Bradley's stated aim for this book is to develop a normative decision theory for bounded agents like us. He thus differs from most normative decision theorists insofar as he does not wish to identify norms that govern the rational decision-making of ideal agents who possess unlimited cognitive resources, and differs from most behavioural economists insofar as he aims to outline how we should make decisions given our limited resources, rather than how we do make decisions given our limited resources.

Bradley not only goes a long way towards accomplishing this ambitious goal, but also develops many ideas along the way that will be of interest to decision theorists and formal epistemologists whose primary preoccupation is not advising human decision-makers. Thus Bradley's book is a definite success. But it's also a hard book – not only because it's impossible to fully appreciate his accomplishment without working through some rather demanding formal argumentation, but also because it's sometimes difficult to see the forest through the trees – i.e. to see how (and in what sense) Bradley's distinct contributions (spread across many chapters) are in service of a unified normative theory of rationality for bounded agents like us.

In what follows, I highlight some of Bradley's contributions on a section-by-section basis, while taking the opportunity to pose some specific questions that are prompted by the details of his work. Along the way, I also reflect on the nature of Bradley's framework more generally, paying special attention to the underlying conception of rationality that it presumes, and the sense in which its face is properly considered human. As I see things, one of the best reasons to read Bradley's book in its entirety is that after doing so, the reader is left with a better sense of what motivates the group of philosophers who work within Richard Jeffrey's broad decision-theoretic framework. My hope is that by reflecting on the precise way in which this approach is geared towards providing advice to actual decision-makers, this review will contribute to our general understanding of this tradition, and will help to refine the sense in which these theorists take the standards of rationality to be constrained by human limitations.

The first five chapters of Bradley's book are largely concerned with describing the strengths and shortcomings of existing decision theories (e.g. von Neumann and Morgenstern Reference von Neumann and Morgenstern1944; Savage Reference Savage1954; Jeffrey Reference Jeffrey1965), introducing the concepts that they deploy (for example, preferences, options, probability and utility), and assessing the role that representation theorems play in their development. Bradley's discussion of these matters is more reflective and self-conscious than is usually found in the literature, and is worth reading for that reason alone. But there are several points that Bradley makes in these chapters that merit special attention.

First, Bradley takes Savage (Reference Savage1954) to task for developing a decision theory that imposes severe restrictions on how decision problems must be framed. He reports that this criticism, shared by other decision theorists in the Jeffrey tradition (e.g. Joyce Reference Joyce1999), is his main reason for adopting Jeffrey's framework. Throughout his discussion of this issue, Bradley places emphasis on the agent who wants to use decision theory in order to figure out what to do. If some decision theory requires the agent to go to great lengths to describe the decision problem in order to apply the theory – lengths that Bradley argues are almost never feasibly travelled in the case of Savage's decision theory – then it won't be very useful for agents who seek guidance about what to do, given their own understanding of the decision at hand. This seems right, but there are other uses of decision theory for which this particular criticism may not be as cutting – for example, when used descriptively to infer an agent's probabilities and utilities, or when used to develop a standard of ideal rationality against which the choices of an agent can be third-personally evaluated. This suggests that Bradley's primary concern is with developing a decision theory that is readily applied first-personally, or, put differently, by agents who seek to determine what they themselves should do.

Second, Bradley signals his intention to develop a decision theory that incorporates distinct kinds of uncertainty in these chapters. According to Bradley, decision-makers are not only confronted with factual uncertainty (i.e. uncertainty about what is the case), but also evaluative certainty (i.e. uncertainty about what should be the case) and modal uncertainty (i.e. uncertainty about what could be the case). Other decision theorists often argue that some of these kinds of uncertainty reduce to the other(s), but, in so doing, typically require that decision problems be framed in a very particular way (for example, such that all of an agent's uncertainty about what should or could be the case is captured by her uncertainty about the state of the world). Regardless of whether this strategy can succeed, it is worth noting that it may be prudent to follow Bradley in separately integrating each of these kinds of uncertainty if the decision theory is to be applied first-personally. If it proves difficult or infeasible for actual decision-makers to describe their decisions in terms of just one kind of uncertainty, then it is useful to have a decision theory that allows the agent to incorporate each kind of uncertainty separately as she frames her decision problem.

Third, when assessing the importance of representation theorems, and, in particular, their role in his preferred Jeffrey–Bolker framework, Bradley argues that what is normally taken as the chief limitation of Bolker's (Reference Bolker1966) representation theorem – i.e. that the axioms governing the preference relation are not sufficient for recovering a precise probabilistic representation of the agent's credences – is actually a feature rather than a bug. Since the game of providing representation theorems is often thought to be about deriving probabilities and utilities from as little as possible (or, alternatively, as simply as possible), this may seem strange. But, while drawing heavily on Joyce's (Reference Joyce1999) representation theorem, Bradley argues that the real lessons of Bolker's theorem are, first, that ascriptions of numerical degrees of belief should be recovered from comparative preference judgements and comparative confidence judgements (rather than just comparative preference judgements), and, second, that there is no methodological primacy for preference over belief. This latter claim may strike some descriptive economists as bizarre since preferences are sometimes treated as observable, while comparative confidence judgements are typically not. But in the context of deploying decision theory first-personally, this isn't so bizarre. When you stop to reflect on how comparatively confident you are that some states of affairs will obtain – for example, whether it's more likely that France or Croatia will win the next World Cup – it's often not very hard, and certainly a lot easier than assigning them non-arbitrary precise probability estimates. Thus from the first person, there appears to be a certain psychological reality to comparative confidence judgements that is not obviously shared by precise probability assignments, and it thus seems reasonable to treat comparative confidence judgements as primitives in the contexts that most concern Bradley.

