The ordered weighted government transparency average: Colombia case

1 Introduction

In decision-making problems, it is noteworthy to analyze attitude based on analogous information analysis and the uncertainty of human behavior [1]. An issue that deserves to be studied within decision problems is the way in which the government generates and presents information to their citizens. This interchange of information between the government and its citizens is called transparency; this term is part of a multidisciplinary concept in which the insertion of political sciences, law, public administration, sociology and, in general, several social sciences is made [2].

Transparency generates in public institutions an understanding of activities and their responsibility and guarantees, through access to information, government supervision and commitment. Furthermore, it is noteworthy that the cultural and attitudinal aspects of citizens are relevant to carrying out an effective transparency process. Indeed, considering the contexts since it is possible to observe the predisposition that citizens have about the actions of the government and creating a perception about them are of the utmost importance.

In Latin America, transparency actions have been included in public policy [49]. This policy agenda involves aspects such as access to information, conflict regulation and interest management and the creation of control agencies. In Colombia, since 2002, the measurement of the transparency index of public entities, known by the acronym ITEP, consists of a civil society initiative that contributes to the prevention of corruption. ITEP allows the establishment of corruption risk levels using existing information on institutional practices and conditions. These levels are defined as the probability of occurrence of corruption activities by government actors in public entities. These processes are associated with the decision making of public authorities and based on the application of norms that tend to improve the quality of life of people [3, 4].

Since the evaluation of the index is based on the use of information from the governments of the day, it is likely to be modified according to the most convenient criteria to show better results. These actions are subject to the degree of importance, management and weight factor given to the information by the decision maker. In this sense, the use of new tools that allow capturing these aspects can help to produce a scenario about the perceptions of transparency levels.

In the literature, there are already proposed methods that allow these types of information to be captured, such as aggregation operators. Aggregation operators allow the gathering of different types of information to reach a conclusion without omitting information during the process [5]. Thus, aggregation operators allow information fusion by combining several values into a single value [5]. Within this research field, several aggregation methods have been proposed. One of them is the ordered weighted average (OWA) operator proposed by [6]. Thus, according to [6], an OWA operator is characterized by a coefficient that is not directly associated with a particular attribute but rather directly associated with an order position.

Using this operator, several propositions have been developed using linguistic information [7, 8], intuitionistic [9], probabilistic [10], distance [11, 12], Choquet integral [13], Bonferroni mean [14], uncertain [15], induced [16], geometric mean [17], power [18], heavy [19], and dependent [20] operators, from which new extensions have been proposed. These operators enable the aggregation of information guided by quantifiers [5] to solve decision-making problems, where the attitudes and criteria of persons have a great influence on the final result.

Based on the above, the main aim of this paper is to propose a new aggregation operator to improve the evaluation of the transparency index. This new proposition is called the prioritized induced ordered weighted average weighted average (PIOWAWA) operator. The main characteristic of this new operator is that it allows considering the degree of importance, reordering and use of two different weighting vectors, one using the weights assigned by the government and another using the information provided by experts related to which elements are most important. The benefit of using this new operator is that it can bring together the idea of transparency of the government and experts in the same formulation to generate a better view and understanding of the situation of a country/state and what changes need to be made.

A mathematical application of this new method is performed for a Colombia transparency case. The findings highlight that according to the operator used, there are significant changes in the ranking. This is because the level of importance that an expert can give to different variables may be greater even if the initial weight of the variables is lower. The main implications are given by using these aggregation operators for the generation of scenarios by considering the changes in the allocation of weights, the level of importance and the ordering of information simultaneously.

This paper is structured as follows: Section 2 presents the main concepts of transparency. Section 3 describes the main concepts of the OWA operator and its extensions. Section 4 presents a new extension of the OWA operator, which is called the prioritized induced ordered weighted average weighted average (PIOWAWA) operator. Section 5 illustrates a mathematical application of a Colombian transparency case through aggregation operators. Section 6 presents the main conclusions.

2 Government transparency

Governments generate information for communication with the population. This allows the functioning of democratic societies [50]. Thus, the interchange of information between the government and its citizens is called transparency. This information is either made public or kept at discretion. This information can comply with government functions; be used as propaganda; and be published as documents, spreadsheets or databases. In this way, the information provides enough variables to the public for attention, interest and skills [21].

