A multi-criteria approach for assigning weights in voting systems

Following a new analytical orientation, this paper proposes an approach for investigating group versus individual trade-offs in general voting situations. From specific particularizations of a general family of p-norms, three social decision rules have been derived. These rules trigger three different systems of weights in the voting system. Once the three rules were obtained and the rationale underlying them justified, a compromise framework integrating all the rules was formulated. The compromise framework is computationally based on goal programming. In this way, it is possible to establish directly trade-offs and balance compromise solutions between the three social rules. Thus, not only is the weighted voting system formulated within a theoretically sound framework, but the applicability of the approach to real political situations is also more flexible and pragmatic.


Introduction
In many voting systems, decision makers (DMs) are not treated equally.This is the case of collective decision-making problems involving stakeholders, where, if one of them has more shares of stock, then her/his vote is weighted more heavily (e.g., the case of the European Union institutions, in which larger member countries obtain more votes).
A weighted voting system is a voting procedure, whereby not all DMs are of the same importance in the final solution.These differences between DMs can be implemented by using different importance weights in a mathematical model.Meanwhile, a weighted voting system extends the idea of "one voter-one vote" to a more general scenario described as "one voter-w votes" where w defines the importance weight of a DM.
In a weighted voting system, the DM's weight can be identified with her/his power in the decision process; i.e., their voting power or power index.In this paper, we shall consider this identification.However, it should be noted that a DM's weight is not always an accurate depiction of that power, and sometimes, a DM with several votes may exert little power.
On the other hand, the analysis of the voting power through the establishment of power indices between competitive organizations is an attractive theoretical problem full of potential applications.In fact, the analysis of how to allocate power between countries, committees, professional societies, etc., is a crucial and complex decision-making problem arising daily in democratic societies.To undertake this task, the basic idea is to attach to each organization a power or voting index.Two classic power indices were proposed by Penrose (1946), Banzhaf (1965) and Shapley and Shubik (1954).Since the publication of these pioneer works, there has been relatively profuse literature dealing in one way or in another with this problem.Without being exhaustive, besides these pioneer power indices, we have the Johnston index (1978) and the Holler-Packel index (1983), among others.Comprehensive reviews of this topic with a general orientation are those of Felsenthal and Machover (1998) as well as Turnovec (1996) for the special case of applying these indices to the decision-making processes in the European Union institutions.
In this work, we were interested on how to assign a system of weights in a weighted voting system, in order to quantify group versus individual trade-offs in voting situations.
A new angle for dealing with this type of issue has been explored.Thus, three social decision rules leading to three different systems of weights have been identified.These rules were derived from the particularization of a general family of topological p-norms.Once the three social decision rules are obtained and interpreted in preferential terms, a theoretical framework is formulated.In this way, not only are the three social rules unified, but it is also possible in a straightforward manner to establish trade-offs and compromises between them, which make the analysis theoretically sound as well as more flexible and pragmatic when considering potential applications.
The paper is organized as follows: After these introductory ideas, in Sect. 2 the theoretical framework is introduced, it being demonstrated that, with a light computational burden, it is possible to obtain "compromise importance weights" with a clear preferential meaning.Section 3, with the help of a simple numerical example, illustrates how the proposed approach works.Finally, Sect. 4 raises the main conclusions derived from this research and points out possible further research.

