Smokers, Psychos, and Decision-Theoretic Uncertainty
Introduction
Recently, there has been significant debate about the nature of decision theory:
whether the correct theory is evidential, causal, or something else again. The
principal problem in this debate is that powerful counterexamples seem to have been
raised to all the major views. The Smoking Lesion seems to be a decisive counterexample
to evidential decision theory; but The Psychopath Button seems to be a decisive
counterexample to causal decision theory.
In response to this, some philosophers have expressed pessimism. Rachael Briggs
argues that “no decision rule can do everything that we want”; 1 Andy Egan
regretfully asserts that he “do[es] not have… a theory to offer”2 that is able to get the
intuitions right in the cases that have been given in the literature. Others instead have
tried to develop new decision theories that satisfy the intuitions,3 but these generally
1 Rachael Briggs, “Decision-Theoretic Paradoxes as Voting Paradoxes,” Philosophical Review, CXIX, 1
(2010): 1-30, at p. 1.
2 Andy Egan, “Some Counterexamples to Causal Decision Theory,” Philosophical Review, CXVI, 1
(2007): 93-114, at p. 113.
3 For example: Frank Arntzenius, “No Regrets, or: Edith Piaf Revamps Decision Theory,” Erkenntnis,
LXVIII, 2 (2008): 277–97; Ralph Wedgwood, “Gandalf’s Solution to the Newcomb Problem,” Synthese,
CXC (2013): 1–33; Johan E. Gustafsson, “A Note in Defence of Ratificationism,” Erkenntnis, LXXV, 1
(2011): 147–50; Huw Price, “Causation, Chance, and the Rational Significance of Supernatural
Evidence,” Philosophical Review, CXXI, 4 (2012): 483–538.
come at the cost of considerable theoretical inelegance, or suffer from
counterexamples of their own.
In this article I propose a new way of making rational sense of our seemingly
divergent intuitions across cases. The key is the idea, briefly suggested by Robert
Nozick,4 that there is an important sense of ‘ought’ (though not the only sense of
‘ought’) according which a decision-maker ought to take their decision-theoretic
uncertainty into account when making decisions. I call the general idea that there are
norms that take into account normative uncertainty metanormativism, and any decision-
theory that takes decision-theoretic uncertainty into account a meta decision theory.
Metanormativism has principally been explored in relation to moral uncertainty
(though the project currently lacks a name, hence my introduction of
‘metanormativism’). The standard view in that literature is that, under moral
uncertainty, one should maximise expected choice-worthiness, or, equivalently, minimise
expected wrongness. 5 Proponents of metanormativism about moral uncertainty don’t
propose maximise expected choice-worthiness as a rival view to first-order moral theories.
Rather, they think that there are different senses of ‘ought’: a first-order moral sense
4 Robert Nozick, The Nature of Rationality (Princeton, N.J.: Princeton University Press, 1994), pp. 43–50.
5 For example: Ted Lockhart, Moral Uncertainty and Its Consequences (Oxford: Oxford University Press,
2000); Jacob Ross, “Rejecting Ethical Deflationism,” Ethics, CXVI, 4: 742–68; Andrew Sepielli, “What
to Do When You Don’t Know What To Do,” in Russ Shafer-Landau, ed., Oxford Studies in Metaethics
(Oxford: Oxford University Press, 2009), pp. 5-28; William MacAskill, “The Infectiousness of Nihilism,”
Ethics, CXXIII, 3 (2013), 508-520. From what I know, decision-theoretic uncertainty has been
mentioned only by Andrew Sepielli, “What to Do When You Don’t Know What to Do When You
Don’t Know What to Do…,” Noûs, XLVIII, 3 (2014): 521-544.
2
of ‘ought’, which is not sensitive to a decision-maker’s moral uncertainty, and a
different (more subjective or less idealized) sense of ‘ought’ that takes moral
uncertainty into account. In the same way, in this article I don’t propose meta
decision theory as a rival to causal decision theory or evidential decision theory.
Rather, a different sort of ‘ought’ is at play — one that is relevant to less idealized
agents than the ‘ought’ of first-order decision-theory.
My intention in this article isn’t to defend metanormativism about decision theory,
though I briefly offer some motivation for it in section II. Rather, my intention is to
show that, if metanormativism about decision theory is true, it has two important
implications for the causal versus evidential decision-theory debate. First, it allows us
to make rational sense of our seemingly divergent intuitions across the Smoking
Lesion and The Psychopath Button. Second, it generates strong new arguments for
preferring the causal approach to decision-theory over the evidential approach.
The structure of my argument is as follows. After quickly describing Newcomb’s
problem and the causal versus evidential distinction in section I, in section II I
introduce and briefly motivate metanormativism about decision-theory, and discuss
Nozick’s views.6 In section III, I give the most natural formulation of meta decision
theory, and show how it gets the right intuitive results in both The Smoking Lesion and
The Psychopath Button. I show how meta decision theory can convincingly explain why
we get the intuitions we do in a way that is far more theoretically elegant than other
accounts that have been proposed in the literature.
6 In contrast to a metanormative view according to which there are norms that are relative to moral
and prudential uncertainty, but not relative to decision-theoretic uncertainty.
3
I then argue that if metanormativism about decision-theory is true, we have strong
grounds for preferring the causal approach to decision theory over the evidential
approach. In section IV, I show that decision-theoretic uncertainty undermines the
intuitive case for evidential decision theory over causal decision theory. In section V, I
show that decision-theoretic uncertainty gives us the resources to construct a
counterexample to the “Why Ain’cha Rich?” argument in favour of evidential
decision theory.
I. Newcomb’s Problem
Newcomb’s problem is typically introduced through the following case:
Standard Predictor7
You have two boxes in front of you, Box A and Box B. Box A is opaque; box
B, transparent. You have the option to take either box A only, or both B and
A. You can see that Box B contains $1000. Box A either contains $1million or
$0. Moreover, someone (“The Predictor”) with an amazing ability to predict
other people’s actions had control over the boxes. If the Predictor predicted
that you would choose Box A only, then he put $1 million into Box A. If the
Predictor predicted that you would choose both boxes, then the Predictor put
nothing into Box A. What should you do?
