Prediction Impairs Deliberation

3.1. Psycho-Button

Much of philosophers’ recent anxiety regarding CDT stems from the counterintuitive nature of its verdicts in Psycho-Button, a problem introduced by (Egan 2007) but presented in sanitized form here:¹²

Psycho-Button: A superintelligence places a button before you. The button is either rigged so that pushing it will credit a hundred dollars into your bank account or so that pushing it will debit two hundred dollars from your bank account. You are offered a choice between pushing or not pushing the button. Finally, you know that the superintelligence has (at some point in the past) rigged the button to credit the money into your account if and only if they predicted you would not push the button.

Let us assume that you take the superintelligence to be a highly reliable (though perhaps still imperfect) predictor of your behavior and thus take your choice to be highly indicative of the prediction made. We shall also suppose that, for purposes of this problem, you only care about money and value it linearly. One formal instantiation of Psycho-Button is then given in the desirability and probability matrices represented in Tables 3 and 4.

Recognizing that the superintelligence’s predictions again constitute the relevant dependency hypotheses, it is easy to calculate both that EDT recommends against pressing the button (i.e., $V(\mathit{\text{Do}}{n}^{’}t)>V(\mathit{\text{Push}})$)¹³ and that CDT recommends the opposite (i.e., $U(\mathit{\text{Push}})>U(\mathit{\text{Do}}{n}^{’}t)$).¹⁴ Psycho-Button thus seems like an advantage for EDT and a liability for CDT. Conditional upon pressing the button, one will (assuming updating by conditionalization) become very confident that the superintelligence has predicted this and thus that pressing is very likely to cause one to lose two hundred dollars. In the common jargon, pushing the button is unratifiable: it appears suboptimal conditional upon its own enactment. The CDT agent is thus bound to regret her decision in this case.

Of course, the same is true for agents who decide not to press the button. Conditional upon this decision, one will also become very confident that the superintelligence has predicted this and thus that pressing would very likely have caused one to gain a hundred dollars. So neither option in this case is ratifiable. Nonetheless, there still seems to be a prudentially relevant asymmetry between the two choices. One will intuitively regret pressing the button much more than one will regret abstaining, which may suggest that abstaining is the rationally obligatory course of action. Even if this common intuition is misguided, however, it seems highly plausible that we should at least avoid insisting that the opposite course of action (i.e., pushing the button) is itself obligatory and, thus, that refraining from pushing the button is not even rationally permissible. But a straightforward application of CDT to the above version of Psycho-Button appears to yield just this unhappy result. Hence, the case certainly seems to constitute a prima facie strike against CDT.¹⁵

3.2. Frustrater

An even stronger objection to CDT stemming from decision instability comes in the form of the Frustrater example cooked up by (Spencer & Wells 2019):¹⁶

Frustrater: A superintelligence places two boxes before you, $A$ and $B$, along with an envelope $E$ containing forty dollars. You are offered your pick of the three. The superintelligence has distributed a hundred dollars between $A$ and $B$, according to their prediction of your choice. If they predicted you’d take $A$, the hundred is in $B$. If they predicted you’d take $B$, the hundred is in $A$. If they predicted you’d take the envelope, they distributed the money evenly between $A$ and $B$, placing fifty dollars in each.

Let us assume that you take the superintelligence to be a highly reliable (though perhaps still imperfect) predictor of your behavior and thus take your choice to be indicative of the prediction made in the relevant ways. We shall also suppose that, for purposes of this problem, you only care about money and value it linearly. One formal instantiation of Frustrater is thus given in the desirability and probability matrices represented in Tables 5 and 6.

EDT recommends taking the envelope here (since $V(E)>V(A)=V(B)$),¹⁷ while CDT recommends taking either box $A$ or box $B$ (since $U(A)=U(B)>U(E)$).¹⁸ Whatever we think of CDT’s recommendation in Psycho-Button, its verdict here certainly seems problematic for the theory. You are confident that if you choose either $A$ or $B$ your move will have been predicted and you will thus gain nothing. However, you know that taking the envelope guarantees you a sure forty dollars. While none of your options here are ratifiable (as in Psycho-Button), taking the envelope certainly seems like the prudent thing to do, yet CDT forecloses the possibility of rationally taking this course of action. Another problem for CDT.

