The Association for Symbolic Logic publishes analytical reviews of selected books and articles in the field of symbolic logic. The reviews were published in The Journal of Symbolic Logic from the founding of the journal in 1936 until the end of 1999. The Association moved the reviews to this bulletin, beginning in 2000.
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Consider the predicament of a policy-maker who must decide whether to implement Solar Radiation Management (SRM) on a global scale even though she is unaware of some of the potential consequences. Ideally, we would like Bayesian decision theory to be able to help the policy-maker with her decision. However, the standard prescriptions apply only to logically omniscient, fully aware agents. The uncertainty introduced by the policy-maker’s limited awareness goes beyond the uncertainty that standard Bayesian decision theory can handle. Or so accepted wisdom has it.
In Beyond Uncertainty, Katie Steele and H. Orri Stefánsson challenge this accepted wisdom. The book begins with the presentation of the standard (Bayesian) model of rational preference and belief that Steele and Stefánsson expand upon throughout. As is typical of philosophers, they prefer a Jeffrey-style framework upon which propositions are the objects of both beliefs and desires, and rational agents’ preferences over outcomes are constrained by the axioms of subjective expected utility (SEU) theory.
In Section 2 this framework is extended to accommodate sequential decision-making and, in Section 3, limited awareness (or, more specifically, anticipated awareness growth) is situated within this extended framework. Since it has standardly been assumed that agents are fully aware of all the possibilities that are relevant to their decisions, some additional formal machinery is required to achieve this. To this end, Steele and Stefánsson introduce the notion of an ‘awareness context’ which allows for the characterisation of a subjective possibility space (i.e., a set of possibilities that is dependent on an agent’s state of awareness). More formally, they define an agent’s awareness context as the set,
$\mathbf {X}$
, of basic propositions that the agent is aware of. Next, they define possibilities as truth functions,
$\omega _i$
, over the basic propositions in
$\mathbf {X}$
, and an agent’s real set of possibilities,
$\mathbf {W_X}$
, as the subset of the putative set of possibilities that contains only those possibilities that the agent deems consistent. From here, each basic proposition
$X_i$
is associated with the collection of
$\omega _i \in \mathbf {W_X}$
for which
${X_i}$
is true; these collections generate a Boolean algebra of sets,
$\mathcal {F_{\mathbf {X}}}$
, in the usual way.
Against this backdrop, awareness growth is characterised as the enlargement of an agent’s subjective possibility space. The idea is that, upon awareness growth, there is a shift in awareness context from
$\mathbf {X}$
to
$\mathbf {X^+}$
where
$\mathbf {X^+} = \mathbf {X} \cup \mathbf {X_j}$
and
$\mathbf {X_j}$
is the set of all basic propositions
${X_j} \notin \mathbf {X}$
that the agent becomes aware of. This in turn corresponds to a shift from
$\mathbf {W_X}$
to
$\mathbf {W_{X^+}}$
and from
$\mathcal {F_{\mathbf {X}}}$
to
$\mathcal {F_{\mathbf {{X^+}}}}$
. Intuitively, one might think of awareness growth as a learning event. However, it should be apparent that it is distinct from standard (Bayesian) learning events wherein an agent gains evidence for a familiar proposition and the space of possibilities shrinks; instead, the space of possibilities expands. Due to this difference, conditionalisation does not apply in cases of awareness growth. Thus, two questions are left open: how should a rational agent represent anticipated awareness growth in their decision problem? And how should they revise their credences upon awareness growth?
The formal machinery that Steele and Stefánsson introduce in Section 3 turns out to be important for the answer that they provide to the second question. In particular, it drives the argument presented across Sections 4 and 5 which says that there is no general (conservative) norm of rationality that governs credence revision upon awareness growth. This argument challenges the popular view in the literature which upholds the norm of Reverse Bayesianism (E. Karni and M.-L. Vierø, ‘Reverse Bayesianism’: A choice-based theory of growing awareness.
American Economic Review,
vol. 103 (2013), no. 7, pp. 2790–810). Roughly, Reverse Bayesianism can be interpreted as saying (in Steele and Stefánsson’s terminology) that, upon awareness growth, a rational agent should update their credences in such a way that they preserve the probability ratios of nonnull events in their original awareness context (i.e., if
$s_1$
is twice as likely as
$s_2$
in the original awareness context,
$s_1$
must be twice as likely as
$s_2$
in the new awareness context). This proposal is ultimately rejected for two reasons. First, Reverse Bayesianism seems to produce counterintuitive results when the propositions that an agent becomes aware of are evidentially relevant to the comparison of propositions of which the agent was already aware. Second, since, on Steele and Stefánsson’s picture, the set of possibilities associated with any basic proposition
${X_i}$
is constituted by truth functions over the basic propositions in X and awareness growth involves a revision of X, it therefore also involves a revision of the set of possibilities associated with any basic proposition
${X_i}$
. As such, on this picture all credences may be affected by awareness growth in some way and any general conservative norm of credence revision, such as Reverse Bayesianism, is misguided.
Whilst Section 5.4 contains a positive upshot to this largely negative argument, the heart of Steele and Stefánsson’s positive proposal lies in their response to the first question, which is spelled out in Sections 6 and 7. Here they first propose replacing maximally specific states with ‘events’ which describe those contingencies that the agent is aware of in as much detail as possible but which may nevertheless fail to fully determine the outcomes of the available acts. Second, they propose incorporating a subjective catch-all which signifies “some abstract proposition standing in for a broad class of contingencies that the agent thinks she may later be in a position to concretize” (p. 74).
If we return to the example of the policy-maker who must decide whether to implement SRM, the idea is to specify the decision problem as follows (Table 1):
Table 1 The policy-makers decision
Where f and g are the available acts (perhaps ‘implement SRM’ and ‘abstain’),
$E_1$
to
$E_n$
are the most fine-grained mutually inconsistent events the policy-maker can think of, and ‘??’ is a subjective catch-all standing in for those contingencies that she suspects she is unaware of.
Represented like this, Steele and Stefánsson argue that the policy-maker’s predicament is somewhat unremarkable and (if certain basic conditions are met) we can treat her as an EU maximiser just like any ordinary reasoner. However, they go on to canvas two norms of rationality—‘Awareness Reflection’ and ‘Preference Awareness Reflection’—which they think should constrain the synchronic credences and desires (respectively) of agents like the policy-maker who anticipate their awareness will grow in rather specific ways.
Whilst the positive proposal spelled out in Sections 6 and 7 leaves several questions open, Steele and Stefánsson successfully lay the foundations for others working within normative decision theory and related areas of economics and computer science to take up these questions and continue the work of characterising the reasoning of rational, but less-than-fully aware, agents.
