## 3 Simulation-Based Evaluation  
At first, rigorous simulation-based investigation is launched to explore the dynamics of the proposed scheme. The performance of the devised mechanism is studied while considering a wide range of parametric settings, including varying sample sizes and the parameter defining the extent of indecisiveness. We consider three samples as, \(n=25,\ 50\) and \(100\), for two values of tie parameters, that is, \(\tau=0.1\) and \(0.2\). These states are then studied for three competing objects, that is, \(m=3\), under a preset preference ordering where \(\theta_{3}>\theta_{2}>\theta_{1}\) and \(\theta_{1}=0.24,\ \theta_{2}=0.36\) and \(\theta_{3}=0.40\). Table 2 presents the artificially generated data sets resulting from the aforementioned parametric settings. Tables 3-5 demonstrate the relevant summaries highlighting the various performance aspects of the schemes. Table 3 provides the Bayes estimates of worth parameters along with the associated 95% credible intervals. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(n\) & \(\mathbf{w_{1.12}}\) & \(\mathbf{w_{2.12}}\) & \(\mathbf{w_{0.12}}\) & \(\mathbf{w_{1.13}}\) & \(\mathbf{w_{3.13}}\) & \(\mathbf{w_{0.13}}\) & \(\mathbf{w_{2.23}}\) & \(\mathbf{w_{3.23}}\) & \(\mathbf{w_{0.23}}\) \\ \hline \multicolumn{11}{c}{\(\tau=0.10\)} \\ \hline
25 & 8 & 16 & 1 & 6 & 18 & 1 & 5 & 19 & 1 \\ \hline
50 & 11 & 37 & 2 & 9 & 39 & 2 & 16 & 33 & 1 \\ \hline
100 & 27 & 72 & 1 & 19 & 73 & 8 & 44 & 51 & 5 \\ \hline \multicolumn{11}{c}{\(\tau=0.20\)} \\ \hline
25 & 5 & 18 & 2 & 6 & 17 & 2 & 10 & 12 & 3 \\ \hline
50 & 15 & 27 & 8 & 12 & 34 & 4 & 19 & 26 & 5 \\ \hline
100 & 19 & 72 & 9 & 24 & 73 & 3 & 37 & 52 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Artificial data under the preset experimental states. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & \(\hat{\mathbf{\theta_{1}}}\) & \(\hat{\mathbf{\theta_{2}}}\) & \(\hat{\mathbf{\theta_{3}}}\) & \(\hat{\mathbf{\tau}}\) \\ \hline \multicolumn{11}{c}{\(\tau=0.10\)} \\ \hline
25 & 0.2549 & 0.3014 & 0.4437 & 0.0865 \\  & (0.2448, 0.2651) & (0.2905, 0.3122) & (0.4301, 0.4571) & (0.0728, 0.1002) \\ \hline \multirow{2}{*}{50} & 0.2414 & 0.3403 & 0.4183 & 0.0911 \\  & (0.2330, 0.2498) & (0.3389, 0.3415) & (0.4091, 0.4275) & (0.0806, 0.1016) \\ \hline \multirow{2}{*}{100} & 0.2491 & 0.3516 & 0.3992 & 0.0986 \\  & (0.2337, 0.2494) & (0.3508, 0.3603) & (0.3964, 0.4022) & (0.0933, 0.1039) \\ \hline \multicolumn{11}{c}{\(\tau=0.20\)} \\ \hline
25 & 0.2689 & 0.3739 & 0.3832 & 0.1881 \\  & (0.2640, 0.2738) & (0.3619, 0.3860) & (0.3610, 0.3954) & (0.1715, 0.2048) \\ \hline \multirow{2}{*}{50} & 0.2428 & 0.3510 & 0.3958 & 0.2128 \\  & (0.2327, 0.2528) & (0.3489, 0.3647) & (0.3896, 0.4275) & (0.1956, 0.2199) \\ \hline \multirow{2}{*}{100} & 0.2421 & 0.3594 & 0.3969 & 0.1926 \\  & (0.2390, 0.2454) & (0.3556, 0.3624) & (0.3939, 0.4098) & (0.1894, 0.2086) \\ \hline \end{tabular}
\end{table} 
Table 3: Estimates of worth parameters and associated 95% credible intervals (in parenthesis) for pre-defined experimental settings. The summaries presented in the above table competently indicate the legitimacy of the proposed model in retaining the predefined underlying ordering of the competing objects, that is, \(\theta_{3}\rightarrow\theta_{2}\rightarrow\theta_{1}\), when ties in the discrete choice data are permitted. Additionally, it is noteworthy that the proposition competently captures the degree of indecisiveness precipitated in the comparative information. This fact is realized regardless of the varying sample sizes and different values of the tie parameter. However, a more profound performance of the approach under discussion is witnessed with an increased sample size. For example, the closest and most precise estimation of both delicacies, that is, the worth parameters and tie parameter, are witnessed for the case of \(n=100\), where the proposed scheme closely estimates the values of both the worth parameters and tie parameter.