A friend and colleague of mine recently sent me some thoughts and questions pertaining to Peirce, probabilities, and single cases. I’d be very interested in getting others’ thoughts on this. Thus, here (with his permission) is an excerpt from my friend’s email:
I was thinking about the problem of the single case today, and I was wondering what you make of it. I’m taking the problem from Peirce, where you will recall, he characterizes it somewhat as follows:
Suppose you stand before God and he presents you with two decks of cards. The first has 25 white cards and 1 black card; the second has 25 black cards and 1 white card. He tells you which deck is which, and says that you may choose which deck to draw a single card from. He also tells you that if you draw a white card, you will be welcomed into heaven but that if you draw a black card, you will be condemned to hell. I think it would be hard to dispute that one ought to draw from the first deck. But, as Peirce says, what consolation can we give to the man who chooses from the first deck but draws the black card?Now, as I understand the literature following the positivists (and maybe Feller), the problem of the single case has been made out to be about the meaningfulness of probability statements with respect to singular events, like drawing once from such a deck. But it seems to me the problem is about rationality. Specifically, it is about the rationality of following or being guided by probabilities in the single case, even when you know what the probabilities are.
So, what do you think? What arguments do frequentists (or propensity theorists) and Bayesians (both subjective and objective) give to connect probability with rational action in cases like this? Is there a standard move in the literature?
So indeed, what do you think? More specifically, I wonder what you think about my colleague’s distinction between Peirce’s version of the problem and the problem that typically goes by that name these days. And I wonder how you would answer the questions concluding his email.
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I wonder if there’s something important missing from the description of the problem. You told us that God has described the contents of each deck, but you haven’t told us anything about the process by which cards are drawn. (Does it somehow favor black cards over white? Does it favor blacks cards only when those cards are surrounded by white cards?) To describe the process as “random” of course begs the question, while to describe it as “you know, mixing the cards around, sticking your hand in, and drawing one out” brings in background data we have about how that kind of process tends to go.
By the way, if the problem does turn out to be about rationality, we’ll have a nice (?) answer to Peirce’s question about our consolation to the man who draws the black card. It’s the same consolation we give anyone who does a rational thing that turns out badly, for instance Williams’s character who thinks the glass contains gin but winds up drinking petrol.
Hi Mike! Here are a few thoughts in response:
First, after reading the passage from Peirce (I’ve included the whole passage in a comment below), I think that my friend was correct about the distinction. As I read him, Peirce takes what is commonly called the Problem of the Single Case (namely, the putative problem that probability statements are meaningless with respect to singular events) as an unarguable fact. But then he shows that this fact leads to problems regarding rationality in isolated cases. So, the problem doesn’t arise with regards to probability statements about single cases (Peirce seems to be perfectly content with the fact that such statements don’t have meaning); the problem arises when one tries to find a rational basis for action in isolated cases.
Second, I’m afraid that the original passage from Peirce doesn’t do much to answer your question about the nature of the drawings. Do you think this lack of detail disarms the problem?
Third, in light of the original passage, it would seem that Peirce already thought of your consolation to the man who draws the black card from the pack containing the larger proportion of red cards … and he doesn’t think much of it: “He might say that he had acted in accordance with reason, but that would only show that his reason was absolutely worthless.” What do you think?
Hi Jonah,
I haven’t read the Peirce carefully myself yet, but the points that you describe suggest an interesting perspective on the interaction of probability and rationality. If he assumes that probability doesn’t exist in single cases, and that rationality requires taking an action with high probability of success (or some more sophisticated requirement that makes rationality depend on some pre-existing notion of probability), then of course there’s a problem here.
But the Bayesian seems to have a natural set of responses. Here’s an example of one way to do it. First, she can draw distinctions between several interpretations of probability. Then, she can concede that single case situations leave some of these interpretations undefined. However, there is only one notion of probability that has a tight connection to rationality, and this one in fact gets the dependence in the other direction – probability is just a way of numerically representing the beliefs and preferences of a rational agent, where “rational agent” must be understood already. Finally, the Bayesian gives arguments that a certain set of axioms are all really criteria of rationality, and then proves a representation theorem. Maybe there can be something like a Principal Principle tying other notions of probability to this one, but the only constitutive link between probability and rationality goes the other direction.