6.1. Maximin CDT

Suppose that you agree with the guiding thought behind CDT that instrumental value is a matter of causal efficacy rather than auspiciousness, but that you also satisfy Open-mindedness and thus adopt maximally indeterminate act credences. For concreteness, let’s say that the decision problem you currently face is a version of Psycho-Button with sharp conditional probabilities as given in Table 4. Each avatar in your representor can apply traditional CDT to arrive at causal efficacy values for pushing and not pushing the button. If all of these avatars agreed concerning the relative causal merits of pushing and not pushing, we could simply say that you should respect their unanimous opinion. (This is what happens for an open-minded agent confronting Newcomb.) But they don’t. Avatars that are relatively confident of your opting to push the button will recommend against that course, while those relatively confident of your opting not to push will recommend the opposite. An open-minded agent’s assessment of the causal efficacy of her options in Psycho-Button is thus indeterminate. What to do?

A popular suggestion for how to go about making decisions in contexts involving this sort of credal indeterminacy is to apply the rule of maximizing minimum expected utility:

Maximin CDT: Rational agents deliberating over a finite action set $\mathcal{A}$ ought to choose an act $a\in \mathcal{A}$ that maximizes $\mathit{\text{mi}}{n}_{P\in \mathcal{P}}\{U_P(a)\}$, where $\mathcal{P}$ is the deliberating agent’s credal representor and $U_P$ is causal utility relative to $P$.³⁶

Maximin CDT clearly agrees with traditional CDT in cases where the latter applies (i.e. when $\mathcal{P}$ is a singleton), but generalizes to cover cases of credal indeterminacy as well. In such cases, Maximin CDT encodes a kind of pragmatic pessimism: if the relative causal expected utility of two acts is indeterminate, assume the worst about each and maximize the minimum. There are plenty of objections one can raise to this as a general decision rule, but, to its credit, when coupled with Open-mindedness, Maximin CDT does yield precisely the correct answers in each of the three cases that have concerned us so far: Newcomb, Psycho-Button, and Frustrater.

Maximin CDT gets Newcomb right because, even for an open-minded agent, the causal expected utility of two-boxing is determinately greater than the causal expected utility of one-boxing. Though Psycho-Button jettisons this determinacy, Maximin CDT still continues to yield the intuitive recommendation of not pushing the button in the case of open-minded agents since for such agents: $mi{n}_{P\in \mathcal{P}}\{U_P(\mathit{\text{Do}}{n}^{’}\mathit{\text{tPush}})\}=0>–500=\mathit{\text{mi}}{n}_{P\in \mathcal{P}}\{U_P(\mathit{\text{Push}})\}$. Even Frustrater, which was the greatest source of trouble for CDT in the absence of Open-mindedness, is handled appropriately by Maximin CDT, which unambiguously recommends taking the envelope, as the causal expected utility of taking the envelope for an open-minded agent is determinately 40, while, relative to distinct avatars, the causal expected utility of taking box A and taking box B can each drop to as low as 0.

So far, so good for Maximin CDT. Unfortunately, however, this rule’s seeming virtues are largely an artifact of the puzzle cases we have so far chosen to set our focus upon. In cases like Psych-Button, each of the available options is self-frustrating or unratifiable. In many problems that have this feature, Maximin CDT seems to yield intuitive results, but there are other problems where its verdicts are more clearly problematic.

Nice Psycho-Button: A superintelligence places a button before you. The button is either rigged so that pressing it will credit a hundred dollars into your bank account or so that pressing it will debit two hundred dollars from your bank account. You are offered a choice between pressing or not pressing the button. Finally, you know that the superintelligence has (at some point in the past) rigged the button to credit the money into your account if and only if they predicted you would press the button.

Let us assume that you take the superintelligence to be a highly reliable (though perhaps still imperfect) predictor of your behavior and thus take your choice to be highly indicative of the prediction made. We shall also suppose that, for purposes of this problem, you only care about money and value it linearly. One formal instantiation of Nice Psycho-Button is then given in the desirability and conditional probability matrices represented in Tables 9 and 10. (Note: Table 10 encodes conditional probabilities for states given acts, rather than unconditional probabilities, so as to avoid contradicting Open-mindedness.)

