Consensus Group Decision Making Under Model Uncertainty with a View Towards Environmental Policy Making

Abstract

In this paper we propose a consensus group decision making scheme under model uncertainty consisting of an iterative two-stage procedure based on the concept of Fréchet barycenter. Each stage consists of two steps: the agents first update their position in the opinion metric space adopting 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 stage. In cases of large heterogeneous groups, the procedure can be complemented by an auxiliary initial homogenization stage, 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.

Appendices

Appendix A: Proof of Proposition 3.3

The proof uses a duality argument. To simplify the exposition we assume that M is compact (or that we focus our attention on a compact subset of M). We define the function $L: M \times {{\mathbb {R}}}_{+} \times {{\mathbb {R}}}_{+}^{N+2} \rightarrow {{\mathbb {R}}}$ by

$$\begin{aligned} L(x,s,\mu _{00},\mu _0, \mu _1, \ldots , \mu _N) = s - \mu _{00} s + \mu _0\left( s -\epsilon _{min}^2\right) + \sum _{i=1}^{N} \mu _i \left( d^2(x,x_i)-s\right) , \end{aligned}$$

and note that

$$\begin{aligned} \max _{{\left( {\mu _{{00}} ,\mu _{0} ,\mu _{1} , \ldots ,\mu _{N} } \right) \in \mathbb{R}_{ + }^{{N + 1}} }} L\left( {x,s,\mu _{{00}} ,\mu _{0} ,\mu _{1} , \ldots ,\mu _{N} } \right) = & \\ \qquad \qquad \qquad \left\{ {\begin{array}{*{20}c} s & {{\text{if}}\;d^{2} (x,x_{i} ) < s,\;i = 1, \ldots ,N \; \& \;0 \le s \le _{{min}}^{2} ,} \\ { + \infty } & {{\text{otherwise}}.\qquad \qquad \qquad \qquad \qquad } \\ \end{array} } \right. \\ \end{aligned}$$

(15)

The variables $\mu _{00},\mu _0, \mu _1, \ldots , \mu _N$ play the role of Lagrange multipliers. Hence problem (5) can be written as

$$\begin{aligned} \text{ Problem } (5)=\min _{x \in M, s \in {{\mathbb {R}}}_{+}} \max _{\mu \in {{\mathbb {R}}}_{+}^{N+2}}L(x,s,\mu ), \end{aligned}$$

(16)

where we used the compact notation $\mu =(\mu _{00}, \mu _0, \mu _1, \ldots , \mu _N)'$. Using the minimax theorem we can exchange the order of the maximum and the minimum to obtain

$$\begin{aligned} \begin{aligned} \text{ Problem } (5)=&\min _{x \in M, s \in {{\mathbb {R}}}_{+}} \max _{\mu \in {{\mathbb {R}}}_{+}^{N+2}} L(x,s,\mu ) \\=& \max _{\mu \in {{\mathbb {R}}}_{+}^{N+2}} \bigg ( \underbrace{ \min _{x \in M, s \in {{\mathbb {R}}}_{+}} L(x,s,\mu )}_{:= D(\mu )} \bigg ). \end{aligned} \end{aligned}$$

(17)

The function

$$\begin{aligned} D(\mu ):=\min _{x \in M, s \in {{\mathbb {R}}}_{+}} L(x,s,\mu ), \end{aligned}$$

(18)

is called the dual function and as seen by (17) can help characterize the solution of the primal problem (5). We now proceed to the calculation of the dual function $D(\mu )$. By rearranging L as

$$\begin{aligned} L(x,s,\mu )=\left( 1+\mu _0 - \mu _{00}- \sum _{i=1}^{N}\mu _i\right) s - \mu _0 \epsilon ^2_{\min } + \sum _{i=1}^{N} \mu _i d^2(x,x_i), \end{aligned}$$

(19)

we can see that the only interesting (non-generate) case corresponds to $\epsilon _1=\epsilon _2=...=\epsilon _N$ with $s>0$ (i.e. $\mu _{00}=0$) and $s=\epsilon _{\min }^2$ (i.e. $\mu _0, \mu _1,\mu _2,...,\mu _N>0$). This leads to the dual function

$$\begin{aligned}{} & {} D(\mu )=\min _{x \in M} \sum _{i=1}^{N} \mu _i d^2(x,x_i), \,\,\, \end{aligned}$$

(20)

$$\begin{aligned}{} & {} 1 - \sum _{1}^{N}\mu _i =0. \end{aligned}$$

(21)

In this case, the set of Lagrange multipliers $\mu _1,\mu _2,...,\mu _N$ can be realized as weights since they belong to $\Delta ^{N-1}$. From now on let us denote $w=(\mu _1,\mu _2,...,\mu _N)'$ and express

$$\begin{aligned} D(w)= \min _{x \in M} \sum _{i=1}^{N} w_{i} d^2(x,x_i). \end{aligned}$$

(22)

The minimum over $x \in M$ is realized at the Fréchet barycenter of ${{\mathbb {M}}}$, with weight vector w (as above). This implies that we should look for the solutions of problem (5) among the Fréchet barycenters of ${{\mathbb {M}}}$. It remains to find the appropriate weights for the barycenter. According to the dual formulation (17) the weights which are related to $\mu >0$ are obtained as the solution of the dual problem

$$\begin{aligned} \max _{\mu \in {{\mathbb {R}}}_{+}^{N+2}} D(\mu ) =\max _{w \in \Delta ^{N-1}} V_{{{\mathbb {M}}}} (w), \end{aligned}$$

where $V_{{{\mathbb {M}}}}(w)$ is the Fréchet variance of ${{\mathbb {M}}}$ when choosing w as the weight vector. Hence, we conclude that the solution of (5) is a Fréchet barycenter with weights chosen so as to maximize the corresponding weighted Fréchet variance.

