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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 "freeloading" (for lack of a better term) on people who already got the vaccine?

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    $\begingroup$ The herd immunity threshold is when the spread goes down by itself, it is not when one cannot get infected anymore. And Nash equilibrium is a solution concept for games, a mathematical model of a strategic situation. Without specifying the game, it says nothing. $\endgroup$ Mar 14 at 0:34
  • $\begingroup$ @MichaelGreinecker Yes, I'm aware that this is a wildly simplistic view of herd immunity - my point was that once vaccines start decreasing the probability of getting the disease, wouldn't the incentive to get the vaccine start decreasing too? Eventually, would there be some equilibrium point where people would no longer have an incentive to "switch" strategies? $\endgroup$ Mar 14 at 1:11
  • $\begingroup$ @MichaelGreinecker You do bring up a good point about me not specifying the game. I'm a complete novice with game theory (just 5 lectures in to the series, which hardly makes me an expert); any advice on how I can develop this better? I would, at a minimum, need to explicitly specify who the players are and what the payoffs for the different strategies that players could follow are, right? $\endgroup$ Mar 14 at 1:28
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    $\begingroup$ You may want to check out the volunteer's dilemma game. Section 4.3 of Osborne (2003) has a detailed example. $\endgroup$
    – Herr K.
    Mar 14 at 16:39
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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.

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    $\begingroup$ It is good that you point out the equilibria depends on the game. However the game you propose seems ill fitted to reality. The disease is not eliminated immediately once herd immunity is reached, thus vaccinations that protect one from bad outcomes can still be individually beneficial even if a small reputational cost is incurred. $\endgroup$
    – Giskard
    Mar 14 at 15:22
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    $\begingroup$ @Giskard, I wasn't trying to produce a general theory of human choice :). All games are oversimplifications of reality (if anything because you have more than 2-3 people who are playing in the real world). Herd immunity doesn't necessarily eradicate the disease (you still have cases of measles,even though >90% of the population is immune), but prevents it from causing an epidemic. Anyway there are people who don't vaccinate themselves or their children against measles or rubella, but would it there were an epidemic. I don't know why, but I can try to model their behaviour with the last game. $\endgroup$
    – didymus
    Mar 14 at 15:33
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Here there are many ways to go. As didymus was saying, it really depends on how one chooses to model the phenomenon. I propose a game where players have different payoff functions. As a matter of fact, having a large number of players with heterogeneous preferences is probably a more realistic model of the phenomenon we are trying to describe and it allows to obtain situations where not everybody get vaccined, but let's try to keep the game as simple as possible. I will assume that the game is simoultaneous, as this is the framework where Nash Equilibrium make sense.

Imagine a game with N players, each of them can choose either VAX or NO-VAX. There is a "herd-immunity" threshold n. Let us divide the players into three groups (A, B and K)

Description of the players:

Assume that a portion A (smaller than N) of them have a payoff function that makes VAX a strictly dominant strategy. These might be people that believe in vaccines no matter what, or people that want to get vaccinated because they are scared of the disease and they want to avoid getting seriously sick.

Assume also that a portion B (small enough that A + B is smaller or equal to N) has NO-VAX as strictly dominant strategy.

Finally, let us say that the rest of the population K (equal to N minus A minus B) is worried about the epidemic aspect of the disease, but these agents don't like the idea of getting vaccined and see the vaccine as a cost (say, -3). Thus, they will take the vaccine if this leads to herd-immunity, but they'd rather not. These agents's payoff functions are as follows:

i) if the total number of people getting the vaccine is above the herd immunity threshold (n), and player i chose NO-VAX, player i gets 10;

ii) if the total number of people getting the vaccine is below herd immunity and player i chose NO-VAX, player i gets 0;

iii) if the total number of people getting the vaccine is above the herd immunity threshold (n), and player i chose VAX, she gets 7;

iv) if the total number of people getting the vaccine is below herd immunity and player i chose VAX, she gets -3;

Nash Equilibrium analysis

Before starting, let us distinguish into different cases. Cases 1, 2 and 3 are immediate and maybe non-interesting, but they show easy ways to obtain Nash Equilibria with heterogeneous behaviors. Skip direclty to case 4 for something more interesting.

  1. A + B = N (simplest case: all players have str dominant strategies)

In this case, the only Nash Equilibrium is: players in the A group play VAX, players in the B group play NO-VAX. (And this means that if A is smaller than the herd-immunity thershold n, then herd-immunity won't be reached)

  1. A + B < N (some players have no dominant strategy) and K + A < n (herd-immunity cannot be reached)

As before, players in A choose VAX and players in B choose NO-VAX. Since herd-immunity cannot be reached, players in the group K won't get any benefit getting vaccined. Since these agents see getting vaccined as a cost, they will all choose NO-VAX.

  1. A + B < N (some players have no dominant strategy) and A > n (herd-immunity is automeatically reached, only with agents in group A)

As before, players in A choose VAX and players in B choose NO-VAX. Here there is a (unique, I believe) Nash Equilbrium where all players in A choose VAX, and all players in B and K choose NO-VAX. Agents in K choose NO-VAX because herd-immunity threshold is reached even without them, so they can enjoy the benefit of herd-immunity without bearing the costs of taking the vaccine. (They get 10 with NO-VAX and 7 with VAX, hence no profitable deviation here)

  1. A + B < N (some players have no dominant strategy) and K + A > n (herd-immunity can be reached)

Here we get to the "interesting" case. As usual, players in group A all choose VAX and those in B all choose NO-VAX. Now, we assumed that n is the number of vaccinated people that would lead to herd-immunity. So, we need still (n - A) player to choose VAX in order to obtain herd-immunity. There is a nash equilibrium where z (= n - A) players of group K choose VAX and the others choose NO-VAX. To be sure this is a Nash Equilibrium, we need to check that there is no profitable deviation for a player, GIVEN the actions of the other players. We can focus just on players in group K, as players in group A and B have strictly dominant strategies.

  • player i is among the z-group (those playing VAX): payoff = 7

         Say that player i deviates and plays NO-VAX. What happens? We take others'  
         actions as given, so there are (z - 1) + A < n  players choosing VAX: 
         herd-immunity is not reached. 
         Player i obtains a payoff of 0 < 7. Hence, player i has no profitable 
         deviation.
    
  • player i is not in the z-group (those playing VAX): payoff = 10 (he's playing NO-VAX)

         Herd-immunity has already been reached. If player i chooses VAX, he gets
         7 < 10. Hence, player i has no profitable deviation.
    
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