## 4 Results
Additionally, analyzing the shapes and crossings of cumulative distributions, in line with the concept of first-order stochastic dominance, may allow us to determine the final rank of each city based on a simulation analysis that accounts for the uncertainty of utility functions. In Figure 4A, we present the cumulative distributions of ranking positions obtained from the simulation for all 83 cities. It is evident that, for some cities, we observe a degenerate distribution, indicating a stable position in the final ranking. This is especially true at the beginning (top three cities) and end of the ranking (last eight), where the cumulative distributions are represented by single vertical lines. In many cases, however, the  
Figure 3: Dependencies between BTOPSIS values for different utility functions. cumulative distributions are more complex, with two-point, three-point, or more extensive distributions. In such cases, as shown in Figure 4B, a more precise recommendation regarding the final rank may require using stochastic dominance analysis. For instance, Vienna and Luxembourg can be ranked fourth or fifth depending on the forms of utility functions. However, the rules of first-order stochastic dominance (FSD) allow us to consider the chances of Luxembourg ranking higher than Vienna as significantly smaller; hence, Vienna can be considered better than Luxembourg. A similar situation occurs when comparing Luxembourg and Munich. The latter will be considered worse. Geneve is consistently ranked seventh and below Munich, so there is no need to apply FSD to reach this conclusion. Based on the results of the FSD analysis, we can construct a dominance relation graph and visualize it using a Hasse diagram (see Figure 5). This diagram will naturally account for situations of incomparability, where the shapes of the cumulative distributions of two cities do not allow us to determine which one is better (dominates in the FSD sense). The Hasse diagram allows us to observe ambiguities in the orderings resulting from the simulation study. For example, we can see that Cardiff and Graz have an unresolved position, alternating between 8 th and 9 th place in the simulated rankings. In terms of FSD, they should be considered incomparable and placed on the same level in the final ordering. A bit further down, the situation is interesting: Rennes should definitely be treated as 10 th, while Copenhagen, Oslo, and Helsinki share the 11 th position (from the simulation, we see they could rank 11 th, 12 th, or 13 th). However, Helsinki differs from the other two. Copenhagen and Oslo should be considered better than Dublin, Hamburg, and Bialystok, meaning there is no basis to conclude that they ranked lower than 13 th in the simulations. Helsinki, however, directly dominates only Bialystok, so in some simulations, it might have ranked lower than Dublin and Hamburg, placing it below them in the ranking. Again, FSD perspective recommends us to consider it at the same rank level as Copenhagen and Oslo, just after Rennes. When interpreting the domination results, it is essential to remember that final recommendations regarding the rank of some towns will depend on the filtration approach used. In this analysis, we applied a top-down filtration, positioning each city as high as possible, just below its immediate predecessor. Consequently, Malmo is considered at the same rank level as Amsterdam. If an alternative bottom-up filtration were used (positioning alternatives as low as possible, just above their immediate successor), Malmo would be placed at the same rank level as Reykjavik. Figure 4: Simulation results—general distributions of rank positions of towns (**A**) and those for towns occupying fourth to seventh position in various replications (**B**). FSD does not allow for a more definitive analysis of the results. One could turn to the concept of second-order stochastic dominance to achieve a presumably more definitive ordering, but this requires assumptions about the risk preferences of the decision-maker making the determination [75].