Appendix B: Proof of Proposition 3.6

The proof proceeds in several steps:

Step 1 We note that problem (7) is equivalent to the minimization problem

$$\begin{aligned} \min _{x \in M} \sum _{i=1}^{N} \psi _{i}\left( d^2(x,x_{i})\right) , \end{aligned}$$

(23)

where $\psi _i=-\ln \phi _i$. This is easy to see, as maximizing P is equivalent to minimizing $-\ln P$. Note that the functions $\psi _i$ are increasing.

Step 2 Let $\partial \phi$ be the subdifferential of a (convex) function $\psi$. The first order condition for ${\bar{x}}$ to be a local minimizer of $\psi$ is that $0 \in \partial \psi ({\bar{x}})$. Applying Assumption 3.5, we assume that for each $i=1, \ldots , N$ we can express $x_i=x_i(\theta _i)$ for some $\theta _i \in H$, and similarly, any point $x \in X$ can be expressed as $x=x(\theta )$ for some $\theta \in H$. To simplify notation we will not revert to the $\theta$ parametric notation but keep our initial notation in terms of x. We apply the first order condition to $\psi (x):=\sum _{i=1}^{N}\psi _i\left( d^2(x,x_i)\right)$, which in parametric form becomes

$$\begin{aligned} \psi (\theta ):=\sum _{i=1}^{N} \psi _i\left( d^2(x(\theta ),x_i(\theta _i)\right) =\sum _{i=1}^{N} \psi _i\left( d^2(\theta ,\theta _i)\right) , \end{aligned}$$

with the latter used as a simplification of the notation. The problem of minimizing over $x \in X$ is then transferred to minimizing over the parameter $\theta \in H$. This yields

$$\begin{aligned} 0 \in \partial \sum _{i=1}^{N} \psi _i\left( d^2(\theta ,\theta _i)\right) = \sum _{i=1}^{N} \partial \psi _i\left( d^2(\theta ,\theta _i)\right) , \end{aligned}$$

(24)

where for the second equality we have used the subdifferential calculus (which holds since $\psi _i$ and $d^2(x,x_i)$ are continuous). We now apply the subdifferential rule for composite functions (see e.g. Corollary 6.72 in Bauschke and Combettes (2017)). This yields that for each $i=1,\ldots , N$ it holds that $z \in \partial \psi _i\left( d^2(\theta ,x_i)\right) (y)$ if there exists $\alpha _i \in \partial \psi _i \left( d^2(y,x_i)\right)$ and $w_i \in \partial d^2(\cdot , x_i) (y)$ such that $z=\alpha _i w_i$. To simplify the exposition let us assume that $\psi _i$ is $C^1$, so that the subdifferential of $\psi _i$ is a singleton consisting of a single value, that of the derivative $\psi _i'\left( d^2(y,x_i)\right) >0$, with its positivity guaranteed by the fact that $\psi$ is increasing. Condition (24) together with the above subdifferential rule implies that

$$\begin{aligned} 0 \in \sum _{i=1}^{N} {\bar{w}}_{i} \partial d^2(\theta ,\theta _i)(\theta ^*), \end{aligned}$$

(25)

where ${\bar{w}}_i=\psi _i'\left( d^2(\cdot , x_i)\right) (\theta ^*) >0$. Defining $w_{i}=\frac{{\bar{w}}_i}{\sum _{i=k}^{N} {\bar{w}}_{k}} > 0$, and dividing (25) by $\sum _{k=1}^{N} {\bar{w}}_{k}$ we obtain

$$\begin{aligned} 0 \in \sum _{i=1}^{N} w_{i} \partial d^2(\theta ,\theta _i)(\theta ^*), \end{aligned}$$

(26)

where $w=(w_1, \ldots , w_N)\in \Delta ^{N-1}$. From (26) we conclude that $\theta ^*$ is the solution of the minimization problem

$$\begin{aligned} \min _{\theta \in H} \sum _{i=1}^{N} w_i d^2(\theta ,\theta _i), \end{aligned}$$

(27)

i.e. $x^*=x(\theta ^*)$ corresponds to a Fréchet barycenter for the choice of weights $w=(w_1, \ldots , w_N) \in \Delta ^{N-1}$.

Step 3 It remains to show that such a choice of weights is feasible. The weights $w=(w_1, \ldots , w_N)$ must be such that if $x^*=x(\theta ^*)$ is the solution of (26) then it must hold that

$$\begin{aligned} w_i =\psi _i'\left( d^2(x^*,x_i)\right) =\psi _i'\left( d^2(\theta ^*,\theta _i)\right) , \,\,\, i=1,\ldots , N. \end{aligned}$$

(28)

Since $x^*=x(\theta ^*)$ is a function of the weights vector $w \in \Delta ^{N-1}$, we can interpret (28) as a system of equations for w, the solution of which in $\Delta ^{N-1}$ will characterize the appropriate weights for the barycenter at which consensus can be reached. Defining the function $w \rightarrow g(w):=x(\theta ^*)$ where $\theta ^*$ is the solution of problem (26), we can rewrite (28) as

$$\begin{aligned} w_{i}=\psi _i'\left( d^2(g(w), x_i)\right) , \,\,\,\, i=1, \ldots , N. \end{aligned}$$

(29)

If the functions $w \mapsto \psi _i'\left( d^2(g(w),x_i)\right)$ are continuous then a solution to (29) in $\Delta ^{N-1}$ is guaranteed by Brouwer’s fixed point theorem. It thus remains to show the continuity of the functions $w \mapsto \psi _i'\left( d^2(g(w),x_i)\right)$. Since $\psi _i'$ are continuous it suffices to show the continuity of the function $w \rightarrow g(w):=x(\theta ^*)$ where $\theta ^*$ is the solution of problem (26).

