*2.1. Problem Formulation*

First, the log-linear model that is used to determine the priorities of the groups of decision makers is explained: in what follows, N (μ, σ) denotes the univariate normal distribution of mean μ and standard deviation σ; Np (μ, ∑) denotes the p-variant normal distribution of mean vector μ and the matrix of variances and covariances ∑; Tp (μ, ∑, υ) denotes the p-variant Student t distribution with mean vector μ, scale matrix ∑ and degrees of freedom υ; Gamma(p, a) denotes the gamma distribution with shape parameter p and scale parameter 1/a; χ2 ν denotes the chi-squared distribution with υ degrees of freedom; I A denotes the indicator function of set A; ∝ indicates 'proportional to' and [Y|X] denotes the density function of the conditional distribution of Y given X.

Let **G** = {D[1], ... ., D[K]} be a group of K homogeneous decision makers (k = 1, ... , K), **A** = {A1, ... , An} be a set of n alternatives and **R**(k) = r (k) ij ; k = 1, ... , K be the nxn paired comparison matrices issued by each decision maker.

We assume, without loss of generality, that the matrices of judgments are complete—all paired comparisons have been made. If some of the rij comparisons are missing, the proposed methodology could be analogously adapted, as shown in reference [6].

We further assume that the decision makers of **G** have homogeneous opinions regarding the priorities of each of the alternatives of **A** so that:

$$\mathbf{y}\_{\vec{\text{ij}}}^{(\mathbf{k})} = \boldsymbol{\mu}\_{\mathbf{i}}^{(\mathbf{G})} - \boldsymbol{\mu}\_{\mathbf{j}}^{(\mathbf{G})} + \boldsymbol{\varepsilon}\_{\vec{\text{ij}}}^{(\mathbf{k})}; \mathbf{k} = 1, \dots, \mathbf{K}; \mathbf{1} \le \mathbf{i} < \mathbf{j} \le \mathbf{n}; \tag{1}$$

with y(k) ij = log r (k) ij , where:


The normalised priorities of the group will be given by the vector **w**(G) = w(G) i ; i = 1, . . . , n v(G)

where w(G) i = i n ∑ i=1 v(G) i ; i = 1, ... , n. Likewise, the level of homogeneity will be determined by the

standard deviation of the errors σ(G) that quantifies the inconsistency level of all decision makers in the group with the priority vector **w**(G).

### *2.2. Estimation of the Local Priorities for the Group*

The estimation of the vector of group priorities **w**(G) uses a Bayesian approach that allows exact inferences about their values. We adopt, as a prior, a normal-gamma distribution given by:

$$\mu^{(\mathcal{G})} = \left(\mu\_1^{(\mathcal{G})}, \dots, \mu\_{n-1}^{(\mathcal{G})}\right)'|\mathbf{r}^{(\mathcal{G})} \sim \mathcal{N}\_{n-1}\left(0, \frac{1}{c\_0 \mathbf{r}^{(\mathcal{G})}} \mathbf{I}\_{n-1}\right) \text{ with } c\_0 > 0 \tag{2}$$

$$\pi^{(\mathcal{G})} = \frac{1}{\sigma^{(\mathcal{G})2}} \sim \text{Gamma}\left(\frac{\mathbf{n}\_0}{2}, \frac{\mathbf{n}\_0 \mathbf{s}\_0^2}{2}\right) \tag{3}$$

that is the standard conjugate distribution used in Bayesian literature [7]. The constants c0, n0 and s2 0 determine the degree of strength of the prior distribution. In the illustrative example we have taken c0 = 0.1 so that the influence of the prior distribution of μ(G) is not significant. The hyper-parameters n0 and s2 0 are determined from the maximum levels of inconsistency σ2max allowed for each decision maker so that

$$\mathbb{P}\left[0 \le \mathfrak{o}^{(\mathcal{G})2} \le \mathfrak{o}^{2}\_{\max}\right] = 1 - \alpha$$

being, 1 − α (0 < α < 1) the level of credibility that we want to achieve. The value of σ2max has been set using the consistency thresholds of the geometric consistency index (GCI) proposed by [8]. In our illustrative example, and given that n = 4, we take σ2max = 0.35 and α = 0.05, which resulted in n0 = 0.1 and s2 0 = 0.0014.

