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No, It’s Not a Prisoner’s Dilemma (the second in a continuing series):

The prisoner’s dilemma is the original counterintuitive hot take. Some social scientists and journalists just looove that dilemma because of how delightfully paradoxical it can be.

But some situations that are described as prisoner’s dilemmas aren’t really.

I discussed one such example in my article, Methodology as ideology: Some comments on Robert Axelrod’s “The Evolution of Cooperation.” For more background on that story, see here, here, and here.

Another example came up recently. Paul Alper points us to a news article entitled, “The Pandemic is a Prisoner’s Dilemma Game,” which states:

People who choose not to be vaccinated effectively get a free ride, reaping the benefits of reduced virus transmission generated by the people who do opt for vaccination. But the free rides generate a collective threat. That is the prisoner’s dilemma . . . Vaccination decisions based purely on self-interest can lead to vaccination coverage that is lower than what is optimal for society overall . . . It boils down to a fundamental problem known as the tragedy of the commons . . . There is a misalignment of individual interests and societal interests.

I don’t buy it. Indeed, much of the discussion recently is about who gets the vaccine first. In the prisoner’s dilemma formulation, it’s good for you if other people get the vaccine, but you personally benefit from not getting it. That doesn’t make sense, given that the vaccine is reducing your risk from coronavirus complications. Sure, I understand that some people are afraid of getting vaccinated, but that’s another story. People have all sorts of fears, but I think people’s general attitude is that they personally want to be vaccinated so they can reduce their risk and then go about their lives without fear.

In short, I don’t see this as a prisoner’s dilemma, and this looks like one more case where journalists and social scientists making something look more subtle than it actually is.

P.S. As usual, Palko was ahead of the rest of us on this one.

52 Comments

  1. Anoneuoid says:

    Id guess most people wary of the vaccine have also been those behaving in ways that increase their risk of exposure. Ie, they are not afraid of covid.

    So do they actually consider others being vaccinated as a benefit to themselves?

  2. Jonathan (another one) says:

    But the Prisoner’s Dilemma doesn’t depend on *general* attitudes. It depends on there being a sufficiently large mass of people whose personal costs of vaccination are sufficiently high that they choose not to get vaccinated and whose risks of disease are reduced sufficiently through mass vaccination that they would prefer to get vaccinated if the other laggards could be dragooned along with them. I agree that the parameters to make that happen probably aren’t right, but you can’t just cite a general tendency to prove that.

    • Phil says:

      The key characteristic of the Prisoner’s Dilemma is that if you do what is best for you personally, conditional on what your counterparty is doing, then you make things worse for your counterparty. And since this is true of both you and your counterparty, you both end up in the worst part of the payoff matrix.

      How does taking the vaccine make things worse off for your counterparty? I understand that you’re saying that the “counterparty” is a “large mass of people” but I don’t see how that changes things. If you take the vaccine, you’re certainly not hurting anyone else; indeed you are helping them.

      • malcolmkass says:

        Well, I think you might be off here.

        “How does taking the vaccine make things worse off for your counterparty?”

        Public goods have that same strategic nature as PD games where if I invest in a public good, my counterpart benefits from that temptation payoff. But I am also helped if my counterpart does decide to invest as well.

        In this case, I think it’s “how does taking the vaccine make you worse off if your counterpart doesn’t take it?”. Vaccination isn’t a PD game because once you are vaccinated, your payoff doesn’t depend on the actions of the other party anymore.

  3. I think a whole bunch of people DO want the vaccine, but it’s nowhere near the say 70-80% needed to keep the virus from being able to spread exponentially. Of course some people will have natural immunity but that clearly wanes over a 3-6mo period.

    If 50% of the population enthusiastically gets vaccinated, it’s not necessarily enough to prevent continual nursing home deaths and elderly people dying from visiting their grandkids etc. So this is more about how big are the groups not a binary issue of yes vs no

    • Jordan says:

      “Of course some people will have natural immunity but that clearly wanes over a 3-6mo period.”

      Can you please share some sources that discuss this?

      I ask because I’m under the impression that immunity likely persists for at least as long as the pandemic itself (so, no less than a year). I’m aware antibodies wane over time, but that’s not the same thing as immunity itself returning to anything like a naive state. If immunity truly waned after six months, wouldn’t we be seeing far higher re-infection rate by now?

