Using Maxwell Distribution to Handle Selector’s Indecisiveness in Choice Data: A New Latent Bayesian Choice Model

Abstract

This research primarily aims at the development of new pathways to facilitate the resolving of the long debated issue of handling ties or the degree of indecisiveness precipitated in comparative information. The decision chaos is accommodated by the elegant application of the choice axiom ensuring intact utility when imperfect choices are observed. The objectives are facilitated by inducing an additional parameter in the probabilistic set up of Maxwell to retain the extent of indecisiveness prevalent in the choice data. The operational soundness of the proposed model is elucidated through the rigorous employment of Gibbs sampling—a popular approach of the Markov chain Monte Carlo methods. The outcomes of this research clearly substantiate the applicability of the proposed scheme in retaining the advantages of discrete comparative data when the freedom of no indecisiveness is permitted. The legitimacy of the devised mechanism is enumerated on multi-fronts such as the estimation of preference probabilities and assessment of worth parameters, and through the quantification of the significance of choice hierarchy. The outcomes of the research highlight the effects of sample size and the extent of indecisiveness exhibited in the choice data. The estimation efficiency is estimated to be improved with the increase in sample size. For the largest considered sample of size 100, we estimated an average confidence width of 0.0097, which is notably more compact than the contemporary samples of size 25 and 50.

1. Introduction

Competent decision making requires a variety of cognitive skills assisting the notion of the search for value information to enhance the working potentials, especially when dealing with a complex multifaceted environment [1]. This process demands comparing and mastering the available choices while simultaneously dealing with the practical limitations [2]. Therefore, the enchanted status of analyzing and modeling choice behaviors in the multidisciplinary research literature is of no surprise. The well-directed historic tour of [3] traced the roots of the comparative notions in seventeenth century France, where [4] advocated the use of comparative models (in a very abstract form) as a method to ensure higher levels of fairness in the electoral process. However, it was the seminal contributions of [5,6] that laid the foundational blocks communicable through mathematical rigors to encapsulate individual differences and associated choice behaviors. The aforementioned efforts instigated the idea of paired comparison (PC) experiments and brought related models into the lime light. Since then, PC methodologies have attracted the attention of many researchers from diverse fields of enquiry ranging from health surveillance to sport analysis. For example, in the past, [7] explored the applicability of PC models to evaluate the performance of industrial accessories. Furthermore, [8,9] elucidated the use of the PC approach to analyze sporting events and predict their outcomes. In the recent past, [10] argued the PC schemes were an alternative to the Likert scale for the ranking of psychological markers and indicators’ evaluation. Moreover, [11,12] competently elaborated on the interlinkages coupling the choice modelling strategies and the numerous variants of rational choice theory governing the preference attitudes in political science, sociology and criminology. For comprehensive accounts documenting the utility of PC methodologies in investigative pursuit, one may also consult [13,14,15].

In general, comparative experiments peruse the complexity of rational decision making by providing comprehension of the vital ingredients, such as attribute-level combinations, repeated choices and utility-based trade-off, by offering mutually exclusive choice alternatives to the selectors or judges. The inherent randomness of individual choices is explained by assuming the probabilistic model, whereas the associated utility is delineated through the stimuli governing the overall choice dynamics. In its simplest form the judges, say $n$, are asked about their preferences while pairwise comparing, say $m$, items, objects or individuals through a simple question, that is, “Do you prefer item $i$ over item $j$?”. One may notice the immediate relevance of the inquiry in all sorts of human behavioral assessment mechanisms. Thus, in a complete two factorial setup, each judge provides $\left(\begin{array}{c}m\\ 2\end{array}\right)$ responses in regard to the above inquiry.

Table 1. Hypothetical choice matrix involving single selector and $m$ objects, Y = yes and N = No.

Objects	1	2	3	4	5	-	$\mathit{m}$
1	-	Y	Y	Y	N	-	Y
2	-	-	N	N	N	-	N
3		-	-	N	Y	-	Y
4			-	-	Y	-	Y
5				-	-	-	N
-					-	-	-
$\mathit{m}$							-

presents the binary string of comparative choice hierarchy recorded by a single selector or judge while conducting a paired comparison experiment.

It is trivial to extend the above reported structure to the incorporation of comparative information accumulated over $n$ selectors.

The numerous delicacies have been introduced through a rich stream of ongoing efforts in the above given simple structure, enabling choice models to deal with the complexities of real phenomena. For more recent, interesting and knowledgeable accounts of the more notable contributions, one may consult [16,17,18,19]. This article fundamentally focuses on the entertainment of selectors’ indecisiveness in choice reporting, at methodological and modeling levels. The issue of handling ties in preference data has long been debated and remains a primary component of the available literature focusing on the analysis of discrete choices. The opinion chaos has rightly been summarized by [20] who noted “the key point is that modelling of ties explicitly can be important, although there is no consensus on how this should be done; no approach apart from ignoring ties appears to be in widespread use”.

