CORRELATION, CAUSATION, AND CHOICE

You stand before two boxes. One is transparent and contains \$1000. The other is opaque. You have a choice. You can ‘one-box’ (that is, take the contents of the opaque box) or ‘two-box’ (that is, take the contents of both boxes). The game is set up so that the contents of the opaque box always depend on the earlier prediction of a remarkably successful predictor. When the predictor predicts that you will one-box, she puts \$1,000,000 inside the opaque box. When she predicts that you will two-box, she puts nothing inside the opaque box. Should you one-box or two-box?

This choice is known by philosophers as Newcomb’s problem. Before confronting Newcomb’s problem, you might have thought it obvious that you should always choose whatever option you expect to leave you better off. But it’s uncontroversial that you should expect to be better off when you one-box (since the predictor is considerably more likely to have put the money inside the opaque box when you one-box than when you two-box), yet it is highly controversial whether you should one-box. The reason, as Nancy Cartwright (1983) puts it, is that even though Newcomb’s problem is specified so that selecting the contents of the opaque box provides you with evidence that you’ll be rich, one-boxing is an ineffective strategy for getting rich since there is undeniably nothing that you can do now to causally influence the contents of the opaque box. Like many cardholding ‘causal-decision theorists’, then, you might maintain that you should two-box—even when you think that one-boxing will leave you better off—for the simple reason that it is irrational to act on a strategy that you know to be ineffective.

Though I am sympathetic to the causal-decision theorist’s idea that our beliefs about causal influence are relevant to rational choice insofar as they help us to distinguish between effective and ineffective strategies, I am unwilling to follow the causal-decision theorists who simply write off Newcomb’s problem as a decision-making context where we’re paid to be irrational. Instead, it seems incumbent upon the causal-decision theorist to somehow reconcile causal-decision-theoretic recommendations with the truism that we should choose whatever option we expect to leave us better off—or, at the very least, the causal-decision theorist should provide a more satisfactory explanation than is typically given for why we should break from this truism.

There are several attempts to provide such an explanation in the literature, but I have long been most impressed with Christopher Meek and Clark Glymour’s (1994) idea that we can secure causal-decision-theoretic recommendations while keeping with the truism simply by treating decision-makers’ options as interventions in a causal graph—where intervening to act can be understood (very roughly) as making yourself act by means that are uncaused by any of the other factors under consideration.

This interventionist story is compelling largely because it aspires to offer a lens through which we can view the dispute surrounding Newcomb’s problem. The idea is that we can fruitfully understand the dispute as turning on the status of a fairly commonsensical view of what genuine choice requires, rather than on whether we sometimes shouldn’t choose the option that we expect to leave us best off. Meek and Glymour’s key insight toward this end is that if we treat our choices as up to us in the sense required by intervening, then the axioms of our best (and independently motivated) theory of the relationship between causal relevance and evidential relevance—the theory of causal Bayes nets—imply that we should treat our choices as uncorrelated with their non-effects (including the Newcomb predictor’s prediction). So, as Meek and Glymour would have it, intervening to two-box is expected to leave Newcomb subjects better off than intervening to one-box (even though one-boxing itself is expected to leave Newcomb subjects better off that two-boxing itself).

But as promising as Meek and Glymour’s take on Newcomb’s problem is, there are two problems with their argument. The main tasks of my BJPS article are to articulate these two problems and then to show how they can be addressed by augmenting the usual interventionist conception of choice with some additional commonsensical constraints on the chances of genuine choices.

The first problem is that while the theory of causal Bayes nets straightforwardly implies constraints on what objective probabilities are compatible with what causal stories, it does not straightforwardly imply constraints on an agent’s subjective credences, and thereby does not straightforwardly imply that Newcomb subjects should regard their interventions as evidentially irrelevant to the predictor’s predictions.