Using Bayes' theorem, and taking into account (1)–(3), we calculate the posterior distribution of (μ(G), τ(G)) whose density is given by:

% μ(G) <sup>τ</sup>(G), **y**(k); k ∈ {1, . . . , K}& ∝ ∏ 1≤i<j≤n % y(k) ij μ(G), τ(G)&% μ(G) <sup>τ</sup>(G)&% τ(G) & ∝ ∝ K ∏ k=1 ∏ 1≤i<j≤n τ(G) 1 2 exp ' −<sup>τ</sup>(G) 2 y(k) ij − μ(G) i + μ(G) j 2 ( τ(G) n−1 2 exp ' −c0<sup>τ</sup>(G) 2 n−1 ∑ i=1 μ(G) i 2( x τ(G) n0 2 −1 exp % −<sup>τ</sup>(G) 2 n0s<sup>2</sup> 0 & <sup>I</sup>(0,∞) τ(G) = = τ(G) JK+<sup>n</sup>−1+n0 2 −1 exp 9 −<sup>τ</sup>(G) 2 9 n0s<sup>2</sup> 0 + K ∑ k=1 ∑ 1≤i<j≤n y(k) ij − μ(G) i + μ(G) j 2 + c0 n−1 ∑ i=1 μ(G) i 2 :: <sup>I</sup>(0,∞) τ(G) = τ(G) JK+<sup>n</sup>−1+n0 2 −1 exp ' −<sup>τ</sup>(G) 2 ' n0s<sup>2</sup> 0 + K ∑ k=1 **y**(k) − **<sup>X</sup>**μ(G) **y**(k) − **<sup>X</sup>**μ(G) + c0 μ(G) μ(G) (( <sup>I</sup>(0,∞) τ(G) (4)

where **y**(k) = y(k) ij ; 1 ≤ i < j ≤ n for k = 1, ... , K and **X** = (xij) (J × (n − 1)) with J = <sup>n</sup>(<sup>n</sup>−<sup>1</sup>) 2 is the regression matrix of model (1) so that:



It follows that:

$$
\begin{split} & \quad \left[ \mathfrak{h}^{(\mathsf{G})} \Big| \mathsf{r}^{(\mathsf{G})}, \Big\{ \mathsf{y}^{(\mathsf{k})}; \mathsf{k} \in \{1, \ldots, \mathsf{K}\} \right\} \Big] \propto \\ & \propto \left( \mathsf{r}^{(\mathsf{G})} \right)^{\frac{\mathsf{K} + \mathsf{n}\_{\mathsf{G}}}{2} - 1} \exp \Big[ - \frac{\mathsf{r}^{(\mathsf{G})}}{2} \Big\{ \mathsf{n}\_{\mathsf{0}} \mathrm{s}\_{\mathsf{0}}^{2} + \sum\_{\mathsf{k} = 1}^{\mathsf{K}} \mathsf{y}^{(\mathsf{k})} \mathsf{y}^{(\mathsf{k})} - \mathsf{m}^{(\mathsf{g}) \prime} (\mathsf{K} (\mathsf{X}^{\prime} \mathsf{X}) + \mathsf{c}\_{\mathsf{0}} \mathrm{I}\_{\mathsf{n} - 1}) \mathsf{m}^{(\mathsf{g})} \Big\} \Big] \times \\ & \propto \Big( \mathsf{r}^{(\mathsf{G})} \Big)^{\frac{\mathsf{m} - 1}{2} - 1} \exp \Big[ - \frac{\mathsf{r}^{(\mathsf{G})}}{2} \Big{ \Big{(} \Big( \mathsf{n}^{(\mathsf{G})} - \mathsf{m}^{(\mathsf{G})} \Big)' (\mathsf{K} (\mathsf{X}^{\prime} \mathsf{X}) + \mathsf{c}\_{\mathsf{0}} \mathrm{I}\_{\mathsf{n} - 1}) \Big( \mathsf{n}^{(\mathsf{G})} - \mathsf{m}^{(\mathsf{G})} \Big) \Big\} \Big] \Big] \mathbf{I}\_{\left( \mathsf{O}, \infty \right)} \Big( \mathsf{r}^{(\mathsf{G})} \Big) \end{split} (5)$$

