False Imprisonment

The frequently insightful and usually accurate Megan McArdle gets this part quite completely wrong in her latest Bloomberg column about ObamaCare:

Democratic politicians and insurers are locked in a prisoner’s dilemma. In this classic game-theory case, you and a professional associate are both arrested for theft. If neither of you talks, then you’ll probably get off. But if just one of you talks, then the person who talks will get a reduced sentence, while the other person has the book thrown at them. If you both talk, then both of you go to jail for a long time. The equilibrium is for both of you to talk, just in case the other guy does .

I sincerely hope that anyone who’s ever taken my Principles of Economics course — or for that matter, any Principles of Economics course — can explain to McArdle how wrong this is, and why.

Exercise for the reader: To what extent, if at all, does this howler undermine the larger point of McArdle’s column?

Click here to comment or read others’ comments.


41 Responses to “False Imprisonment”

  1. 1 1 Marcelo

    Wow, that is incredibly embarrassing. With that one article she has singlehandedly undone an ungodly number of man-hours of teaching by economics professors. Not only because she butchered the prisoners’ dilemma, but also because her terrible strategic reasoning undermines the presumption that economic agents are rational.


  2. 2 2 prior probability

    Goodness gracious … she mangled the entire dilemma! The person who snitches ( or “defects” in game theory parlance ) walks or gets a very light penalty when the other prisoner remains silent

  3. 3 3 Prior Probability (Enrique)

    PS … See also my paper “the parable of the prisoners” on ssrn describing the origins of the PD and its many uses in fields as diverse as biology, law, and philosophy (not to mention economics):


  4. 4 4 Phil

    I see only one thing she got wrong: she says that if neither talks, they both get off. But, actually, if neither talks, they both get a light sentence. It’s only if one talks and the other doesn’t that they one who talks gets off.

    Her way, talking makes your situation worse if the other party doesn’t talk. So, neither will talk.

    Is that what you’re getting at? Is this a “howler”? I guess that depends on the answer to the “exercise for the reader”.

  5. 5 5 Tristan

    Sorry, could someone clarify what exactly is wrong with that description? I’ve read about the prisoner’s dilemma in many, many different forms throughout the years, and that doesn’t sound too far off base to me.

  6. 6 6 Steve Landsburg

    Marcelo (#1): Yes. The minor problem is that she’s invented some new game which is not the Prisoners’ Dilemma, and called it the Prisoners’ Dilemma. The major problem is that she’s thoroughly mis-analyzed her new invented game.

  7. 7 7 Tristan

    @prior probability: that’s exactly the way the game she described works.

    Her description:
    “If neither of you talks, then you’ll probably get off.” => (1, 1)

    “But if just one of you talks, then the person who talks will get a reduced sentence, while the other person has the book thrown at them.” => (2, -1) / (-1, 2)

    “If you both talk, then both of you go to jail for a long time.” => (0, 0)

    That’s my interpretation, and it maps to a game with a nash equilibrium where they both snitch, just as she describes. I repeat, what is the glaring mistake?

    Is it perhaps that you are assuming that “probably getting off” is preferable to “a reduced sentence”? Because that’s not necessarily true.

  8. 8 8 Dave B

    I don’t think she has invented a new game. What she has described sounds like a Stag Hunt. Her analysis isn’t completely ridiculous since there are two Nash equilibria. However a Stag Hunt shouldn’t really be a dilemma since obviously the best strategy is to stay silent (hunt the Stag). In reality the decision to talk (hunt the Hare) can be driven by the fear that the other person won’t analyse the problem correctly. Once you have that fear it becomes self reinforcing since you now have a reason why the other person may talk.

  9. 9 9 Harold

    It is difficult to weigh terms such as “throw the book at” and “go to prison for a very long time”. Whilst this is sloppy, no interpretation can save this one. In PD the rational response is to defect, because whatever the other person does, you are better off if you defect.

    In McArdles example, if the other person keeps silent, and you also keep silent you “probably get off”. If you defect, you “get a reduced sentence”. I think the former is preferable to the latter, and this alone sinks it as an example of PD. If the other person defects and you keep silent, you “get the book thrown at you”. If the other person defects and you also defect you “go to jail for a long time”. It is not clear which is worse, so we cannot say if it fits the PD.