This article is mainly divided into five partitions. Section 2 is dedicated to documenting the mathematical foundations of the proposed procedure, whereas Section 3 provides a rigorous account of the simulation-based evaluations of the proposal while mimicking numerous experimental states. Section 4 delineates the applicability of the suggested approach while analyzing drinking water brands’ choice data. Lastly, the main findings are discussed along with some future possible research avenues in Section 5.

2. Methods and Materials

2.1. Preliminaries

Let us consider that full factorial pairwise comparison set up is launched to generate comparative information among $m$ objects by $n$ judges, where pair of stimuli elicits a continuous discriminal process. The latent preference hierarchy between $i\prime th$ object and object $j$ is then thought to follow one dimensional Maxwell distribution over the consistent support in the population, such as

$$f\left(x_i\right)=\sqrt{\frac{2}{\pi }}{x}_i^2\frac{e^{\frac{-x_i^2}{2{\theta }_i^2}}}{{\theta }_i^3},x_i>0, {\theta }_i>0,$$

and

$$f\left(x_j\right)=\sqrt{\frac{2}{\pi }}{x}_j^2\frac{e^{\frac{-x_j^2}{2{\theta }_j^2}}}{{\theta }_j^3},x_j>0, {\theta }_j>0.$$

(1)

Here, ${\theta }_i$ and ${\theta }_j$ are scale parameters of the hallmark structure connecting the probability density of the particle’s kinetic energies to the temperature of the system while taking into account the configurational fluctuations of the system [21]. It is noteworthy that the traditional competency of Maxwell formation in encapsulating the atomic velocity distribution with the assumption of lacking potentials provides natural foundations to model preference stimuli while considering utility as a latent phenomenon. In the case of binary string of choice alternatives, the interest lies in the deduction of preference probabilities as a function of worth parameters dictating the comparative utility precipitation of competing objects. Mathematically, probability of preferring object $i$ over object $j$ that is, $\left(i\to j\right)$, remains quantifiable, such as $p_{i.ij}=P\left(X_i>X_j\right)$. Similarly, $p_{j.ij}=P\left(X_j>X_i\right)$ represents the preference probability of $\left(j\to i\right)$, as a function of estimated worth parameters [22]. Recently, [23] provided the simplified form of preference probabilities such as

$$p_{i.ij}=1+\frac{2}{\pi }\left[\frac{{\theta }_i{\theta }_j\left({\theta }^2{}_i-{\theta }^2{}_j\right)}{{\left({\theta }^2{}_i+{\theta }^2{}_j\right)}^2}-arctan\left(\frac{{\theta }_i}{{\theta }_j}\right)\right]$$

and

$$p_{j.ij}=1+\frac{2}{\pi }\left[\frac{{\theta }_i{\theta }_j\left({\theta }^2{}_j-{\theta }^2{}_i\right)}{{\left({\theta }^2{}_i+{\theta }^2{}_j\right)}^2}+arctan\left(\frac{{\theta }_j}{{\theta }_i}\right)\right]$$

(2)

2.2. Proposed Model

In preference data, when ties are permitted, selectors are indeed offered three potential responses while confronting the task of choosing $\left(i\to j\right)$, those are “yes”, “no” or “no preference”. Thereby, it is to be noted that in the case of a balanced factorial comparative experiment, the selector responses distinguishing each pair follow trinomial distribution. For notational purposes, the preference probability of $\left(i\to j\right)$ is denoted as $p_{i.ij}$, where $p_{j.ij}$ represents preference for $\left(j\to i\right)$. The probability highlighting the extent of indistinctiveness or indecisiveness while comparing both competing objects is reported as $p_{o.ij}$, that is $\left(i=j\right)$ when no preference is given between the paired items under comparison. The accommodation of ties is proceeded in accordance with [24] proposition ensuring intact utility of [25] choice axiom allowing the occurrence of imperfect choices as follows

$$p_{o.ij}=\tau \sqrt{\left(p_{i.ij}\right)\left(p_{j.ij}\right)}$$

(3)

where $\tau >0$ is the constant of proportionality representing the tie parameter and independent of the $\left(i,j\right)$ pair, whereas preference probabilities, $p_{i.ij}$ and $p_{j.ij}$, are as given in Equation (2). Moreover, $p_{i.ij} \mathrm{and} p_{j.ij}\ne 0,1$ and $p_{o.ij}+p_{i.ij}+p_{j.ij}=1$. It is noteworthy that the proportionality functional in Equation (3) ensures that the probability of no preference is dependent upon the extent of distinguishability of pairs. Furthermore, the use of geometric formulation permits representation of the compared item on a linear scale when logarithmic function is applied.