      • Phil says:

        I, too, thought reinfections are fairly rare, suggesting that natural immunity for most people lasts at least 6 months. But maybe there’s more recent news on this that I have missed.

        • Don Baccus says:

          Confirmed reinfections are very rare. If immunity really disappeared in 3-6 months, we’d be seeing a lot of reinfections.

          While it is true that antibody levels drop fairly rapidly after recovery, long-term immunity doesn’t really come from keeping your body awash in antibodies from every disease you’ve been exposed to for the rest of your life.

          Memory B cell immunity is where longer-term immunity comes in:

          “Memory B cells (MBCs) is a B cell sub-type formed within germinal centers following primary infection. MBCs can survive for decades and repeatedly generate an accelerated and robust antibody-mediated immune response in the case of re-infection (also known as a secondary immune response).”

          Every thing I’ve read indicates that there’s no reason why memory B cells are being generated as part of the immune response.

      • Dave says:

        I was also under the impression that immunity is expected to last ~12 months for those who were infected. Even after antibodies decline, T-cell immunity et al kick in for awhile. I’m sure there’s still some uncertainty around it, but here’s a paper from November where they provide evidence that T-cell immunity lasts at least 6 months: https://www.bmj.com/content/371/bmj.m4257

    • Joshua says:

      A total of vaccinated + infected and recovered reduces the rate of spread (and chances of encountering an infected person) even short thst total is short of reaching a “herd immunity threshold.

    • When I said “wanes” I didn’t mean “disappears” just reduces. I personally know of a couple people who have had the virus twice, so it can’t be *that* uncommon. It seems to be that when people have a first case that’s very mild the second case is more easily contracted. (probably first case didn’t produce a major immune response)

      Here is the kind of stuff I’ve read: https://www.cnn.com/2020/10/26/health/covid-19-immunity-wanes-large-study-finds/index.html

      • Dave says:

        “Researchers who sent out home finger-prick tests to more than 365,000 randomly selected people in England found a more than 26% decline in Covid-19 antibodies over just three months.”

        Yeah, that’s focused on antibodies. I guess time will tell about the T cells and B cells, and also the extent to which immunity is effective across strains.

  4. Jordan says:

    “Sure, I understand that some people are afraid of getting vaccinated, but that’s another story.”

    I think this is the story though. That’s what makes it reasonable to consider the prisoner’s dilemma framing.

    The risk of complications from the vaccine is non-zero, so the value of taking the vaccine necessarily diminishes as immunity accumulates in the population. You seem to be arguing (understandably) that the risks are so vanishingly low that the tipping point where it makes sense to skip the vaccine becomes absurdly extreme (e.g., you’re the last person on earth without immunity). However, what makes it a “dilemma” in a person’s mind is the perceived risk. If people perceive the vaccine as much more risky than it actually is, then the decision-making process itself could be fairly described in the tragedy of the commons / prisoner’s dilemma framework.

    All that said, I still lean toward your conclusion that the attention given to this kind of framing is more about selling ads, gaining followers, signalling intelligence, etc. than about having an important discussion around the reasons why people might skip the vaccine.

  5. Chebyshev says:

    The other possibility is that, after having seen the mortality data, a large % people don’t consider covid19 deadly at all.
    So taking a vaccine may not make any sense to them.

    • Chris Wilson says:

      “Having seen misleading analyses and tribal in-group commentary on mortality data, they have concluded it is not deadly”
      There I fixed it for you!

      • Jordan says:

        Don’t be silly. Those with the lowest risk are perfectly reasonable in assuming they will be fine if they become infected. That doesn’t mean the risks of Covid for them are lower than the risks of the vaccine nor does it mean there aren’t other good reasons to get the vaccine. It just means that risk of death really is low for some groups of people. No distortion of the evidence is necessary to lead people to this belief.

        • Joshua says:

          Their risk for serous illness is less small and their risk for a pretty inconvenient illness is less small than that. Plus their risk of infecting grandma ain’t all that small.

          Further, their relative risk of death is significantly higher even if their absolute risk of death won’t be. How many people won’t get a vaccine but will have health insurance to protect against the costs of, say, hospitalization from an auto accident?