For the open-minded agent, Nice Psycho-Button again presents a problem involving indeterminate causal efficacy judgments. However, in this example, your available choices are self-fulfilling rather than self-frustrating. For example, conditional upon pushing the button, you are likely to walk away with a hundred dollars more than you would have gotten by not pushing it. It now seems intuitive that pushing the button is rationally permissible and perhaps even obligatory. Yet Maximin CDT treats this case exactly the same as the original Psycho-Button: it is still refraining that uniquely maximizes minimum causal expected utility. We thus have a relatively clear case of Maximin CDT under-(and likely over-)generating rational permissibility verdicts. This seems enough to set aside Maximin CDT as a generally applicable account of rational choice.³⁷ Is there a simple alternative decision rule that might allow Open-mindedness to more fully save our intuitions regarding these cases?

6.2. Hierarchical CDT

There is. Or so it seems to me, though I should concede up front that I am not entirely confident of how theoretically well-grounded the proposal ultimately is. Its main virtue is simply that it does, by my lights at least, yield intuitively correct (im)permissibility verdicts across the range of decision problems involving act-state dependence. To state this second rule, let’s first define that an act $a$ has a causal efficacy value determinately greater than that of act $b$, written $a\succ b$, for an agent with credal representor $\mathcal{P}$, just in case the causal expected utility of $a$ is greater than that of $b$ relative to each $P\in \mathcal{P}$. An agent’s $\succ$-maximal options are those that are maximal with respect to the partial ordering on acts induced by $\succ$.³⁸ The decision rule we can now propose supplements the determinate verdicts of CDT with those of standard EDT in a hierarchical fashion:

Hierarchical CDT: Rational agents deliberating over a finite action set $\mathcal{A}$ ought to choose any $V$-maximal member of the set of $\succ$-maximal acts in $\mathcal{A}$.

As with Maximin CDT, in the presence of sharp act credences, Hierarchical CDT is nothing other than standard CDT. But, in the case of agents with imprecise credences (e.g., open-minded agents), Hierarchical CDT effectively instructs agents to first apply causalist reasoning and rule out all determinately causally inferior options (e.g., taking one box in Newcomb), and then, amongst the surviving options, choose one that maximizes desirability.

This theory handles all the problems so far considered (again assuming Open-mindedness). It uniquely yields refraining from pushing the button in Psycho-Button, pushing the button in Nice Psycho-Button, and taking the envelope in Frustrater. And it can, I think, be given a sensible enough rationale, by causalist lights, even though I lack the sort of complete justificatory story that one could feel confident in proposing as a fully satisfying theoretical grounding. From the causalist standpoint, rational choice aims at bringing about the best outcomes one can given the causal structure of the world and one’s uncertainty regarding it. That is, a causalist agent wants to perform acts of greatest possible causal efficacy with respect to realizing her ends, i.e., acts that maximize $U$. It thus stands to reason that when the causal efficacy value of one act is determinately inferior to another (e.g., as in Newcomb), the determinately lesser act may safely be removed from deliberative consideration. But when an agent’s credal state fails to fix any determinately $U$-optimal acts, this elimination of causally inferior acts is insufficient to draw deliberation to its close. How should an agent choose amongst acts whose relative causal efficacy values are indeterminate? It seems to me that turning to considerations of auspiciousness à la EDT, while inappropriate as an account of the fundamental aim or orientation of rational choice, is nonetheless at the very least as reasonable a way as any to make decisions when more fundamental causal considerations fall silent.

But we can say more. There is perhaps an analogy to be drawn here between the role played by evidentialist considerations in Hierarchical CDT and the role played by intentions in fixing the rationality of behavior according to some action theorists, most notably (Bratman 2012).³⁹ As Bratman and others rightly emphasize, intentions lack the central action-licensing significance of more fundamental decision guiding attitudes like degrees of belief and value judgments. The mere fact that one intends to $\varphi$ does nothing to render $\varphi$ing rational if $\varphi$ing is not independently recognizable as an instrumentally effective course of action. However, it would be wrong to infer from this that intentions have no role to play in setting bounds for rational action. Supposing that multiple available courses of action are judged rationally adequate relative to one’s beliefs and aims, prior intentions to act in certain ways can play a critical role in arbitrating amongst competing practical possibilities. Bratman has convincingly argued that ascribing intention such a role in the theory of rational action allows agents to coordinate their behavior, both intertemporally and interpersonally, in ways that would otherwise be closed to them.⁴⁰ If this view is correct, instrumentally rational action has a hierarchical structure to it: most fundamentally, rationality in action is determined by the acting agent’s degrees of belief and value judgments (i.e., by considerations of expected utility), but, at a secondary-level, such rationality is also constrained by considerations of intention and planning.