Chapters 6 and 7 are the first in which Bradley develops his own decision-theoretic machinery. Drawing on several already published papers, Bradley introduces new axioms that enable the Jeffrey–Bolker framework to extend to various kinds of conditional attitudes (or attitudes towards prospects that are premised on various kinds of suppositions) and to various kinds of conditional prospects (prospects that are themselves conditional, and that are often expressed in natural language as indicative or subjunctive conditionals).

One of Bradley's contributions here is to characterize the distinction between evidential supposition (i.e. the kind of supposition associated with evidential decision theory) and interventionist supposition (i.e. the kind of supposition associated with causal decision theory). Though Bradley's characterization is compelling if we grant that there are two distinct kinds of supposition, I would have liked to hear Bradley's take on Meek and Glymour's (Reference Meek and Glymour1994) position that ‘interventionist supposition’ and ‘evidential supposition’ need not be treated as two distinct kinds of supposition, but can rather be treated as evidential supposition on two distinct kinds of information – namely, the intervention to bring about the state of affairs in question, and the state of affairs absent intervention. If Meek and Glymour are right, then the reason that ‘interventionist supposition’ seems to run afoul of the conditions that are supposed to characterize evidential supposition is that we conflate interventions and non-interventions – i.e. when we engage in the so-called ‘interventionist supposition’ of x simpliciter, we actually evidentially suppose that x is brought about via intervention.Footnote ¹

Regardless, Chapters 6 and 7 provide the foundations for a decision theory that yields highly influential existing decision theories as special cases, and that accommodates his theory of conditionals, which impressively manages to preserve Adams’ Thesis (which Bradley treats as the thesis that the probability of the indicative conditional, if x, then y, is equivalent to the conditional probability of y given x) without falling prey to Lewis's notorious (Reference Lewis1976) triviality results.

Lewis's results have led many philosophers to believe that the probability of the indicative conditional, if x, then y, cannot be equivalent to the conditional probability of y given x, despite the fact that the assertability of the conditional corresponds to the conditional probability. Bradley, by contrast, argues that we can preserve the equality if we, first, make sure not to conflate evidential supposition and conditional probability (which, he argues, are equivalent in the case of factual prospects, but not when both conditional and unconditional prospects are in play), and, second, abandon Lewis's assumption that degrees of belief must be revised by conditionalization. This is fine as an initial diagnosis of what's at issue with the triviality results, but Bradley owes us a story about the semantics of conditionals that explains why things get weird when both factual and conditional prospects are in play. This is his first task in Chapter 8.

Bradley extends standard possible world semantics by introducing an extra dimension to the possibility space – specifically, by having worlds serve not just as candidates for the way things are or could be, but also as candidates for the ways things are or could be under the supposition of some condition. These ‘possible counteractual worlds’ are then supposed to make conditional sentences true in just the same way that possible actual worlds make factual sentences true – i.e. ‘if x, then y’ is made true at any possible counteractual x-world in which y is true. This extra dimension of possibility space enables Bradley to provide a truth-conditional semantics for conditionals according to which the truth-values of conditionals are not exclusively determined by the (non-counteractual) facts, while preserving Adams’ Thesis for indicative conditionals (provided that the probability of x is independent of what is or would be the case under the evidential supposition that x). This is no small feat, especially considering that Bradley argues in supporting papers (Bradley Reference Bradley2011, Reference Bradley2012) that the triviality results do not threaten his semantics given some plausible constraints on belief measures that can only be articulated from within his multidimensional possibility space.

This may sound like inside baseball, and it may be unclear how or why the development of a semantics for conditionals with these desirable technical properties could be in service of a decision theory with a human face. Bradley's response, I think, would be that this enables him to incorporate modal uncertainty into his decision theory, and that human decision-makers confront modal uncertainty. But because the semantics is still quite complicated – especially insofar as it is harder to think through whether a given conditional is true in his semantics than, for example, in the standard Stalnaker (Reference Stalnaker, Harper, Stalnaker and Pearce1981) semantics – this challenge might have some teeth.

