Abstract
We discuss Herzberg’s (Theory Decis 78(2):319–337, 2015) treatment of linear aggregation for profiles of infinitely many finitely additive probabilities and suggest a natural alternative to his definition of linear continuous aggregation functions. We then prove generalizations of well-known characterization results due to (J Am Stat Assoc 76(374):410–414, 1981). We also characterize linear aggregation of probabilities in terms of a Pareto condition, de Finetti’s notion of coherence, and convexity.
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Notes
Recall that if X is a set and \({\mathcal {X}}\) is an algebra of subsets of X, then to say that a probability \(P: {\mathcal {X}} \rightarrow [0,1]\) is finitely additive means that \(P(\cup _{i=1}^n A_i) = \sum _{i=1}^n P(A_i)\) for all \(n \in {\mathbb {N}}\) and pairwise disjoint \(A_1,\ldots ,A_n \in {\mathcal {X}}\). If, in addition, \(P(\cup _{i=1}^\infty A_i) = \sum _{i=1}^\infty P(A_i)\) whenever \(A_1, A_2,\ldots \in {\mathcal {X}}\) are pairwise disjoint and \(\cup _{i=1}^\infty A_i \in {\mathcal {X}}\), then P is countably additive.
Herzberg adopts McConway’s framework, according to which aggregation functions are defined for all algebras over \(\Omega \) as opposed to a fixed algebra \({\mathcal {A}}\). Although our definition may appear less general, it is clear that our results apply in the McConway–Herzberg framework by systematically quantifying over algebras of \(\Omega \). The framework with a fixed algebra \({\mathcal {A}}\) is both standard (Genest and Zidek 1986; Stewart and Ojea Quintana 2018) and notationally simpler.
Our presentation follows Stewart and Ojea Quintana (2018).
Note that [Q] is nonempty and need not be a singleton. See Rao and Rao (1983, Chapter 3).
See Aliprantis and Border (2006, 11.2).
See, for example, Schirokauer and Kadane (2007).
Wagner (1982, Theorem 7) discovered the equivalence of (b) and (c) independently of McConway using essentially the same proof technique.
To see this, it may help to observe that, by the linearity of the integral, F is Pareto if and only if for all step functions s, \(\int _\Omega s {\text {d}}P_i \ge 0\) for all \(i \in I\) implies \(\int _\Omega s {\text {d}}F(\underline{P}) \ge 0\). A detailed study of results like Proposition 1 can be found in Cassese (2018). Related results appear in Dubins (1975) and Lane and Sudderth (1984).
See Berti et al. (2013, p. 50) and references therein.
A referee points out that connections between coherence and Harsyani’s utilitarian aggregation theorem have been explored in Diecidue and Wakker (2002) and Diecidue (2006). These papers differ from the present one in two important respects. First, their results are stated for finite I, whereas I is arbitrary in Definition 8. Second, our notion of coherent aggregation is more closely analogous to coherence in de Finetti’s sense.
\({\mathcal {D}}\) denotes a directed set. See Aliprantis and Border (2006, 2.4) for an overview of nets.
If \(x \in X\), then \(\delta _x \in \Delta (2^X)\) denotes point mass at x, which is defined by \(\delta _x(A) = 1_A(x)\) for all \(A \subseteq X\).
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Nielsen, M. On linear aggregation of infinitely many finitely additive probability measures. Theory Decis 86, 421–436 (2019). https://doi.org/10.1007/s11238-019-09690-y
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DOI: https://doi.org/10.1007/s11238-019-09690-y