7 In all the cases I give in this paper, I stipulate that the relevant correlations (e.g. between the Predictor
and the money in the box, and between smoking and having a lesion) are perfect correlations. I discuss
whether this aspect of my examples is problematic in section III.
4
Representing the decision-problem in a table, we have:
Money in both boxes Money in one box only
Take one box only $1,000,000 $0
Take both boxes $1,001,000 $1,000
There are two distinct but each seemingly compelling available lines of reasoning.
First, I could reason that if I take Box A only, then I’m almost certainly going to get
$1million. In contrast, if I take both boxes, I’m almost certainly going to get only
$1000. So I should take Box A only. Such reasoning motivates evidential decision theory
(EDT). According to EDT, one should choose the option with the maximal evidential
expected value, where the evidential expected value of an action is defined as the sum,
over all possible outcomes, of the value of the outcome, given that you perform that
action, multiplied by the probability of the outcome conditional on you performing
that action. According to this account, in Standard Predictor you should one-box. The
precise formalization of this view will not matter for the purposes of this paper, but
one simple way to formalize the view is as follows:
��� � = � �! � �(�! & �)
!!!!
5
In the above equation, C is the decision-maker’s credence function, and A, B, C (etc)
are actions that are available to the decision-maker. O1, O2 … On are propositions
that describe a way the world might be. V is the decision-maker’s value function: for any
outcome Oi, V(Oi), takes a real number that measures how valuable Oi is to the
decision-maker.
The above gave a line of reasoning that favoured evidential decision theory. But there
is an alternative line of reasoning. I could reason that the Predictor has already put
the $1million in Box A, or decided against doing so. My choosing both boxes can’t
change that. And, no matter what amount of money is in Box A, I’ll get an additional
$1000 if I take both boxes. So I should take both boxes. Such reasoning motivates
causal decision theory (CDT). According to CDT, one should choose the option with the
highest causal expected value, where the causal expected value (CEV) of an action is
defined as the sum, over all outcomes, of the value of that outcome multiplied by the
probability of the outcome counterfactually conditional on one’s action. There are
very many ways to formalize CDT, but these won’t matter for my purposes, so I will
use the following simple formulation: 8
!
��� � = �(� ⇒ �! )�(�! & �)
!!!
8 A very similar formulation is given, for example, in Arif Ahmed, “Causal Decision Theory: A
Counterexample,” Philosophical Review, CXXII, 2 (2013): 289-306.
6
In this equation, ‘⇒’ denotes the counterfactual conditional: that is, a conditional of
the form “if I were to perform A, Oi would happen”. According to this account, in
Standard Predictor you should two-box.
Different people’s intuitions vary strongly in response to the Standard Predictor. So, in
attempting to adjudicate between causal and evidential decision theory, other cases
are normally used. But before moving on to them, I’ll introduce and motivate the idea
of meta decision theory.
II. Meta Decision Theory
Given the trenchant disagreement between intelligent and well-informed philosophers,
it seems highly plausible that one should not be certain in either causal or evidential
decision theory. In light of this fact, Robert Nozick briefly raised an interesting idea:
that perhaps one should take decision-theoretic uncertainty into account in one’s
decision-making.9 He noticed that our intuitions in Newcomb problems seem to be
stakes-sensitive. That is, it seems that we can generate clear counterexamples to both
EDT and CDT simply by playing around with the Standard Predictor case. By altering
the stakes, we can alter our intuitions. Consider, first, the following case:10
High-Stakes Predictor I (HSP-I)
9 Nozick, The Nature of Rationality, op. cit., pp. 43–50, though Toby Ord and I in conversation
independently came up with this idea before discovering that Nozick had suggested it.
10 This example and the next are structurally the same as examples given by Nozick — I’ve just altered
them a little bit to make the case even stronger.
7
Box A is opaque; Box B, transparent. If the Predictor predicts that you choose
Box A only, then he puts one wish into Box A. With that wish, you'd save the
lives of 1 million terminally ill children. If he predicts that you choose both
Box A and Box B, then he will put nothing into Box A. Box B — transparent
to you — contains a stick of gum. You have two options only: choose Box A,
or choose both Box A and Box B.
Representing this in a table:
Wishes in both boxes Wishes in one box only
Take one box only 1,000,000 lives Nothing
Take both boxes 1,000,000 lives + gum Gum
In this case, intuitively, should you one box, or two box? Though it can be difficult
not to let theory cloud one’s judgment, my intuitive view is clearly that if someone
two-boxes in that case, they made the wrong decision. So do we have a slam-dunk
argument in favour of EDT? Unfortunately not. Consider the following case:
High-Stakes Predictor II (HSP-II)
Box C is opaque; Box D, transparent. If the Predictor predicts that you choose
Box C only, then he puts one wish into Box C, and also a stick of gum. With
that wish, you save the lives of 1 million terminally ill children. If he predicts
8
that you choose both Box C and Box D, then he will put nothing into Box C.
Box D — transparent to you — contains an identical wish, also with the
power to save the lives of 1 million children, so if one had both wishes one
would save 2 million children in total. However, Box D contains no gum. One
has two options only: choose Box C only, or both Box C and Box D.
Representing this in a table:
Wishes in both boxes Wishes in one box only
Take one box only 1,000,000 lives + gum Nothing
Take both boxes 2,000,000 lives + gum 1,000,000 lives
In this case, intuitively, should you one box, or two box? My intuitive view is clear: if
someone one-boxes in the above case, they made the wrong decision.
