A Theory of Best Choice Selection through Objective Arguments Grounded in Linear Response Theory Concepts

Abstract

In this study, we propose how to use objective arguments grounded in statistical mechanics concepts in order to obtain a single number, obtained after aggregation, which would allow for the ranking of “agents”, “opinions”, etc., all defined in a very broad sense. We aim toward any process which should a priori demand or lead to some consensus in order to attain the presumably best choice among many possibilities. In order to specify the framework, we discuss previous attempts, recalling trivial means of scores—weighted or not—Condorcet paradox, TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), etc. We demonstrate, through geometrical arguments on a toy example and with four criteria, that the pre-selected order of criteria in previous attempts makes a difference in the final result. However, it might be unjustified. Thus, we base our “best choice theory” on the linear response theory in statistical physics: we indicate that one should be calculating correlations functions between all possible choice evaluations, thereby avoiding an arbitrarily ordered set of criteria. We justify the point through an example with six possible criteria. Applications in many fields are suggested. Furthermore, two toy models, serving as practical examples and illustrative arguments are discussed.

1. Introduction

Nowadays, statistical physics has found wide open research topics outside of its classical aims in economics and sociology nowadays [1,2,3,4,5]. Thus, consider the interplay between sociology and physics: sociophysics.

Forget Hobbes, Quetelet, Comte, Verhulst, and others there should be no need to point out, as an introduction or justification for this paper, that Galam is a pioneer of modern sociophysics [6,7] (see also the relevant papers [8,9,10,11,12,13,14,15] of interest for consideration in this paper).

One of Galam’s goals was to set a framework, provide techniques, and search for conclusions on the dynamics of opinion formation in various societies. Galam’s studies lead to the finding of the conditions for consensus (kind of an equilibrium), chaotic states, and intermediary complex phases in multi dimension diagrams. However, it is not quite apparent that his findings pertain to what people would refer to or call “the best choice”. But, what is the best choice? It should be admitted that what one would call the best choice is a highly relative and subjective state or concept. The answer contains ingredients from both personal (“selfish”) and global (“self-effacing”) points of view. Moreover, the dependence on exogenous and endogenous conditions is quite significant, but such considerations are left for political discussions elsewhere. We consider that the final states in Galam’s studies or models, and subsequent studies by many, result from a too heavily weighted stochastic set of constraints or hypotheses. Opinion dynamics surely result from individual “votes” (meaning choices) due to herding or because of contrarians [9,12,16]. Nevertheless, do the dynamics imply a good choice? Or worse, is the choice (meaning vote) the best choice? That seems to be a crucial point, not only at election times in democracies, but also in choices like (we limited the references to a few papers) in the media [17], including music genrefication networks [18]; in economics [19], including regional studies [20] and drawdown market price sizes at speculative times [21], in academia, including scholarly journal rankings [22], research networks clustering [23], world universities ranking [24,25] or samba schools ranking [26,27]; and in sport [28,29,30,31].

More explicitly, in the academic domain, what is the best choice when hiring or promoting a colleague, when appointing a vice-chancellor, or when selecting teams for research grants? In the sport domain, the ranking of football teams or of cyclist racing teams is obtained through apparently objective numbers, but the rules can often (or even always) be debated and challenged [32,33]. The same remark holds for the Nobel Prize, the Pulitzer Prize, the Goncourt Prize, the Oscar or Cesar Awards which are given through highly subjective, not objective, criteria; not discounting facing a choice between roads going from X to Y, for going on holidays or to a restaurant, etc.? What is the best equipment or car or cell phone to buy? What is the best food, from a health point of view? All these questions mix subjective and objective criteria.

This boils down to the fundamental and practical question: what are the criteria needed for reaching the best choice? In other words, how should one conclusively rank a set of “things”, “people” or “teams” in a constrained set of criteria?

This leads us to remember that the “final choice” leads to a somewhat paradoxical situation, the Condorcet paradox [34,35,36], for example. Recall that the Condorcet method [34,35,36] is a voting system that will always select the candidate that voters prefer over each other candidate, when comparing between them one at a time. Further considerations on comparing preferred choices lead to Arrow’s incompatibility theorem [37]. Moreover, the order of criteria might lead to ambiguities [38].

