## Anti-vax and The Prisoner's Dilemma? Different games to model the "anti-vax movement"

*in*Literature

You know me... when the term "Prisoner's Dilemma" comes up, I appear.

I wonder if anyone has looked at anti-vaxxers in the context of the prisoner's dilemma and game theory yet?

— skullsinthestars (@drskyskull) February 10, 2015

Game theorists and philosophers ahoy. Isn't this vaccine brouhaha an absolutely perfect articulation of the Prisoner's Dilemma? — Hugh Laurie (@hughlaurie) February 8, 2015

This isn't necessarily a mind-bending concept. There's even been a PNAS article on this exact thing, alongside four different blog posts (that I found; there's probably more) attacking the anti-vax "movement" for being everything including immoral and irrational while using The Prisoner's Dilemma as an analog for people who are anti-vax.

I describe two games that somewhat model the choice patterns of anti-vaxxers.

### Rosenbaum's Vaccination Dilemma

Will Rosenbaum had the most "gameified" version of the vaccination dilemma, which I reiterate here:

Let's just assume that vaccines are more harmful than choosing not to vaccinate. We can assign a "harm value" to each variable:

Vaccination = 4 harm points

No vaccination = 1 harm point

However, the fact is that not vaccinating your child puts the entire population you encounter at risk. Rosenbaum says that for every person who doesn't vaccinate adds "k" harm points to the entire population -- where *k* = the amount of people who don't vaccinate.

So, step two of this game would factor in the additional *k* harm points:

Vaccination = 4 + *k* harm points

No vaccination = *1 + **k* harm points

Rosenbaum states that you can choose to selfishly not vaccinate -- incurring the least amount of harm to yourself -- but the more people don't vaccinate, then the more harm they do to each other. Thus, the more people who don't vaccinate may think they are choosing a more healthy approach, but in reality they are choosing the worst possible outcome.

To visualize this game, I made some nifty game graphs:

In this game, two people choose to vaccinate and four choose to not vaccinate. Since 4 people don't vaccinate, *k* increases to 4. Comparing one individual who doesn't vaccinate (5 points total) to someone who does (8 points total) shows that it does "less harm" to not vaccinate, but the grand total of harm points equals to 28.

In a different game, all 6 people choose to vaccinate. Since no one chooses the anti-vax option, there is no additional harm to the population (which makes sense, because a completely vaccinated population puts 0 people at risk for disease). The total harm points are 4 points less than the harm points in the initial game.

But what happens when the *next* generation chooses to do what you just did?

In the second generation of the first game, you ideally carry-over the points you had in the previous generation then add your choice score (4=vax, 1=no-vax) and the population *k* score. In this case, someone in the second generation decides to not vaccinate, increasing the *k* from 4 to 5. In turn, this inflates the population's harm score to 75 points.

In the second generation of the second game, you again carry-over the points you had in the previous generation, add your choice score, and finally add the population *k* score. Just like in the other 2nd generation game, a second generation decides to switch to no-vaccine. This increases the *k* score from 0 to 1. However, because less people are putting each other at risk, the total harm points equal 51.

Clever game, Rosenbaum. Clever.

If we want to expand on this game, consider "end game criteria". The goal of the game is to keep every column (let's call that a "family") "alive" with the least amount of harm over the most amount of generations. Obviously, if everyone is harmed to some maximum extent, they die -- thus, the end of the game. Let's consider that 120 harm points for the entire population. If a family sums to 20 points, that family dies. You play until every family dies. After every generation, total up the amount of family lines that are still alive. Your grand total once everyone is dead is how many "survival points" your game earned.

In any game, if everyone vaccinates, then every family survives for 5 generations. In a game of 6 families, that's 24 survival points (you don't count the generation they die on because they are dead).

In a game of 6 families, if every family doesn't vaccinate in any generation, everyone dies on the third generation. That would be 12 survival points.