What’s going on in these two cases? From one perspective, they are structurally
identical. In both cases, EDT recommends one-boxing, because one-boxing has the
higher evidential expected value. In both cases, CDT recommends two-boxing,
because two-boxing has the higher causal expected value (and, indeed, dominates
one-boxing). From another perspective, however, they are very different. In HSP-I,
one’s decision is of huge consequence, according to EDT. From its perspective, the
difference in value between one-boxing and two-boxing is the difference in value
9
between saving a million innocent lives and getting a free stick of gum. For CDT,
however, one’s decision in HSP-I is fairly trivial. The decision about whether to one-
box or two-box is merely the decision about whether to get a free stick of gum or not.
In contrast, in HSP-II, the decision is of huge consequence for CDT. The decision
between one-boxing and two-boxing is the decision about whether to save a million
innocent lives. Whereas, the decision in HSP-II is fairly trivial for EDT: it merely
concerns whether to get a free stick of gum or not.
As Nozick noticed, this sort of stakes-sensitivity is suggestive of the idea that our
intuitions are governed at least in part by uncertainty over both CDT and EDT. We
feel the force of both sorts of decision theory, and so we have credence in both of
them. And then, when making decisions, we hedge our bets, going with CDT when
the relative stakes are sufficiently high for CDT, and going with EDT when the
relative stakes are sufficiently high for EDT.11 I call this idea Meta Decision Theory
(MDT).12 According to MDT, one should maximize meta expected value, where the meta
expected value (MEV) of an action is defined as the sum, over all decision theories, of
the probability of that decision theory multiplied by the value of that action on that
decision theory. Or, formally (and again with the caveat that there are many possible
ways to formalize this idea):
11 Of course, this is not the only possible explanation for why our intuitions switch in the two cases. In
sections II and III, I consider and ultimately reject alternative explanations of this phenomenon.
12 A terminological clarification: I’ll use “Meta Decision Theory” (capital letters) or MDT to refer to
the specific view that one ought to maximize expected choice-worthiness over decision theories. I’ll use
“meta decision theory” to refer to any decision theory that claims that what one ought to do (in the
relevant sense) is determined in part by one’s credences in first-order decision theories.
10
!
��� � = � �! �! (�)
!!!
In this formula, D1, D2, … Dn each refers to a decision theory, and �! (�) is the value
that Di assigns to A. In section V, I distinguish between the causal version of MDT
and the evidential version of MDT. However, until that point the distinction will not
matter for my purposes, so I state MDT simply in terms of unconditional
probabilities.13
On the reasonable assumption that we have at least small positive credence in each of
EDT and CDT, MDT would make sense of the stakes-sensitivity suggested above.
Because HSP-I is so much higher-stakes according to EDT than according to CDT,
even very small credence in EDT would make one-boxing have the higher MEV.
The same is true vice-versa for HDP-II.
13 A couple of other notes on this. First, we should of course have non-zero credence in decision
theories other than CDT and EDT, such as Benchmark Theory Wedgwood, “Gandalf’s Solution to
the Newcomb Problem.”, and so uncertainty about these other theories will also have to be taken into
account. In order to keep things simple however I will leave these alternative decision theories to the
side. Second, one might worry whether meta decision theory suffers from the problem of intertheoretic
comparisons. However, the problem of intertheoretic comparisons is substantially easier in the case of
EDT and CDT than it is between different moral theories. EDT and CDT both agree on what a
decision-maker should do in all the many cases where �� � ⇒ �! = ��(�|�! ). We can use this
agreement to normalize the two theories.
11
These high-stakes predictor cases make me think that some version of meta decision
theory is true. However, oddly Nozick himself ultimately rejects that idea, in favour of
a subtly different idea. He says:
I suggest that we go further and say not merely that we are uncertain about
which one of these two principles, [CDT] and [EDT], is (all by itself) correct,
but that both of these principles are legitimate and each must be given its
respective due. The weights, then, are not measures of uncertainty but
measures of the legitimate force of each principle. We thus have a normative
theory that directs a person to choose an act with maximal decision-value.14
And also:
Theorists of rationality have been intent upon formulating the one correct and
complete set of principles to be applied unreservedly in all decision situations.
But they have not yet reached this—at any rate, we do not have complete
confidence that they have. In this situation, won’t a prudent and rational
individual hedge her bets? I want to say more, namely, that no one of the
principles alone is wholly adequate— it’s not simply that we have yet to find
the knockdown argument for the one that is correct.15
14 Nozick, The Nature of Rationality, op. cit., p. 45.
15 Ibid., pp. 46–47.
12
That is, as I understand him, Nozick rejects what I call meta decision theory in favour
of what might be called decision-theoretic pluralism (DTP).16 Whereas MDT is not a rival
to CDT or EDT, DTP is a rival first-order theory.
What’s odd about Nozick’s suggestion is that, even though MDT seems to be the
natural explanation of our stakes-sensitive intuitions, he gives no argument for
preferring DTP to MDT (apart, perhaps, from the cryptic suggestion that MDT
wouldn’t be “normative”). We already know that we’re decision-theoretically
uncertain, and that expected utility theory is in general the best way to handle
uncertainty. This is enough to make MDT plausible, and MDT is enough to explain
our stakes-sensitive intuitions. There therefore seems to be nothing to gain by
suggesting that DTP is true. So DTP seems unmotivated.
Moreover, DTP is not merely unmotivated: it also has two major problems that MDT
lacks. First, DTP has multiple explanatory gaps. Why weigh EDT against CDT in
one way rather than another? MDT has a principled answer to this — namely, that
the weights are one’s credences — whereas DTP does not. And why should the values
EDT and CDT assign to acts be additively separable? Again, MDT has an
explanation of this — that taking an expectation requires values across states to be
16 The analogy is with pluralist moral theories. Someone who maximizes expected choice-worthiness
under uncertainty about whether only wellbeing, or both knowledge and wellbeing, are of value looks a
lot like someone who is conforming with a first-order moral theory that assigns both wellbeing and
knowledge value. In the same way, someone who follows MDT looks a lot like someone who is
conforming with a first-order decision theory that gives weight to both casual expected value and
evidential expected value.
