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Consensus group decision making under model uncertainty with a view towards environmental policy making

Phoebe Koundouri, Georgios I. Papayiannis, Electra V. Petracou, Athanasios N. Yannacopoulos

TL;DR

This work tackles consensus formation under model uncertainty in environmental policy contexts by developing a two-stage, Fréchet barycenter–based framework where agents iteratively update locally and a moderator proposes a global barycenter to achieve agreement. It provides both geometric and probabilistic justifications for using Fréchet barycenters as consensus points, and introduces an evolutionary learning scheme with a time-varying interaction network to model opinion dynamics and convergence. The contributions include a rigorous barycenter-based interpretation of consensus, a dynamic learning algorithm with clustering as a practical extension, and an application to the social discount rate and future-contingency modeling in environmental economics. Empirical demonstrations on SDR and contingency models show the method’s robustness to heterogeneity and model uncertainty, with clear implications for policy design and climate-economics valuation.

Abstract

In this paper we propose a consensus group decision making scheme under model uncertainty consisting of an iterative two-stage procedure and based on the concept of Fréchet barycenter. Each step consists of two stages: the agents first update their position in the opinion metric space by a local barycenter characterized by the agents' immediate interactions and then a moderator makes a proposal in terms of a global barycenter, checking for consensus at each step. In cases of large heterogeneous groups the procedure can be complemented by an auxiliary initial homogenization step, consisting of a clustering procedure in opinion space, leading to large homogeneous groups for which the aforementioned procedure will be applied. The scheme is illustrated in examples motivated from environmental economics.

Consensus group decision making under model uncertainty with a view towards environmental policy making

TL;DR

This work tackles consensus formation under model uncertainty in environmental policy contexts by developing a two-stage, Fréchet barycenter–based framework where agents iteratively update locally and a moderator proposes a global barycenter to achieve agreement. It provides both geometric and probabilistic justifications for using Fréchet barycenters as consensus points, and introduces an evolutionary learning scheme with a time-varying interaction network to model opinion dynamics and convergence. The contributions include a rigorous barycenter-based interpretation of consensus, a dynamic learning algorithm with clustering as a practical extension, and an application to the social discount rate and future-contingency modeling in environmental economics. Empirical demonstrations on SDR and contingency models show the method’s robustness to heterogeneity and model uncertainty, with clear implications for policy design and climate-economics valuation.

Abstract

In this paper we propose a consensus group decision making scheme under model uncertainty consisting of an iterative two-stage procedure and based on the concept of Fréchet barycenter. Each step consists of two stages: the agents first update their position in the opinion metric space by a local barycenter characterized by the agents' immediate interactions and then a moderator makes a proposal in terms of a global barycenter, checking for consensus at each step. In cases of large heterogeneous groups the procedure can be complemented by an auxiliary initial homogenization step, consisting of a clustering procedure in opinion space, leading to large homogeneous groups for which the aforementioned procedure will be applied. The scheme is illustrated in examples motivated from environmental economics.
Paper Structure (27 sections, 2 theorems, 76 equations, 5 figures, 3 tables, 2 algorithms)

This paper contains 27 sections, 2 theorems, 76 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. State of the art and aims and contribution of the present work
  3. State of the art and brief literature review
  4. Scope and contribution of the present work
  5. Opinion space as a metric space and the Fréchet barycenter as consensus point
  6. Opinion space as a metric space
  7. The Fréchet barycenter as consensus point
  8. A geometric characterization of the barycenter as consensus point
  9. Barycenters as the most likely points of consensus
  10. An evolutionary learning approach for reaching a consensus
  11. Motivation
  12. The evolutionary scheme
  13. A numerical experiment
  14. Application in Environmental Economics: Convergence to a Common Social Discount Rate
  15. Motivation
  16. ...and 12 more sections

Key Result

Proposition 3.3

The solution of problem 19-9-2023 corresponds to a Fréchet barycenter of ${\mathbb M}$, for a selection of weights depending on $\epsilon_i$, and chosen so as to maximize a weighted version of the corresponding Fréchet variance.

Figures (5)

  • Figure 1: Illustration of the agents' anchor opinions (different colour indicates different cluster), the local consensus points and the derived consensus points (marked in red) with the proposed evolutionary learning schemes for all three scenarios considered.
  • Figure 2: Convergence illustration to the barycenter by the one-stage process depicting for all agents: (a) distance from the consensus curve (left) and (b) acceptance probabilities with respect to the running barycentric curve
  • Figure 3: The achieved SDR-consensus curves achieved by the one-stage scheme (left), the two-stage process (center) and their comparison (right).
  • Figure 4: The simulated scenario for the beliefs concerning the probability model for the contingency
  • Figure 5: Achieved consensus from the one-stage scheme (left), the two-stage scheme (center) and their comparison (right) concerning the probability model that describes the contingency (in terms of quantiles) from the Uniform Beliefs scenario (upper panel) and the Impatient Agents scenario (lower panel).

Theorems & Definitions (6)

  • Example 3.1: Scenarios in environmental decision making
  • Example 3.2: Metric spaces of curves: Social discount term structure
  • Proposition 3.3
  • Remark 3.4
  • Proposition 3.6
  • Example 3.7