Step 4 We will use the following definitions: Let $\left( w^{(n)}\right)$ be a sequence in $\Delta ^{N-1}$ and define the sequence of functionals $\psi ^{(n)}: M \rightarrow {{\mathbb {R}}}$, by

$$\begin{aligned} \psi ^{(n)}(x)=\sum _{i=1}^{N} w^{(n)}_i d^2(x, x_i). \end{aligned}$$

If needed (which is not required here) we may express this in terms of the parametric representation $x=x(\theta )$, as a functional on the space of parameters H.

Step 5 We claim the following: Consider a sequence $\left( w^{(n)}\right) \subset \Delta ^{N-1}$, such that $w^{(n)} \rightarrow w$ in $\Delta ^{N-1}$, and a sequence $\left( x^{(n)}\right) \subset M$ such that $x^{(n)} \rightarrow x$ in M. Then, $\psi ^{(n)}\left( x^{(n)}\right) \rightarrow \psi (x)$ in ${{\mathbb {R}}}$, where $\psi (x)=\sum _{i=1}^{N} w_{i} d^2(x,x_i)$. To see that, we calculate

$$\begin{aligned} \left| \psi ^{(n)}(x^{(n)}) -\psi (x) \right|= & {} \left| \sum _{i=1}^{N} w_{i}^{(n)} d^2\left( x^{(n)},x_i\right) - \sum _{i=1}^{N} w_i d^2(x,x_i) \right| \\= & {} \left| \sum _{i=1}^{N} \left( w^{(n)}_i -w_i\right) d^2\left( x^{(n)},x_i\right) + \sum _{i=1}^{N} w_{i} \left( d^2\left( x^{(n)},x_i\right) -d^2(x,x_i) \right) \right| \\\le & {} \sum _{i=1}^{N}\left| w^{(n)}_i -w_i\right| d^2\left( x^{(n)},x_i\right) + \sum _{i=1}^{N} \left| d^2\left( x^{(n)},x_i\right) -d^2(x,x_i)\right| \rightarrow 0, \end{aligned}$$

with the first term tending to 0 since $w^{(n)}\rightarrow w$ and the second term tending to 0 since $x^{(n)} \rightarrow x$.

Step 6 We also claim the following: Consider the sequence $\left( x_{*}^{(n)}\right) \subset M$ of minimizers of the sequence of functionals $\left( \psi ^{(n)}\right)$ (i.e. for every $n \in {{\mathbb {N}}}$, $x^{(n)}=\arg \min _{x} \psi ^{(n)}(x)$). Then, if $x^{(n)} \rightarrow x$ in M, it follows that $x =\arg \min _{x \in M} \psi (x)$. To show this, consider any $z \in M$ and a sequence $\left( z^{(n)}\right) \subset M$ such that $z^{(n)}\rightarrow z$ in M. Then,

$$\begin{aligned} \psi (x)&{\mathop {=}\limits ^{\text{ by } \text{ step } \,\, 5}}&\lim _{n} \psi ^{(n)}(x^{(n)}) = \lim \inf _{n} \psi ^{(n)}\left( x^{(n)}\right) \\&{\mathop {\le }\limits ^{ \psi ^{(n)}\left( x^{(n)}\right) \le \psi ^{(n)}\left( z^{(n)}\right) }}&\lim \inf _{n} \psi ^{(n)}\left( z^{(n)}\right) \le \lim \sup _{n} \psi ^{(n)}\left( z^{(n)}\right) {\mathop {=}\limits ^{by step 5}} \psi (z), \end{aligned}$$

hence, by the fact that z is arbitrary, $x = \arg \min _{x \in M} \psi (x)$.

Step 7. We are now in position to show the continuity of the barycenter with respect to the weights, i.e. the continuity of the map $w \rightarrow g(w):=x\left( \theta ^*\right)$ where $\theta ^*$ is the solution of problem (26). Let $(w^{(n)} \subset \Delta ^{N-1}$ be a sequence of weights and for every $n \in {{\mathbb {N}}}$, let $x_{*}^{(n)}$ be the barycenter of ${{\mathbb {M}}}$ for the weight vector $w^{(n)}$, i.e. the minimizer for the functional $\psi ^{(n)}$. Assume that the sequence $(x^{(n)})$ (or a subsequence) has a limit $x \in M$. This can be achieved by a compactness argument (or weak compactness). Then, by step 6, x is the minimizer of $\psi$, hence the barycenter for the weight vector w. Hence, the map g is continuous. This concludes the proof.