Thus, by using tie adjusted formation supported by the liberty of imperfect choices and preference probability sum, it remains verifiable that the preference probability of $\left(i\to j\right)$ remains simplified as

$$p_{i.ij}=\frac{\pi {\left({\theta }_i^2+{\theta }_j^2\right)}^2+2\left\{\left({\theta }_i^3{\theta }_{j-}{\theta }_i{\theta }_j^3\right)-{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}}{\pi {\left({\theta }_i^2+{\theta }_j^2\right)}^2+\tau \sqrt{\left[\pi {\left({\theta }_i^2+{\theta }_j^2\right)}^2+2\left\{\left({\theta }_i^3{\theta }_{j-}{\theta }_i{\theta }_j^3\right)-{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}\right]\left[2\left\{\left({\theta }_i{\theta }_j^3-{\theta }_i^3{\theta }_j\right)+{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}\right]}}$$

(4)

Similarly, governed by the one dimensional Maxwell distribution, the preference probability of $\left(j\to i\right)$ is calculated as

$$p_{j.ij}=\frac{2\left\{\left({\theta }_i{\theta }_j^3-{\theta }_i^3{\theta }_j\right)+{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}}{\pi {\left({\theta }_i^2+{\theta }_j^2\right)}^2+\tau \sqrt{\left[\pi {\left({\theta }_i^2+{\theta }_j^2\right)}^2+2\left\{\left({\theta }_i^3{\theta }_{j-}{\theta }_i{\theta }_j^3\right)-{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}\right]\left[2\left\{\left({\theta }_i{\theta }_j^3-{\theta }_i^3{\theta }_j\right)+{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}\right]}}$$

(5)

Furthermore, the extent of indistinguishability is estimable such as

$$p_{0.ij}=\frac{\tau \sqrt{\left[\pi {\left({\theta }_i^2+{\theta }_j^2\right)}^2+2\left\{\left({\theta }_i^3{\theta }_{j-}{\theta }_i{\theta }_j^3\right)-{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}\right]\left[2\left\{\left({\theta }_i{\theta }_j^3-{\theta }_i^3{\theta }_j\right)+{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}\right]}}{\pi {\left({\theta }_i^2+{\theta }_j^2\right)}^2+\tau \sqrt{\left[\pi {\left({\theta }_i^2+{\theta }_j^2\right)}^2+2\left\{\left({\theta }_i^3{\theta }_{j-}{\theta }_i{\theta }_j^3\right)-{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}\right]\left[2\left\{\left({\theta }_i{\theta }_j^3-{\theta }_i^3{\theta }_j\right)+{\left({\theta }_i^2+{\theta }_j^2\right)}^{2}ta{n}^{0-1}\left(\frac{{\theta }_j}{{\theta }_i}\right)\right\}\right]}}$$

(6)

In the case of $m$ competing objects and $n$ selectors, let $\mathit{w}=\left(w_{0,ij},w_{i,ij},w_{j,ij}\right)$ represent the vector comprehending the observed preferences when object $i$ is competing with object $j$. Additionally, $n_{ij}$ denotes the total number of possible comparisons in $r$ replications by $n$ selectors, where $i\ne j;i\ge 1,j\le m$. Then the likelihood function of realized choice data generated through the complete factorial set up consists of w trials with the permission of ties, is written as

$$l\left(\mathit{w}, \mathit{\theta },\mathit{\tau }\right)={{\displaystyle \prod }}_{i<j=1}^m\frac{n_{ij}!}{w_{0.ij}!w_{i.ij}!( n_{ij}-w_{i.ij}-w_{0.ij})!}{p}_{0.ij}^{w_{0.ij}}{p}_{i.ij}^{w_{i.ij}} p_{j.ij}^{ n_{ij}-w_{i.ij}-w_{0.ij}},0<{\theta }_i<1 \mathit{and} 0<\tau <1.$$

(7)

Here, the issue of identifiability is resolved by ensuring ${{\displaystyle \sum }}_{i=1}^m{\theta }_i=1$, and ${\theta }_i$ is associated worth parameter attached with $i\prime th$ object dictating the degree of preference of the object, where $i=1, 2, \mathrm{\dots }, m$.

