AHP-Group Decision Making: A Bayesian Approach Based on Mixtures for Group Pattern Identification

This paper proposes a Bayesian estimation procedure to determine the priorities of the Analytic Hierarchy Process (AHP) in group decision making when there are a large number of actors and a prior consensus among them is not required. Using a hierarchical Bayesian approach based on mixtures to describe the prior distribution of the priorities in the multiplicative model traditionally used in the stochastic AHP, this methodology allows us to identify homogeneous groups of actors with different patterns of behaviour for the rankings of priorities. The proposed procedure consists of a two-step estimation algorithm: the first step carries out a global exploration of the model space by using birth and death processes, the second concerns a local exploration by means of Gibbs sampling. The methodology has been illustrated by the analysis of a case study adapted from a real experiment on e-democracy developed for the City Council of Zaragoza (Spain).

The authors wish to express their thanks to three anonymous referees for their helpful observations and suggestions on two earlier versions of this paper which we believe have served to significantly improve the quality of this revised text. They are also grateful to Stephen Wilkins for his help in preparing the final draft of the manuscript.

Authors and Affiliations

1. Grupo Decisión Multicriterio Zaragoza, Dpto. Métodos Estadísticos, Facultad de Ciencias Económicas y Empresariales, Universidad de Zaragoza, Gran Vía 2, Zaragoza, 50005, Spain

   Pilar Gargallo, José María Moreno-Jiménez & Manuel Salvador

Corresponding author

Correspondence to José María Moreno-Jiménez.

Partially funded under the research project Electronic Government. Internet-based Complex Decision Making: e-democracy and e-cognocracy (Ref. PM2004-052) approved by the Regional Government of Aragon (Spain) as part of the multi-disciplinary projects programme.

Appendix A

Posterior distribution of θ

In order to make the calculation of the posterior distribution easier, the auxiliary vectors \({\{{\bf z}_{k}=(z_{ k1},\ldots,{z}_{km})^{\prime}; k\,=\,1,\ldots,r\}}\) are introduced. These vectors indicate the component of the mixture (3) to which the decision maker D _k belongs, in such a way that \({{\bf z}_{ k}= {\bf e}_{\ell,m}}\) with probability \({\pi_{\ell};\ell =1,\ldots,{m}}\).

Using the Bayes Theorem the joint posterior distribution of \({{\varvec \theta}}\) and \({{\bf z}=({\bf z}_{1},\ldots,{\bf z}_{ r})}\), \({\left({\varvec \theta} ,{\bf z}\right)\left|{{\bf y}}^{(1)},\ldots,{\bf y}^{(r)}\right.}\), will be given by

where \({I_{A}(x)=1}\) if \({x\in A}\) and 0 otherwise, \({S=\{{{\varvec \Sigma}}_{(n-1)\times{(n-1)}}}\) half-defined positive and symmetric} and \({{{\varvec \prod}}_{\bf m}=\left\{{{\varvec \pi}}\in{\bf R}^{m}:\sum_{\ell =1}^{m}{\pi_\ell}=1\ { \rm and }\ \pi _\ell\geq 0;\ell=1,\ldots,m\right\}}\).

Given that distribution (A.1) has no tractable analytical form, we use MCMC methods in order to draw a sample that allows us to make inferences on the different components of the parameter vector \({{\varvec \theta}}\). In the following paragraph we describe the algorithm used to draw this sample.

Appendix B

Algorithm to draw a sample from the posterior distribution (A.1)

Note that, for each value of m, we have a two-level family of hierarchical models, given by the following equations:

where the parameter vector \({{\varvec \theta}}\) is \({{{\varvec \theta}}= ({{\varvec \theta}}_{1}, {m},{{\varvec \theta}}_{2,{m}}}\)), with \({ {{\varvec \theta}}_1=\left\{\left({\tau^{(k)},{{\varvec \mu}}^{(k)}} \right),\;k=1,\ldots,r\right\}}\) and \({{{\varvec \theta}}_{2,m}=\left\{ {\left({\pi_\ell,{{\varvec \mu}}}_{G}^{(\ell)},{{\varvec \Sigma}} _{G}^{(\ell)}\right),\;\ell=1,\ldots,{m}}\right\}}\) whose dimensionality changes according to the value of m.

Therefore, there is a space of models underlying the problem, \({\{{\bf M}_{m}; m\,=\,1, 2, \ldots\}}\), where \({{\bf M}_{m}}\) is given by equations EQ₁ and EQ_2,m . With the aim of exploring this space and obtaining samples of the posterior distribution (A.1), we have followed the methodology based on birth–death Markov processes developed by Stephens (2000).

In order to explain the algorithm, we define the set \({\Omega =\bigcup\limits_{{m}=1}^\infty {\Omega_{m}}}\) where

is the parameter space of the distribution G for a fixed number m of mixture components (3). Therefore, Ω constitutes the model space in which the process of global exploration is developed.