In a game of 6 families, if three families always vaccinate and three families never vaccinate, the vaccinated families die on generation 3 and the anti-vax families die on generation 5. That's 18 survival points.

The con here is that the anti-vax families live longer at the expense of families who vaccinated. In reality, families who vaccinate should be the ones who survive longer. But the main factor for this is that we are considering this game from the view point of *anti-vax equates to less harm*, which is not actually true.

### Jamming it into The Prisoner's Dilemma: The Vaccination Version

The Prisoner's Dilemma in a repeated-game form works based off of a points matrix.

You can make one of two choices: to cooperate or to defect. Your partner can also make the same choices: to cooperate of to defect.

We can visualize the four different outcomes, with some example points, like so:

Choosing to cooperate introduces risk, while defecting has the least amount of risk. However, as your partner catches on to your strategy, they too may choose the less risky route and defect, decreasing the amount of points you earn total.

We can apply some logic to this game: the less you cooperate, the less you risk and therefore the more likely you are to earn more points than your opponent. However, the less you cooperate, the lower your chances of maximizing your outcome. For example, someone who always defects is predictable and therefore the chances of me defecting is very likely because I am predicting when you will try to make me a "sucker" (aka, when I earn 0 points and you earn 5 points).

On the other hand, the more you cooperate, the more you risk but also the more likely you are to earn points. For example, someone who is cooperating makes me more inclined to cooperate as well because joint-cooperation earns more than joint-defection, and I want to maximize my points.

Let's consider cooperation as the vaccination choice and defection as the no-vaccine choice.

The reward and punishment payoffs make sense because if you and your partner vaccinate, then you are rewarded (maybe because you aren't spreading illness), whereas if you both don't vaccinate, you are not rewarded. However, the payoff is better to not vaccinate because you can sucker a vaccinated person.

Logically, this doesn't make exact sense because really there is no benefit to not vaccinate. Vaccination saves lives. And choosing not to vaccinate is not a benefit to you (or in reality, your child) because that exposes your child to disease and illness -- definitely not something to get points for.

Really, you should gain points for vaccinating when someone doesn't because you are preventing your child from death. We could modify the payoff matrix a little to look like this:

In this matrix, you can see that if you vaccinate, you can either earn 5 points or 3 points. However, if you don't vaccinate, you can either earn 1 point or no points.

Really, this doesn't become a game anymore because the best choices are lumped into one category: to vaccinate.

But that makes sense. Because vaccination isn't a game. And vaccination can only benefit you.

### Take home: In a game model, you can choose to not vaccinate now and be "rewarded" but over time, you lose.

Two things are problematic with trying to jam vaccination into a game model:

1) A game implies strategy. In a two-choice game (in this case, to vaccinate or to not vaccinate), your ability to manipulate your opponent into vaccinating while you don't is considered a potentially optimal strategy. However, unless your end goal is to ultimately kill off everyone around you (sort of like Rosenbaum's Vaccine Dilemma), the strategy of being antivax in a game model presents no longevity. It is a win-now, lose-later strategy, which in most cases comes back to haunt you (e.g. taking out a loan you can't pay back can lead to ruined credit and the inability to buy a house or car or take out future loans).

2) A game implies competition. There are very few games where the population is playing for the same goal. Even in "dealer games" where you are taking money from a dealer/casino/bank, you are still competing and ultimately trying to "beat" the dealer. Vaccination isn't about winning or losing. It's about survival and quality of life. Like in figure 7, vaccination should really be considered a "no-brainer". You can either win or win more. You can consider "winning" as preventing the spread of a disease and "winning more" as encountering a carrier for something you were vaccinated for. You are no longer competing against people who don't vaccinate because vaccination will always earn you more points than not vaccinating. It is literally a foolish enterprise to try and compete with a clearly losing strategy in a game model where only one strategy wins.

If you couldn't add up properly to 36 instead of 28 or viceversa to 20 instead of 24, I can't really understand how you could go on with a whole theory on that... Science's mysteries