2.3. Incorporating Prior Information

In this era of next generation computing hardware, the utility of prior information for the execution of more sound and knowledgeable policy interventions has gain unprecedented momentum in scientific rigors. It is noteworthy that under the considered formation, two priors are required, one to explain stochastic behavior of worth parameters and second for the capsulation of tie parameter. For demonstration, we consider informative prior in the form of Dirichlet prior for the elaboration of worth parameters as follows

$$p_D\left(\mathit{\theta }\right)={{\displaystyle \prod }}_{i=1}^m\frac{\Gamma \left(d_1+d_2+\dots +d_m\right)}{\Gamma d_1\dots \Gamma d_m}{\theta }_i^{d_i-1},0<{\theta }_i<1$$

(8)

Similarly, Gamma prior is considered for the conceptualization of tie parameter as under

$$p_G\left(\tau \right)=\frac{a_2{}^{a_1}}{\Gamma a_1}{\tau }^{a_1-1}{e}^{-a_2\tau },0<\tau <1$$

(9)

Here, $d_i$, ${\mathrm{a}}_1$ and ${\mathrm{a}}_2$ are the hyperparameters of the prior structure. The motivation behind the use of Dirichlet prior remains intact under the notion of parsimony, as it employs fewer number of hyperparameters and thereby is thought to be providing more concise estimates. Furthermore, the Gamma prior remains attractive for tie parameter as both distributional spaces are bounded over $\left(0, 1\right)$ range, and thus offers natural support to the estimation efforts.

2.4. Posterior Distribution and Estimation of the Worth Parameters

By using the prior distributions given in Equations (8) and (9) along with likelihood function of Equation (7), the joint posterior distribution is deducted as below

$$p\left({\theta }_1,{\theta }_2,\dots ,{\theta }_m|\mathit{w}\right)=\frac{1}{k}{\displaystyle \prod }_{i<j=1}^m{\theta }_i^{d_i-1}{\tau }^{a_1-1}{e}^{-a_2\tau }{p}_{0.ij}^{w_{0.ij}}{p}_{i.ij}^{w_{i.ij}}{p}_{j.ij}^{ n_{ij}-w_{i.ij}-w_{0.ij}},$$

(10)

where $k={{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^{1-{\theta }_1}\dots {{\displaystyle \int }}_{{\theta }_m=0}^{1-{{\displaystyle \sum }}_{i=1}^{m-1}{\theta }_i}{{\displaystyle \prod }}_{i<j=1}^m{\theta }_i^{d_i-1}{\tau }^{a_1-1}{e}^{-a_2\tau }{p}_{0.ij}^{w_{0.ij}}{p}_{i.ij}^{w_{i.ij}}{p}_{j.ij}^{ n_{ij}-w_{i.ij}-w_{0.ij}}d{\theta }_{m-1}\dots d{\theta }_{2}d{\theta }_1$ and represents the normalizing constant. The deduction of marginal posterior distributions requires the resolve of complex integrations involved in the expression of joint posterior distribution. The objective is attained by the launch of Gibbs sampling—popular approach of Markov chain Monte Carlo methods [26]. In general, Gibbs sampling proceeds by assuming $P\left(\underset{\_}{\theta };\underset{\_}{x}\right)$ be the joint posterior density, where $\underset{\_}{\theta }=\left({\theta }_1,{\theta }_2,\dots ,{\theta }_m\right)$. The conditional densities of worth parameters are then given by $P\left({\theta }_1\mathrm{I}{\theta }_2,{\theta }_3\dots ,{\theta }_m\right),P\left({\theta }_2\mathrm{I}{\theta }_1,{\theta }_3\dots ,{\theta }_m\right)\dots P\left({\theta }_m\mathrm{I}{\theta }_1,{\theta }_2\dots ,{\theta }_{m-1}\right)$. The Gibbs sampler now initiates by assuming initial values upon worth parameters such as $\left({\theta }_2{}^{\left(0\right)},{\theta }_3{}^{\left(0\right)},\dots ,{\theta }_m{}^{\left(0\right)}\right)$ and conceptualizes the conditional distribution of ${\theta }_1$ such that $P\left({\theta }_1{}^{\left(1\right)}\mathrm{I}{\theta }_2{}^{\left(0\right)},{\theta }_3{}^{\left(0\right)},\dots ,{\theta }_m{}^{\left(0\right)}\right)$. The iterative procedure continues until the convergence occurs. For demonstration purposes, the expression for the marginal posterior distribution of worth parameter associated with $m\prime th$ object, that is ${\theta }_m$, is solved as follows

$$\begin{gathered} p\left({\theta }_m|\mathit{w}\right)=\frac{1}{k}{{\displaystyle \int }}_{{\theta }_1=0}^{1-{\theta }_m}\dots {{\displaystyle \int }}_{{\theta }_m=0}^{1-{{\displaystyle \sum }}_{i=1}^{m-2}{\theta }_i-{\theta }_m}{{\displaystyle \prod }}_{i<j=1}^m{\theta }_i^{d_i-1}{\tau }^{a_1-1}{e}^{-a_2\tau }{p}_{0.ij}^{w_{0.ij}}{p}_{i.ij}^{w_{i.ij}}{p}_{j.ij}^{ n_{ij}-w_{i.ij}-w_{0.ij}}d{\theta }_{m-1}\dots d{\theta }_1, \\ 0<{\theta }_m<1 \end{gathered}$$

(11)

The $\left(1-\alpha \right)100\%$ credible intervals attached with ${\theta }_m$, say $C_r$, are obtained numerically by solving the given expression

$${\displaystyle \int } p\left({\theta }_k:\mathit{x},w,\tau \right)d{\theta }_m=1-\alpha ,$$

(12)

which is a subset of ${\theta }_m\prime s$ parametric space. Figure 1 below presents the flow diagram summarizing the working of the proposed scheme along with algorithmic advancements.