Appendix C: Proof of the Claim in Example 3.7

We recall (see e.g. Bhatia et al. (2019)) that between two measures $P_i \sim N(\mu _i, S_{i})$, $i=1,2$, the Wasserstein distance $W_{2}^{2}(P_1,P_2)$, admits the closed form

$$\begin{aligned} W_2^2(P_1,P_2)=\Vert \mu _1 -\mu _2\Vert ^2 + Tr\bigg (S_1 + S_2 - 2 \left( S_1^{1/2} S_2 S_1^{1/2}\right) ^{1/2}\bigg ). \end{aligned}$$

(30)

Moreover, given a set of probability measures ${{\mathbb {M}}}$ consisting of Gaussian measures $P_i$, $i=1, \ldots , M$, and a weight vector $(w_1, \ldots , w_{K})$, the corresponding Wasserstein barycenter $P_{B}$ is a Gaussian measure $P_{B} \sim N(\mu , S)$ with $\mu = \sum _{i=1}^{K} w_{i} \mu _{i}$, and S being a matrix that satisfies the equation

$$\begin{aligned} 0=I-\sum _{k=1}^{K} w_{k} \left( S_{k} \# S^{-1}\right) \,\, \Longleftrightarrow \,\, S = \sum _{k=1}^{K} w_{k} \left( S^{1/2} S_{k} S^{1/2}\right) ^{1/2}, \end{aligned}$$

(31)

where the notation $A \# B$ is used to denote the geometric mean between two positive definite symmetric matrices given by

$$\begin{aligned} A \# B = A^{1/2}\left( A^{-1/2} B A^{-1/2}\right) ^{1/2} A^{1/2} = B \# A. \end{aligned}$$

Without loss of generality we will assume that $\mu _k =0$, $k=1,\ldots , K$ (else simply center the measures). We will also consider problem (7) on ${{{\mathcal {N}}}}({{\mathbb {R}}}^{d}) \subset {{{\mathcal {P}}}}({{\mathbb {R}}}^{d})$, the subset of Gaussian measures on ${{\mathbb {R}}}^d$. With the above information problem (7) can be expressed as

$$\begin{aligned} \max _{S} \Psi (S):= \max _{S} \sum _{k=1}^{K} \psi _{k} \bigg ( Tr(S_k) + Tr\left( S - 2 g_{k}(S)\right) \bigg ), \end{aligned}$$

(32)

where $\psi _k =\ln \phi _k$ and $g_{k}(S):= \left( S_{k}^{1/2}S S_{k}^{1/2}\right) ^{1/2}$. Problem (32) is an optimization problem on the set of positive definite symmetric matrices. It can be treated by considering the Fréchet derivative of the functional in (32) with respect to S. Using the rules of Fréchet differentiation and assuming sufficient smoothness for the functions $\psi _{k}$ we have that for any deviation $S+ \epsilon Z$ from the matrix S the action of the Fréchet derivative $D\Psi (S)$ on any matrix Z yields

$$\begin{aligned}{}[D\Psi (S)]Z = \sum _{k=1}^{K} \psi _{k}'\left( W_{k}\right) Tr\left( Z - [Dg_{k}(S)]Z\right) , \end{aligned}$$

(33)

where we use the simplified notation

$$\begin{aligned} W_{k}= Tr(S_k) + Tr\left( S - 2 g_{k}(S)\right) . \end{aligned}$$

Moreover, define the quantities

$$\begin{aligned} \Lambda _{k} = \psi _{k}'(W_{k}) \in {{\mathbb {R}}}_{+}, \end{aligned}$$

where the positivity of $\Lambda _k$ is guaranteed by the properties of the functions $\psi _{k}$. Following Bhatia et al. (2019), we can compute

$$\begin{aligned} Tr (Dg_{k}(S) Z) = Tr\left( \left( S_{k} \# S^{-1}\right) Z \right) , \end{aligned}$$

so that (33) yields (using the linearity of trace) that

$$\begin{aligned}{}[D\Psi (S)]Z =Tr\left[ \left( \sum _{k=1}^{K} \Lambda _k\right) I - \left( \sum _{k=1}^{K} \Lambda _{k} \left( S_{k} \# S^{-1}\right) \right) Z\right] . \end{aligned}$$

The first order condition for the solution of (32) is $[D\Psi (S)]Z=0$, for all possible perturbations Z of the covariance matrix S. Upon defining

$$\begin{aligned} w_{k} = \frac{\Lambda _{k}}{\sum _{j=1}^{K} \Lambda _{j}} \in [0,1], \,\,\, k=1, \ldots , K, \end{aligned}$$

the first order condition becomes

$$\begin{aligned} Tr\left[ \left( I -\sum _{k=1}^{K} w_{k} \left( S_{k} \# S^{-1}\right) \right) Z \right] =0, \,\,\, \forall \, Z, \end{aligned}$$

which implies that the solution of (32) corresponds to a Gaussian measure with covariance matrix S such that

$$\begin{aligned} I -\sum _{k=1}^{K} w_{k} \left( S_{k} \# S^{-1}\right) =0 \,\, \Longleftrightarrow \,\, S = \sum _{k=1}^{K} w_{k} \left( S^{1/2} S_{k} S^{1/2}\right) ^{1/2}, \end{aligned}$$

(34)

i.e. $P^{*}$ is the barycenter of ${{\mathbb {M}}}$ with a selection of weights $w_{k}$, endogenously obtained by the preferences on the agents towards their anchor point (in other words their bargaining power).

Note that Eq. (34), although formally the same as Eq. (31) has a fundamental difference from (31). In (34) the coefficients $w_{k}=w_{k}(S)$, i.e. are depending on S, whereas in (31) the coefficients $w_{k}$ are constants. It remains to show that Eq. (34) admits a solution. To show that we define the operator T, by $S \mapsto T(S):=\sum _{k=1}^{K} w_{k} \left( S^{1/2} S_{k} S^{1/2}\right) ^{1/2}$. It can be shown that this operator maps the closed convex set ${{{\mathcal {K}}}}=\left\{ S \in {{\mathbb {R}}}_{+}^{d \times d} \,\, \mid \,\, c_1 I \le S \le c_2 I\right\}$, where $c_1,c_2 \ge 0$ and by $\le$ we denote the natural ordering $S_1 \le S_2 \,\, \Longleftrightarrow S_1 -S_2 \ge 0$ (meaning $S_1-S_2$ positive definite) onto itself. The set ${{{\mathcal {K}}}}$ is convex, and the map T is continuous, so by the Brouwer fixed point theorem T has a fixed point, therefore (34) admits a solution.