This global exploration process consists of the choice of the movement and its implementation. There are two types of possible movements: incorporation (birth) of a new component of the mixture (3) or removal (death) of one of the components. We define births and deaths on Ω as follows:

A birth or incorporation of a new component \({\left({\pi_{{m}+1} ,{{\varvec \mu}}}_{G}^{({m}+1)},{{\varvec \Sigma}}_{G}^{({ m}+1)}\right)}\) is said to occur when the process jumps from \({\Omega_{m}}\) to \({\Omega_{{m}+1}}\) where:

The death or removal of the \({\ell}\) th component of \({\Omega_{m}}\), \({\left( {\pi_\ell,{{\varvec \mu}}_{G}^{(\ell)},{{\varvec \Sigma} }}_{G}^{(\ell)}\right)}\) is said to occur when the process jumps from \({\Omega_{m}}\) to \({\Omega_{{m}-1}^{\left({-\ell} \right)}}\) where: \({\Omega_{{m}-1}^{\left({-\ell}\right)=}}\)

The algorithm consists of two steps that are alternated in each iteration. In the first step, a global exploration of the model space \({\{{\bf M}_{m}, m\,=\,1, 2, \ldots\}}\) is carried out by means of birth–death point processes with reversible jumps developed by Stephens (2000). In the second, a local exploration for a model \({{\bf M}_{m}}\) with m fixed is carried out by applying Gibbs sampling (Robert and Casella 1999) to the model \({{\bf M}_{m}}\) during a number I of iterations fixed beforehand by the analyst.

The scheme of the algorithm is as follows:

Step 0: Start

The maximum number of algorithm iterations, \({\hbox{IT}_{\rm max}}\) and the number of iterations for the Gibbs sampling, \({\hbox{IT}_{\rm GS}}\), used for carrying out a Bayesian analysis of each model explored by the algorithm, are fixed. \({m^{(0)}}\) is extracted from a \({\hbox{Poisson}(\lambda_{0})}\) and a set of objects

is obtained using, for example, the prior distribution (4)–(8). Thereafter, the auxiliary vectors \({{\bf z}^{(0)}=\left\{{{\bf z}_1^{(0)} ,\ldots,{\bf z}_{r}^{(0)}}\right\}}\) with \({{\bf z}_{k}^{(0)}=\left({{z}_{k1}^{(0)} ,\ldots,{z}_{{km}^{(0)}}^{(0)}}\right)^{\prime}}\) ; \({k=1,\ldots,r}\) are generated in such a way that \({{\bf z}_{ k}^{(0)}}\) is equal to the \({\ell}\) th coordinate vector of \({{\bf R}^{{m}^{(0)}}}\) with probability \({\pi_\ell^{(0)}}\) ; \({\ell=1,\ldots,{m}^{(0)}}\) and, from these \({\{{{\varvec \mu}}^{(0,{k})};\,{k}=1,\ldots,{r}\}}\) are generated by means of the normal distribution \({N_{{ n}-1}\left({\sum_{\ell=1}^{{m}^{(0)}} {{z}_{{ k}\ell}^{(0)}{{\varvec \mu}}}_{G}^{(0,\ell)}} ,\sum_{\ell=1}^{{m}^{(0)}}{{z}_{{k}\ell}^{(0)} {{\varvec \Sigma}}}_{G}^{(0,\ell)}\right)}\). The iterations counter it is initialised (it = 1) and the following steps are repeated until \({\hbox{it} > \hbox{IT}_{\rm max}}\).

Step 1: Local exploration by means of Gibbs sampling

Steps 1(a) to 1(f) are executed during IT_GS iterations

Step 1(a): Draw \({\left\{{\tau^{({{\rm it},k})};{k}=1,\ldots,{r}} \right\}}\) from

Step 1(b): Set \({{m}^{({\rm it})} = {m}^{({\rm it}-1)}}\) and draw \({\{{{\varvec \mu}}^{({{\rm it},k})};{k}=1,\ldots,{r}\}}\) from \({N_{{ n}-1}( {\bf MED}^{({k})}{\bf VAR}^{({k})})}\) where

Step 1(c): Draw \({\left\{{{\varvec \Sigma}}_{G}^{({\rm it},\ell)};\ell =1,\ldots,{m}^{({\rm it})}\right\}}\) from \({\hbox{IW}(n_{\ell},{\bf D}_{\ell})}\) with \({n_{\ell}= {n}_{0}+1+\sum_{{k}=1}^{r} {{z}_{{k}\ell}^{({\rm it}-1)}}}\)

Step 1(d): Draw \({\left\{{{\varvec \mu}}_{G}^{({\rm it},\ell)};\ell =1,\ldots,{m}^{({\rm it}1)}\right\}}\) from \({{N}_{{ m}^{({\rm it}-1)}}\left({{\bf MED}_{G}^{(\ell)},{\bf VAR}_{G}^{(\ell )}}\right)}\) where

Step 1(e): Draw \({\left\{{\bf z}_{k}^{({ \rm it})};{k}=1,\ldots,{r}\right\}}\) from \({\hbox{Mul}(1,{p}_1^{({k})} ,\ldots,{p}_{{m}^{({\rm it})}}^{({k})})}\) where

Step 1(f): Draw \({{{\varvec \pi}}^{({ it})}=\left({{{\varvec \pi}}_1^{({\rm it})},\ldots,{{\varvec \pi}}_{{ m}^{({\rm it})}}^{({\rm it})}}\right)^{\prime}}\) from Dirichlet \({\left({1+\sum_{{k}=1}^{r}{{z}_{{k}1}^{\rm (it)} },\ldots,1+\sum_{{k}=1}^{ r}{{z}_{{km}^{({\rm it}-1)}}^{({\rm it})}}}\right)}\). Set it = it + 1.