Therefore, the value of transparency is framed in the integrity and effectiveness of the government. However, information asymmetry becomes an obstacle that diminishes the abilities of directors to allow monitoring [22]. Thus, the importance and repercussions of government transparency as a mechanism that helps to provide information related to policies, capacities and, in general, data on a democratic society are presented below.

3 Methodology

An interesting extension of the OWA operator can be performed if the main characteristics of definitions 2-4 are combined in a single operator. This new operator is called the prioritized induced ordered weighted average weighted average (PIOWAWA) operator. The idea of this operator is that in group decision making, not all experts have the same degree of importance in the results (POWA operator), the reordering step is not performed according to the maximum or minimum criteria but instead through induced values (IOWA operator), and two different weighting vectors are used (OWAWA operator). The definition is as follows.

Definition 6. A PIOWAWA operator of dimension n is a mapping PIOWAWA : R ⁿ xR ⁿ  → R that has an associated weight vector w of dimension n, where w _j  ∈ [0, 1] and ∑ j = 1 n w j = 1 , so that:

PIOWAWA ( 〈 u 1 , b 1 〉 , 〈 u 2 , b 2 〉 , … , 〈 u n , b n 〉 ) = ∑ i = 1 q ∑ h = 1 n i b j e ˆ ij C ij ( x ) ,(5) where b _j is the jth element that has the largest value of u _i ; u _i is the induced order of variables; e ˆ ij is the corresponding weight of the jth criteria in the ith category, where i = 1, …, q, j = 1, …, i _i ; each argument b _i has an associated weight (WA) d _i with ∑ i = 1 n d i = 1 where d i ∈ [ 0 , 1 ] , d j ˆ = β w j + ( 1 - β ) d j , where β ∈ [0, 1] and d _j is ordered according to b _j ; and C _ij  (x) measures the satisfaction of the jth criteria in the ith group by alternativex ∈ X, for each i = 1, …, q, j = 1, …, i _i .

The properties of the PIOWAWA operator are as follows.

a) Commutativity: Assume f is the PIOWAWA operator; f (〈u _i , b _i 〉, …, 〈u _n , b _n 〉) = f (〈u _i , c _i 〉, …, 〈u _n , c _n 〉).

b) Monotonicity: Assume f is the PIOWAWA operator; if |u _i , b _i | ≥ |u _i , c _i | for all i _i , then f (〈u _i , linebreak b _i 〉, …, 〈u _n , b _n 〉) ≥ f (〈u _i , c _i 〉, …, 〈u _n , c _n 〉).

c) Bounded: Assume f is the PIOWAWA operator; then, min{ b _i  } ≤ f (〈u _i , b _i 〉, …, 〈u _n , b _n 〉) ≤ max { b _i  }.

d) Idempotency: Assume f is the PIOWAWA operator; if |u _i , b _i | = b for all i, then f (〈u _i , b _i 〉, linebreak  … , 〈u _n , b _n 〉) = b.

It is important to note that the PIOWAWA operator can be reduced to another simpler operator according to the complexity of the problem to be analyzed, and some examples are as follows:

a) if u _i  = 1/n, then the operator becomes the prioritized ordered weighted average weighted average (POWAWA) operator.

b) if in d j ˆ = β w j + ( 1 - β ) d j , β = 1, then the operator becomes the prioritized induced ordered weighted average (PIOWA) operator.

c) if in d j ˆ = β w j + ( 1 - β ) d j , β = 0, then the operator becomes the prioritized induced weighted average (PIWA) operator.

Because of the importance of government transparency, the use of the PIOWAWA operator and other OWA operators to identify different scenarios based on the reordering of the weights that incorporate the traditional formulation used in Colombia is proposed. With the use of these operators, it will be possible to identify how much different changes can increase or decrease the transparency score and to propose changes in public policies. The steps that are needed to use the operators in government transparency are as follows.

Step 1. The information to be evaluated is needed. In this paper, the Colombian governorship transparency ranking for 2015-2016 was used (see Annex 1).

Step 2. The actual weights that are used to calculate the final score (see Table 1).

Step 3. The next step is to determine another weighting vector that will be used. To do this, three different experts on transparency in Colombia were asked to provide their opinions about the weights related to each subfactor (see Table 2). In this case, the experts are people from the actual government who have more than 10 years in positions related to governmental transparency and that have helped in the generation of new policies and rules in governmental transparency.

Another important factor that needs to be determined is the importance of each weighting vector. In this case, the weighting vector from Step 2 will have a value of 40% and the weights provided by the experts will have a value of 60%.