I suspect that Peirce might point to examples like this as suggesting that any free-standing notion of rationality is inherently problematic, but that seems like an unnecessarily skeptical conclusion. I suppose the justification I give above, if taken as a complete description, gives a radical subjective Bayesian picture, but there are probably ways to modify it to suggest other pictures as well.
Thanks for the thoughts Kenny! It’s probably worth pointing out here too that the Peirce quote is being considered out of context, and that Peirce almost certainly intends this passage to be read in the wake of his defenses of a frequency interpretation of probability (he was a great admirer of Venn’s Logic of Chance if I remember correctly) and his disdain for the sort of Bayesian, personalist view that you have in mind. Nonetheless, your points are very well taken!
I hope this won’t come out too garbled, I’m shooting from the hip, as it were.
Peirce’s own solution to the puzzle about rationality is that we have to expand our interests beyond our individual selves and identify with a larger community — a community whose interests are not limited as ours are. This appears to be a frequentist way out, i.e. show how our single-case decisions are part of a larger class for which both rationality and probability make sense. So, I think, his argument goes something like this: you should draw from the mostly red deck because if all the members of the community draw from the mostly red deck, then the community will be better off (better off than if they drew from the mostly black deck).
I don’t think this solution is satisfactory, but I don’t really know why not. On the other hand, I’m not persuaded by the Bayesian account either. And I don’t think that making probability depend on rationality weakens the objection. I mean, what stops the ultra-pessimistic conclusion: rationality is not worth having?
Consider the four cases that can arise in Peirce’s story.
(1) I draw from the mostly red deck and draw a red card.
(2) I draw from the mostly red deck and draw a black card.
(3) I draw from the mostly black deck and draw a red card.
(4) I draw from the mostly black deck and draw a black card.
Drawing a red card from the mostly red deck is not surprising, and it seems to be the rational thing to do given the stakes. But is the outcome really a product of the player’s *rationality*? It is still down to luck at some level. I don’t know what to say to the player who draws a black card from the red deck. Similarly, it seems to me that a rational person censures drawing from the mostly black deck. But what does such censure matter to the player who draws a red card from the black deck? Surely that player’s felicity isn’t lessened by being lucky rather than smart.
After a little hunting, I found the original passage from Peirce (from “The Doctrine of Chances,” CP 2.652):
I’m not sure I see how the card draw described by Peirce is unique in a way that is problematic for statistical probabilities. The event constitutes, presumably, a standard card draw from a shuffled deck of 26 cards. I don’t see that even von Mises would have had a problem assigning a statistical probability for drawing the uniquely colored card in the described situation. Every event, even statistical ones, have features that make them unique, features we leave out of those that qualify them for the reference class. What happens after the card is drawn, what consequences the result has, seems to me to be irrelevant (just as it’s not usually relevant _who_ draws the card). What am I missing?
In a sense, you’re not missing anything. In another sense, you’re missing the whole problem. As the original question went and as Jonah emphasized, the problem is about rationality, not about the meaningfulness of probability in the given scenario. The invocation of probability makes the case interesting, because in cases where there everything is non-probabilistic, doing what is rational guarantees success. Whereas, when an action is chancy, one might do the “rational” thing and still fail. Imagine you are playing Monopoly. You own the orange properties just before Free Parking, and all the other players are visiting Jail or within three spaces of it. The rational thing to do (I think) is to leverage as much money as you can into houses or hotels on those properties. But doing so doesn’t guarantee that you will win the game. Maybe all the other players will get lucky and miss your properties.
The question, then, is about why we say that picking from the mostly white deck is the rational thing to do, or why we say buying houses/hotels is the rational thing to do. And a big part of answering the challenge is figuring out what to say to someone who has done the rational thing and lost big.
Does that make sense or am I confused? Incidentally, I’d still like to know how people think the puzzle is to be resolved. What do you think, Jonah?