Three social rules
Let us consider the following general setting.There is a general voting scenario in which a known number of social groups are involved (i = 1, 2, …, m).The size of each social group is also known and determined by the number of members (n 1 , n 2 , …, n m ).It is also assumed that the sizes of the groups surrogate the social influence or importance of each one in the corresponding voting process.Now the following basic question is posed: What voting power, measured by a power index, should be associated with each social group considered?Obviously, the answer to this question depends upon the social rule used.Some basic social rules underpinned with good theoretical properties are the following: Rule 1. "One person one vote."According to this rule, the power index I i of the generic ith social group will be equal to: Rule 2. "One social group one vote."According to this rule, the power index I i of the generic ith social group will now be equal to: It is rather direct that not only can the above two rules be unified but that, also, another intermediate rules can be obtained by resorting to the following family of p-norms: It is straightforward that if in (3) we make p = 1, the "one person one vote" rule is obtained, and if we make p ∞, then the "one group one vote" rule is derived.In short, p-norm plays the role of a balancing factor between the two above social rules; that is, as the value of p-norm increases, then more importance is attached to the "group" in detriment of the "individual" and vice versa.For the balancing interpretation of p-norms within a preferential aggregation context, see the seminal works by Yu (1973) and Yu (1985, pp. 66-92).
On the other hand, it can be intuited that the opposite poles "individual versus group" (p = 1 and p ∞, respectively) can present application problems in real scenarios.In fact, the solution for p = 1 could be biased against the interests of the small-sized groups, whereas solutions for p ∞ can clash with the interest of large-sized groups.For these reasons, in some real cases, the voting power of each social group is established with the help of a norm with an intermediate value of metric p between 1 and ∞.For instance, the bylaws of the International Federation of Operational Research Societies (IFORS) establish that "the member's voting strength" is the square root of its qualified membership (i.e., this society resorts to formula (3) for p 2).In this way, a certain compromise between the interests of the countries with a large number of members and of the countries with a small number of members is sought.
In this paper, a method pursuing a similar objective of a balance between the interests of individuals and groups is proposed by following a different theoretical direction.In fact, it will not resort to changes in the value of metric p, but will combine several social decision rules within a coherent and computable theoretical framework.
The two rules commented above are considered socially worthwhile and are consequently incorporated into the new framework.But, additionally, the following new social decision rule has been introduced: Rule 3. "The maximum deviation or discrepancy between rules 1 and 2 is minimized." The rationale underlying this new rule is quite intuitive and simple.It can be understood as being a compromise between the interests of the individuals versus the interests of the groups.In fact, if the deviation between rules 1 and 2 is a large one, then there is a big discrepancy between the interests of the individuals versus the interest of the groups that is obviously not good, contrario sensu a small discrepancy means a proximity between both opposite interests that can be considered socially positive in order to obtain a solution accepted by all the groups.The calculation of the generic power index I i according to this new rule is tantamount to the calculation of the maximum deviation within a mathematical programming context, which leads in our case to the following linear programming model [see for technical details Steuer (1989, chaps 14 and 15), and González-Pachón and Romero (2011)].
Min D Subject to: where D represents mathematically the maximum deviation, i.e., in our context the optimum value of variable D measures the discrepancy of the social group presenting the largest difference between the two solutions corresponding to the rules "one person one vote" and "one group one vote," respectively.A sensible attitude would be to accept that if the value of D is smaller, it would be easier to match the interests of the large and small groups.On the other hand, the value of D will always be strictly positive except in the particular cases that the sizes of all the groups are equal.In this exceptional situation, the solutions provided by the three social rules will coincide.

The mathematical model
So far in order to establish the voting power of a set of social groups, we have formulated three "social rules" each one with a clear preferential interpretation and enjoying good properties.The next step in our analysis consists in formulating a framework able to unify the three rules as well as to determine trade-offs and compromises between them.This task can be accomplished in a quite straightforward way by formulating the following adaption of an extended goal programming (EGP) model (see Romero 2001; González-Pachón and Romero 2016):