The same concern about inside baseball may apply to Bradley's Chapter 9 discussion of how various existing decision theories can be formulated from within his framework, and of the way in which they are related to each other. But this discussion is of definite interest to the insiders. For example, anyone steeped in the debate between causal decision theorists and evidential decision theorists will be excited to see that both causal decision theory and evidential decision theory can be formulated within Bradley's general framework by construing the agent's options differently.Footnote ² Moreover, Bradley contributes to the humanization project in this chapter by providing a new perspective on when (and under what assumptions) existing decision theories capably countenance widespread human decision-making patterns as rational. For example, it is standardly thought that the preferences expressed by subjects confronted with the Ellsberg paradox are inconsistent with Bayesian decision theory, but in Bradley's framework, Savage's decision theory by itself is consistent with these preferences, even though Savage's theory and von Neumann and Morgenstern's theory (or Savage's theory and the Principal Principle) are jointly inconsistent with these preferences.

Bradley advertises Chapter 10 as where he responds to the challenge that his framework (as it stands at this point of the book) sticks its neck out too little because it requires only that agents be consistent. This may give the impression that Bradley's ambition in this chapter is to develop the normative framework so that it's sufficiently demanding, despite its human face. But while Bradley does effectively extend the prescriptive reach of his framework by arguing for norms that specify how agents should update their attitudes in light of their evidence – norms that go above and beyond consistency – my opinion is that the human face of Bradley's approach is never more on display than in this chapter. This is because he proposes updating norms that apply to agents who are consciously trying to figure out for themselves how confident they should be in various prospects, given their prior beliefs and learning experiences, rather than updating norms that operate over a rich set of state descriptions that are not within the agent's power to consciously entertain. (Most Bayesians take the latter approach.)

In the last section of the book, Bradley's explicit aim is to further develop the decision theory so that it applies to agents who lack full awareness and are not maximally opinionated (in the sense, for example, that their probability estimates cannot be expressed by a single probability function). Especially interesting is his introduction of (un)awareness into decision theory and belief revision. Along these lines, Bradley develops tools to distinguish an agent who is unopinionated (or, more specifically, who hasn't settled on a particular attitude to adopt) towards some prospect, x, from an agent who is simply unaware of x because she has never entertained x. To my mind, this again puts front and center Bradley's focus on developing a normative framework for agents who are determining for themselves what to believe or do (since (un)awareness plays a distinctive role for the deliberating agent). Though Bradley believes that agents can be rational while unopinionated or unaware, there are various ways, according to Bradley, that agents can be unopinionated or unaware and irrational. For example, he holds that agents who are unopinionated about something should be coherently extendable – i.e. such that they could (hypothetically) reach a coherent set of judgements by becoming opinionated.

In order to develop the theory of rationality so that it properly constrains these distinct attitudes of (un)awareness and (un)opinionation, Bradley formulates normative requirements in terms of (non-singleton) sets of probabilities and desirabilities, but differs from some other so-called ‘Imprecise Bayesians’ insofar as he always treats further opinionation as rationally permissible – viz., if it is permissible to adopt a set of probability functions as one's credal state, then it is also permissible to adopt any particular set member. He also (i) surveys various decision rules that allow agents to make up their minds when unopinionated, and (ii) argues, like Levi (Reference Levi1986) before him, that Imprecise Bayesianism can be squared with lots of human behaviour (for example, the Ellsberg data) if we recognize that pragmatic considerations must bridge the gap between the preferences of unopinionated agents and their choices. In the last chapter of the book, he attempts to bridge this gap just a bit by introducing a higher-order confidence relation between sets of probabilities that agents can use to help determine what to do.

So in what precise sense does Bradley present a normative framework for bounded agents like us? Bell et al. (Reference Bell, Raiffa and Tversky1988) famously introduced a distinction between descriptive, normative and prescriptive approaches to decision theory. Given everything that we've said thus far, it may seem as though Bradley takes the prescriptive approach – i.e. that his book aims to answer the (Bell et al. Reference Bell, Raiffa and Tversky1988: 9) question, ‘How can real people – as opposed to imaginary, super-rational people without psyches – make better choices in a way that does not do violence to their deep cognitive concerns?’

Though this does seem like a fair characterization of the book, Bradley's approach to answering this question definitely differs from the approach outlined by Bell et al. This is because Bradley's aim is to develop norms whose content should be endorsed by rational agents, but that differ from norms of ideal rationality insofar as they apply to contexts of deliberation as human agents can themselves represent them, while the project that Bell et al. describe is that of developing decision-making strategies that will, if applied by actual agents, lead to optimal performance on average, even if they don't encode the true mandates of rationality. Thus Bradley's strategy is to relax classical Bayesian accounts of rationality, and to extend the domain over which they govern (by, for example, incorporating conditional prospects), while the Bell et al. project involves developing decision-making strategies that can conflict with the mandates of rationality, but that help limited agents to make good choices by providing simple advice that tends to lead to success.

My suspicion is that many philosophers working in decision theory share Bradley's project (at least some of the time), and I am grateful to Bradley for writing a book that clarifies the nature of this project by showing that there is a prescriptive theory of rationality to be developed that deserves its title insofar as it intended to be used by actual human decision-makers, but that does not just consist of developing effective decision-making strategies.