          • Jordan says:

            “That doesn’t mean the risks of Covid for them are lower than the risks of the vaccine nor does it mean there aren’t other good reasons to get the vaccine. It just means that risk of death really is low for some groups of people.”

            My caveat covered your point.

        • Chris Wilson says:

          You answered your own rebuttal. I was making a joke to point out the point of large scale vaccination efforts is to achieve a population level outcome.
          Besides, most of the people who downplay the deadliness of this disease are not basing that on rational analysis of the data, but rather the factors I mentioned.

          • Jordan says:

            “Besides, most of the people who downplay the deadliness of this disease are not basing that on rational analysis of the data, but rather the factors I mentioned.”

            Given the fact that ordinary people are fully capable of making rational judgments with data as straightforward as mortality data, I’m going to go with the more parsimonious explanation that people simply looked at the data. Unless you have some evidence that suggests we ought to adopt your brainwashing theory, I’m gonna stick with my priors on this one.

            • Joshua says:

              > Given the fact that ordinary people are fully capable of making rational judgments with data as straightforward as mortality data, I’m going to go with the more parsimonious explanation that people simply looked at the data.

              You think it’s parsimonious to say that because they’re capable thst means they do it?

              You’re kidding. Right? The parsimonious assessing is that they don’t bother to evaluate data. They just go with what fits their biases. We have tons o’ examples.

      • Chebyshev says:

        You ain’t Bayesian, are you?

        • Chris Wilson says:

          Huh?
          It was mainly a joke about the quarters I have seen the lack of concern coming from. Joshua is way over-reacting above. My point is that vaccines require a large fraction of population to get them irrespective of their personal risk calculus in order to work as intended. This is a deadly virus for certain segments of society and widespread vaccination among otherwise low risk, healthy people is a key part of how we can protect them. Especially since we largely failed at deploying our other tools to kill this thing in the egg like we should have done.

          • Joshua says:

            Joshua? Or Jordan?

            • Chris Wilson says:

              Sorry Joshua! I meant Jordan. Ah the perils of communicating on the internet :) On the whole though this blog has a great quality of conversation, even through disagreements…

          • Jordan says:

            So when you said:

            “Having seen misleading analyses and tribal in-group commentary on mortality data, they have concluded it is not deadly”

            What you really meant was:

            “Vaccines require a large fraction of population to get them irrespective of their personal risk calculus in order to work as intended. This is a deadly virus for certain segments of society and widespread vaccination among otherwise low risk, healthy people is a key part of how we can protect them. Especially since we largely failed at deploying our other tools to kill this thing in the egg like we should have done.”

            ???

            If you’d just said this the first time, then maybe I wouldn’t have assumed that you were saying that young healthy people who believe their risk is low must have been brainwashed by their tribe.

    • elin says:

      1 of every 1000 people in the US has died in just about 9 months. That is somewhat comparable to measles prevaccine (a bit higher).

      “In 1912, measles became a nationally notifiable disease in the United States, requiring U.S. healthcare providers and laboratories to report all diagnosed cases. In the first decade of reporting, an average of 6,000 measles-related deaths were reported each year.

      “In the decade before 1963 when a vaccine became available, nearly all children got measles by the time they were 15 years of age. It is estimated 3 to 4 million people in the United States were infected each year. Also each year, among reported cases, an estimated 400 to 500 people died, 48,000 were hospitalized, and 1,000 suffered encephalitis (swelling of the brain) from measles.”

      So it’s really on the same scale in terms of numbers infected but higher on deaths. So yeah, we can “look at the figures” and say that this is a disease that we would routinely vaccinate for and have in the past with great success except in populations that don’t vaccinate.

      https://www.cdc.gov/measles/about/history.html

  6. Radford Neal says:

    It’s not a prisoner’s dilemma because not getting vaccinated is an incorrect decision if you find out that other people are not getting vaccinated. In contrast, in the prisoner’s dilemma, you defect even if you know that that other person has defected.

    I’m assuming here that the risk from side effects of the vaccine is small enough that it’s desirable to get vaccinated if nobody else has been. If that’s not true, it’s just a bad vaccine, which nobody should take. And I’m assuming that the risk from side effects is large enough that there’s some fraction of the population being vaccinated past which it’s desirable to not be vaccinated yourself. If not, it’s just a risk-free vaccine that everyone should take.