Appendix D: Extensions to the Evolutionary Algorithm

The evolutionary scheme presented in this section can be further extended.

1.1 Appendix D.1 A Two Stage Scheme Involving a Clustering Step for Opinion Homogenization and Group Formation

As indicated by the numerical experiments in Sect. 4.3, large degree of inhomogeneity of the positions of the agents in opinion space, especially in cases of large groups, may lead to delay in the convergence to consensus. In certain cases (i.e. in cases of emergency etc), such delays may be unwanted. A way to avoid situations like this is to find ways of grouping the N agents in K as far as possible homogeneous subgroups,⁶ each one characterized by a representative opinion ${\bar{x}}_{k}$, $k=1,\ldots , K$, and then performing the consensus procedure described in Sect. 4.2 among them.

To this end we propose the following version of the celebrated K-means clustering algorithm. We consider the opinions of the large groups as elements $\{x_1, \ldots , x_{N}\}$ of the opinion metric space (M, d). The idea is that like opinions will form clusters in this metric space. Upon being able to identify these clusters, we may form a coarse graining of the group into sub-groups of opinions, which can be treated as homogeneous groups given the level of coarse graining. Mathematically, this corresponds to breaking the large group G into k subgroups $G_{i}$, $i=1,\ldots , k$, such that $G = \bigcup _{i=1}^{k} G_{i}$, and $G_{i} \cap G_{j} = \emptyset$ for $i \ne j$, with the opinions $x_{\ell } \in G_{i}$ being as homogeneous as possible. As discussed above, homogeneity of a subgroup will be understood in terms of the Fréchet function of the subgroup, whereas a relevant measure for the center of the group will be the Fréchet barycenter of the subgroup. This scheme can be applied for any relevant metrization of the opinion space M (see e.g. examples in previous section), for the case of the Wasserstein space see Papayiannis et al. (2021). The proposed clustering algorithm to be implemented in the opinion space is summarized in Algorithm 2.

Algorithm 2

K-Means Clustering Scheme in the Opinions Space

At the convergence of the algorithm, K clusters of opinions are determined, centered at the points ${\bar{x}}_{k}$, $k=1, \ldots , K$ in opinion space (M, d). Each of these clusters can be understood as a more or less “homogeneous” group of agents in terms of opinions. Denoting the groups by $G_{k}$, $k=1, \ldots , K$, we expect our clustering algorithm to perform well in segregating the general group of agents G into subgroups if the Fréchet variance of each subgroup $V_{k}:= \min _{z \in M} F_{G_{k}}(z)$ is comparatively low. Recall that the Fréchet variance of a subset $G_{k} \subset M$ can be also understood as an indicator of its homogeneity.

Note that the above algorithm can be expressed in terms of an optimization problem of the form

$$\begin{aligned} \min _{\begin{array}{c} {\bar{x}}_{k} \in M, \\ k=1, \ldots , K \end{array} } \sum _{j=1}^{K} \sum _{i=1}^{N} a_{ij} d^2(x_{i}, {\bar{x}}_{k}), \end{aligned}$$

(35)

where

$$a_{{ij}} = \left\{ {\begin{array}{*{20}c} 1 & {{\text{if}}\;j = \arg \min _{\ell } d(x_{i} ,\bar{x}_{\ell } ),} \\ 0 & {{\text{otherwise}}.\qquad \qquad \quad \;\,} \\ \end{array} } \right.$$

(36)

In other words, the elements $a_{ij}$ provide information as to the membership of the point i to the cluster j, taking the value 1 if i belongs to cluster j and 0 otherwise. The K-means algorithm solves this problem by the following two-step procedure iterating Steps A and B till convergence:

• A. Given the centers ${\bar{x}}_{j}$, calculate $a_{ij}$ solving the minimization problem (36). This generates a membership matrix $A=(a_{ij}) \in {{\mathbb {R}}}^{N \times K}$ containing binary entries, with each column k of A denoting the composition of the group $G_{k}$.

• B. Given the solution for $a_{ij}$ from Step A, the new centers are determined by solving (35). Note that this step breaks down into K decoupled problems, each one involving the minimization of the Fréchet function for each $G_{k}$, or equivalently finding the Fréchet mean of the group, which is recognized as the center of the corresponding cluster. The objective’s value at the minimum will then be the sum of the Fréchet variances of the clusters $\sum _{k=1}^{K} V_{G_{k}}$.

We close this section by summing up the two-stage group decision process to the following steps:

• 1. Collect and map all opinions of the group as points $z_i$, $i \in G$ into the appropriate opinion space ${\mathcal {M}}$.

• 2. Perform a clustering procedure in opinion space as proposed in Algorithm 2 to form K groups in opinion space.

• 3. Identify the group agreement point as a barycenter of the set of opinions ${{\mathbb {M}}}=\{z_{B,1}, \ldots , z_{B,K}\}$, i.e. as a barycenter of barycenters.