Group versus individual interests
Constraints: Sign restrictions 5) is a straightforward structural adaptation of an extended goal programming formulation.However, the following comments can clarify the meaning of the basic elements of the model.Thus, it is important to note that λ 1 and λ 2 play the role of control parameters.Thus, when λ 1 1 and λ 2 0, the voting power minimizing the deviation between the interests of individuals and groups (i.e., minimum value of variable D) is obtained, when λ 1 0 and λ 2 1, the best power index from the point of view of the interests of the individuals is obtained, and finally when λ 1 λ 2 0, the best option from the perspective of the groups is obtained.For values of the control parameters λ 1 and λ 2 such as: compromises between the above three power indices, if they exist, will be obtained.In other words, these control parameters can be interpreted as the marginal rate of substitutions between the interests of individuals, as well as groups and discrepancies between them.On the other hand, α i , β i , η i , and ρ i are the usual negative and positive deviation variables common in every type of goal programming model, measuring possible underachievements or overachievements with respect to the targets or right-hand values.By construction, in this particular case all the deviation variables are unwanted and as such must be incorporated into the achievement function of the model.
Two technical features about model ( 5) can be highlighted.First, the right-hand side values of the different goals are in some cases very different.In this situation, the solutions provided by the model could be biased as more importance is given to the goals with higher target values (i.e., the larger groups).In order to avoid this potential problem, the deviation variables in the achievement function were divided by their respective targets; that is, instead of minimizing absolute deviations, the model minimizes percentage deviation (see Jones and Tamiz 2010 pp. 34-39).Second, as it was indicated above all the goals of model ( 5) are double-sided; that is, neither underachievement nor overachievement is wanted.This technical point derives from the logic underlying the model, but also precludes the possibility of obtaining nonefficient solutions in a Paretian sense [see and Jones and Tamiz (2010, Chapter 6)].
The meaning and role of the set of conditions F require a clarification.Thus, the set F represents the formulation of the conditions to be imposed in a particular scenario.It is obvious that this type of condition is of a clear "ad hoc" nature, but also increases the flexibility of the proposed theoretical framework.For instance, let us assume a situation in which it is accepted that if n i is larger than n j (i.e., the size of group ith is larger than the size of group jth), the power index associated with the former group cannot be lower than the power index associated with the latter one.In this type of situation, the set of conditions F will be read as follows: ε being a small perturbation term.This auxiliary term can be fixed in different ways.Thus, it can be made equal in all the cases, proportional to the differences of sizes between groups, etc.The computational character and the technical details of this proposal are clarified in the next section with the help of a numerical illustration.

Numerical illustration
In order to illustrate how the method proposed in the previous section works, the following simple numerical example is introduced.Let us assume that there are four social groups with the following number of members or sizes: n 1 700, n 2 200, n 3 90 and n 4 10 Hence, the power indices from the perspective of the interests of the individuals (Rule 1) will be equal to: From the perspective of the interest of the groups (Rule 2) we have: Finally, the vector of discrepancies between both perspectives will be equal to: [0.45, 0.05, 0.16, 0.24] It is assumed that after a negotiation the following perturbation terms, to some extent proportional to the sizes of the groups, are accepted: By applying model ( 5) to these data, the following model with the typical structure of an extended goal programming formulation is obtained (Romero 2001):

Goals
Sign restrictions By solving the above parametric linear programming model, the results shown in Table 1 were obtained.
The first three rows of Table 1 represent the voting power indices for each group, when only one of the three social decision rules considered is taken into account.The last three rows present the additional solutions derived from the parametric analysis implemented.These solutions can be considered surrogate compromises of the first three.Hence, in this illustrative example three intermediate or compromise solutions have been obtained.Obviously, the number of these types of solutions is an empirical question, depending on other things of the size of the problem analyzed.In short, according to the analysis undertaken the voting power indices of the four social groups considered in this example present the following ranges of variation: 0.262 ≤ I 1 ≤ 0.700, 0.154 ≤ I 2 ≤ 0.255, 0.090 ≤ I 3 ≤ 0.250, 0.010 ≤ I 4 ≤ 0.241 We wish to remark that the values of the perturbation terms used in order to define the set F of conditions is optional and is of a clear "ad hoc" character.On the other hand, we should insist that the inclusion or not of this type of condition as well as the fixation of its numerical values is the only information to be negotiated between the groups in order to implement the proposed method.

Concluding remarks
The results derived from this research have interest for the following reasons: (a) The proposed approach allows the establishment of voting power indices that are not based on a single social decision rule, but on a coherent structure of competing social decision rules.(b) All the proposed rules have a clear preferential meaning.
On the other hand, these rules can be straightforwardly measured just by knowing the number of members of each social group.(c) The implementation of the method entails a very meagre computational burden.In fact, it is only necessary to solve a parametric linear programming model of a moderate size.
Finally, it would seem to be of interest to point out a possible connection between the voting power analysis undertaken in this paper and the applied literature in group decision making.Thus, in the latter, the weights attached to each social group involved in the decision-making process are normally proportional to the size of the group (Rule 1), or it is of the same value for each group (Rule 2).In other cases, a sensitivity analysis is also undertaken in order to test the robustness of the rankings obtained to possible changes in the values of the respective group weights.The method proposed in this paper might be useful for rationalizing the determination of the group weight values as well as for organizing the implementation of a sensitivity analysis.In fact, the output provided by the proposed voting power model does not generally represent a single vector of group weight values but a collection of interval values that can orientate more efficiently a possible sensitivity analysis.

Table 1
Voting power indices for each social group