    (Of course, in practice, people will disagree about the risk, but that’s a completely different phenomenon.)

    Neither is this necessarily a tragedy of the commons type situation, as long as the number of people who have been vaccinated is public knowledge. With perfect rationality (I know…) people will initially get vaccinated, until the fraction vaccinated is high enough that the risk of side effects leads the remaining people to decline vaccination.
    It seems to me that this may well be the optimal result.

    My only reservation on that is if that level of vaccination isn’t quite enough to push R less than 1, whereas it would be pushed less than 1 if some number of additional people greater than one were to get vaccinated, and the disease would then be wiped out. I could imagine then that each individual’s rational decision is to not get vaccinated, even though if many of them coordinated to all get vaccinated they’d wipe out the disease and all be better off. But I think one would have to work out the details further to be sure that they’d really be rational to refuse the vaccine in this circumstance, given that if R remains above 1, they’d be at nearly the original risk of getting the disease.

  7. Zhou Fang says:

    I guess it could be a prisoner’s dilemma if the powers that be charge $(sufficently high) a dose for the vaccine, I guess.

    • malcolmkass says:

      I think you need to assume something else because the payoffs from vaccination still will not depend on the actions of your counterpart. Maybe the vaccination is individually costly, but with social benefits of economic recovery.

  8. Dave says:

    It’s a free riding situation with prisoners’ dilemma incentives if the reason people aren’t getting vaccinated is because they expect the vaccination of others to benefit them with some protection but think there are costs/risks to getting it themselves.

    If they are just general anti-vaxxers, or they don’t think it will work at all, then it’s not a prisoner’s dilemma for them.

    Once we get past the health care workers and front line workers, I think a prisoner’s dilemma probably will explain the reluctance of some, but people will judge the payoffs differently, and hopefully enough will get it early.

    Also, to those who think the vaccine is risky, they will see option value in waiting. They may be wary that a two month trial isn’t enough to reveal longer run side effects, but four months from now, when those trial participants have had it for six months, perhaps they will feel more confident. So there’s an element of sequential decision making here too, but that’s not really a repeated prisoners’ dilemma because those who vaccinate early won’t have to do so again (for awhile anyway).

    In short: yes and no.

  9. Jim Ancona says:

    I agree with Andrew, this isn’t a Prisoner’s Dilemma. It just that vaccination has a positive externality–each person vaccinated pushes us towards herd immunity. Herd immunity is a public good, i.e. it’s non-excludable and non-rivalrous, so it’s likely to be under-provided in a free market, because the person being vaccinated only gets the private benefit (they’re less likely to get Covid). They get (or don’t get) the benefit of herd immunity whether they’re vaccinated or not.

    I’m not sure how much of an issue this is likely to be in practice. With a 90% effective vaccine with apparently minimal side-effects, the private benefits are large and the costs are low. Right now it’s a moot point–way more people want the vaccine than there are doses available. If convincing people to get vaccinated is still a problem when we have enough vaccine available, we could use incentives to try to make up for the externality. For example, we could allow people who have been vaccinated to do things that are otherwise restricted, like drinking in bars or attending sports events and concerts. We could also just pay people to get vaccinated, or alternatively, fine them if they don’t.

    • Andrew says:

      Jim:

      Yes, perhaps part of the confusion is that people think “public good” . . . “tragedy of the commons” . . . “prisoner’s dilemma”! I was annoyed at the above-quoted news article partly because it was written in the scientist-as-hero paradigm and partly because I’ve seen this prisoner’s dilemma fallacy before—it was the subject of my undergraduate thesis, many years ago.

  10. Joshua says:

    I’m guessing that more people are going to get a vaccine than originally said they wouldn’t.

    >… Sure, I understand that some people are afraid of getting vaccinated

    That said, my guess is that not an insignificant # of people are going to choose against getting a vaccine, not because of a fear about its safety, but because of an ideological orientation.

    This isn’t just some decontextualized decision-making exercise. Some not insignificant # of people don’t want to get a vaccine because they have an investment in a political ideology that rejects the need for a vaccine – because the pandemic is a part of a plot to destroy capitalism. And because it’s part of Bill Gates’ plan to get rich by forcing people to sacrifice their freedoms.