2.5. Bayes Hypothesis Testing

Now, we proceed towards the evaluation of statistical significance of the underlying comparative hierarchy. In Bayesian framework, the task is accomplished by quantifying the posterior probabilities and resultant Bayes factors associated with concerned hypothesis. The complementary hypothesis streaming pair of objects is given as

$$H_{ij}:{\theta }_i\ge {\theta }_j .H_{ji}:{\theta }_j<{\theta }_i.$$

The posterior probabilities deciding upon the existent discrepancies remain calculable as

$${\varphi }_{ij}={{\displaystyle \int }}_{\zeta =0}^1{{\displaystyle \int }}_{\eta =\zeta }^{\left(1+\zeta \right)/2}P\left(\zeta ,\eta {\rm I}\omega \right)d\eta d\zeta ,$$

(13)

where $\eta ={\theta }_i$, $\zeta ={\theta }_i-{\theta }_j$. It is trivial to show that ${\varphi }_{ji}=1-{\varphi }_{ij}$. The Bayes factor now remains estimable with straightforward operation such that $BF={\varphi }_{ij}/{\varphi }_{ji}$. Generally accepted criterion nominating the degree of significance while employing Bayes factor is given as

• $BF\ge 1$, support $H_{ij}$
• ${10}^{-0.5}\le \mathrm{BF}\le 1$, minimal evidence against $H_{ij}$
• ${10}^{-1}\le BF\le {10}^{-0.5}$, substantial evidence against $H_{ij}$
• ${10}^{-2}\le BF\le {10}^{-1}$, strong evidence against $H_{ij}$
• $BF\le {10}^{-2}$, decisive evidence against $H_{ij}$.

2.6. Limitations of the Proposed Scheme

It is important to note the limitations of the proposed mechanism at this stage. Our newly developed model is demonstrated to be workable for the adjustment of the response of “no choice or tie”. However, the existence of ties in comparative data is not the only challenge. The choice data may suffer from two other sources of contamination that are: (i) the order of presenting the objects, most commonly known as order effect, and (ii) the tendency to report socially desirable responses while hiding the true status of the matter. It is noteworthy that the origin of these complications is different from the documented response of no choice. The indecisiveness arises from either indistinguishable nature of the competing objects or from the judge(s) lacking the capability to distinguish the objects, whereas desirability bias generates due to the fondness of the respondent(s) towards being socially acceptable or approved. On the other hand, order effect indicates the lack of consistency of the judge(s). Keeping the inherent differences in mind, it is anticipated that new post hoc strategy capable of entertaining aforementioned complexities, one by one or simultaneously, is desirable in future. It can be reported that the devised scheme is capable of catering ties in its present formation. However, the treatment of aforementioned challenges is an attractive future research scope.

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, $n=25, 50 \mathrm{and} 100$, for two values of tie parameters, that is, $\tau =0.1 \mathrm{and} 0.2$. 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. Artificial data under the preset experimental states.

$\mathit{n}$	${\mathit{w}}_{1.12}$	${\mathit{w}}_{2.12}$	${\mathit{w}}_{0.12}$	${\mathit{w}}_{1.13}$	${\mathit{w}}_{3.13}$	${\mathit{w}}_{0.13}$	${\mathit{w}}_{2.23}$	${\mathit{w}}_{3.23}$	${\mathit{w}}_{0.23}$
$\tau =0.10$
25	8	16	1	6	18	1	5	19	1
50	11	37	2	9	39	2	16	33	1
100	27	72	1	19	73	8	44	51	5
$\tau =0.20$
25	5	18	2	6	17	2	10	12	3
50	15	27	8	12	34	4	19	26	5
100	19	72	9	24	73	3	37	52	11

presents the artificially generated data sets resulting from the aforementioned parametric settings.

Table 3. Estimates of worth parameters and associated 95% credible intervals (in parenthesis) for pre-defined experimental settings.