where **m**(G) = -K-**X X** + c0In−<sup>1</sup>−<sup>1</sup>**<sup>X</sup>** K∑k=1 **<sup>y</sup>**(k).Therefore, it follows from (5) that:

$$\left\{\boldsymbol{\mu}^{(\mathcal{G})} \middle| \boldsymbol{\tau}^{(\mathcal{G})}, \left\{\mathbf{y}^{(\mathcal{k})}; \mathbf{k} \in \{1, \dots, \mathcal{K}\} \right\} \sim \mathcal{N}\_{\mathbf{n}-1} \left(\mathbf{m}^{(\mathcal{G})}, \frac{1}{\boldsymbol{\tau}^{(\mathcal{G})}} \middle| \mathcal{K}(\mathbf{X}^{\prime}\mathbf{X}) + \mathbf{c}\_{0} \mathbf{I}\_{\mathbf{n}-1} \right)^{-1} \right\} \tag{6}$$

$$\mathbf{r}^{(\mathbf{G})} \Big| \left\{ \mathbf{y}^{(\mathbf{k})}; \mathbf{k} \in \{1, \ldots, \mathbf{K}\} \right\} \sim \text{Gamma}\left(\frac{\mathbf{n}\_0 + \mathbf{J}\mathbf{K}}{2}, \left(\frac{\mathbf{n}\_0 + \mathbf{J}\mathbf{K}}{2}\right) \mathbf{s}^{(\mathbf{G})2}\right) \tag{7}$$

where s<sup>2</sup>(G) = n0s20<sup>+</sup> K∑ k=1 *<sup>y</sup>*(k) *<sup>y</sup>*(k)−**<sup>m</sup>**(G) (K(**<sup>X</sup> <sup>X</sup>**)+c0In−<sup>1</sup>)**m**(G) n0+JK .

Integrating with respect to τ(G) in (5), it follows that:

$$\mu^{(\mathcal{G})} | \left\{ \mathbf{y}^{(\mathcal{k})}; \mathbf{k} \in \{1, \dots, \mathcal{K}\} \right\} \sim \mathcal{T}\_{\mathbf{n}-1} \left( \mathbf{m}^{(\mathcal{G})}, \mathbf{s}^{(\mathcal{G})2} \left( \mathbf{K} (\mathbf{X}^{\prime}\mathbf{X}) + \mathbf{c}\_{0} \mathbf{I}\_{\mathbf{n}-1} \right)^{-1}, \mathbf{n}\_{0} + \mathbf{J} \mathcal{K} \right) \tag{8}$$

From the posterior distributions (7) and (8), point estimates and credibility intervals of **w**(G) and σ(G) can be obtained using the posterior median and the corresponding quantiles.

In the case of σ2(G), and taking into account that from (7) τ(G) ∼ <sup>χ</sup>2n0+JK (n0+JK)s(G)<sup>2</sup> , a 100 (1 − α)%, the Bayesian credibility interval for σ2(G) is given by 9s<sup>2</sup>(G)(n0+JK) <sup>χ</sup>2n0+JK,1− α2 , s<sup>2</sup>(G)(n0+JK) <sup>χ</sup>2n0+JK, α2 : where <sup>χ</sup>2<sup>ν</sup>,<sup>α</sup> denotes the (1 − α)th quantile of the distribution <sup>χ</sup>2ν.