13
additively separable — whereas DTP does not. And, finally, why even think that
there would be different sorts of “decision-theoretic value”? Decision-theoretic
pluralism is very different from other sorts of pluralism about value: typical pluralist
theories make sense of different values because different values supervene on different
sorts of stuff. In contrast, the different decision-theoretic values that Nozick suggests
arise merely out of how uncertainty is taken into account. So Nozick’s account does
not gain plausibility from the plausibility of pluralism about value in general.
Second, DTP misrepresents what’s going on in the stakes-adjusted Newcomb cases.
To see this, consider a variation on his cases.17
Four-box Predictor
Box A and Box C are opaque; Box B and Box D, transparent. The Predictor
has a 100% success rate at predicting which box or boxes you’ll choose. You
have the following four options:
(1): Take A and C only
(2): Take A, B and C
(3): Take A, C and D
(4): Take A, B, C and D
17 I thank [REDACTED] for this suggestion.
14
If the Predictor predicts you will take Box B, he will put nothing in Box A. If
he predicts you will not take Box B, he will put into Box A a wish with value of
1 million children’s lives.
If the Predictor predicts you will take Box D, he will put nothing in Box C. If
he predicts you will not take Box D, he will put into Box C a wish with the
value of 1 million children’s lives.
Box B — transparent to you — contains a stick of gum. Box D — also
transparent to you — contains a wish with the value of 1 million children’s
lives and also a stick of gum.
Representing this in a table:
15
Wish in neither Wish in A, but Wish in C, but Wish in both A
A nor C not C not A and C
Take A and C Nothing 1 million lives 1 million lives 2 million lives
Take A, B and 1 bit of gum 1 million lives 1 million lives 2 million lives
C + 1 bit of gum + 1 bit of gum + 1 bit of gum
Take A, C and 1 million lives 2 million lives 2 million lives 3 million lives
D + gum + a bit of gum + 1 bit of gum + 1 bit of gum
Take A, B, C 1 million lives 2 million lives 2 million lives 3 million lives
and D + 2 bits of gum + 2 bits of gum + 2 bits of gum + 2 bits of gum
The astute reader might have noticed that someone in a Four-box Predictor situation is
just someone who faces both HSP-I and HSP-II at the same time. The very astute
reader might have noticed that this therefore constitutes a “Jackson Case” under
decision-theoretic uncertainty: a case in which one ought (in some sense) to do
something that one knows one ought (in some other sense) not to do.18
18 Where “Jackson case” refers to Frank Jackson, “Decision-Theoretic Consequentialism and the
Nearest and Dearest Objection,” Ethics, CI, 3 (1991): 461–82.
16
According to CDT, one ought to perform act (4). According to EDT, one ought to
perform act (2). But we should think that, in at least some sense of ‘ought’, what the
decision-maker ought to do is act (3). By performing (4) rather than (3), one risks
losing the opportunity to save 1 million children for the sake of a stick of gum. (This
was the motivation for one-boxing in HSP-I.) By performing (1) rather than (3), again
one risks losing the opportunity to save 1 million children for the sake of a stick of
gum. (This was the motivation for two-boxing in HSP-II.) And if one performs act (2)
rather than (3), one takes both risks at the same time. So one should perform act (3),
and take Boxes A, C and D: that’s the only safe bet. And it’s the only choice that
seems consistent with our intuitions in both HSP-I and HSP-II.
In the above situation, the correct thing to say, I think, is that, in some sense of ‘ought’
(the sense that first-order decision theories are talking about), one ought to perform
either act (2) or act (4), but that, in another sense of ‘ought’ (the sense that is relative
to decision-theoretic uncertainty), one ought to perform act (3). That’s the appraisal
that MDT gives of the situation. But that’s not the appraisal that Nozick’s view gives.
According to Nozick’s view, all there is to say is that one ought to perform (3), because
that’s what the true decision theory (that is, DTP) claims: it’s simply false, in any sense,
that one ought to choose either (2) or (4). And that seems to misrepresent what’s really
going on in our appraisal of the Four-Box Predictor.
For these reasons, for the rest of the paper I will set Nozick’s view to one side, and
instead assume that MDT is the most plausible rational explanation of the stakes-
sensitivity of our intuitions. Nozick quickly moved on from the suggestion, and as far
17
as I know it has not been pursued elsewhere.19 But that’s unfortunate, because MDT
has important implications that have not been noticed.
III. The Smoking Lesion and the Psychopath Button
First, MDT allows us to resolve an apparent conflict in our intuitions.20 My suggestion
is that the divergence in our intuitions across cases in the literature can be understood
as hedging between EDT and CDT, in a way that is mandated by MDT.
Consider The Smoking Lesion:21
The Smoking Lesion
19 Though Sepielli (“What to Do When You Don’t Know What to Do When You Don’t Know What
to Do….”, op. cit.) considers the view in the context of discussing the regress problem.
20 That is not to say that it can resolve the divergence in all of our intuitions. In particular, there is a
class of cases that it seems to me are not about whether evidential or causal decision theory is true, but
rather are about whether the usual formulation of causal decision theory accurately captures the idea
of ‘causing the best consequences’. I place Andy Egan’s time-traveller and Oracle cases (“Some
Counterexamples to Causal Decision Theory,” op. cit.) in this category, as well as Arif Ahmed’s
‘Nomological Gamble’ example (“Causal Decision Theory: A Counterexample,” op. cit.). I take these
examples to violate the letter of CDT, but not its spirit (though Ahmed goes on to argue that no way of
formalising CDT in such a way that it gets the right answer in his case is compatible with free choice.
That is an interesting, but very different, argument).
21 Here I use the formulation given in Egan, “Some Counterexamples to Causal Decision Theory,” op.
cit., p. 94.
18
Susan is debating whether or not to smoke. She believes that smoking is
strongly correlated with lung cancer, but only because there is a common
cause — a lesion that tends to cause both smoking and cancer. Once we fix
the presence or absence of this condition, there is no additional correlation
between smoking and cancer. Susan prefers smoking without cancer to not
smoking without cancer, and she prefers smoking with cancer to not smoking
with cancer.