Analogously, the proponent of Hierarchical CDT has a similarly layered view of rational choice. Fundamentally, it is determined by causal considerations, but evidential considerations may still play a secondary role in rational arbitration. There may, in fact, be more than just a faint analogy here between Hierarchical CDT and the view that intention can act as a filter on rationally admissible choices. In the decision problems considered hereto, agents have not been afforded the opportunity to form intentions to pre-commit to particular choices causally prior to the predictions of the superintelligence. But we can nonetheless ask ourselves what intentions they would have been rational (in a causalist sense) to form in this regard, if they could have done so. Invariably in these examples, the recommendation of CDT would be to pre-commit to acting in accord with EDT. For example, in Newcomb, one wants to lead the superintelligence to place a million dollars in the opaque box, so pre-committing to one-boxing is advisable on causalist grounds. In Psycho-Button, one knows that pre-committing to pushing the button will lead the superintelligence to configure the button so as to debit two hundred dollars from your account, and so the only sensible thing to do is to pre-commit to refraining. And so on. Thus, in these particular cases, the prescription to maximize desirability among causally admissible options is tantamount to deferring to suitable counterfactual pre-commitment intentions when the verdicts of causal expected utility maximization are ambiguous.⁴¹

As indicated, I don’t pretend that a story like this constitutes a fully satisfying theoretical foundation for Hierarchical CDT. Too many questions remain. For example, even granting that actual intentions can act as rational filters in contexts of evaluative indeterminancy, why accord the same status to counterfactual ones? Moreover, while it seems intuitive that EDT coincides with appropriate causalist pre-commitment dispositions in the sorts of cases that concern us, I haven’t worked out this suggestion in any precise detail, let alone offered any general proof of the hypothetical identification. Still, I think we have said enough to warrant taking Hierarchical CDT seriously as a candidate for open-minded application of CDT. Additionally, as we shall see below, adopting this method of resolving evaluative indeterminacy allows us to satisfy certain plausible rationality principles that other approaches (e.g. Deliberational CDT) have notably foundered upon. My central argument on the approach’s behalf, however, remains the fact that, when coupled with Open-mindedness, Hierarchical CDT saves the relevant phenomena, so to speak, by neatly systematizing a wide range of our intuitive rational permissibility judgments.

The import of all of this for our defense of Open-mindedness should be clear: If Hierarchical CDT is the right way for agents to make decisions in the face of credal indeterminacy, then a strong pragmatic argument can indeed be made in favor of Open-mindedness. Open-minded agents following Hierarchical CDT never seem to make worse decisions than their more opinionated, sharp-credenced counterparts, and sometimes do much better. Or rather, this is close to the truth. Hierarchical CDT still requires, in my view, a modest revision before I can fully stand behind my claim that it constitutes the correct account of rational choice. But this revision is most easily introduced in the context of comparing it with other exisiting accounts of rational choice, and so I shall defer it until §7.3.

7.1. Deliberational CDT Revisited

The advantage of coupling CDT with the Open-mindedness postulate relative to supplementing it with deliberational considerations of the sort suggested by authors like Skyrms, Arntzenius, and Joyce, should by now be apparent. The former can allow us to account for the rationality of such actions as taking the envelope in Frustrater, while the latter cannot. At the same time, embracing Open-mindedness is compatible with preserving the central virtues of Deliberational CDT, most notably its satisfaction of Causal Dominance and its attention to the causal information gleanable from hypothetical decisions (albeit that deliberative and open-minded, hierarchical CDT agents pay attention to such causal information in quite different ways).

7.2. Graded Ratifiability

Rather than drawing radical epistemic lessons like Open-mindedness, some philosophers have taken the cluster of decision problems discussed so far to motivate constructing entirely new decision rules for assessing the choiceworthiness of acts. Perhaps most famously, (Harper 1986), building on an idea of (Jeffrey 1983), suggests supplementing standard CDT with a ratifiability condition: maximize $U$ only amongst ratifiable options, i.e., options that maximize $U$ conditional on their own performance. The obvious objection to such a proposal is that some decision problems (including Psycho-Button, Frustrater, etc.) lack ratifiable options. Harper replies that such problems only lack ratifiable options if we rule out the possibility of mixed strategies, that is, random choices employing (internal or external) chance devices. When such genuine randomization is possible and unpenalized, this move suffices to render Harper’s proposal applicable. But if these conditions can’t be met (e.g., if randomizing is either impossible or costs something), ratifiable options may again disappear, rendering the ratifiability criterion of little use.⁴²