    • Chebyshev says:

      There are simpler explanations. One does not have to resort to conspiracy theories of people believing Bill Gates taking away freedoms or ruling the world.

      My hypothesis is that it is a simple risk or nuisance calculation for a majority of people not taking vaccines. Media always blows up crazies on both the right and the left.

      Is it a norm to get a flu vaccine? What % of people did not take the (annual) flu shot in 2019? Plenty of grad school educated people don’t flu shots. They are not anti-vaxx. Just not worth the trouble.

      • Jordan says:

        I generally agree with you that the more parsimonious explanations are that people are simply looking at the data, but I don’t think the comparison to the flu is quite right here. Many young healthy people with the flu experience nothing more than the sniffles for a few days, often without feeling a need to stay home from work/school.

        Although young healthy people are unlikely to die from Covid, there is a much higher chance that it interferes with their lives in ways that outweigh the inconvenience of a vaccine. Given the prevalence of data suggesting that Covid sucks even for young healthy people, I think its possible that a lot of young healthy people believe the costs of taking the time to get vaccinated are outweighed by the benefits of not feeling like shit for several days on some random date in the future.

        Personally, I’d rather spend one hour to get the vaccine than spend a few days in bed feeling like shit. It also means that if I need to go to the hospital for another reason (e.g., a car crash), I won’t get Covid while recovering from that other thing. I guess my point is that a young healthy person can come to the conclusion that the vaccine is a good idea perfectly rational reasons.

      • Joshua says:

        > There are simpler explanations. One does not have to resort to conspiracy theories of people believing Bill Gates taking away freedoms or ruling the world.

        Simpler explanations are available. One doesn’t have to…..

        But we know that a not insignificant # of people that believe there’s a conspiracy afoot. There is a real cohort that believe in the “Plandemic” and that this is all a hoax to “reset” the global and polirical world order.

        I doubt that very many people calculate the relative odds here. And if you don’t think that COVID presents any real risk, that it’s just basically the sniffles that the risk is inflated by false positives from PCR tests and mis-attributed causes of death and docs trying to get extra money, etc., etc., then there’s just no reason to get a vaccine being pushed by big Pharma.

        Don’t try to generalize about how people weigh odds be generalizing feom your own analysis or that of people who frequent the comments section of a statistics blog.

      • Joshua says:

        > Is it a norm to get a flu vaccine? What % of people did not take the (annual) flu shot in 2019?

        Practically speaking, flu vaccine has virtually zero political/ideological weight. There are some antivaxers who extend their views about other vaccines to flu vaccines, but there’s little political signal that surfaces.

        COVID is a political/ideological proxy.

  11. anon says:

    | Vaccinate | Not Vaccinate
    Vaccinate | 1, 1 | 0, 2
    Not Vaccinate | 2, 0 | -1, -1

    I’m sure most people are familiar with the classic prisoner’s dilemma table, but I thought it would be helpful to put up a table here to make things concrete. I think Andrew’s explanation for why this scenario is incomplete. We can include fears of vaccination and still show that this isn’t a PD.

    In the table above:
    * If everyone else vaccinates and you do too, you get 1 utility, since you will be free of COVID. If you don’t vaccinate you get 2 utility since you dont have to confront your fear of needles and are now safe from COVID. Therefore, if other people vaccinate you should not vaccinate.
    * If no one else gets vaccinated, you get 0 utility from vaccination since your protection is imperfect in the hordes of unvaccinated carriers, and you got jabbed with a needle. You get -1 utility if you don’t vaccinate and no one else does because now your covid risk is not reduced at all. Therefore, if no one else vaccinates, you should vaccinate.

    This is not a prisoner’s dilemma. For it to be one, *regardless of what everyone else does*, it has to be better for you to choose the socially worse outcome (i.e. not vaccinate). In this case, for it to be a prisoner’s dilemma, it would need to be worse for you to vaccinate in a world where no one else does than it would be for you not to. That would only be the case if you think that the value to you of the vaccine is unconditionally negative, and if that is the case, then there’s no need for the prisoner’s dilemma at all! No one should get vaccinated ever.

    Various disclaimers, utility values above are ordinal and hypothetical provided for theoretically illustrative purposes only.

