# The Vaccine Game

In my previous post I advanced the idea that people turn to moralism and from there to government coercion when faced with a free-rider problem. I didn’t provide a detailed, concrete example of the kind of free-rider problem I had in mind but David Foster provided me one: vaccinations.

Let’s approach this as would game theorists by creating an abstract game which we will call “The Vaccination Game.” The players in the game are parents trying to provide their children maximum protection from harm in all matters related to vaccinations.

All acts have benefits and risk, and vaccinations are no exceptions. The vaccination game is defined by 3 numbers:

D: The chance of catching a disease if unvaccinated
R: The chance of having a negative reaction to the vaccine itself
V: Percentage of the population that is vaccinated
NV: Percentage of the population that is not vaccinated.
P: The propagation threshold, i.e., the percentage of the population that must remain unvaccinated in order for the disease to have a chance of spreading through the population.

Lets assume that vaccinations are perfect, i.e., after vaccination the chance of ever contracting the disease is zero. (In reality, this is never the case.)

Notice that D,V,NV and P are all linked together but that R stands independent.

A rationally selfish parent faces two choices. First, they can get their child vaccinated and assume the risk of R, a negative reaction to the vaccine. Second, they can skip the vaccination, avoiding R but assuming the risk of disease, D.

No matter what happens, R remains the constant. A certain predictable number of children who take the vaccine will have a negative reaction that will result in an outcome as or more severe than the disease itself.

D, however, fluctuates wildly based on one factor: How many other parents also forgo vaccination. D is proportional to NV. As long as only a few parents skip vaccination, P>NV and the unvaccinated child avoids both disease and the risk of an adverse reaction to the vaccine. As the percentage of unvaccinated people in the population rises, the chances an unvaccinated person will contract the disease increases. When NV>P, D increases rapidly and not being vaccinated may suddenly become very dangerous.

While P>NV, parents who do not vaccinate get a free ride on parents who do. By assuming the risk of vaccination, vaccinating parents parents drive NV.

### 17 thoughts on “The Vaccine Game”

1. With rational actors given perfect information, a parent would give the vaccination when D ‘greater than’ R, otherwise not. So the equilibrium value is D = R. (With D appropriately normalized over lifetime.)

I suppose parents are “free riding” when they don’t vaccinate and D ‘less than’ R, but you could also argue that it is Pareto efficient to allow this. All those who don’t vaccinate are better off, and none of those who do vaccinate are any worse. And when D ‘greater than’ R, parents who don’t vaccinate are just acting irrationally.

So basically, the existence of the vaccine pegs the disease risk to R. I don’t see much of a moral dimension to the vaccine choice unless the vaccine or the information are imperfect. Of course, they both are.

2. Artd0dger,
With rational actors given perfect information…
That is rather the rub. To make the theoretical rational decision a parent would have to know how many other parents would choose not to vaccinate but others will vaccinate based on how many other vaccinate and so on and so on. A real world actor will never have the information needed to make a rationally optimal decision so the abstraction is useless in describing real world behavior.
…it is Pareto efficient to allow this.
Good point but irrelevant. From the perspective of a centralized planner or someone interested in some overarching common good that might make a good argument but from the perspective of a parent seeking to reduce risk for their child it means nothing. People won’t say, “my child takes risk to protect your child but its okay because it creates a Pareto optimum for society on whole.”

3. Shannon,
Maybe I’m a bit thick, but I’ve never understood how the vaccine example constitutes a collective action problem. The parent who gets her child vaccinated accepts the R risk, and is completely unaffected by the choice of other parents. There is no real sense in which she can claim a transgression in “free riding,” because what doesn’t affect her is really none of her business. The parent who doesn’t get his child vaccinated accepts the D risk, whatever it may turn out to be. He is affected by the choices of other parents, but he has no business complaining about it because he was unwilling to take the reciprocal R risk. His actions strictly increase the D risk, but only to other D parents like himself.

Perhaps there are good enough real-world reasons to justify coercive vaccination (like the epidemic nature of D), but I still don’t see how “free riding” is among them.

4. Shannon, I’ve just read the original post and comments, and I think vaccines would be a good example of what you’re talking about if you include a non-zero probability D2 of getting the disease even when vaccinated, as you mentioned in those comments. Otherwise the “cost” incurred by the free riders on the vaccinated is only 2nd order — for example, living in a community ravaged by a disease to which you are immune. BTW, I’m not at all trying to defend skipping vaccinations — I just got a flu shot today. But I think if they work well the moral dimension of an individual’s choice is quite diminished, leaving mostly the stupidity dimension.

5. ArtD0dger,

Lets define V as the percentage of the population who does receive vaccinations. So, 1-V=NV or D varies in the inverse of V. We can think of D as controlled by either NV or V.

In other words, the more people who accept vaccination and the associated risk, then the lower the chance an unvaccinated person will get the disease. At some point (when the unvaccinated population

6. Comment Deleted for Pretending to Know What Causes Autism.

7. Comment Deleted for Intelligently Arguing Against Someone Who Believes They Know What Causes Autism.

8. Hee hee.

9. The fact that you people refuse to discuss the themerisol-autism link renders any further discussion of the merits of vaccination moot. It is criminally irresponsible to push people into vaccination programs against their will.

10. Kurt9,

The fictional thermerisol-austism link doesn’t alter the basic dynamic. If it were true it would merely make non-vaccinating parents even bigger free-riders since they would gain the protection from infectious diseases without having to run the risk of conventional reactions to vaccinations and thermerisol induced autism.