To see this, consider the example of the duelling meteorologists: Imagine that you’re certain that whether it rains on 31 July in New Orleans is completely causally unconnected from whether there is bad air quality in Shanghai on 31 July. Suppose further that you’ve just consulted the forecasts of two meteorologists (Jim and Kim) and that you know that one of them is right about both the chance of rain in New Orleans and the chance of bad air quality in Shanghai, but you aren’t sure who.  Jim and Kim agree that the chance of rain in New Orleans is independent from the chance of bad air quality in Shanghai, but Jim regards both chances as relatively high (say, 0.8) while Kim regards both chances as relatively low (say, 0.4). Now, on the morning of 31 July, you look out your window in New Orleans window to discover that it’s raining. Should your glance out the window increase your confidence that there is bad air quality in Shanghai?

While it’s plausible that you should regard the objective chances of the weather in New Orleans and Shanghai as uncorrelated because they are causally isolated from one another, it’s implausible that their causal isolation gives you reason to treat them as evidentially irrelevant to each other. After all, you know that either Jim or Kim has correctly identified the chances, and your discovery of rain in New Orleans gives you reason to trust Jim over Kim (since Jim’s chance estimate for rain was substantially greater than Kim’s). Thus, your glance through your New Orleans window provides you with some evidence that the chance of bad air quality in Shanghai is high, thereby licensing you to increase your confidence in the bad air quality itself.

This means that the theory of causal Bayes nets should not be interpreted in terms of implying direct constraints on rational subjective credences, and instead should be understood as implying direct constraints on objective chances. In the context of Meek and Glymour’s take on Newcomb’s problem, the upshot of this is that it’s not clear how or whether the theory of causal Bayes nets actually manages to provide decision-makers with reason to treat their interventions as evidentially irrelevant to their non-effects, despite Meek and Glymour’s claims to the contrary.

The second problem is that even when a decision maker is certain about the underlying chances, the theory of causal Bayes nets does not imply that we should treat our interventions as irrelevant to their non-effects in decision-making contexts where we have foreknowledge that some effect of our intervention will obtain. This doesn’t rear its head in the context of Newcomb’s problem, but it does arise in other ‘exotic’ decision-making contexts that split causal-decision theorists and evidential decision theorists.

Consider the birthweight problem: Suppose that you’re pregnant and that an oracle has just informed you that your baby will unfortunately be born underweight. You recently learned that when a baby is born underweight, their prognosis is significantly better if their mother smoked during gestation than if she did not. This is because the odds of survival are better when the newborn is underweight because its mother was a smoker than because the newborn suffers from some genetic condition. Even if you find smoking to be a bit unpleasant, should you take up the habit?

When confronted with the birthweight problem, it’s clear that causal-decision-theoretic reasoning favours abstaining from smoking (if you mind smoking at all), since your choice of whether to smoke exerts no causal influence over whether your newborn will suffer from a genetic condition—or, put differently, since there is nothing that you can do now to change your newborn’s genetic makeup by deciding to smoke. But it is predicted by the theory of causal Bayes nets that you should regard whether you intervene to smoke as correlated with whether your newborn will suffer from the genetic condition, given the oracle’s foreknowledge that your baby will be underweight. After all, it’s a live possibility that your newborn will be born underweight because of your choice to smoke, and this would be good news insofar as it’d decrease the odds that your newborn would be born underweight because of some genetic condition. But since this good news would not reflect any control that you have over the genetic condition, it’s not the sort of news that gives causal-decision theorists reason to countenance smoking as an effective strategy for influencing the newborn’s weight.

Do these problems spell doom for Meek and Glymour’s approach to securing causal-decision-theoretic recommendations? I argue that if we augment the usual interventionist conception of choice with some additional constraints on the chances of choices, then we can rescue the general approach. But we will see that the generality of the rescue attempt depends on just how much we’re willing to assume about the chances of choices.

