated. Each player has some payoff.
Having spent time with a few eager students of game theory myself, I sense that the &quot;real&quot; question you might have wanted to ask was:

What is the best way to motivate people to get inoculated, given that they have varying attitudes towards the vaccine or vaccination in general.*

To answer that question from a normative perspective, i.e. assuming everyone is rational in the economic sense, we actually need the theory not of games but of mechanism design. Game theory's primary concern is to predict the outcome(s) of some given strategic interaction, regardless of whether those outcomes are desirable (e.g. mutual defection is the predicted, albeit inefficient, outcome in Prisoner's Dilemma). On the contrary, the goal of mechanism design is to implement some desirable outcome via a strategic interaction; that is, to come up with a suitable game whose equilibrium is the outcome desired by the mechanism designer. See Dixit et al. (5th ed, ch. 14-15) for an easy introduction, or Tadelis (2013, ch.14) for a more rigorous treatment at an advanced undergraduate level.
Of particular relevance here is a special case of mechanism design called information design, where the designer (e.g. a government or public health authority) uses a suitable information revelation strategy (e.g. when and what data to publish) to persuade the public to get inoculated, given that they hold a variety of initial beliefs about the efficacy of the vaccine and will rationally update their beliefs based on information released by the designer. Information design is still at the forefront of economic research, so the linked JEL paper, which I consider above the undergraduate level, is probably as introductory a summary on this topic as you can get for now.
There is also a positive approach to address the above question. By positive I mean the  approach is founded on the acknowledgement that people are not perfectly rational and are predisposed to act in a certain way when appropriately primed. A prominent example is the nudge theory, which draws on insights from psychology and economic experiments to frame choice problems in such a way that people are more likely to pick the option favored by the so-called &quot;choice architect&quot; (the behavioral economic counterpart of a mechanism designer).
Lastly, although there is behavioral game theory as such, I feel that it's not as relevant to your question because much of the literature is devoted to explaining the gaps between actual play and predicted play of some canonical games.

*I'm not quite sure what you mean by &quot;appointment setting&quot; beyond its logistic aspect, so I will ignore it.