Forgoing vaccination today is only non-suicidal because most people do get vaccinated. That massively reduces the chances that a non-vaccinated person will get diseased.

Think of it this way: The contemporary chances of a child getting autism is (IRRC) 1 in 166. The chances of an unvaccinated person getting a vaccinated disease in a population that is 98% vaccinated is something like 1 in 10,000. The chances of a person getting a fatal communicable disease in a 2% vaccinated population is 1 in 4.

Even if every case of autism in the world was a side effect of vaccination, people would be insane to forego vaccination unless they knew that the vast majority of others would assume the risk of vaccination and protect the unvaccinated as a side effect! If you knew that most people would not get vaccinated for any reason, then not vaccinating your own child would give them worse odds of survival than playing russian roulette with a six shooter.

It is criminally irresponsible to push people into vaccination programs against their will.

Well, that’s the question isn’t it? If I place my children at risk and you benefit from it, do I have any moral right to ask you to share the risk? I think this dilemma explains a lot of moralistic behavior. The target behavior of the moralism causes no immediate harm to other and if only a few engage it, it does little harm but if the behavior spreads disaster results.

11. Shannon,

Vacinated kids do not get the diseases that they are vaccinated against. Only the kids who are not vaccinated get these diseases. The vaccinated kids do not share any risk. There is no freerider problem here. You are making an issue over nothing at all.

12. Kurt9,

Vacinated kids do not get the diseases that they are vaccinated against

But they do run the risk associated with vaccination. More to the point, their vaccination interrupts the diseases transmission chain making every unvaccinated person less likely to encounter the disease. If enough people get vaccinated, the risk to the unvaccinated drops to near zero, possibly even less than the risk of a vaccine reaction. I think that people who run the risk of vaccination resent those who receive the benefits of having others vaccinated with also running the risk.

13. Shannon,
I don’t think themerisol has anything to do with this post since you have stipulated an idealized scenario, and any such risk would be encapsulated in the R probability of having a negative reaction to the vaccine. However, I still don’t see how this idealized scenario constitutes a genuine free rider problem. Your 11:43 pm post seems to have been cut short…

This discussion introduces the free rider problem in the context of an n-prisoner’s dilemma. The problem is in this situation, the “cooperate” response (i.e., take the vaccine) yields the same outcome regardless of the responses of the other players, so there really is no prisoner’s dilemma. There is only a self-interested individual action, coincidentally promoting an end that is not part of the individual’s intention. The logic of the situation dictates that individual action will drive the overall risk to R, so a collective decree is probably not even particularly utilitarian.

There are plenty of examples of genuine free rider problems, but there are also many people who seem to be disposed to use even the flimsiest allegations of externalities or market failures to justify curtailing individual liberty. This sounds more like the latter to me. Saying “people who run the risk of vaccination resent those who receive the benefits of having others vaccinated” as though resentment itself might morally justify intervention seems – somewhat uncharacteristic of what I usually read here.

14. ArtD0dger,

…so there really is no prisoner’s dilemma.

As my immediately previous post showed, the safety of unvaccinated children depends highly on the number of vaccinated children in the population. I think that is where the conflict arises.

Saying “people who run the risk of vaccination resent those who receive the benefits of having others vaccinated” as though resentment itself might morally justify intervention seems – somewhat uncharacteristic of what I usually read here.

I wasn’t trying to justify but just to describe. I am interested in the set of problems in which if a small number of individuals engage in a behavior then no harm results but if a large number do then harm does result. I think some behaviors we might qualify under the heading of moralism evolved to control such behaviors. Vaccination is just an example and perhaps a poor one since the risk of vaccination is very low and benefit is very high.

15. Shannon,

Thanks for touching on a hot topic to get my mind off the election. Thimerosal is still found in the flu vaccine, so if you have kids please don’t do that to them, especially if they are under 2. If anything because of the neuro development of a child, I would not recommend any vaccine before the age of 2 (80% of brain development occures before the age of 2). Many cases of Autism seem to occure during vaccine shot around the 18 month time period.

There has NEVER been a published study conducted by the CDC or the multitude of drug companies on the safetly of vaccines on our children. The biggest thing you can do to protect your child is make sure their immune system is functioning correctly. (proper diet, plenty of exercies, sunlight, correct spine alignment, and plenty of sleep)

Risk of Autism 1 : 150 (or higher)
Risk of Diabetes 1 : 400 (big increase in the last 40 year)

Compare this to the risk of serious complication of the other dieases that vaccines are suppose to protect you against. (but may not)

16. Mike Drew,

Did you even read this post or any of Shannon’s other posts on this topic?

17. Mike Drew,

(1) Thimerosal has been eliminated in childhood vaccination in many countries starting with Sweden in (iirc) 1992. Over 15 years of research has shown no alteration in autism rates. Case closed.

(2) Even the original study that started the hysteria was found to be badly flawed and perhaps fraudulent. [pdf]

(3) Even if Themersoal did cause every case of autism every in history of humanity and even if any vaccinations caused every case of autism, the benefits of vaccination would still outweigh cost by two orders of magnitude.

(4) If don’t want to vaccinate yourself I won’t force you. All I ask is that you wear some kind of emblem, visible at a distance, so that I can avoid you when you inevitably become a walking biological weapons factory and delivery system.

I know from previous experience that people like you hold your belief as a matter of faith and that no evidence I can advance will convince you otherwise. If you want to have a discussion on this matter you must in your next post answer this question:

What evidence could conceivable convince you that vaccinations DID NOT cause autism?

If you can’t answer that question, then you hold your beliefs out of faith and further discussion is pointless. I will delete all of your future post until you answer it.