$\mathit{n}$	${\widehat{\mathit{\theta }}}_1$	${\widehat{\mathit{\theta }}}_2$	${\widehat{\mathit{\theta }}}_3$	$\widehat{\mathit{\tau }}$
$\tau =0.10$
25	0.2549 (0.2448, 0.2651)	0.3014 (0.2905, 0.3122)	0.4437 (0.4301, 0.4571)	0.0865 (0.0728, 0.1002)
50	0.2414 (0.2330, 0.2498)	0.3403 (0.3389, 0.3415)	0.4183 (0.4091, 0.4275)	0.0911 (0.0806, 0.1016)
100	0.2491 (0.2337, 0.2494)	0.3516 (0.3508, 0.3603)	0.3992 (0.3964, 0.4022)	0.0986 (0.0933, 0.1039)
$\tau =0.20$
25	0.2689 (0.2640, 0.2738)	0.3739 (0.3619, 0.3860)	0.3832 (0.3610, 0.3954)	0.1881 (0.1715, 0.2048)
50	0.2428 (0.2327, 0.2528)	0.3510 (0.3489, 0.3647)	0.3958 (0.3896, 0.4275)	0.2128 (0.1956, 0.2199)
100	0.2421 (0.2390, 0.2454)	0.3594 (0.3556, 0.3624)	0.3969 (0.3939, 0.4098)	0.1926 (0.1894, 0.2086)

,

Table 4. Posterior probabilities and resultant Bayes factors associated with competing hypothesis.

$\mathit{n}$	Posterior Probabilities			Bayes Factor
	${\mathit{H}}_{12}$	${\mathit{H}}_{13}$	${\mathit{H}}_{23}$	${\mathit{B}}_{12}$	${\mathit{B}}_{13}$	${\mathit{B}}_{23}$
$\tau =0.10$
25	0.1085	0.0001	0.0026	0.1217	0.0001	0.0026
50	0.0377	0.0002	0.0234	0.0392	0.0002	0.0240
100	0.0126	0.0001	0.0527	0.0128	0.0001	0.0557
$\tau =0.20$
25	0.0006	0.0004	0.3549	0.0006	0.0004	0.5502
50	0.0039	0.0001	0.0414	0.0039	0.0001	0.0432
100	0.0005	0.0001	0.1876	0.0005	0.0001	0.2309

and

Table 5. Posterior preference probabilities of pairwise competing objects when ties are permitted.

$\mathit{n}$	${\mathit{p}}_{1.12}$	${\mathit{p}}_{2.12}$	${\mathit{p}}_{0.12}$	${\mathit{p}}_{1.13}$	${\mathit{p}}_{3.13}$	${\mathit{p}}_{0.13}$	${\mathit{p}}_{2.23}$	${\mathit{p}}_{3.23}$	${\mathit{p}}_{0.23}$
$\tau =0.10$
25	0.3789	0.5804	0.0405	0.1871	0.7797	0.0330	0.2607	0.7021	0.0370
50	0.2819	0.6782	0.0398	0.1885	0.7766	0.0348	0.3555	0.6022	0.0421
100	0.2802	0.6768	0.0429	0.2201	0.7399	0.0398	0.4001	0.5534	0.0464
$\tau =0.20$
25	0.2295	0.6953	0.0751	0.2189	0.7069	0.0740	0.4427	0.4712	0.0859
50	0.3229	0.5846	0.0924	0.2523	0.6607	0.0869	0.3708	0.5344	0.0947
100	0.2757	0.6431	0.0811	0.2197	0.7044	0.0757	0.3857	0.5273	0.0868

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.

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, ${\theta }_3\to {\theta }_2\to {\theta }_1$, 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. Moreover, the decreased width of the credible interval shows the precision of the procedure with which it remains capable of estimating the preset experimental states. This realization is further highlighted in Figure 2 depicting the prevalent variability in the Bayes estimates through side-by-side box plots. One may notice that minimal variation is attributed with a larger sample size.

The significance of the utility differences of the competing objects are demonstrated by quantifying the posterior probabilities and related Bayes factors for the complementary hypothesis. The results are summarized in Table 4. The establishment of a predefined preference ranking and its associated significance is observable from the calculated posterior probabilities and Bayes factors. Regardless of the varying sample sizes and the extent of indecisiveness, we witnessed maintained ordering such as, ${\theta }_3\to {\theta }_2\to {\theta }_1$; however, with different extents of associated significance. In general, we estimate that for $\tau =0.10$, substantial evidence exists indicating that ${\theta }_2\to {\theta }_1$, and decisive evidence is observed highlighting ${\theta }_3\to {\theta }_1$, whereas strong evidence establishes the significance of ${\theta }_3\to {\theta }_2$. This observation is realized for all considered sample sizes. Furthermore, in the case of $\tau =0.20$, strong evidence is attached with ${\theta }_2\to {\theta }_1$ and ${\theta }_3\to {\theta }_1$ along with strong indications of the instance of ${\theta }_3\to {\theta }_2$.

The estimated posterior probabilities of the preferences while pairwise comparing all three objects are assembled in Table 5. The outcomes seal the consistent behavior of the proposed mechanism.