1.2 Appendix D.2 Updating the Discounting Parameter

Possible extensions of the scheme proposed in Sect. 4.2 where the sensitivity parameter $r_i$ could be time-varying can be conceived. For example a possible evolution scheme for the parameter could be described as

$$\begin{aligned} r_i(t+1) = L_i(t, {\mathbb {M}}(t)),\,\,\,\, \forall i\in G, \end{aligned}$$

(37)

considering $L_i$ as the loss function of the agent i depending on the states of the current opinion set ${\mathbb {M}}(t)$, i.e. indicating the loss (under the assumption that $L_i\ge 0$) taking into account the time t and the level of homogeneity in the opinion set ${\mathbb {M}}(t)$. For instance, a possible choice could be the subjective rule

$$\begin{aligned} r_i(t+1) = \left\{ \begin{array}{ll} 0, &{} \text{ if } d^2(\mu _i(0), \mu _j(t)) \le \epsilon _i, \forall j,\\ e^{-\rho _i t} \sum _{j\in {\mathcal {N}}_i} w_{ij}(t) d^2(\mu _i(0), \mu _j(t)), &{} \text{ if } d^2(\mu _i(0), \mu _j(t)) > \epsilon _i, \forall j, \end{array} \right. \end{aligned}$$

(38)

where $\rho _i > 0$ expresses the agent’s preferences concerning a fast resolution of the problem while $\epsilon _i>0$ denotes the agents desire to deviate from her/his anchor preferences $\mu _i(0)$. In this setting, the preferences concerning the time upon which a consensus should be reached governs the determination of the general time-preferences parameter as t grows.

Appendix E: On the Choice of Metrics and the Calculation of the Fréchet Mean

1.1 Appendix E.1 Choice of Metric and the Fréchet Mean

The choice of metric depends on the opinion space. For instance, in the Example 3.2 concerning the social discount rate and the valuation of future uncertain costs, a possible choice of metric space could be determined in thew following way. There are two important components of the opinion space (a) the yield curve characterizing the discount rate and (b) the probability distribution of the future risk to be evaluated. Then, the natural choice of the opinion space is a Cartesian product of two metric spaces $M=M_1 \times M_2$, where $M_1$ corresponds to the metric space of possible yield curves and $M_2$ corresponds to the metric space of possible probability models for the future risk. Let us consider each one separately.

1.1.1 The Space of Discount Rate Curves

As we denoted, $M_1$ is a space of curves $f: [0,T] \rightarrow {{\mathbb {R}}}_{+}$, where T is the time horizon in question. Not any curve is a representation of a suitable term structure model, so we restrict ourselves to parametric families of curves, that are consistent with economic reasoning. One possible choice for $M_1$ could be the parametric family of curves ${{{\mathcal {R}}}}$. A suitable example can be the set defined in (4). To simplify the notation, we will denote all relevant parameters as $\theta \in \Theta$, where $\Theta$ is a suitable parameter space (e.g. $(y_1,\phi )$ in the case of ${{{\mathcal {R}}}}$ in (4)). We will denote any element of $M_1$ as $\Phi (\cdot ; \theta )$, meaning a function $t \mapsto f(t)=\Phi (t; \theta )$, parameterized by some $\theta \in \Theta$ and a suitable function $\Phi : [0,T]\times \Theta \rightarrow {{\mathbb {R}}}$. By definition we can identify any element $f \in M_1={{{\mathcal {R}}}} \subset L^{2}(0,T)$ by an element $\theta \in \Theta \subset {{\mathbb {R}}}^{d}$. We will denote by $\Phi ^{-1}$ the mapping ${{{\mathcal {R}}}} \rightarrow \Theta$, that assigns to any $f \in {{{\mathcal {R}}}}$ a relevant $\theta \in \Theta$, such that $f(\cdot )=\Phi (\cdot ; \theta )$. We do not necessarily require this mapping to be single valued (but we require $\Phi$ to satisfy this property).

As stated above, $M_1={{{\mathcal {R}}}}$ is not a linear space, since linear combinations of functions $f \in {{{\mathcal {R}}}}$ are not necessarily elements of ${{{\mathcal {R}}}}$. There are various choices for the metric on $M_1$. One possible choice of metric on $M_1={{{\mathcal {R}}}}$ could be in terms of a suitable metric for $\Theta \subset {{\mathbb {R}}}^{d}$, i.e.

$$\begin{aligned} d_{M_1}(f_1, f_2)= d_{\Theta }(\Phi ^{-1}(f_1), \Phi ^{-1}(f_2)) = d_{\Theta }(\theta _1, \theta _2), \end{aligned}$$

(39)

where $\theta _i$ are such that $f_{i}(\cdot )=\Phi (\cdot ; \theta _i)$, $i=1,2$, and $d_{\Theta }$ is a suitable metric for $\Theta \subset {{\mathbb {R}}}^{d}$ (a possible choice being an $\ell _{p}$ metric, e.g. the Euclidean metric). The nonlinear nature of the transformation $\Phi$, turns $d_{M_1}$ into a metric on $M_1$ which is not directly related to a metric derivable from a norm on $L^{2}(0,T)$, eventhough it may be related to a norm in the parameter space $\Theta$.

