**4. Case Study**

In September 2006 the Government of Aragón and the Zaragoza City Council advanced the 'Zaragoza Plan for Sustainable Mobility', which led to the construction of the city's Tram Line 1 that was completed in 2013. The construction was controversial and in the following municipal elections, the political parties presented their proposals for improving transport in the city. Representatives of the main political parties (PSOE, PP and CHA) attended the university to explain their preferred alternative. The case study concerns a citizen participation project in a local community which contemplated alternatives put forward by the political parties during the municipal elections for the extension of the city tram network.

Eleven students (K = 11) from the 'Electronic-Government and Public Decisions' course of the Faculty of Economics and Business at the University of Zaragoza (Spain) were involved in its implementation.

There were 4 alternatives:


As the construction of Line 1 had a high economic cost, and the selection of any of first three alternatives assumes a significant investment, a fourth alternative (no cost) was included. The selection problem was solved using the Analytic Hierarchy Process [2]. The hierarchical model comprised four levels (the goal, 3 criteria, 9 attributes and 4 alternatives).

The results of the Investment Cost attribute have been used to illustrate the proposed methodology. The prior distribution parameters were c0 = 0.1, n0 = 1 and s20 = 0.0014 (corresponding to take σ2max = 0.35 and α = 0.05) and β = 0.05. The number of partitions was equal to the 678,570 that were processed in 98.84 s of CPU by a Toshiba Ultrabook KIRA with Intel (R) Core™ i7-4510U CPU @ 2.00GHz 2.60 GHz (64 bits) and 8 Gb of RAM.

The resulting number of groups was equal to five and the most probable composition was: **G**1 = {D1, D2, D3, D7}, **G**2 = {D4, D10}, **G**3 = {D5, D6}, **G**4 = {D9}, **G**5 = {D11}. The results obtained are shown in Tables 1–3. More specifically, Table 1 contains the posterior medians of the priorities of each alternative for each decision maker and each group. Table 2 shows the posterior probabilities that each alternative would be the most preferred, corresponding to the Pα distributions. Table 3 gives the probabilities for each ranking corresponding to the Pγ distributions. The values were calculated from (7)–(8) and (11)–(12), as described in Section 2.2, using S = 10,000 simulations.

So, for example, group G1 made up of decision makers D1, D2, D3 and D7, gives the highest priority (0.4502) to alternative A4, followed by the alternatives A2 (priority 0.3076), A3 (0.1518) and A1 (0.0879) (see Table 1). This ranking is also reflected by its Pα and Pγ distributions, which give the maximum posterior probabilities to the A4 alternative (97.31%, see Table 2) and the ranking 4231 (96.94%, see Table 3). Even though the individual opinion of D3 is different to the rest of the members of the group (their preferred alternative is A2 and the ranking is 2431), the consistency of the group G1 (0.2790) is good, being lower than the maximum level of inconsistency 0.35. This is due to the high priority of D3 for alternative A4 (0.2705) and the similarity of their priorities to alternatives A1 and A3 which means that G1 can be considered as a homogeneous group.

From the tables, it can be observed that the compositions of the groups are very much determined by their similarity to the most preferred alternative. The decision makers from groups **G**1 and **G**3 mostly prefer the alternative A4, those from group **G**2 prefer alternative A1, those from **G**3 prefer alternative A3 and those from **G**4 prefer alternative A2 (see Tables 1 and 2). However, groups **G**1 and **G**3 differ in the rankings (see Table 3). The decision makers from the group **G**1 tend to prefer the 4231 ranking, while those from the group **G**3 prefer 4123. Nevertheless, preferences within each group are not completely homogeneous; in group **G**1, decision-maker D3 shows a greater preference for alternative A2. This

opinion is shared with decision maker D11 of group **G**4, although it assigns a priority 0.2705 to alternative A4 that justifies its inclusion in group **G**1. Something similar happens in group **G**3, in which decision maker D5 shows a greater preference for alternative A1, although it assigns a non-negligible priority (0.3613) to alternative A4 and to the ranking 4123, hence its inclusion in group **G**3.

The consistency levels of the actors and most of the groups are acceptable (<0.35). The only exception is G3 whose consistency is estimated as 0.4071, but with a 95% credibility interval [0.20, 1.43] that does not reject the consistency hypothesis σ<sup>2</sup>(G3) ≤ 0.35.

The ambiguities of the opinions are revealed if we analyse the partitions selected by Occam's window with the smaller number of groups, included in Table 4 and Figure 2. Figure 2 incorporates, in each node, the groups of decision makers that are classified together in all the selected partitions and includes a link between two nodes if their components are classified together in some of the partitions. Most of the decision makers are linked because of their preferences for alternatives A2 and A4, the latter being the alternative which most decision makers support. Only D4, D9 and D10 are isolated because of their preferences for alternatives A1 (D4 and D10) and A3 (D9).

**Table 1.** Priorities and consistency for each decision maker and each group.


in**bold** the highest priority.

**Table 2.** Alpha distributions for each decision maker and each group.


in **bold** the probabilities higher than 20%.


**Table 3.** Gamma distributions for each decision maker (Di) and each group (Gj) (probabilities higher than 20% are in bold).

There are 4 homogeneous groups: {D1, D2, D7} that prefer alternative A4 and the ranking 4231; {D5, D6} position alternatives A2 and A3 in the last places and they show a non-negligible preference for A4; {D4, D10} prefer alternative A1 and put A3 in last place; decision maker {D9} prefers A3. Decision makers D3, D8, D11 are in more intermediate and ambiguous positions. In the case of D3, this is due to the greater preference for A2 (shared with D11) and the non-negligible preference for A4 (which places them close to the group {D1, D2, D7}). In the case of D8, the intermediate position is due to their preferences for A4 and A2, in that order, which places them close to the group {D1, D2, D7}, as well as to the rejection of A3, which places them close to the group {D5, D6}.

In order to achieve as broad an agreemen<sup>t</sup> as possible, alternative A4 could be suggested, given that a majority of decision makers (in groups G1 and G3) showed a preference for it. The negotiation should be aimed at convincing decision makers D4, D9, D10 and D11.

With regards to the practical implications of the Bayesian procedure proposed in this work, it is worth mentioning that, as in AHP, these applications are numerous, especially in matters of strategic planning where the number of actors is not usually very high.

**Table 4.** Partitions selected by the Occam window.


† The ratio calculates the quotient of the posterior probability for the most probable model and that for the model corresponding to each partition.

**Figure 2.** Relations of opinions existing in D.