In this case, intuitively Susan should smoke. But, problematically, EDT recommends
against smoking. The Smoking Lesion has been taken to be a fatal counterexample to
EDT. However, if our intuitions are explained in part by MDT, then our intuition
regarding The Smoking Lesion should change simply if we alter the stakes. And it seems
that it is. Consider the following case.
Stakes-Adjusted Smoking Lesion
The lesion is not correlated with mere lung cancer. Rather, the lesion causes
people both to smoke before they are 35 and to burst into flames on their 35th
birthday, enduring several hours of agony, before dying (even though smoking
does not cause the spontaneous self-combustion). Moreover, let us suppose that
Susan isn’t really that fussed about smoking. She hasn’t been inclined to
smoke previously, but she’s feeling whimsical today and so has a slight
preference for smoking that cigarette. It’s the day before her 35th birthday.
Should she smoke?
19
In this case, it seems very clear to me, intuitively speaking, that Susan should not
smoke, even though CDT would recommend smoking.22 Simply by altering the stakes,
we have transformed an apparent counterexample to EDT into an apparent
counterexample to CDT. This stakes-sensitivity is exactly what MDT would predict.
Next, consider The Psychopath Button:23
The Psychopath Button
Paul is debating whether to press the “kill all psychopaths” button. It would,
he thinks, be much better to live in a world with no psychopaths. And Paul is
almost certain that he is not a psychopath. Unfortunately, Paul is quite
confident that only a psychopath would press such a button. Paul very
strongly prefers living in a world with psychopaths to dying. Should Paul press
the button?
In this case, intuitively Paul should not press the button. But, problematically,
according to CDT Paul should press the button. Again, however, if MDT explains
our intuitions, then our intuitions about The Psychopath Button should be stakes-sensitive.
And it seems that they are. Consider the following modification of the case:
22 Some causal decision theorists I have spoken with have bitten the bullet in this case. But I have a
very hard time believing that such a response is genuinely a basic intuition, rather than a judgment that
has been tainted by one’s theoretical commitments.
23 Again I just modify slightly the formulation given in Egan, “Some Counterexamples to Causal
Decision Theory,” op. cit., p. 97. This case was initially presented in Egan’s paper, but was suggested to
him by David Braddon-Mitchell.
20
Stakes-Adjusted Psychopath Button
It is 1890, and Paul knows that Baby Hitler is a psychopath, knows that only
one other person (which may be him) is a psychopath, and knows of the
atrocities that will happen in the following sixty years if Hitler survives.
Moreover, let us suppose that Paul has a terminal illness. He will surely die
within a few hours. He wants to have those last few hours alive and, being a
selfish sort of person, mildly prefers having those hours to killing Hitler.
However, because it’s so morally important to kill Hitler, he only has a very
mild preference for living those few hours at the cost of Hitler’s survival. Now,
what should Paul do?
It seems very plausible to me that he should push the button. But, if so, then, again,
our intuitions about this case have switched merely by altering the stakes involved.
Again, this is exactly what MDT predicts.
We can make this explanation more precise. Let’s define the relative stakes ratio, in two
option cases, as the ratio of (EEV(right action according to EDT) – EEV(wrong action
according to EDT) to (CEV(right action according to CDT) – CEV(wrong action
according to CDT)). If MDT is correct, then it’s how this ratio changes that should
affect our intuitions.
Now, I’d personally be roughly indifferent between a guarantee of $1000 and 1%
chance of $1 million, so, given my preferences, the relative stakes ratio in Standard
Predictor is approximately 99:1. The Smoking Lesion is taken to be more favourable to
21
CDT than Standard Predictor is. So, if our intuitions roughly track MDT’s
recommendations, then we should expect the relative stakes ratio to be less than 99:1.
And that’s what we find. If I ask myself, for example, whether Susan would be willing
to take up smoking even at the cost of causing a 1% chance of moving from the low-
risk group for lung cancer (which non-smokers belong in) to the high-risk group for
lung cancer (which regular smokers belong in), I imagine her being willing to take that
cost. I imagine her only becoming indifferent at around 10%, suggesting that, when
presented with the case, I intuitively assess the relative stakes ratio as only being about
10:1. This is in line with MDT’s prediction.
The Psychopath Button is taken to be more favourable to EDT than Standard Predictor is.
So, if our intuitions roughly track MDT’s recommendations, we should expect the
relative stakes ratio in The Psychopath Button to be greater than 99:1. And that’s what
we find. If I ask myself, for example, whether Paul would be willing to kill all
psychopaths even at the cost of a 1% chance of causing his own death (perhaps he has
a gun with 99 empty chambers but one loaded chamber, and pointing the gun at his
head and pulling the trigger is the only way to kill all psychopaths), I imagine him not
being willing to take that risk. I certainly wouldn’t do it. Even if I thought it was ok to
murder innocents for a greater good (!), and even if I thought that killing all
psychopaths would be a net good, I still value my own life too much to make that sort
of sacrifice. But if that’s correct, then I have intuitively judged the relative stakes ratio
to be greater than 99:1, which is what MDT predicts.24
24 Egan gives another case, the Murder Lesion. It seems to me that, again, the reason our intuitions in this
case favour EDT is because of the relative stakes. However, the relative stakes don’t seem to be quite as
22
One might object that, in the ‘high-stakes’ and ‘stakes-adjusted’ cases given above, we
can explain the divergence in our intuitions by appeal to empirical uncertainty.