Step 2: Global exploration of the model space.

Step  2(a): Set \({{{\theta}}^{({\rm it})} ={{\theta}}^{({\rm it}-1)},\,{\bf z}^{({\rm it})}={\bf z}^{({\rm it}-1)},\,{{\Omega}}_{{m}^{({\rm it})}}^{({\rm it})}=\left\{{\left({{{\pi}}_\ell^{({\rm it})},{{\varvec \mu}}_{G}^{({\rm it},\ell)},{{\varvec \Sigma}}_{G}^{({\rm it},\ell )}}\right);\ell=1,\ldots, {m}^{({\rm it})}}\right\}}\) and t = 0.

Step 2(b): If \({{m}^{({\rm it})} = 1}\), go to Step 2(d). Otherwise, calculate the death rate for each component of the mixture

where \({{L}({{\Omega}}_{{m}})=\prod\limits_{k=1}^{r}{\left({\sum_{\ell =1}^{m}{\pi_\ell\varphi \left({ {{\varvec \mu}}^{({k})};{{\varvec \mu}}_{G}^{(\ell)}, {{\varvec \Sigma}}_{G}^{(\ell)}}\right)}}\right)}}\) is the likelihood function for the set of points \({{{\Omega}}_{m}}\), where \({({\bf x};{{\varvec \mu}},{{\varvec \Sigma}})}\) is the density function of a \({N_{{n}-1}({{\varvec \mu}},{{\varvec \Sigma}})}\) evaluated in \({{\bf x}\in {\bf R}^{n-1}}\).

Step 2(c): Simulate the time of the next jump \({t=t + v}\), where v is sampled from an exponential distribution with mean \({\frac{1}{\lambda _0+\delta}}\) where \({\delta= \sum_{\ell=1}^{{m}^{({\rm it})}}{\delta_\ell}}\) is the total death rate of the process. If \({t > \hbox{it}}\), return to Step 1 of the algorithm. Otherwise, go to Step 2(d).

Step 2(d): Simulate the type of jump. Here, we distinguish two cases:

Step 2(d1): If \({{m}^{({\rm it})}= 1}\), a birth happens following the procedure (B.1).

Step 2(d2): If \({{m}^{({\rm it})} > 1}\), a birth happens following the procedure (B.1) with probability \({\frac{\lambda_0}{\lambda_0 +\delta}}\), and the death of the components \({\ell}\) th following the procedure (B.2) happens with probability \({\frac{\delta_\ell }{\lambda_0+\delta}}\). In this case, the values of \({\left\{{ {\bf z}_{k}^{({\rm it})}:{\bf z}_{k}^{({\rm it}-1)}={\bf e}_\ell}\right\}}\) are again reassigned by applying Step 1(e).

In both cases, if the jump carried out is a birth, set \({{m}^{({\rm it})}={m}^{({\rm it})}+1}\), and if it is a death, set \({{m}^{({\rm it})}= {m}^{({\rm it})}{-}1}\). Return to Step 2(b).

As a consequence of the algorithm a sample of the distribution (A.1) is obtained:

where it₀ is the estimated number of iterations necessary for the process convergence to the stationary distribution (A.1) and s is the number of lags to neglect the serial autocorrelation. This value can be estimated following the usual procedures in the literature (Robert and Casella 1999, Cap. 8).

This sample can be used to draw inferences on the different components of \({{{\varvec \theta}}}\). In particular, an estimation of distribution G is given by \(E[G({{\varvec \mu}})\vert{\bf y}]\). This expectation can be calculated using the Blackwell-Rao estimator (Casella and Robert (1996)) given by

From (B.4) it is possible to calculate estimations of the distributions of the priorities \({\{w_{i}; i\,=\,1,\ldots,n\}}\) (with or without normalisation), as well as the posterior distribution of the preference structures by means of Monte Carlo methods. From these preference structures it can be inferred if there is consensus between the decision makers, or if there are several modes which make clear the existence of several opinion groups. In this latter case, the groups would be located by using the samples of the individual priorities \({\{{{\varvec \mu}}^{({{\rm it},k})}; \hbox{it}=\hbox{it}_{0}, \ldots,\hbox{IT}_{\rm max}, k\,=\,1,\ldots,r\}}\) and/or the indicators \({{\{{\bf z}^{\rm (it)}; \hbox{it}=\hbox{it}_{0},\ldots,\hbox{IT}_{\rm max}\}}}\). This could be carried out using classification algorithms. Alternatively, we could use perceptual maps which reflect the individual preferences for each alternative (see the example described in Section 4).

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