In light of the potential inapplicability of the absolute ratifiability criterion, (Barnett 2022) draws our attention to the notion of graded ratifiability. Even in decision problems where no strictly ratifiable options exist, some options may still appear less ratifiable than others. For example, as noted in our initial discussion of Psycho-Button, while you will regret both pushing and refraining, conditional on each act’s own performance, there is a sense in which you will regret pushing much more than you will regret refraining. After all, pushing is likely to cause you to lose two hundred dollars, while refraining only indicates missing out on the opportunity to receive a hundred. Formally, Barnett defines an act $A$’s degree of ratifiability, relative to an alternative $B$, as $U_A(A)–U_A(B)$, where $U_A$ is simply causal expected utility computed relative to $P(\cdot |A)$, i.e., the agent’s credence function conditioned upon $A$. It is straightforward to verify that, so understood, the degree of ratifiability of pushing in Psycho-Button (relative to refraining) is indeed less than the degree of ratifiability of refraining (relative to pushing).

A number of authors maintain, with Barnett, that the correct measure of choiceworthiness in binary decision problems is degree of ratifiability. The decision rule that instructs maximizing this quantity is, for example, an implication of such decision theories as Wedgwood’s (2011) Benchmark Theory, Podgorski’s (2022) Tournament Decision Theory, and Gallow’s (2020) Managing the Improvement News, which can all be viewed as attempts to generalize the graded ratifiability rule beyond binary problems.⁴³ The details of these generalizations are not terribly important for our purposes, though we may note that all of these theories neatly handle the cases of decision instability that originally motivated concern for CDT (e.g., they recommend taking the envelope in Frustrater), while avoiding some of the problematic verdicts of EDT (e.g., in Newcomb).

If any of the above graded ratifiability theories are correct, we lose our present argument that prediction actively impairs rational deliberation. However, this is only because, like EDT, none of these theories’ choice rules employ quantities that logically fix act probabilities. For example, in the binary case, the degree of ratifiability of one act $A$ relative to another $B$ is only a function of $U_A(A)$ and $U_A(B)$, both of which are independent of the probabilities, if such exist, of $A$ and $B$. So, under graded ratifiability theories, act probabilities evade the charge of practical liability but are open to critique along more traditional lines. In particular, on these views, act probabilities fall prey to Spohn’s charge of epiphenomenalism or practical impotence. So, even if I am wrong that the correct fundamental measure of choiceworthiness is given by causal expected utility and this measure is instead given by some generalization of graded ratifiability, the deliberation crowds out prediction thesis may yet remain a reasonable position.⁴⁴

That said, I believe there are good reasons to prefer coupling CDT with Open-mindedness to replacing it with a graded ratifiability theory. For starters, while graded ratifiability theories yield intuitive answers regarding many puzzling decision problems, they don’t seem to do so universally. In particular, I have in mind problems like Nice Psycho-Button.⁴⁵ Computing the degrees of ratifiability of pushing and refraining in this problem, we find that pushing is strictly less ratifiable than refraining, even though both options are ratifiable in the absolute sense and pushing seems at least intuitively like a rationally permissible (if not obligatory) course of action.⁴⁶ This puts graded ratifiability theories at odds not only with my proposal, but also with both EDT and (standard and deliberational) CDT, which, in at least some cases, recommend pushing the button in Nice Psycho-Button.⁴⁷ These theories are thus non-conservative in the sense of violating:

Conservatism: If an act is recommended by EDT and possibly recommended by Deliberational CDT, then it is at least possibly rationally permissible.⁴⁸

Given that EDT and CDT are our most popular and best developed theories of rational choice, built upon substantially different measures of choiceworthiness, Conservatism plausibly takes their agreement regarding the (potential) rational permissibility of an act to be a reliable indicator of the act’s (potential) rational permissibility. EDT and Deliberational CDT themselves obviously satisfy this condition trivially, as does Hierarchical CDT. But graded ratifiability theories violate Conservatism by uniquely singling out refraining as rational in Nice Psycho-Button.