    However, since her (erroneous) conclusion is that the incentive is for both to defect, which is the same conclusion as from the proper PD, I am not sure if the mangling makes any difference to her wider argument (although I have not yet read her column).

    I had understood that the key thing about the PD is that a rational response by both parties does not maximise social welfare. Social welfare is maximised only if either both parties are irrational or cooperation is somehow imposed. Classic example is an arms race. I recently speculated that voting is also an example.

  10. 10 10 Steve Landsburg


    I had understood that the key thing about the PD is that a rational response by both parties does not maximise social welfare. Social welfare is maximised only if either both parties are irrational or cooperation is somehow imposed.

    That’s one key thing. The other key thing is that both parties have dominant strategies, i.e. the decision to confess is the only rational choice whether or not you believe the other guy is confessing.

  11. 11 11 David R. Henderson

    Steve, I notice that no commenter has answered the question you posed. I would read her article and try to, but I’ve got to head off to work.

  12. 12 12 Sam Wilson

    “A loud-voiced part of their constituents, if organized for votes, may easily outweigh the whole.” -Arthur Cecil Pigou

    I can’t figure out what she means with the defection analogy. “Insurers need legislators to hold this law together, particularly the individual mandate.” I don’t understand what it means to stand strong here. In game theory, to select a strategy implies taking action. What action precisely would insurers take?

    Desiring an outcome is not the same as acting to achieve that outcome. The players in this game are politicians and organized constituents. Insurance firms are bystanders.

  13. 13 13 Alex

    Dave B #8 I agree it’s a Stag Hunt. I would imagine that the risk dominance of hunting hare (defection) is more inherent to politics than payoff dominance. I’d say the only thing McArdle got wrong (within the scope of her intention) was the name of the game.

    Also note that stag hunts have mixed equilibria too, in which the odds never favor the stag (cooperation). And if political decisions smack of anything more than risk avoidance, it’s sheer randomness…

  14. 14 14 David R. Henderson

    I know I said I must get going to work but I want to answer Harold above. It is NOT the case that they fail to maximize social welfare. What’s true is that they fail to maximize their own welfare. In the classic Prisoner’s Dilemma, if the guys really are guilty and the crime was horrendous enough, social welfare could well be maximized if they go to prison for quite a well. Moreover, in the part of economics where we often apply the PD, namely why cartels tend to break down, cartels breaking down is a good thing for society: it gets us closer to maximizing social welfare.

  15. 15 15 Nick

    Even though she mislabeled the game as the prisoner’s dilemma, she then goes on to describe the political situation in terms of her mislabeled game and draw conclusions consistent with the game. So I wouldn’t call this a ‘howler’.

  16. 16 16 Ryan P

    I don’t think it undermines her point at all. It’s true that the theoretical game she’s referring to isn’t a PD but something like a modified assurance game, and it’s also true that she doesn’t describe the optimal strategy quite right. But the real life game she’s describing also isn’t a PD game but a modified assurance game with uncertainty about payoffs (or “Nature’s” moves). So swap out the theory with a different game that better describes the real case she’s discussing, and the whole thing is fixed.

  17. 17 17 Alan Gunn

    I agree that it doesn’t undermine the rest of the argument, but I don’t think the rest of the argument is convincing. It assumes that if the insurance companies sell insurance below cost for another year and if the Democrats in Congress don’t bail, the whole program can survive. I think the program was doomed from the start. Its funding mechanisms were based on fantasy (especially the one about how we’ll get half a trillion bucks from Medicare), it required both the federal and the state governments to do a lot of new things close to perfectly, the penalties for not buying expensive insurance can easily be avoided by arranging things so you don’t get a tax refund, etc. Even if they’d had able people working full time on getting the web site up, the whole contraption would have collapsed. We’re probably lucky that they botched it so early, before even more damage was done.

  18. 18 18 Kirk

    Leaving the IRS with no effective enforcement means for unpaid penalties was a HUGE mistake in ensuring the law’s success, but politically the “Don’t get insurance and the IRS puts a lien on your house and levies your bank account” idea would have been suicide for the law’s passage.