The discussion, and the subsequent answers, should pertain to a comparison of the evaluation methods according to criteria [39,40]. Most of these are based on previous achievements, even though their forecasting value, not mentioning consistency for future achievements or impact, are far from certain.

This drastically annoying deduction seems to stem from the plethora of “parameters”, i.e., possible criteria. Practically, one turns toward aggregation processes [41,42,43], going from multi dimensions toward a single number. This makes life complicated enough when one turns toward the modelling. Thus, one wishes to have some indubitable argument; often, that means having rigorous mathematical theorems. However, this is often hard to implement, in particular for laymen (or lay women, or lay others). Therefore, mathematical arguments might be by-passed through physics concepts which allow metaphors and analogies, like in modern statistical mechanics or like in Galam’s view of social thought dynamics, for example, on networks.

Therefore, after outlining elements for discussion from information theory, arguments from geometry, and surely arguments on complex systems and sociophysics, we propose a powerful argument grounded in linear response theory (LRT): in LRT, the coefficients (magnetic susceptibility, transport coefficients, etc.), measured in the laboratory and estimated theoretically, are discussed and defined through the correlations between the (fluctuations of the) dependent variables. Thus, we suggest calculating all correlation functions implying the relevant variables in the sociophysics research topics of interest, in particular when searching for opinion formation, as due to Galam. This idea seems to be missing in earlier considerations. We claim that this idea is leading to a more objective hierarchy of values in opinion formation choice, i.e., topics.

Within this set of considerations, a study framework can be conceived together with applications, thereby leading to the following structural content of this study.

Section 2 contains a brief review of “old” and “modern” techniques for selecting and ranking agents or events, like the rank-size “laws”. We discuss such attempts, recalling trivial “means of scores”, be they weighted or not, the Condorcet paradox, etc. We recall preference aggregation techniques, like the maximum likelihood rule (MLR) (Section 2.1), and the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) (Section 2.2).

However, the aggregation problems have two different aspects: rank aggregation or score aggregation. These are briefly distinguished in Section 2.3.

In particular, in Section 2.4, we start from geometrical aspects for ranking scores, or measures, and for later aggregating multi-valued measures. Furthermore, we demonstrate through such geometrical arguments that the order of criteria might make a difference on the final result, but this might be unjustified. It is easily observed that, if only three criteria are used, the procedure leads to an undebatable result. We show on a set of four criteria for a toy model that the order is drastically relevant and influences the outcome after aggregation.

We base our best choice theory in Section 3, upon the linear response theory. We indicate that one should be calculating correlations functions between all possible choice evaluations. Appendix A recalls the fundamentals of a linear response theory in statistical physics.

We conclude with some emphasis, if it is needed, on the finding that the best ranking is obtained from a method based on rigorous arguments for obtaining a final score and ranking. We are aware that subjective arguments often influence the final decision. We suggest applications in a few domains in Section 4. Two examples can be found in Appendix B.

2. Modern Ideas and Methods

In this Section, we consider ideas and methods for the ranking and selecting of the best outcome due to inequalities between events and agents, also called “variables”, thereby assuming that a hierarchy takes places and leads to the best choice [44].

Before recalling modern ideas, let us remind the reader that the selection process leads to some so-called rank-size (RS) displays or tables. Indeed, RS analysis is the basic way of measuring disorder in a population: the largest “size” gets the first rank, and a hierarchy, from which an ordering through inequality is deduced [44]. This is a hierarchical description. In fine, this leads to considering whether empirical laws follow patterns, thereby suggesting models.

The RS law derives from the analytical form presented by the variables, often ranked in descending order as a function of the (discrete) index i, giving a “rank” to each of the $g_i$, $i=1,\cdots ,N$ variables. The cumulative law ${\sum }_i{g}_i$ leads to the cumulative concentration distribution curve (CCDC). The sum ${\sum }_i^N{g}_i$ over all the i elements can serve as a normalisation value. One immediately obtains the “normalised” CCDC. When the normalised CCDC goes over the 80% threshold, this defines the Pareto rank, $r_P$. The Pareto principle expects this rank $r_P$ to be equal to $N/5$. Thus, the RS method is convenient, and better suitable for large populations, i.e., when the size or/and rank ranges can be large. This is rarely the case when only a few selection criteria are implied.