Step 4. In this step, the information about the induced values has to be determined by each expert (see Table 3).

Step 5. In this step, the weights that will be assigned to each expert are determined. In this case, the weights were assigned based on the total years of experience of the expert divided by the total years of experience of all experts. The results give the following percentages: e₁ = 0.45, e₂ = 0.35, e₃ = 0.20.

Step 6. With all the information, we proceed to the use of different aggregation operators. In this specific case, the operators that will be used are the OWAWA and IOWAWA operators. The results are presented in Table 4.

Step 7. With the results of Table 4, the unification of the results based on the individual information of each expert will be done based on the weights provided in Step 5. The results are presented in Table 5.

Step 8. In this step, the analysis of the results will be performed. First, it is important to note that the changes in the Top 10 states based on the different aggregation operators are presented in Table 6.

With the information provided in Table 6, it is possible to see that Antoquia is always the most transparent state in Colombia, but we also found some interesting changes. For example, in Cundinamarca, 6 of the operators are ranked 6th, one is ranked 4th and one is ranked 2nd. The main reason for these changes is that in the IOWAWA operator, expert 2 gives more importance to the disclosure of procedures and citizen services, anti-corruption policies, measures and strategies and systems of attention to citizens, which are subfactors whose initial weights are less than 6%. In this sense, it is possible to visualize how much a score can change if a subfactor has less or more importance than that in the scores provided by the government.

This variation is important to consider because sometimes states can develop strategies to increase their scores, such as considering only the subfactors with higher weights, and this will make them climb higher at the expense of subfactors that sometimes are less important according to the weights.

This is why it is important to use different aggregation operators to observe the reality that is happening in a country because sometimes based only on a single formulation, not all scenarios can be seen, and valuable information can be omitted. For example, if we calculate the variance of the results based on the initial score and the different aggregation operators, it can be observed that the average of the variation is 10.22 (see Annex 1). For example, Atlántico ranges from 58.79 to 68.15, Boyaca ranges from 61.97 to 71.62, and Quindio ranges from 65.71 to 74.62.

With this analysis, it can be affirmed that the same information can produce multiple scenarios based on the aggregation that is added or changes in the order that the arguments and weights are assigned. This is why it is important for the government, enterprises and, in general, decision makers to use aggregation operators and visualize different scenarios, which will help them to have a better understanding of problems to be solved and make better and more accurate decisions. The main idea of presenting different rankings and results is that the policy-making process is conducted with the information that is available; however, if the results are not as high or as low as the initial results indicate, the actions that the government will take can vary or change and improve the benefits for the governed. This is where the use of different aggregation operators becomes indispensable to visualize these situations.

The objective of this paper is to present a new extension of the OWA operator that can be applied to government transparency ranking. This new operator is called the prioritized induced ordered weighted average weighted average (PIOWAWA) operator.

This new operator combines the IOWA, POWA and OWAWA characteristics in the same formulation. This operator aids in solving problems in group decision making. When not all experts determine the same degree of importance in the results, the reordering step is not performed according to the maximum or minimum criteria but instead through induced values, and two different weighting vectors are used.

The application of this operator is carried out in the creation of scenarios on the levels of transparency in Colombia. Thus, identifying how much different changes can increase or decrease the transparency score is presented to propose changes in public policies. The findings highlight that according to the operator used, there are significant changes in the ranking. For example, Antoquia is always the most transparent state in Colombia, and Cundinamarca in 6 of the operators is ranked 6th, in one is ranked 4th and in one is ranked 2nd. Thus, it is shown how much a score can change if a subfactor has less or more importance than that in the scores provided by the government. This variation deserves attention since a strategy can be carried out to increase the index when considering subfactors with higher weights.

The main implications are given by using these aggregation operators for the generation of scenarios by considering the changes in the allocation of weights, the level of importance and the ordering of information simultaneously. In addition, this approach allows capturing the attitudes of experts when considering their opinions in a more dynamic and complex operator. Likewise, it is shown that aggregation operators, such as the one proposed, are useful for visualizing different scenarios for a better understanding of problems to be more resilient and precise in decision making.

In future investigations, this operator can be extended to other techniques such as the Bonferroni mean, Choquet integral, Muirhead mean, Maclaurin symmetric mean and logarithmic operators [51, 52]. Likewise, it can be applied to other cases of decision-making problems, such as personnel selection and investment.