Another possible choice could be a metric compatible with the vector space in which the nonlinear set ${{{\mathcal {R}}}}$ is naturally embedded, a choice for which could be the Hilbert space $L^{2}(0,T)$. Hence a suitable metric for $M_1$ could be as follows: For any two $f,f' \in M_1= {{{\mathcal {R}}}}$ there exists a pair $\theta , \theta ' \in \Theta$ such that f can be identified by $t \mapsto \Phi (t; \theta )$ and $f'$ can be identified as $t \mapsto \Phi (t; \theta ')$. Then we may define the metric

$$\begin{aligned} d_{M_1}(f,f')= \bigg ( \int _{0}^{T} |\Phi (t, \theta ) -\Phi (t, \theta ')|^2 dt \bigg )^{1/2} =: {\hat{d}}(\theta , \theta '). \end{aligned}$$

(40)

Note that while this metric formally coincides with the $L^2$ norm, the space $(M_1, d)$ is not a vector space on account of the nonlinearity of $M_1$, eventhough it is endowed with a metric compatible with the norm of the vector space in which $M_1$ is embedded in. This metric is typically used for spaces of curves of a given parametric representation, i.e. for spaces of curves of the general form ${{{\mathcal {R}}}}:=\{ t \mapsto \Phi (t; \theta ) \,\,: \,\, t \in [0,T], \,\, \theta \in \Theta \}$, as for example in the shape invariant model which is commonly used in functional data analysis (see e.g. Bigot (2013), or Papayiannis et al. (2023) and references therein). Other choices are of course possible, for alternative choices we refer the reader to Srivastava and Klassen (2016), where a detailed discussion of related issues is provided, see also Steyer et al. (2023) for recent applications.

For the numerical illustrations presented in this paper, we opted for the choice (39), rather than (40), mainly for two reasons: (a) Since often decision making concerning scientific issues is model based, consensus upon a model is naturally reduced to consensus on the parameters of the model and (b) the consensus path in the finite parameter space is easier to illustrate and visualize. On the other hand, experiments were also performed using the alternative metric (40), and the results were qualitatively and quantitatively similar.

For any of the above choices, the Fréchet mean in $M_1$ is now defined as follows:

$$\begin{aligned} f_{B}=\arg \min _{f \in M_1} \bigg ( \sum _{i=1}^{N} w_i d^2(f, f_i ) \bigg ) =\arg \min _{\theta \in \Theta } \sum _{i=1}^{N}w_{i} {\hat{d}}^2(\theta , \theta _i), \end{aligned}$$

(41)

where ${\hat{d}}$ is defined as in (40). It is important to note that as defined, $f_{B} \in {{{\mathcal {R}}}}$, and is not merely an element of the embedding space of ${{{\mathcal {R}}}}$, which is $L^2(0,T)$. Moreover, we must stress that the averaging obtained in (41) is a nonlinear averaging of the yield curves $f_{i} \in {{{\mathcal {R}}}}$, and not the standard linear averaging $f_{A}=\sum _{i=1}^{N} w_{i} f_{i}$ in the embedding space $L^{2}(0,T)$, with $f_{A}$ typically having the property $f_{A} \not \in {{{\mathcal {R}}}}$, hence being not acceptable as a suitable yield curve.

1.1.2 The Space of Probability Models for the Future Risk

We now consider the space of possible distributions of future risks. This a space of probability distributions on ${{\mathbb {R}}}^d$, so a suitable choice of opinion space would be $M_2= {{{\mathcal {P}}}}({{\mathbb {R}}}^d)$, the space of probability measures on ${{\mathbb {R}}}^d$. This is again a space that does not admit a vector space structure. A suitable metrization is in terms of the Wasserstein metric presented in Example 3.1. In the case where the risks can be represented by a single random variable (i.e. in terms of their pecuniar value only) we can consider distributions on ${{\mathbb {R}}}$. These can be represented as distribution functions F, hence $M_2$ can be identified with the space of distribution functions. This is clearly not a vector space, but may be embedded on a suitable vector space of functions (e.g. measurable functions). As it turns out, it is not the actual distribution function which is important in the metrization of ${{{\mathcal {P}}}}({{\mathbb {R}}})$, but rather its generalized inverse, the quantile function $Q:=F^{-1}$. For $M_2={{{\mathcal {P}}}}({{\mathbb {R}}})$, the Wasserstein metric can be conveniently expressed in terms of quantiles of the distributions; for any two $P, P' \in {{{\mathcal {P}}}}({{\mathbb {R}}})$ with quantiles $Q=F^{-1}, Q'=F^{' -1}$, respectively, the chosen Wasserstein metric would be

$$\begin{aligned} d_{M_2}(P,P')=\bigg ( \int _{0}^{1} (Q(s)-Q'(s))^2 ds \bigg )^{1/2} =\bigg ( \int _{0}^{1} \left( F^{-1}(u)-F^{' -1}(u)\right) ^2 du \bigg )^{1/2}. \end{aligned}$$

The corresponding Fréchet mean is well defined and unique (if at least one of the distributions is absolutely continuous with respect to the Lebesgue measure see Agueh and Carlier (2011)) so that

$$\begin{aligned} P_{B}=\arg \min _{P \in {{{\mathcal {P}}}}({{\mathbb {R}}})} \bigg ( \sum _{i=1}^{N} w_{i} d_{M_2}^2(P_i, P) \bigg ), \end{aligned}$$

which leads to a representation for $P_{B}$ represented by a distribution function $F_{B}$ defined by the quantile averaging scheme

$$\begin{aligned} F_{B}^{-1} =\sum _{i=1}^{N}w_{i} F_{i}^{-1} \,\,\, \Longleftrightarrow \,\,\, F_{B}=\bigg ( \sum _{i=1}^{N}w_{i} F_{i}^{-1} \bigg )^{-1}. \end{aligned}$$

(42)

The representation (42) is clearly a nonlinear representation for the mean, unlike the linear mean

$$\begin{aligned} {\bar{F}}=\sum _{i=1}^{N}w_{i} F_{i}, \end{aligned}$$

which is commonly used in model averaging or consensus formation.