To this end, consider what I call ‘chance transparency’, namely, the idea that the chances of genuine choices must be transparent to the decision maker, in the sense that the decision maker must be certain of the chances that they will choose a particular way as they settle for themselves what to do. Chance transparency is thus not in the business of saying that the chance of any choice must be any particular number, but instead just says that the number (whatever it is) must be the same in every chance distribution entertained by the decision maker. This idea has some intuitive appeal insofar as it captures the popular view that we have ‘non-observational’ knowledge of how we’re likely to choose as we settle for ourselves what to do when we make a genuine choice (since we can understand such knowledge in terms of immediate access to the underlying chances). Moreover, it straightforwardly solves the problem posed by the duelling meteorologists example, since we can prove that when decision makers are sure about what causes what but unsure about the underlying chances, then they should regard their choices (or interventions) as evidentially irrelevant to their non-effects, provided that their choices all get assigned the same probability in the objective chance distributions under consideration.

So that solves the first problem, but what about the second? Here, chance transparency doesn’t fare so well. Chance transparency is trivially satisfied, and so doesn’t help, whenever the decision maker is sure about the underlying objective chances, including in decision-making contexts wherein the decision maker is in possession of exotic evidence. When choice gets exotic, we need bigger guns.

Chance indifference is the idea that genuine choice requires that the chance of every option (or intervention) be the same as every other option. Thus chance indifference goes beyond chance transparency by imposing constraints on the particular numbers assigned to the chances of choices—that is, it states that the chance of each choice be assigned equal probability. This may jibe with common sense insofar as it captures the idea that when a choice is yours, the existing chances don’t compel or incline you to choose in any particular way. But chance indifference may also sound alarm bells since it seems that we often come to decision-making contexts disposed to choose in particular ways (for example, because of our upbringing, because of our genetic makeup, because of our personality, and so on). When decision-making is like this, it is not genuine according to chance indifference. So it may seem overly strict to side with chance indifference by maintaining that the chances of choices cannot nudge us towards choosing any particular option or options.

Either way, unlike choice transparency, chance indifference capably solves both problems. It solves the first problem in just the same way that chance transparency does, because any decision maker who represents their own choice as satisfying chance indifference will likewise represent their choice as satisfying chance transparency (since every objective probability distribution under consideration will agree that the chance of every option is the same). But it also solves the problem posed by the birthweight problem, by providing us with a rationale for an approach to updating on foreknowledge that enables the interventionist to distinguish between effective and ineffective strategies even when choice is exotic. I’ve discussed the details of this updating procedure elsewhere (Stern 2021), but the basic idea is that we can secure causal-decision-theoretic reasoning by updating on the exogenous intervention to bring about the foreknown evidence, rather  than the foreknown evidence itself.

The justification for this updating procedure is supposed to be that updating on the interventions to bring about the foreknown evidence is compatible with the genuineness of choice, while updating on the foreknown evidence itself is not. But until now, it has remained unclear why the former but not the latter update should be compatible with the genuineness of choice—especially since the interventionist is independently already committed to the view that genuine choices are up to the decision maker in the sense required by modelling them as interventions. This is where chance indifference can earn its keep. If we side with chance indifference over chance transparency and thereby posit the additional numerical constraints on the chances of genuine choices, we introduce the threat of evidence compromising the genuineness of choice by forcing the chances of choices to depart from their enforced (indifferent) values. One way to generally block this threat is to update on the interventions of foreknown evidence, rather than the foreknown evidence itself.

So, what should we believe about the chances of choices? The answer to this question depends on more than the decision-theoretic issues I’ve surveyed here. But to the extent that it’s valuable to secure causal-decision-theoretic recommendations while sticking with the truism that we should choose whatever option we expect to leave us better off, there appears to be reason to adopt some significant constraint on the chances of choices. If there is principled reason to set aside a decision theory’s application to exotic choice (for example, because foreknowledge is too exotic to care about!), then chance transparency arguably provides us with everything the interventionist ever wanted. But if we should be concerned with a decision theory’s application to exotic choice (as many philosophers contend), then chance indifference is attractive.

Of course, depending on your take on the nature of agency, you might reply that some or all of these constraints on choice are too demanding (including perhaps the original interventionist constraint). I have nothing against this response. Instead, to my ear, this sounds like the beginnings of a novel argument against causal-decision-theoretic recommendations from within the interventionist approach to decision theory. And if that’s where my article takes us, then that’s fine by me.