    • Kasey says:

      Well it seems clear that we won’t reach 100% vaccine coverage, so you should really change “everyone vaccinates” to “most people vaccinate”. Given this, it seems dubious that there would be a higher payoff to not vaccinate given that others do.

  12. The Other Guy says:

    Sounds like the chicken game. Each wants the other to vaccinate but not do it themselves. Maybe that partially explains the belligerence of anti-vaxxers. Being perceived as crazy is useful in a game of chicken, as I recall from middle school.

  13. Steve says:

    I think that a prisoner’s dilemma can arise here. Suppose the vaccinations have proceeded without a hitch. Now, we need one more person to obtain herd immunity, you think to yourself, someone else will get vaccinated and I and society will be protected without my having to get pricked in the arm. But, everyone else is just like me, and so the last group that needs the vaccine to put us over the top fails to get the vaccine.

    I think one lesson to be learned is that public health officials should not speculate about when we are reaching herd immunity. I wish they would retire the term “herd immunity.” There really is no such thing. All that is happening is that the spread slows substantially. Polio is still out there. If you don’t have the polio vaccine, you can get polio although the chances in the US right now are extremely small. If people think that herd immunity is the same as actual immunity, then yes, I do think that they may rationally opt for the socially worst outcome as the population approaches herd immunity. But at this point, there is no dilemma.

    • Andrew says:

      Steve:

      Yes, I agree that the term “herd immunity” is misleading. Slowing the spread is not the same as immunity.

    • Mark Palko says:

      Way out of my field here (not sure if I’m using the terms correctly), but I believe to be a prisoner’s dilemma, the single Nash equilibrium would be where no one gets vaccinated, put another way if you don’t get vaccinated, there’s no advantage for me getting vaccinated. (in reality, if you’re an anti-vaxxer, I really want to get the shot)

      The 2×2 cooperate/betray grid doesn’t line up with the pay-off grid for the vaccinated/not grid, though it does qualify as a free rider problem.

      But yes, the current use of “herd immunity” causes more confusion than anything else.

  14. paul alper says:

    I thought that the prisoner’s dilemma included the fact that each prisoner is unaware of what the other is doing. In the case of Covid vaccination, we presumably are informed of what others have done. Is this discrepancy sufficient to ruin the comparison?

  15. Min says:

    “In the prisoner’s dilemma formulation, it’s good for you if other people get the vaccine, but you personally benefit from not getting it. That doesn’t make sense, given that the vaccine is reducing your risk from coronavirus complications.”

    Don’t you know the song refrain:

    “I may be sick, but Lord I’m free”?

    I wish that was a joke.

    • Chebyshev says:

      Song of life:

      “Oh i am alive, alive
      Masked and imprisoned
      But i am alive, alive
      Jim is grateful forever
      His granda Mary
      In isolation is alive, alive
      ‘Cause I am masked
      Oh i am alive, alive
      Mary is alive, alive.”

      Merry christmas!

  16. J Green says:

    Consider the situation of a country deciding whether or not to cooperate with other countries by sharing access to vaccines.

    The case for that situation being a prisoner’s dilemma was made a few weeks ago:
    https://www.bloomberg.com/opinion/articles/2020-12-05/now-that-vaccines-are-coming-what-about-poor-countries
    “A lab at Northeastern University in Boston has modeled two counterfactual scenarios of what would have happened if a vaccine had been available in March 2020. In one, the first two billion doses are snapped up by rich countries, while only the remaining billion are allocated among all others. In the second, all three billion are distributed from the start to all countries in proportion to their populations.
    In the first or “uncooperative” case, the vaccine would have averted 33% of global deaths through Sep. 1. In the second or “cooperative” scenario, it would have prevented 61%. That’s a lot of lives saved — even in countries that would have had the vaccine in either scenario.
    The situation is therefore a bit like the famous Prisoner’s Dilemma in game theory. If all countries cooperate, the world can achieve an optimal outcome and defeat the pandemic soon and decisively. If they don’t cooperate, Covid will drag on and there’ll be many more deaths. The dilemma is that each individual country also has an incentive to “cheat,” relying on others to do the sharing while snatching all the doses it can. But this leaves the others even worse off than if no one cooperated.”

    The article also gives estimates of the $ costs and $ savings associated with cooperating and with not cooperating.

    Is that a prisoner’s dilemma?

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