In case where the future risk is not represented in a satisfactory way by a single number, we can consider $M_2$ as the space of probability measures on ${{\mathbb {R}}}^d$, denoted by ${{{\mathcal {P}}}}({{\mathbb {R}}}^d)$. This is not a vector space also, and can be metrized again using the Wasserstein metric (defined in Example 3.1). In this setup there are in general no closed form solutions for the Wasserstein metric. However, the Wasserstein barycenter, is well defined and unique if at least one of the measures $P_i$ is absolutely continuous with respect to the Lebesgue measure on ${{\mathbb {R}}}^{d}$, so that

$$\begin{aligned} P_{B}=\arg \min _{P \in {{{\mathcal {P}}}}({{\mathbb {R}}}^d)} \bigg ( \sum _{i=1}^{N} w_{i} d_{M_2}^2(P_i, P) \bigg ). \end{aligned}$$

In the special case of Location-Scatter families, where each probability measure is parameterized as $P_{i}=LS(\mu _i, S_{i})$, where $\mu _i \in {{\mathbb {R}}}^{d}$, $S_{i} \in M^{d\times d}_{+}$ the Wasserstein distance between any two measures $P_1,P_2$ is given by (30), and the corresponding Wasserstein barycenter is a probability measure from the same family

$$\begin{aligned} Q_{B}=LS(\mu _B.S_{B}), \end{aligned}$$

(43)

with covariance matrix satisfying the equation

$$\begin{aligned} S_{B}=\sum _{i=1}^{N} w_{i} \left( S_{B}^{1/2} S_{i} S_{B}^{1/2}\right)^{1/2} . \end{aligned}$$

(44)

Clearly this is far from the standard linear averaging on the vector space of in which distribution functions are naturally embedded.

The full opinion space in the social discount factor example is $M_{1}\times M_{2}$, endowed with the metric

$$\begin{aligned} d((f,P),(f',P'))=d_{M_1}(f,f') + d_{M_2}(P,P'). \end{aligned}$$

As mentioned above, the Fréchet mean is not always uniquely defined. In the examples used here, i.e. the Wasserstein case where at least one of the probability measures are absolutely continuous with respect to the Lebesgue measure, or for certain examples of manifolds, the uniqueness of the Fréchet mean has been established (see e.g. Arnaudon et al. (2012), Arnaudon and Miclo (2014), Agueh and Carlier (2011)). However, the proposed methodology is not directly affected if the chosen opinion space is such that the uniqueness of the Fréchet mean is not guaranteed. In such cases, a local minimizer for the Fréchet functional could serve very well as a local consensus point, and the proposed scheme could easily be applied for the determination of local consensus points. The situation can be considered analogous to the situation concerning Nash equilibria, which may not be unique, but nonetheless even local Nash equilibria can be a useful concept in decision making.

1.2 Appendix E.2 Algorithms for Determining the Fréchet Barycenter

The algorithmic determination of the Fréchet barycenter depends crucially on the choice of the relevant metric space setup. We comment here for the choice of metric space, for the example of the consensus concerning the social discount rate, using the same notation as in the previous subsection.

For the $M_1$ part we use the finite dimensional representation of the relevant set of curves ${{{\mathcal {R}}}}$, in order to calculate the Fréchet barycenter. In particular, given a set of yield curves $\{f_1, \ldots , f_N\} \subset {{{\mathcal {R}}}}$, we use their parameterization in terms of the parameters $\{\theta _1, \ldots , \theta _N\} \subset \Theta$, and express the (infinite dimensional) optimization problem

$$\begin{aligned} f_{B}=\arg \min _{f \in M_1} \bigg ( \sum _{i=1}^{N} w_i d^2(f, f_i ) \bigg ), \end{aligned}$$

(45)

in terms of the finite dimensional optimization problem

$$\begin{aligned} \theta _{B}=\arg \min _{\theta \in \Theta } \sum _{i=1}^{N}w_{i} {\hat{d}^2}(\theta , \theta _i), \end{aligned}$$

(46)

which then characterizes $f_{B}$ in terms of $f_{B}= \Phi (\cdot , \theta _{B})$. Problem (46) is treated using standard finite dimensional techniques, e.g. first order optimization methods, or standard higher order methods, e.g. quasi-Newton methods. For the numerical experiments presented here we have used a mixture of gradient based methods and quasi-Newton methods for the study of problem (46) leading to the determination of the Fréchet mean.

For the $M_2$ part, in the case where the loss distribution is one dimensional, the determination of the barycenter comes directly from the representation (42). The only demanding task from the point of view of computation in (42) is the determination of the relevant quantiles. However, for the application in mind, a suitable class of distributions is the class of generalized extreme value distributions as in (14), for which the quantiles are available in closed form analytically, and for which the quantile averaging (42) reduces to simple parameter averaging. In the case where the loss distribution is multi dimensional, we may restrict our attention to the class of Location Scatter family, and imply the representation of the Wasserstein barycenter in terms of the representation (43), with $S_{B}$ given by (44). Then, the only computationally demanding part is the determination of $S_{B}$, in terms of the solution of the matrix Eq. (44). This can be easily performed numerically in terms of the fixed point scheme

$$\begin{aligned} S_{B,k+1}=\sum _{i=1}^{N} w_{i} \left( S_{B, k}^{1/2} S_{i} S_{B, k}^{1/2}\right) . \end{aligned}$$

(47)

which quickly converges to the required value of the covariance matrix $S_{B}$, corresponding to the Wasserstein barycenter.