For completeness, nevertheless, let us mention that Marfels [45] distinguishes several types of concentration ratios according to weighting schemes and their structure, which can be discrete or cumulative [46]. Beside such ratios, inequality aspects are often discussed in order to touch upon socio-economic concerns [47].

Furthermore, one can recall that, when the goal is to find a compromise between the various rankings, the statistical median is thought to be the most appropriate solution [48,49]. However, in situations when decision making should be a way of compromising between conflicting decisions, the use of MLR, discussed in the following Section 2.1, is justified.

2.1. Maximum Likelihood Rule

Recall that a choice demands some ranking. This is usually done by combining ordered preference lists into a single consensus value, as in a reviewing procedure [49,50]. We outline the MLR, which is based directly on rankings and not on the scores.

In brief, methods of preference aggregation, such as the MLR, are based on the concept of pairwise preference notions [51], which are much used in economics and in opinion formation, alongside the simple majority voting rule, without a discussion of the scores. It is also known as the Kemeny [52] rule (or the Kemeny–Young method [48]). In order to go beyond the limit of the method, one introduces a variant, taking into account the behavioural argument of “blindness to small changes” (BSC) [53].

Mathematically, an MLR ordering is defined as one that minimises the total number of discrepancies among all the reviewers in their pairwise preferences between all options. It can be viewed as a voting scheme that determines not just a single chosen winner, but an entire ordered list. Therefore, the MLR ordering generally satisfies the most possible reviewers regarding their stated rankings of options. The ordering does not use whatever information about how much higher a candidate is ranked over another, but only a relative ordering (it is worth pointing out that MLR is a Condorcet method).

Practically, one can say that the method counts the pairwise preferences and applies the majority rule over each of them.

Examples abound on applying this rule and finding “solutions” [54]. One short illustration is given in Appendix B.

The second step of the method is counting the number of times where one candidate is ranked over another. In doing so, the MLR emphasises that the first-ranked choice is preferable compared to all other options in individual pairwise comparison. Similarly, the MLR second ranked choice would be be preferred to all other options (except the first ranked), and so on.

In order to complete this Section, we may consider that the MLR implies that candidates with the same score are considered to belong to the same “indifference set”. If this occurs, one decides to list the “candidates” (specifying the “agents” or “events”) one after the other, e.g., in alphabetical order, in such a set. One may consider other ways to manage equal scoring, but there is no need to foresee any special treatment here to take care of ex aequo positions in the ranking. Nevertheless, in most evaluation procedures, a linear order is requested for the final ranking; the most unwanted point is when two “agents” are ex aequo on the first rank. In general, in order to achieve such a goal, or resolve such a dilemma, extra information is a posteriori added for discriminating equally scored candidates. This might be unjustified but this “solution” is left for more suitable considerations and is beyond outside of the scope of the present study.

2.2. Technique for Order Preference by Similarity to Ideal Solution

The TOPSIS, was originally developed by Hwang and Yoon [55], with further pertinent developments, e.g., in Refs. [56,57,58,59,60], is a method of compensatory aggregation that compares a set of alternatives by identifying weights for each criterion, normalising scores for each criterion and calculating the geometric distance between each alternative and the ideal alternative, which is the best score in each criterion.

TOPSIS works attractively in various areas; see, e.g, [61].

2.3. Score Rather Than Rank Aggregation

However, some fundamental emphases between processes in the preferential ordering problems must be made through distinguishing rank aggregation from score aggregation. Often, the ranks are not known, but are deduced from scores. However, as pointed out, one does not always know how the scores are obtained (recall the number of stars in the Michelin guide or the Gault and Millau scores for restaurants, or the ranking order on Trip Advisor). In such cases, scores are incompatible and sometimes incomprehensible; ranking only makes some sense, allowing for ex aequos.