To calculate a credibility interval for w(G) i (1 ≤ i ≤ n) the Monte Carlo method is applied by extracting a sample μ(G,s) = μ(G,s) 1 ,..., μ(G,s) <sup>n</sup>−1 ; s = 1, . . . , S from (8) and calculating a sample of the posterior distribution of **w**(G), **W**(G) = **w**(G,s) = w(G,s) 1 ,...,<sup>w</sup>(G,s) n ; s = 1, . . . , S with exp%μ(G,s) i& 

w(G,s)i = n∑ j=1 exp%μ(G,s) j & where μ(G,s)n = 0. From this sample a credibility interval for w(G)i is given 

by %w(G) i α2 , w(G) i -1 − α2 &where w(G) i (α) is the αth quantile of the sample **W**(G). 

Alpha distributions could also be calculated **PG**<sup>g</sup> α = P**G**<sup>g</sup> <sup>α</sup>,1,...,P**G**<sup>g</sup> α,n with:

$$\mathbf{P}\_{\mathbf{a},\mathbf{i}}^{\mathbf{G}} = \mathbf{P}\Big|\mathbf{w}\_{\mathbf{i}}^{(\mathbf{G})} = \max\_{1 \le j \le n} \mathbf{w}\_{\mathbf{j}}^{(\mathbf{G})} \Big| \left\{ \mathbf{y}^{(\mathbf{k})}; \mathbf{k} = 1, \dots, \mathbf{K} \right\} \Big|; \mathbf{i} = 1, \dots, \mathbf{n} \tag{9}$$

and gamma distributions with:

$$\mathbf{P}\_{\mathbf{Y},\mathbf{Y}\_{\mathbf{h}}}^{\mathbf{G}} = \mathbf{P}\left[\mathbf{w}\_{\mathbf{y},\mathbf{h}}^{\left(\mathbf{G}\right)} \le \dots \le \mathbf{w}\_{\mathbf{y},\mathbf{h}}^{\left(\mathbf{G}\right)}\Big|\left\{\mathbf{y}^{\left(\mathbf{k}\right)};\mathbf{k} = 1,\dots,\mathbf{K}\right\}\Big|;\mathbf{h} = 1,\dots,\mathbf{n}!\tag{10}$$

where γh = ( γh,1, ... , γh,n) is the hth permutation of the elements of A sorted according to the lexicographical order. The approximate calculation of these probabilities is from the sample **W**(G) by means of the expressions:

$$\boldsymbol{\hat{P}}\_{\mathbf{x},\mathbf{i}}^{\mathbf{G}} = \frac{1}{\mathbf{S}} \sum\_{s=1}^{\mathbf{S}} \mathbf{I}\_{\mathbf{w}\_i^{(\mathbf{G},s)} = \max\_{1 \le j \le n} \mathbf{W}\_j^{(\mathbf{G},s)}}(\mathbf{s}) \tag{11}$$

$$\mathcal{P}\_{\mathcal{Y},\mathcal{Y}\_{\mathbf{h}}}^{\mathbf{G}} = \frac{1}{\mathcal{S}} \sum\_{\mathbf{s}=1}^{\mathcal{S}} \mathcal{I}\_{\mathbf{w}\_{\mathcal{Y}\_{\mathbf{h}}1}^{(\mathcal{G},\mathbf{s})} \leq \dots \leq\_{\mathcal{W}\_{\mathcal{Y}\_{\mathbf{h},\mathbf{n}}}^{(\mathcal{G},\mathbf{s})}}(\mathbf{s})}(\mathbf{s}) \tag{12}$$

These distributions report on preferences as well as the most preferred alternative and ranking for the group.

### **3. Identification of Homogeneous Groups of Actors**

The procedure for estimating the priorities of a group, as detailed in the previous section, is based on the hypothesis of the similarity of the opinions of the decision makers. However, it is quite possible that there will be different opinions. In this case, and in order to facilitate a subsequent negotiation process, it is useful to identify the different opinions within the group and the actors that support them; this section presents a systematic procedure for doing this. It utilises a Bayesian oriented tool selection model based on the use of the Bayes factor as a selection element.