A further reason to prefer the approach to rational choice sketched here to those of Barnett, Wedgwood, Podgorski, and Gallow, concerns the well-known difficulties faced by these authors when attempting to generalize the rule of maximizing degree of ratifability beyond binary decision problems to ones involving three or more options. Barnett suggests the principle that one option $A$ ought to be preferred to another $B$ just in case $A$’s ratifiability relative to $B$ is greater than $B$’s relative to $A$. While this does allow us to construct a preference relation over options in more-than-binary decision problems, a preference relation so constructed is liable to include cycles, rendering the thought of maximizing according to it senseless. Recognizing this, Barnett, offers no general decision rule for many option problems, content to leave us with only constraints on preference rather than choice. That this is an unsatisfying state to leave decision theory in is attested to by Wedgwood, Podgorski, and Gallow’s creative efforts to develop richer theories that offer generally applicable choice recommendations.

However, each of these valiant attempts ends up contradicting what we early on (in §2) identified as the fundamental aim of rational decision making (namely, causally promoting good outcomes) in virtue of violating Causal Dominance. While these theories each satisfy Causal Dominance in binary decision problems (they were, after all, developed in part with an eye toward satisfying this principle in Newcomb), this guarantee is lost in the context of decision problems involving at least three options. Clever examples due to (Ahmed 2012) and (Spencer & Wells 2019) suffice to bring this out in the case of each of Wedgwood, Podgorski, and Gallow’s theories. Since this fact is already well known and rehearsing the proof of it would require introducing the targeted theories in more detail, I will content myself with merely registering the complaint.^49,50

Of course, some will object to my judgment that Causal Dominance is any kind of virtue for a decision theory to satisfy. Such objectors will include not just partisans of EDT who reject causalist verdicts even regarding Newcomb but also some theorists broadly sympathetic to causalist arguments in that case. For example, Spencer and Wells, though committed to the rational necessity of taking both boxes in Newcomb, view their Semi-Frustrater case, which can be used to illustrate Causal Dominance violations on the part of some graded ratifiability theories, as a counterexample to Causal Dominance and not as a counterexample to the theories whose Causal Dominance violations it witnesses.⁵¹

Semi-Frustrater: A superintelligence places two buttons before you, one black and one white. Exactly one button is rigged so that pushing it will credit a hundred dollars to your bank account, while the other is rigged so as to have no effect. You may press only one button, with either your right or your left hand. Pressing with your right hand will credit an extra five dollars to your account. The superintelligence has (at some point in the past) rigged the white (black) button to credit the money into your account if and only if they predicted you would push the black (white) button. The superintelligence is extremely reliable when it comes to predicting right hand movements but no better than chance when it comes to left hand movements.⁵²

Making the usual assumptions about your values and beliefs, we may formally instantiate a version of Semi-Frustrater via Tables 11 and 12.

Supposing that you are an open-minded causalist, we can make two judgments here thanks to Causal Dominance: namely, $\mathit{\text{WhiteRight}}\succ \mathit{\text{WhiteLeft}}$ and $\mathit{\text{BlackRight}}\succ \mathit{\text{BlackLeft}}$. Hierarchical CDT can thus only license pressing with your right hand. But Spencer and Wells take it to be intuitive that pressing with your left hand is more rational than pressing with your right due to the comparative unpredictability of the former forms of action. Since some graded ratifiability theories recommend pressing with the left hand in Semi-Frustrater,⁵³ one might thus claim that their liability to violate Causal Dominance in certain non-binary choice problems is a virtue rather than a vice. From the perspective of the thoroughly causalist spirit that animates this paper, however, this would be a mistake. As suggested in §2, taking both boxes ought to be rationally favored over taking only one by an agent facing Newcomb precisely because such an agent knows that the objective value of taking both is strictly greater than that of taking one. For exactly the same reason though, an agent facing Semi-Frustrater ought to favor pressing the white (black) button with her right hand to pressing the white (black) button with her left hand, since the former action is known to yield a greater outcome than the latter. Given these pairwise judgments about rational favorability, I don’t find it especially intuitive that either of the left hand options must turn out to constitute the overall most rational choice to make in Semi-Frustrater.⁵⁴