  19. 19 19 Harold

    14. Yes, you are right. I meant that the total welfare of the participants is not maximised by the rational choice. In some instances this will be the same as social welfare if everyone is a participant.

  20. 20 20 Chris

    If I’m reading her game correctly, not talking is the _dominant_ strategy!? If I backstab the other guy successfully, I get PENALIZED? As long as your partner is not a moron, you know that he will realize this as well, of course!

    The comments above seem to acknowledge that her game is not a PD, but then fail to actually analyze the game.

  21. 21 21 DMS

    I’m confused why there is so much confusion. The “howler” is in the language “if neither of you talks, then you’ll probably get off” (call this PGO), which I take to be unambiguously better than “if just one of you talks, then the person who talks will get a reduced sentence” (call this RS). That is PGO is much better than RS, when it should be reversed. The prisoner’s dilemma exists when the payoff case to me of I Defect/You Don’t is slightly BETTER than We Both Don’t Defect. She has it backward – I should Probably Get Off (PGO) when I Defect/You Don’t and have a Reduced Sentence (RS) when We Both Defect. Otherwise the equilibrium doesn’t result in a strategy of always defecting as the optimal case.

  22. 22 22 DMS

    sorry, it should be “…and have a Reduced Sentence (RS) when We Both Don’t Defect.”

  23. 23 23 DMS

    I should have added that I may still be the confused one on Steve’s question. As I see it, once you cut her some slack and forget that it isn’t a true PD, her payoff matrix seems to support her argument, i.e. there will be a highly unstable equilibrium of cooperation, until such time as there is a whiff of defection by either side, which will immediately cause the other side to drop out as well. Am I missing something?

  24. 24 24 Jack

    Her payoff matrix is accurate. Her reasoning for why the players choose to talk is what’s inaccurate. It’s not “in case the other guy talks” – it’s because they are better off by talking no matter what the other guy does.

  25. 25 25 Jack

    And I don’t think her reasoning hurts her article – I think trying to compare a Prisoner’s Dilemma to the health care insurance debacle does. I don’t think one should assume that the players of the health care game are rational, and think it’s even worse to assume we could know what the payoff matrix looks like.

  26. 26 26 Ken B

    I’m with Dave B in 8 and Alex in 13: it looks like a Stag Hunt. Here is William Poundstone on the game http://www.heretical.com/pound/staghunt.html

    Her Analysis of the stag hunt isn’t bad. As has been noted it’s probably a mixed equilibrium which is best but can you explain that briefly, and why really in an article like this would you even try? All she’s really trying to do is highlight the incentive to defect. I think she did so. Her big error then is in naming the game. So hands up those who before they read this post really knew the difference between prisoners dilemma and Stag hunt?

  27. 27 27 Ben

    If Ken B is right (and it seems like he is) and her only mistake that Steve wants to highlight is that she misnamed the Stag Hunt the Prisoner’s Dilemma then note that she technically did no such thing. She did not say they’re locked in The Prisoner’s Dilemma; she said they’re locked in a prisoner’s dilemma. The situation she described is a dilemma and prisoners experience it. Therefore her description is also technically accurate as well as being accurate in spirit.

  28. 28 28 Ken B

    Oh Ben, that’s a good catch. She used lower case and the indefinite article. I’m embarrassed not to have noticed that myself.

  29. 29 29 James Kahn

    I don’t know if Ken B is being sarcastic in 28, but she refers to it as “this classic game-theory case,” and proceeds to describe a scene which is (apart from the payoffs) the one that motivates the Prisoners’ Dilemma. So that doesn’t get her off the hook. Perhaps all she did was get the name wrong, but that’s enough to create a lot of needless confusion among her readers, at least those familiar what the Prisoners’ Dilemma is. For one thing, we’re forced to spend time figuring out what she really meant, which is deadly for a writer.

  30. 30 30 Ken B

    No sarcasm. Stag Hunt IS a classic of game theory, and I have heard people describe families of games under the rubric “prisoner’s dilemma games”, referring to games where defection can pay leading to suboptimal results. It’s simply a case of most of us jumping to conclusions, because we’ve been exposed to the conventional name for a certain particular game.

