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Aggregation of Coherent Experts’ Opinions: A Tractable Extreme-Outcomes Consistent Rule

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Fusion Methodologies in Crisis Management

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Abstract

The paper defines a consensus distribution with respect to experts’ opinions using a multiple quantile utility model. We show that the Steiner point (Schneider, Isr J Math 2:241–249, 1971) is the representative consensus probability. The new rule for aggregation of experts’ opinions, which can be simply evaluated by the Shapley value, is prudential and coherent.

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Notes

  1. 1.

    Whether opinions are expressed as probability measures, densities, mass functions, or odds, subjective probability distributions are used to form a consensus distribution through linear opinion pools (Stone 1961), linear opinion pools that only satisfy the marginalization property (DeGroot and Montera 1991), logarithmic opinion pools that satisfy external Bayesianity (Winkler 1968), or generalized logarithmic opinion pools (Genest et al. 1986) that use mathematical aggregation models.

  2. 2.

    Crès et al. (2011) focus on the maxmin expected utility model so that “the decision maker’s valuation of an act is the minimal weighted valuation, over all weight vectors of the experts’ valuations” (a weight pessimistic scenario). Gajdos and Vergnaud (2013) characterize preferences that exhibit independently aversion towards imprecision and conflict. They aggregate different evaluations through a multiple weights model where “a decision maker will satisfy the conflict aversion hypothesis whenever her degree of conflict aversion is higher than her degree of imprecision aversion.”

  3. 3.

    A capacity υ is convex if \(\upsilon (A \cup B) +\upsilon (A \cap B) \geq \upsilon (A) +\upsilon (B)\),\(\forall A,B \in 2^{S}\).

  4. 4.

    The Steiner point of a polytope, also called Steiner curvature centroid, is the weighted average of its vertices, in which the weight for each vertex is proportional to its outer angle. Details are in Schneider (1971, 1993).

  5. 5.

    The Shapley value is an operator that assigns an expected marginal contribution to each player in a coalitional game with respect to a uniform distribution over the set of all permutations on the finite set of players. Details are in Gajdos et al. (2008).

  6. 6.

    Details are in Basili and Chateauneuf (2011).

  7. 7.

    The dual or conjugate capacity \(\overline{\upsilon }\) is defined by \(\overline{\upsilon }(A) = 1 -\upsilon (A^{C})\ \forall A \in 2^{S}\).

  8. 8.

    Proposition 1 is a direct consequence of Proposition 3 in Basili and Chateauneuf ( 2011).

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Authors and Affiliations

  1. DEPS and SEM, University of Siena, Piazza San Francesco 7, 53100, Siena, Italy

    Marcello Basili

  2. IPAG Business School and PSE-CES, University of Paris-I, Paris, France

    Alain Chateauneuf

Authors
  1. Marcello Basili
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  2. Alain Chateauneuf
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Correspondence to Marcello Basili .

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Editors and Affiliations

  1. State University of New York at Buffalo, Buffalo, New York, USA

    Galina Rogova

  2. State University of New Yorkat Buffalo, Buffalo, New York, USA

    Peter Scott

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Basili, M., Chateauneuf, A. (2016). Aggregation of Coherent Experts’ Opinions: A Tractable Extreme-Outcomes Consistent Rule. In: Rogova, G., Scott, P. (eds) Fusion Methodologies in Crisis Management. Springer, Cham. https://doi.org/10.1007/978-3-319-22527-2_19

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  • DOI: https://doi.org/10.1007/978-3-319-22527-2_19

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22526-5

  • Online ISBN: 978-3-319-22527-2

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