Is such a strongly causalist orientation compatible in spirit though with the intuitive judgments I endorse in cases like Psycho-Button and Frustrater? Or, put differently, is there a motivational tension between insisting upon both right handed pressing in Semi-Frustrater and envelope taking in Frustrater? Not to my mind. A key point of this essay has been that if we augment strongly causalist commitments with endorsement of the deliberation crowds out prediction thesis and its explication in terms of Open-mindedness, we are forced to conclude that (unlike in Semi-Frustrater) none of the options in Psycho-Button or Frustrater is of determinately greater utility than any other, so none is uniquely choiceworthy according to the most fundamental measure of choiceworthiness. But that does not imply that none is uniquely choiceworthy, for there may be secondary measures of choiceworthiness that serve to establish the relative advisability of options whose merits are inadequately distinguished by the lights of the fundamental measure. This is exactly what Hierarchical CDT proposes: a layered account of rational choice featuring a fundamental causal measure of choiceworthiness supplemented with a secondary evidential measure. If this tiered account of rational choice is correct, we should see nothing strange about pairing right handed pressing in Semi-Frustrater with common sentiment about cases like Psycho-Button and Frustrater. Of course, while these theoretical judgments align with my own intuitive reactions to these cases, I recognize that they will not align with those of every reader, each of whom is free, as always, to weigh the costs and benefits of the various proposals on offer for themself.⁵⁵

7.3. General Ratifiability

There is an alternative, quite different generalization of ratifiability due to (Gustafsson 2011) that goes by the name general ratifiability. An act $A$ is generally ratifiable just in case there is no act $B$ such $U_C(B)>U_C(A)$ for all available acts $C$. That is, an option is generally ratifiable if there is no alternative option that will foreseeably enjoy greater utility regardless of what choice is ultimately made. Causally dominated acts are never generally ratifiable, and, hence, taking only one box in Newcomb is neither ratifiable nor generally ratifiable. However, in other decision problems, many options that are not ratifiable are nonetheless still generally ratifiable (e.g., pushing the button in Psycho-Button). In finite choice problems, Gustafsson shows that generally ratifiable options, unlike ratifiable ones, are always guaranteed to exist, without invoking mixed strategies.

This enables Gustafsson to suggest that, as a first approximation, the correct theory of rational choice is given by:

General Ratifiability: Rational agents deliberating over a finite action set $\mathcal{A}$ ought to choose any $V$-maximal member of the set of generally ratifiable acts in $\mathcal{A}$.

This theory yields all the intuitive verdicts in the decision problems so far noted and satisfies Causal Dominance and Conservatism. In fact, I think Gustafsson’s theory (with the below qualifications), unlike EDT or theories of graded ratifiability, constitutes an extensionally correct account of rational choice. This is so, I suggest, because it actually agrees with Hierarchical CDT in the case of open-minded agents! An act is generally ratifiable just in case an open-minded agent would not deem it deteminately inferior to another with respect to causal efficacy. Hence, for open-minded agents, maximization of desirability amongst generally ratifiable acts is nothing other than maximization of desirability amongst $\succ$-maximal acts. Gustaffsson and I have arrived at the same destination by different routes.

I am thus inclined to view the relationship of Gustafsson’s proposal to my defense of Open-mindedness as symbiotic rather than antagonistic. The validity of Open-mindedness coupled with the recognition of $U$ as the correct fundamental measure of choiceworthiness can underwrite an explanation as to why only generally ratifiable acts are potential candidates for rational selection, while the soundness of Hierarchical CDT accounts for the secondary role of desirability maximization in the statement of General Ratifiability. Gustafsson’s proposal should thus not be seen as a competitor to Hierarchical CDT or as an escape route for those hoping to reject Open-mindedness.

Gustafsson’s insightful discussion of General Ratifiability does, however, helpfully reveal some previously alluded to flaws in our initial statements of both Open-mindedness and Hierarchical CDT that call for revision. Gustafsson cites Arntzenius as providing the following counterexample to General Ratifiability:

Three Boxes: A superintelligence offers you your pick of three boxes: $A$, $B$, and $C$. If the superintelligence predicted that you would opt for $A$, they placed two dollars in $A$ and one dollar in $B$. If the superintelligence predicted that you would opt for $B$, they placed four dollars in $A$ and three dollars in $B$. If the superintelligence predicted that you would opt for $C$, they placed one dollar in $A$ and two dollars in $B$. $C$ has been left empty regardless.