  31. 31 31 JohnW

    I believe there are only three interesting 2-person-symmetrical games like that: Prisoner’s Dilemma, Stag Hunt, and Chicken. If there are any others that make sense and are non-obvious, I have not come across them.

    I think anyone who is writing about one of them should at least have a passing familiarity with the others.

  32. 32 32 Martin-2

    “Probably get off” sounds like the best deal to me, so that means no-one talks and everyone wins. Hooray!

  33. 33 33 Harold

    In Stag Hunt, the equilibrium is for both to cooperate. In PD, it its for both to defect. In McArdles example, she claims the equilibrium is for both to defect, which would be the case for PD, but some claim that she has set it up as a Stag Hunt. It is mangled.

    I also don’t think she gets off the hook even if it was a correct depiction of a stag hunt. Stag hunt is not traditionally set up with prisoners, and it doesn’t make much sense to set it up that way.

  34. 34 34 Al V.

    As everyone else has pointed out, her example is obviously not PD. However, I don’t see this as a Stag Hunt either. In a Stag Hunt, once I choose to hunt the hare, my outcome is determined regardless of what the other person chooses. If an insurer defects from ObamaCare, its relative success is determined not only by what course the politicians choose, but also by what other insurers choose. This seems like a Stag Hunt, but with many hunters playing at once.

  35. 35 35 Henri Hein

    @27: Ken B is always right.

  36. 36 36 Henri Hein

    I’m confused about PD. I have heard it talked about in different ways. Here is how I understand it, and I am pretty sure lots of people understand it this way.

    Al and Bob are arrested for bank robbery. There is no evidence against them. If neither talks, both go free. If Al says, “Yes, we really did rob the bank, but Bob held the gun and did all the shouting. I was just the lookout.” Al gets 6 months and Bob gets 5 years. If both talk, the DA is going to say, “You cannot both have been the lookout” and goes for 5 years each. I don’t understand how it can always be better to talk.

    But either way, I have no idea what is has to do with congress and insurers.

  37. 37 37 Harold

    #36 Henri Hein. There is one important bit you missed off – if you both stay silent the police will prosecute you both for a lesser crime. If you cooperate on your own, you get off scott free.
    So in you scenario, by turning States Evidence on his own, Al get nothing and Bob gets 5 years.
    If Bob also betrays, both get 3 years
    If both stay silent, the each get one year for a lesser offence.

    Looking at it from Al’s point of view as one of the players. Say Bob stays silent. If Al stays silent he gets 1 year on a lesser charge, if Al betrays, he gets off. Say Bob partner betrays Al. If Al stay silent I get 5 years. If Al betrays as well he gets 3 years.

    So rational self interested parties end up with 3 years, when they could have got one year only by cooperating.

  38. 38 38 Harold

    #37 – When I said if you cooperate on your own you get off scott free, I meant if you cooperate with the police. In the terms of PD this is actually defection.

  39. 39 39 Henri Hein

    Thanks Harold. I’ll have to check Powell’s book now, because that is not how I remember him presenting it.

  40. 40 40 iceman

    Seems pretty clear from the article she was trying to compare this to a classic PD, but what she describes doesn’t quite fit that OR a stag hunt, for reasons others have mentioned separately:

    1)She says you come out worst if the other guy defects and you don’t, which is true for PD but not SH (where if you defect you come out the same whatever the other guy does). This provides some reason to believe the thing could fall apart.
    2) She also says the cooperating case is better for both than being the only defector — i.e. defecting is not a dominant strategy — which is true for SH but not PD. This weakens her case from #1.

    In the Washington example she gives, they imposed the regulations but never enforced the mandate, i.e. the govt defected before the game started so that had no chance.

    BTW I admit I was not familiar with stag hunts before (the game theory kind that is).

  41. 41 41 iceman

    Oops I said 1) wrong above – my understanding is you do also come out worst in the SH if the other guy defects and you don’t, hence the risk the deal falls apart. But she also seems to say being the only defector is better for you than if both defect, which would seem to create a stronger “race for the exit”, and this is true for PD but not SH (where the value of defecting doesn’t depend on what the other guy does).

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