4. Application—Preference of Drinking Water Brands

We now proceed by demonstrating the applicability of the devised model by studying the choice data of three drinking water brands commonly available in market. The pairwise comparative data with permitted ties were collected from fifty local residents of Islamabad, Pakistan, by inquiring about their preferred brand among (i)—Aquafina (AQ), (ii)—Nestle (NL) and (iii)—Kinley (KN). One may notice that in this situation $n=50$ and $m=3$.

Table 6. Preference counts pairwise choices of the drinking water brands.

$\mathbf{Pairs} \left(\mathit{i},\mathit{j}\right)$	${\mathit{w}}_{\mathit{i}.\mathit{i}\mathit{j}}$	${\mathit{w}}_{\mathit{j}.\mathit{i}\mathit{j}}$	${\mathit{w}}_{0.\mathit{i}\mathit{j}}$
$\left(\mathrm{AQ}, \mathrm{NT}\right)$	17	27	6
$\left(\mathrm{AQ}, \mathrm{KL}\right)$	26	19	5
$\left(\mathrm{NT}, \mathrm{KL}\right)$	28	18	4

displays the collected data, whereas Figure 3 depicts the distribution of counts in the relevant predefined classifications.

We start the exploration by first eliciting the hyperparameters using confidence levels by the use of the joint posterior distribution of Equation (10) and defining the prior predictive distribution as under

$$p\left(w_{i.ij}, w_{0.ij}\right)={{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^{\infty }{{\displaystyle \prod }}_{i<j=1}^m{Q}_{ij}{\theta }_i^{d_i-1}{\left(1-{\theta }_i\right)}^{d_j-1}{\tau }^{a_1-1}{e}^{-a_2\tau }{p}_{0.ij}^{w_{0.ij}}{p}_{i.ij}^{w_{i.ij}} p_{j.ij}^{ n_{ij}-w_{i.ij}-w_{0.ij}}d\tau d{\theta }_i$$

Here, $Q_{ij}=\frac{n_{ij}!}{w_{0.ij}!w_{i.ij}!( n_{ij}-w_{i.ij}-w_{0.ij})!}\frac{\Gamma \left(d_i+d_j\right)}{\Gamma d_i\Gamma d_j}\frac{a_2{}^{a_1}}{\Gamma a_1}$. Ref [27] proposed the elicitation of hyperparameters through the function as follows

$$\Psi \left(\mathit{c}\right)=argmi{n}_c{{\displaystyle \sum }}_{i=1}^k\left|{\left(CCL\right)}_i-{\left(ECL\right)}_i\right|,$$

where $\mathit{c}$ is the set of elicited hyperparameters, whereas $k$ represents the number interval required to elicit the hyperparameters. Additionally, ${\left(CCL\right)}_i$ and ${\left(ECL\right)}_i$ are the confidence level and elicited confidence level, respectively, characterized with the specific hyperparameter. By exploiting the joint posterior distribution, the confidence levels are given as

$${{\displaystyle \sum }}_{w_{1.12}=0}^2{{\displaystyle \sum }}_{w_{0.12}=0}^{1}p\left(w_{1.12}, w_{0.12}\right)=0.05,{{\displaystyle \sum }}_{w_{1.12}=3}^5{{\displaystyle \sum }}_{w_{0.12}=0}^{1}p\left(w_{1.12}, w_{0.12}\right)=0.07,$$

$${{\displaystyle \sum }}_{w_{1.13}=0}^2{{\displaystyle \sum }}_{w_{0.13}=0}^{1}p\left(w_{1.13}, w_{0.13}\right)=0.05,{{\displaystyle \sum }}_{w_{1.13}=3}^5{{\displaystyle \sum }}_{w_{0.13}=0}^{1}p\left(w_{1.13}, w_{0.13}\right)=0.07,$$

$${{\displaystyle \sum }}_{w_{2.23}=0}^2{{\displaystyle \sum }}_{w_{0.23}=0}^{1}p\left(w_{2.23}, w_{0.23}\right)=0.05,{{\displaystyle \sum }}_{w_{2.23}=3}^5{{\displaystyle \sum }}_{w_{0.23}=0}^{1}p\left(w_{2.23}, w_{0.23}\right)=0.07.$$

The elicited values of the hyperparameters for both priors, that is, Dirichlet prior for worth parameters and Gamma prior for the tie parameter, are $d_1=2.5012$, $d_2=2.6595$, $d_3=2.7001$, $a_1=2.1086$ and $a_1=5.4596$.

Table 7. Summaries of the analysis of the drinking water choice data.

