Section
3.2 shows that choosing
\(A^+\) over
B is not only consistent with
some coherent completion of
\(\succ \),
\(u^*\), but that it is better than outcome rational to do so. Meanwhile, Peterson’s (
2015) weak money pump argument shows that Hare’s (
2010) prospectism criterion does not ensure outcome rationality on its own. Nevertheless, prospectism can be reconciled with outcome rationality by bringing more of the information contained in the set of coherent completions
\(\mathcal {U}^*_{O,\succ }\) to bear upon the choice between
\(A^+\) and
B. Specifically, this can be accomplished with the
ensemble of rankings consistent with
\(\mathcal {U}^*_{O,\succ }\).
4.2 Ensemble approach
Ensemble methods are used extensively in diverse areas in which either (a) multiple individuals’ beliefs, forecasts, or preferences are aggregated for collective action or (b) a single decision-making entity develops multiple different explanations, hypotheses, or models that it aggregates for the purpose of making a specific inference. The term ‘ensemble method’ is used in machine learning and artificial intelligence, while the same collection of aggregation techniques—by voting or by averaging—are referred to by a variety of labels across the different areas in which they are employed.
As in the contexts of artificial intelligence, multiple forecasters, and Bayesian model uncertainty, the problem of choosing an option under incomplete preferences can be partitioned into two steps, the first of which applies Epicurus’ (c. 342–270 BCE) Principle of Multiple Explanations:
If more than one theory is consistent with the observations, keep all theories.
6 Construction of the set of all coherent completions of preferences
\(U^*_{O,\succ }\) is the corresponding step in prospectism. It is in the second step where the presently proposed ‘ensemble prospectism’ departs from Hare’s (
2010) seminal formulation. In a manner consistent with ensemble methods more generally, ensemble prospectism utilizes the full range of coherent completions of the decision maker’s incomplete, partial preferences as enumerated in
\(U^*_{O,\succ }\), rather than any (arbitrary) individual element of this set, to guide choice. The task of rationally guiding choice in the context of incomplete preferences can be viewed as a problem of inference about the decision-maker’s underlying, as-yet unknown preferences. Due to this close parallelism, ensemble methods are not only a convenient source of apposite terminology, but also a source of apposite solutions.
Because of the sparseness of information in this context, determining tight bounds on the prior over the ensemble of possible rank orderings is problematic. Approaches based on Bayesian Model Averaging—or, equally, weighted voting procedures—are, therefore, not suitable in the present context. This leaves a large collection of potentially applicable unweighted voting procedures (Brams and Fishburn
2002; Brandt et al.
2013). The problem is simplified from that typically addressed in social choice, however, because strategic voting is not a consideration within the present context.
The second step can be formalized as the application of a voting rule to the ensemble of rankings
\(S^*_{O,\succ }\) associated with
7 the set of all coherent extensions of
\(\succ \),
\(U^*_{O,\succ }\). Given the number of choice options |
O| and the restriction(s) imposed by the strict preferences (if any) contained in
\(\succ \), the number of different rankings present in the ensemble is given by
\(n=|S^*_{O,\succ }|\). The set of all non-empty subsets of the choice options may be written as
\(\mathcal {F}(O)\), which we will see below may be understood as the set of feasible voting-rule outcomes. We specialize the definition of a voting rule to the ensemble of rankings as follows.
A voluminous literature has developed a veritable zoo of voting rules, and a decision maker might in principle employ any one of a large number of different voting rules. For ensemble prospectism to be immune to weak money pump arguments, the ensemble-voting rule must satisfy additional requirements. First, the decision maker must not have the possibility to switch from one voting rule to another within a time frame that would make it possible for the different properties of distinct voting rules to introduce decision-sequence-level intransitivity. Second, the voting rule must not be a member of the class of stochastic (or lottery) voting rules, as such rules could also introduce decision-sequence-level intransitivity. Third—and for the very same reason—the voting procedure should be resolute, yielding a unique winner, or else if it is not resolute, and yields a set of co-winners, the tie-breaking rule applied must not be random. In the definition below, the second and third requirements above are jointly invoked with the ‘completely non-stochastic’ qualification.
Table 2
Application of Borda rule to mildly sweetened partial preferences
\(A^+\!\!, B, A\)
| 2 | 1 | 0 |
\(B, A^+\!\!, A\)
| 1 | 2 | 0 |
\(A^+\!\!, A, B\)
| 2 | 0 | 1 |
Borda count | 5 | 3 | 1 |
Many voting rules satisfy these requirements, including, e.g., simple and common (positional) scoring rules such as the plurality rule, the anti-plurality rule, and Borda’s rule. Among all positional scoring procedures, Borda’s rule is least susceptible to paradoxes and other pathological behavior, including being the only scoring rule that will never award a Condorcet winner the lowest cumulative score (Brandt et al.
2013). Where Borda’s rule and the other positional procedures do fall short is in susceptibility to strategic manipulation—which, fortunately, is not a consideration in the present context, just as it is not a consideration in contexts, where Borda’s rule has been used to combine inferences from diverse methods to improve inferential performance (Marbach et al.
2012). Under the Borda rule each of the
n rankings present in
\(S^*_{O,\succ }\) awards a score (or ‘points’) from a maximum of
\(|O|-1\) to the highest-ranked option, through to 0 for the lowest-ranked option (
\(|O|\!-\!1, |O|\!-\!2, ..., 1, 0\)). Each option’s
Borda count is the sum of its scores across all
n ranking profiles, and the Borda rule chooses the option with the highest Borda count. Indeed, all |
O| options may be ordered (completely and transitively) by their respective Borda counts, which here we may denote
\(\succ _{\scriptscriptstyle \!BC}\).
In the example below, we revisit the ensemble of possible rank orderings
S detailed in Eq. (
4.1). We present its ensemble-prospectism solution, operationalized for illustrative purposes with a ‘Borda’ ensemble-voting rule.