According to this explanation, in HSP-I we get the one-boxing intuition because we
can’t really imagine ourselves to be certain that the Predictor will get it right purely
through prediction. In any situation we can imagine, so the objection goes, there will
remain some residual uncertainty that choosing the one box causes there to be a wish
biased towards EDT as they are in The Psychopath Button. This might explain why The Murder Lesion is
not as convincing a ‘counterexample’ to CDT as The Psychopath Button is. MDT allows us to explain a
couple of other puzzles as well. First, James Joyce (“Regret and Instability in Causal Decision Theory,”
Synthese, CLXXXVII, 1 (2012): 123-45, at p. 125) says that The Psychopath Button ‘is not original with Egan,’
because structurally similar cases were given in Paul Weirich, “Decision Instability,” Australasian Journal
of Philosophy, LXIII, 4 (1985): 465–72; Allan Gibbard, “Weakly Self-Ratifying Strategies: Comments on
McClennen,” Philosophical Studies, LXV, 1–2 (1992): 217–25; Judea Pearl, "The Curse of Free-Will and
the Paradox of Inevitable Regret," Journal of Causal Inference, I, 2 (2013):255-7. Joyce takes the shared
structural similarity to be that all are cases where every option (prima facie) unratifiable according to
CDT. But, if this is right, then why weren’t these earlier cases taken to be grave counterexamples to
CDT, in the way that some at least have taken The Psychopath Button to be? The answer is with the
stakes. The Psychopath Button is not similar to the earlier cases with respect to the relative stakes ratio. And
it’s the relative stakes ratio that gives The Psychopath Button its bite. Second, MDT enables us to explain
is why our intuitions in Standard Predictor seem to favour EDT significantly more if the Predictor is
infallible, rather than merely highly accurate (as Jordan Howard Sobel discusses in “Infallible
Predictors,” The Philosophical Review, XCVII, 1 (1988): 3–24). The answer is twofold. First, increasing the
probability of the $1 million further biases the stakes in EDT’s favour. Second, as a matter of
psychology, we tend to overvalue a “sure thing” (which is why, for example, the Allais paradox arises):
so the move from 99% certainty to 100% certainty biases the stakes in favour of EDT by considerably
more than merely the value of an additional 1% chance of $1 million.
23
in the one box, and that’s how the Predictor pulls off his trick. Similarly, in Stakes-
Adjusted Smoking Lesion, perhaps we simply can’t imagine ourselves not to have some
credence that smoking causes bursting into flames. In either case, if we’ve got even
small credence in that empirical hypothesis, then both CDT and EDT will
recommend one-boxing in HSP-I and not-smoking in the Stakes-Adjusted Smoking
Lesion. One can attempt an analogous explanation with respect to HSP-II and The
Stakes-Adjusted Psychopath Button.
Speaking personally, my intuitions are sufficiently robust that we could replace the
stick of gum with the lives of ten thousand children and I would have the same view
that one should one-box in HSP-I and two-box in HSP-II. (The same is true for
adjustments to the stakes in Stakes-Adjusted Smoking Lesion and Stakes-Adjusted Psychopath
Button.) Given this, empirical uncertainty doesn’t seem to be a very good explanation
of my intuitions. However, there is a stronger response, which is that these extreme
cases aren’t strictly necessary to the use of MDT as an explanatory hypothesis for why
our intuitions favour CDT in The Smoking Lesion and EDT in The Psychopath Button. All
we need to show is that the relative stakes are more heavily biased towards CDT in
The Smoking Lesion than they are in the Standard Predictor, and are more heavily biased
towards EDT in The Psychopath Button than they are in the Standard Predictor. And that’s
exactly what my discussion of the relative stakes ratio accomplished.
So MDT seems to do well in terms of giving a rational grounding for our seemingly
conflicting intuitions. In fact, I think that it’s the best account of our intuitions in these
cases that I know of.
24
Consider, in contrast, the response to The Psychopath Button suggested by James Joyce,
as part of a defense of CDT.25 The idea is that, as Paul decides to perform one action
rather than another, he immediately gains evidence about whether he is a psychopath.
Given his initial credences, pushing the button has the higher causal expected value.
But as soon as he begins to decide to push the button, he gains evidence that he is a
psychopath, and his credences should change. And with those new credences, not-
pushing the button has the highest causal expected value. But as soon as he begins to
decide to not-push the button, he gains evidence that he is not a psychopath, and
suddenly pushing the button has the higher causal expected value again. Eventually,
his credences over whether he’s a psychopath or not end up in equilibrium, with the
expected causal value of both pushing the button and of not-pushing the button as the
same.
This response is interesting. However, It seems to me that my explanation of our
intuitions is significantly better than Joyce’s.26 This is for three reasons, presented in
order of increasing importance.
25 Joyce, “Regret and Instability in Causal Decision Theory,”, op. cit. His view is very similar to that of
Arntzenius (“No Regrets,” op. cit.), and both draw heavily on work by Brian Skyrms, The Dynamics of
Rational Deliberation (Cambridge, Mass.: Harvard University Press, 1990). What I will say in response to
Joyce applies fairly straightforwardly to Arntzenius’s view.
26 Of course, my explanation is not inconsistent with Joyce and Arntzenius’s explanation. What I’m
questioning is not whether their account is true but whether it’s a satisfactory explanation of our
divergent intuitions across these cases.
25
First, Joyce’s account can’t explain our intuitions in HSP-I and the Stakes Adjusted
Smoking Lesion. In order to explain our intuitions in those cases he would have to
appeal to some other explanation. In contrast, my account can both take the
intuitions at face value, rather than having to offer a speculative debunking argument,
and can offer one unified explanation for our varying intuitions, rather than having to
offer two distinct explanations.
Second, his account doesn’t get the intuitions right in very similar cases. Suppose, for
example, that the button is wired up to Paul’s brain, so that as soon as he begins to
intend to push the button, all psychopaths are killed. His beliefs therefore aren’t able
to achieve equilibrium. In this case, CDT really would recommend that he intend to
push the button. But it seems that this minor alteration to the case doesn’t affect our
intuitive appraisal of what Paul should intend to do.