Let us once more assume that you take the predictor to be highly reliable, value money linearly, etc. Given these assumptions, both $A$ and $B$ qualify as generally ratifiable and hence according to General Ratifiability, considerations of desirability ought to be employed in deciding between them. But $V(B)>V(A)$, assuming a sufficiently reliable predictor, and hence General Ratifiability ends up recommending $B$. Similarly, if an open-minded agent faces Three Boxes, she will not be able to conclude that either $A$ or $B$ has determinately superior utility and, hence, Hierarchical CDT will recommend deciding in favor of $B$ on grounds of auspiciousness. But this seems wrong. The only way $B$ could possibly yield a better return than $A$ would be if the superintelligence predicted that you would take the empty box $C$, which is a senseless (i.e., generally unratifiable, causally dominated) option. Given that you won’t take such a senseless course of action, and should be able to recognize as much, choosing $B$ appears inferior to choosing $A$.

Gustafsson suggests modifying General Ratifiability to account for cases like this by introducing a recursive hierarchy of general ratifiability notions. Say that an act $A$ is generally ratifiable₀ just in case it is generally ratifiable and say that $A$ is generally ratifiable_n just in case it is generally ratifiable with respect to the subset of available options that are generally ratifiable_n-1. We can use this hierarchy to define the generally ratifiable_∗ acts as acts that are generally ratifiable_n, for all natural numbers $n$. Gustafsson then proposes that rational choice is ultimately determined by:

Iterated General Ratifiability: Rational agents deliberating over a finite action set $\mathcal{A}$ ought to choose any $V$-maximal member of the set of generally ratifiable_∗ acts in $\mathcal{A}$.

This solves the Three Boxes problem, since, even though $B$ is $V$-maximal, only $A$ is generally ratifiable_∗. Fortunately, we can extend the open-minded causalist story that undergirded General Ratifiability into a story that can undergird General Ratifiability_∗ as well if we allow for suitable modifications of Open-mindedness and Hierarchical CDT.

Beginning deliberation as an open-minded causalist, you should be able to recognize that $C$ is determinately causally inferior to its rivals. However, upon updating on this information, $A$ can then be recognized as having determinately greater utility than $B$. This suggests that perhaps Open-mindedness was stated a bit too strongly: it is only at the start of practical deliberation that a rational agent should adopt maximally imprecise credences. However, once she can see, prior to deliberation’s close, that an act is certainly unchoiceworthy relative to the fundamental standard of choiceworthiness (i.e., causal efficacy), it is admissible to update on this information and factor it into her subsequent deliberation. To make this precise, suppose that an agent with credences given by representor $\mathcal{P}$ confronts a choice set $\mathcal{A}$. Say that $a{\succ }_{0}b$, where $a,b\in \mathcal{A}$, just in case $U_P(a)>U_P(b)$ for all $P\in \mathcal{P}$, and let $A_n\subset \mathcal{A}$ be the disjunction of options that are not maximal with respect to ${\succ }_n$. Now define $a{\succ }_{n+1}b$ to hold just in case $U_P(a)>U_P(b)$ for all $P\in \mathcal{P}{(\cdot | ({\vee }_{i=0}^n{A}_i)}^C)$.⁵⁶ Finally, say that $a{\succ }_{\ast }b$ just in case $a{\succ }_{n}b$ for all natural numbers $n$.

These new definitions should leads us to restate Hierarchical CDT as:

Hierarchical CDT_∗: Rational agents deliberating over a finite action set $\mathcal{A}$ ought to choose any $V$-maximal member of the set of ${\succ }_{\ast }$-maximal acts in $\mathcal{A}$.

Since, given Open-mindedness, the ${\succ }_{\ast }$-maximal acts are simply those that are generally ratifiable_∗, this statement of Hierarchical CDT_∗ again allows us to view the theory proposed here as ultimately in harmony with Gustafsson’s.

To reiterate, from the vantage point we have now reached, Open-mindedness is best seen as a constraint upon an agent’s credences at the start of her deliberations. Once an agent realizes that her practical reasons (i.e., her utility judgments) tell decisively against a given course of action, $A$, there is nothing wrong with the agent contracting her credence in $A$ down to zero. Indeed, this is exactly what a rational agent ought to do. In light of such a realization of determinate suboptimality, deliberation has already done its work with respect to $A$ and is no longer in any danger of being inappropriately crowded out or impaired by prediction. But adoption of sharp act credences prior to such a conclusion remains a dangerous game that can significantly hinder the practical deliberation of otherwise rational agents, as illustrated by cases like Frustrater, etc.