Thanks Jonathan; I think that that’s the clearest statement of the puzzle on the blog so far. That said, I’m not sure at all that there can be a settled resolution to this problem, and I suspect that the “resolution” that many would opt for is precisely the one that Peirce finds so distasteful. Something like:
Well, sorry it didn’t work out for you, but at least you acted in accord with reason. Unfortunately, the rational act was not linked with certainty to the more desirable outcome; the act was fully rational (in light of the respective probabilities and utilities) but that doesn’t mean that the desired outcome was fully certain. Now have fun in hell!
Again, not much of a consolation I suspect! But as soon as probabilities enter the picture, it seems that we must allow for the most rational act not always leading to the most desired outcome.
One other thing. Of course, the question – in the light of my last comment – then becomes the following: “yeah, but what do you mean that the act was fully rational if it didn’t lead to the desired outcome in the real world?” And the answer is something like: “I mean that it was tied to the highest probability for the most desirable outcome in the real world.” But this answer then takes us right to the problem of the single case as understood more commonly today: “yeah, but what could you possibly mean by the probability in this single case?” So the puzzle raised by Peirce is indeed closely related to the problem of the meaningfulness of probability statements pertaining to single cases. Do you agree?
I don’t know. As I mentioned in a reply below, it may be that Peirce was running together the two kinds of single-case problem. But I think the best way to understand the problem here is as a challenge to justify the practice of maximizing expected utilities. (I think, as it turns out, that Peirce produces such a justification. But I also think that justification is pretty lame.) The problem, then, isn’t really about the meaningfulness of probabilities but about why it is rational to maximize *expected* utility, when this does not come with a guarantee about *actual* utility.
Thanks, Jonathan, for pointing me in the right direction, I do understand the proposed problem better now.
I had the impression that the event was supposed to be problematic for statistical (frequentist) probabilities in the same way that an event such as China winning the race to land a man on Mars is, but this is evidently not the case. Card draws qualify as a “mass phenomenon” (to use von Mises term), allow, in principle, for unlimited repetition, we supposedly have actual statistical data supporting our assigning the probability 1/26 to the event of drawing the black card from the red deck, and so on.
But of course von Mises (I don’t know Peirce’s theory very well so use him as the prototypical frequentist) had another problem with assigning probabilities to particular events, even statistical events, in that it seemed to him that the probability could be anything depending on the choice of reference class. I suppose the event in Peirce’s thought experiment could be regarded as problematic in this sense. But it’s not perfectly obvious to me that it is. We are allowed, on the frequentist picture, to say that the probability of drawing some particular card from some shuffled deck of 26 cards is 1/26. To me this seems to go far in justifying our claim about what is the rational choice in the described situation.
Peirce says something funny in one place: “But in the case supposed, which has no parallel as far as this man is concerned…” This seems to suggest that only statistical data that has been personally and directly acquired qualifies. This isn’t what other frequentists had in mind at all. Frequentist probabilities are supposedly objective, and the underlying data can be shared. Bracketing the reference class problem, I am “frequentistically justified” in believing that the probability of drawing a particular card from a shuffled deck of 26 is 1/26, even if I personally only ever make one such draw.
Yeah, I think you’re right about what you say here. In this sense, I don’t think you’re missing anything at all. Maybe I should have been clearer about that above!
Peirce says funny things in a lot of places, though more often than not, what he says turns out to be interesting. It may be that he had too strong an empiricism/nominalism about probabilities at this stage in his thinking. And it may be that he was running together the two kinds of single-case problems.
I have nothing substantive to add, but would note that this is a timely question.
Peirce seems to be concerned about the psychological impact on the petitioner who despite his best efforts at applying reason and the presumed very high propensity of the single trial to go his way nevertheless loses the game and suffers an undeserved disastrous consequence. With appropriate tailoring, Peirce’s scenario is analogous to the recent financial manipulations involving high risk mortgages and associated derivative financial products. Except, of course, instead of the financial players (who gambled that a very low probability event wouldn’t occur in a single “trial” and lost) deservedly going to psychological and financial hell, (relatively) innocent bystanders are instead. Unlike the bankers, Peirce’s petitioner apparently isn’t well-connected with members of heaven’s admission committee who can pull a few strings on his behalf notwithstanding his bad luck.