Third and finally, Joyce’s account doesn’t capture the intuition even in the original
case. Once deliberational equilibrium is reached, pushing the button has the same
expected value as not-pushing the button. But that’s not capturing the intuition,
which is clearly in favour of it being a mistake to push the button, rather than it being
permissible to push the button. Joyce makes some very brief suggestions, based on the
heuristics and biases literature, concerning why we might think that the intuition is
not reliable in this case, and, in general, I am perfectly happy to sacrifice fit with the
intuitive data for the sake theoretical elegance. But if we have an independently
motivated explanation of why those intuitions are rational, then we should prefer that
explanation to the debunking explanation, unless the debunking explanation is on
very strong ground indeed. So we should prefer MDT’s explanation to Joyce’s.
26
Joyce’s is not the only alternative explanation in town. Ralph Wedgwood has
introduced ‘Benchmark Theory,’ which gets the right answer in both The Smoking
Lesion and The Psychopath Button.27 But it suffers from intuitive counterexamples too.28
Johan Gustafsson proposes a decision theory that captures the intuitions in both The
Smoking Lesion and The Psychopath Button cases.29 But that proposal comes at a cost of
considerable theoretical inelegance: importing an idea of “iterated general ratifiability”
that does not seem independently motivated. Another potential explanation comes
from Huw Price, who suggests we should understand causality in subjectivist terms, so
that evidential probability and causal probability are, despite appearances, the
same.30 Again, however, this comes at major theoretical cost, depending on the truth
of particular positions in the metaphysics of both causation and free will.31 And if an
alternative account explains our divergent intuitions without using such heavy
philosophical machinery, as the MDT account does, then we should prefer that
alternative explanation.
27 Wedgwood, “Gandalf’s Solution to the Newcomb Problem,” op. cit.
28 See, for example, Briggs, “Decision-Theoretic Paradoxes as Voting Paradoxes,” op. cit.
29 Gustafsson, “A Note in Defence of Ratificationism,” op. cit.
30 Price, “Causation, Chance, and the Rational Significance of Supernatural Evidence,” op. cit.
31 Price himself acknowledges this, when he says: “As we have seen, the EviCausalist relies heavily on
the idea that the epistemic viewpoint of an agent is distinctive in certain ways. Roughly, it requires that
agents see their own actions as “uncaused,” at least in the midst of deliberation about those actions.
This not only binds the fate of the EviCausalist, at least in some sense, to that of free will; it also means,
potentially even more uncomfortably, that EviCausalism becomes a rope that binds causation to the fate
of free will — no problem, perhaps, if these notions turn out to share the same fate, but a problem if
they do not.” Ibid., p. 536 (italics in the original).
27
In general, we already know (i) that we are decision-theoretically uncertain; and (ii)
that expected utility theory is in general the best way to accommodate uncertainty. So,
even independently of its ability to explain our divergent intuitions, we should think
that there is an argument for thinking that MDT is true. It explains our divergent
intuitions without using any ad hoc philosophical machinery. There is therefore a
strong argument via Occam’s razor for preferring the MDT explanation to any other
explanation that does not have the same independent plausibility.
So I think that MDT provides the best explanation of our apparently inconsistent
intuitions. Now let’s turn to further implications of this view, and see how it gives
grounds to undermine the two best arguments in favour of EDT.
IV. Undermining the intuitive argument for EDT
One way to argue in favour of EDT is via appeal to cases. EDT looks appealing for
people, like me, who think that you should one-box in the standard Newcomb
problem, and for those who are particularly concerned by The Psychopath Button. So it
looks like EDT it at least fairly well supported by the intuitive data.
Considerations of decision-theoretic uncertainty undermine this argument. The
relative stakes ratio in the standard Newcomb problem depends on one’s level of risk-
aversion with respect to money, but for any normal agent is heavily biased in favour
of EDT. For me the relative stakes ratio is approximately 99:1, so if I had only 1.1%
or higher credence in EDT, then, by MDT’s lights, I should one-box in the standard
Newcomb problem. So, far from providing an argument for thinking that the
28
evidential approach is the best approach, the intuitions merely show that we should
have at least a small credence in EDT. Indeed, because intuitions in the standard
Newcomb case are unclear, and because the Smoking Lesion favours CDT even
though in that case the stakes are still biased towards EDT, it seems that the credence
in EDT that is warranted by appeal to intuitions about particular cases is not very
large at all.
The fair way to adjudicate, on intuitive grounds, between EDT and CDT, would be
to consider cases where the stakes are evenly balanced. Such a case would look as
follows:32
Money in both boxes Money in one box only
Take one box only $20 $0
Take both boxes $30 $10
Even I — who used to self-identify as a stark-raving one-boxer33 — get almost no
intuition in favour of one-boxing in this case. So EDT is not the intuitive view. In fact,
I only start to get one-boxing intuitions once the amount that might be in the opaque
32 I use small amounts of money, so that we can safely assume that utility is approximately linear with
respect to money in this case.
33 Anecdote: when I was first presented with The Smoking Lesion case I thought it was supposed to be an
argument in favour of EDT.
29
box is twenty times as great as the amount that is certainly in the transparent box. So,
as far as the argument from intuition goes, I should have no more than a small
credence in EDT. So the intuitive argument for EDT is far weaker than it would have
first seemed.
Appeal to intuitions has been used as one major argument in favour of EDT. The
other argument is the “Why Ain’Cha Rich?” argument. Let’s consider that now.
V. A Counterexample to “Why Ain’Cha Rich?”
When I introduced meta decision theory, I used unconditional credences. But we
could formulate both evidential and causal versions of meta decision theory.
According to causal meta decision theory (CMDT) we should maximize causal meta
expected value (CMEV) where:
!
���� � = � � ⇒ �! �! (�)
!!!
Again using ‘⇒’ to denote the counterfactual conditional. It should be clear that for
all A, D, Pr(� ⇒ �)= Pr � . Acting one way rather than another can’t affect which
decision theory is true. So nothing is lost by simply using unconditional credences:
!
���� � = � �! �! (�)
!!!
30
In contrast, according to evidential meta decision theory (EMDT) we should
maximize evidential meta expected value (EMEV), where:
!
���� � = � �! |� �! (�)
!!!