Estimates			Bayes Factor
${\widehat{\mathit{\theta }}}_{\mathit{i}}$		${\widehat{\mathit{\tau }}}_{\mathit{i}\mathit{j}}$		${\mathit{B}}_{\mathit{i}\mathit{j}}$
${\widehat{\theta }}_{NL}$	0.37153	${\widehat{\tau }}_{NL,AQ}$	0.12504	$B_{NL,AQ}$	16.181
${\widehat{\theta }}_{AQ}$	0.32643	${\widehat{\tau }}_{NL,KL}$	0.13252	$B_{NL,KL}$	56.306
${\widehat{\theta }}_{KL}$	0.30204	${\widehat{\tau }}_{AQ,KL}$	0.12878	$B_{AQ,KL}$	3.062

presents the Bayes estimates of the worth parameters dictating the choice hierarchy of the comparative data gathered from the drinking water experiment along with the associated evidence of significance. The proposed scheme establishes the choice ranking as such: ${\widehat{\theta }}_{NL}\to {\widehat{\theta }}_{AQ}\to {\widehat{\theta }}_{KL}$, indicating that most of the individuals selected in the sample preferred the Nestle brand, followed by Aquafina, whereas Kinley was the least preferred. This hierarchy can also be anticipated from the observed data. Moreover, the preference ordering is observed to be statistically significant with strong evidence associated with the instances of ${\widehat{\theta }}_{NL}\to {\widehat{\theta }}_{AQ}$ and ${\widehat{\theta }}_{NL}\to {\widehat{\theta }}_{KL}$, whereas we witnessed substantial evidence attached with ${\widehat{\theta }}_{AQ}\to {\widehat{\theta }}_{KL}$. These realizations are further supported by the posterior probability estimates of the comparative preferences, given in

Table 8. Posterior preference probabilities of comparative choices.

Preference Probabilities
$p_{NL.NL,AQ}$	0.54696	$p_{NL.NL,KL}$	0.59222	$p_{AQ.AQ,KL}$	0.51621
$p_{AQ.NL,AQ}$	0.39331	$p_{KL.NL,KL}$	0.34921	$p_{KL.AQ,KL}$	0.42357
$p_{0.NL,AQ}$	0.05973	$p_{0.NL,KL}$	0.05856	$p_{0.AQ,KL}$	0.06022

.

5. Discussion and Conclusions

The realization and confrontation of choices is unavoidable in every aspect of daily life. Thereby, the methods facilitating the fundamentals of choice dynamics have a long and well cherished history in the multidisciplinary research literature. It has been competently argued in the existing literature that the foundations of rational decision making stand on the essential elements of (i) utility: the latent factor derived by the choice axiom [28] and (ii) consistency: the extent of judgment following the axiom using inferential soundness [29]. Therefore, the search for such methods capable of entertaining both fundamentals simultaneously, has attracted noticeable attention in research circles [30]. However, the issue of handling the indecisiveness of selector(s) in reporting choice data remains long standing. There is undoubted consensus over the misleading nature of indecisive responses, but, unfortunately, the available literature lacks in its ability to demonstrate feasible solutions, especially at the methodological level. Motivated by the aforementioned factors, this article proposes a new choice model in conjunction with the Bayesian paradigm when the judge or selectors have the opportunity of reporting a “no preference” response. The proposed scheme is fundamentally advantageous in reducing the forced response bias. It is anticipated that by permitting the occurrence of ties while reporting the preferences, the selector(s) are offered extended flexibility to be able to report their true status. Moreover, by devising a workable approach, the scheme enables the investigator to estimate the influence of ties in the reported data through methodological sound pathways. The applicability of the suggested approach is affirmed on multiple fronts, such as through mathematically derived expressions, and by the launch of rigorous simulations and being demonstrated empirically. The outcomes of the research substantiate the legitimacy of the proposed mechanism, especially on four frontiers. Firstly, it is witnessed that the newly suggested model delicately maintains the inherent ordered structure of the observed choice data. This is observed with respect to all considered sample sizes and the varying extent of choice parameters. Secondly, the proposed scheme enables us to estimate the degree of indecisiveness prevalent in the comparative information. Furthermore, in concordance with asymptotic theory, the estimated subtitles become more obvious with an increased sample size. Lastly, the suggested model assists the rational decision making notions by providing sound inferential aspects facilitated through the Bayesian framework.

At this stage, it is obligatory to report the limitations of the newly developed model that offers attractive research pursuits for the future. Along with ties, the choice data show numerous concerns of practical significance such as the order of presentation of the competing objects (order effect) and socially-motivated preferences (desirability bias). If not treated appropriately, the aforementioned contaminations pose serious threats to the validity of the modeling strategies by producing misleading results. It is noteworthy that this research encapsulates the issue of ties and is not capable of entertaining other documented complexities. In future, it will be interesting to further elaborate the proposed procedure for the accommodation of order effects and desirability bias.