These two views will almost never come apart: it’s a very rare situation when acting
one way or another gives you evidence for one decision theory rather than another.
But it’s not impossible for the two to come apart. And if we look at those admittedly
rare cases we can construct a counterexample to the “Why Ain’cha Rich?” argument.
According to the “Why Ain’cha Rich?” argument, the average return of one-boxing
exceeds the average return of two-boxing. Moreover, everyone can see that the
average return of one-boxing exceeds the average return of two-boxing: so one-
boxing foreseeably gives us more of what we want than two-boxing does. And, so the
argument goes, a decision theory cannot be correct if it recommends an option that
foreseeably gives you less of what you want than some other option does. So CDT
can’t be correct.
In response, the defender of CDT can say that Newcomb’s cases are unusual: these
are cases where a devious person has set things up to reward irrational behavior. So it
isn’t surprising that irrational people like those who act in accordance with EDT end
up richer. However, to date the defender of CDT hasn’t been able to come up with a
31
convincing case where one gets rewarded for not following EDT.34 And that seems
problematic.
However, if we are comparing CDMT and EMDT, things are different. Once we
allow decision-theoretic uncertainty into the picture, we can construct a case where
performing the action EMDT recommends foreseeably makes one poorer. So there is
no longer an asymmetry between the causal and evidential approach, and the “Why
Ain’cha Rich?” argument loses its force. Here’s the case:
The Meta Newcomb Problem
Sophie faces two boxes, as follows:
Wishes in both boxes Wishes in one box only
One box 2 million lives35 0 lives
Two box 3 million lives 1 million lives
34 David Lewis (“`Why Ain’cha Rich?’,” Noûs, XV, 3 (1981): 377–80) argues that it’s impossible. In his
introduction to decision theory, Weatherson summarizes the literature on this as follows: “it turns out
to be very hard, perhaps impossible, to construct a problem of this sort for evidential decision theorists.”
(p. 89; available at brian.weatherson.org). Arntzenius (“No Regrets,” op. cit.) has proposed an example,
but it is debatable whether the example is successful or is even coherent. For discussion of that case, see
Arif Ahmed and Huw Price, “Arntzenius on ‘Why Ain’cha Rich?’,” Erkenntnis, LXXVII, 1 (2012): 15–30.
35 Again using “lives saved” rather than dollars because linear value over number of lives saved is more
plausible than linear value over dollars.
32
Sophie’s beliefs are as follows. She has 51% credence in EDT and 49%
credence in CDT. Before taking her action, she is almost certain that there are
wishes in both boxes. However, conditional on her two-boxing, she is almost
certain that there is a wish only in the transparent box.
Given these credences:
VEDT(One Box) = ~1 * 2 million lives + ~0 * 0 lives = ~2 million lives
VEDT(Two Box) = ~0 * 3 million lives + ~1 * 1 million lives = ~1 million lives
VCDT(One Box) = ~1 * 2 million lives + ~0 * 0 lives = ~2 million lives
VCDT(Two Box) = ~1 * 3 million lives + ~0 * 1 million lives = ~3 million lives
So the meta decision problem looks as follows:
Value, given EDT Value, given CDT
One box ~2 million ~2 million
Two box ~1 million ~3 million
However, Sophie places great weight, epistemically, on what people actually
do in Newcomb cases (rather than what people claim their intuitions are about
what they would do in such cases). She thinks that the actions of typical
human agents in Newcomb cases provide very good evidence in favour of
33
CDT or EDT. And she believes she is a typical human agent. So how she acts
will affect her credences in the two decision theories. If she one-boxes, she will
update in favour of EDT, and come to have 52% credence in EDT and only
48% in CDT. If she two-boxes, she will significantly update in favour of CDT,
and come to have 60% credence in CDT and 40% credence in EDT.
What should Sophie do? To answer this, let’s work out the expected values. We have:
VCMDT(One Box) = 0.51×2 + 0.49×2 = 2
VCMDT(Two Box) = 0.51×1 + 0.49×3 = 1.98
So CMDT recommends one-boxing. And we have:
VEMDT(One Box) = 0.52×2 + 0.48×2 = 2
VEMDT(Two Box) = 0.4×1 + 0.6×3 = 2.2
So EMDT recommends two-boxing.
So if we take into account decision-theoretic uncertainty, then the causal theory can
tell you to one-box while the evidential theory tells you to two-box. In the above case,
if Sophie follows EMDT she foreseeably ends up saving fewer lives than if she follows
CMDT. So, unlike at the first order, one cannot construct a “Why Ain’cha Rich?”
argument in favour of EMDT over CMDT.
This makes it seem very plausible that the correct meta decision theory is causal. But
we can go a bit further than that. It would seem odd if the correct meta decision
theory were causal while the correct first-order decision theory is evidential. It seems
34
plausible that our views about which variety of first-order decision-theory is correct
and which variety of meta decision theory is correct should be at least roughly
coherent. So evidence about which meta decision theory is true seems also to give
evidence about which first-order decision theory is true. So, even if we can’t construct
a counterexample to “Why Ain’cha Rich?” for EDT, the fact that we can construct
such a counterexample for EMDT weakens, at least to some degree, the “Why
Ain’cha Rich?” argument in favour of EDT.
So, as well as being able to provide an explanation of our divergent intuitions across
cases, considerations relating to meta decision theory allow us to generate novel
arguments against EDT. So meta decision theory (and the idea of metanormativist
more generally) seems to be a powerful tool in the causal versus evidential debate.
VII. Conclusion
In this article I’ve argued that meta decision theory has two important implications.
First, it can explain the apparent divergence in our intuitions between the Standard
Predictor, The Smoking Lesion, and The Psychopath Button. Second, it undermines both the
intuitive argument in favour of EDT, and, to some extent, the “Why Ain’cha Rich?”
argument as well. Considerations of decision-theoretic uncertainty are therefore a
powerful tool for use in debates between causal and evidential decision theory — a
tool that gives the causal approach a significant new advantage.
35