However, if scores are known from a reliable set of measures and for a given set of criteria, some objective construction can follow, leading to some meaningful score aggregation. A scoring function can be defined as $f^{\left(X_j\right)}\left(s_i\right)$, with $s_1,s_2,s_3,$…$,s_m$, being the scores on each i-th criterion, for a given “agent” $X_j$, with $j=1,$…$,n$.

The aggregation results, from the set of n “events”, “agents” and the m scoring criteria, sorted, for example, in decreasing order, lead to finding the top-k “candidates” according to the scoring function, imposing $f\left(\left[s_i\right]\right)<f\left(\left[s_i^{\prime }\right]\right)$. The objective is straightforward: to compute the top-k “agents” with the minimum cost. While the target is apparent, the methods can be quite different, depending on the choice of f. The very first problem stems from the sorting out algorithm. Let us point to the Fagin algorithm [62], to the medrank algorithm [63] and to the threshold algorithm [64]. For some completeness, another method is the Borda count [65]: for each ranking, one assigns a score D equal to the number of objects it defeats. The total weight of D is the number of points it accumulates from all rankings.

However, these rank or score ranking methods can be challenged because they are missing a key ingredient, i.e., the selection order of the criteria.

2.4. On the Sequence of Criteria: A Geometrical Perspective

In brief, many (if not all) earlier studies ignore a crucial factor: the sequence of criteria. It can be pointed out and straightforwardly illustrated that the ascertainment of criteria has an effect on ranking values; this is common knowledge to anyone having participated in surveys [66,67] and/or selection processes.

Suppose that there are three criteria giving a numerical value (a, b and c, respectively) for the agent or event. One can define a coordinate system with three axes stemming from some origin O in equivalent directions (such that the angle between each axis is, therefore, $2\pi /3$) and plot the values on each “criterion axis”. Next, one way to aggregate the three values is to consider them as three sides of a triangle, and calculate the triangle area, which becomes the “score”.

In this case, one may recall that it would be necessary, in order to find the area of a triangle with three sides, that one uses the Heron’s formula: indeed, the area S of a triangle with three known sides, a, b and c, is calculated from

$$S=\sqrt{s(s-a)(s-b)(s-c)},$$

(1)

where s is the semi-perimeter of the triangle, i.e., $s=(a+b+c)/2$.

On the other hand, knowing two sides b and c around an angle A, the formula to calculate the area of a triangle is given by $(1/2)bc·sinA$. Obviously,

$$S=bc/2\mathrm{if}A=\pi /2,$$

(2)

while

$$S=(\sqrt{3}/4)bc\mathrm{if}A=2\pi /3(\mathrm{or}\pi /3).$$

(3)

In the latter case, this means that whatever the order of criteria, the total area, i.e., the sum of the three triangles areas, is invariant and equal to $S=(\sqrt{3}/4)(bc+ab+ac)$.

However, the matter is different when there are more than three criteria or sustaining axes. Consider the case of four criteria. Let the axes form a coordinate system with four axes in symmetric directions. The permutation of axes leads to six different polygons. A toy example, with $a=1$, $b=2$, $c=3$ and $d=4$, can be seen in Figure 1. The four inner triangles in each polygon are rectangular triangles for which each area is easily obtained (the total area is given in Figure 1 in arbitrary units); it can be well noticed that, according to symmetry, only three different areas are relevant. Therefore, this leads to three different sizes if the area is considered to be the aggregated number for ranking the agents or events.

When the number of criteria increases, from which many more polygons can be drawn, the area of such polygons can always be decomposed into a number of triangles, for which each area can be calculated using Equation (1), remembering that, using the length of two sides (a and b) of the known angle (C) between them, in any triangle, one can easily calculate the third side length of the triangle through the Al-Kashi formula, i.e.,

$$c^2=a^2+b^2-2ab·cosA.$$

(4)

3. The Problem and Its Solution

Thus, the way to proceed is as follows. First, take a survey with as many possible criteria that are needed for a population, whatever its size. One may suppose, without loss of generality, that the choice is based on Likert scales, whatever the useful range $[0,5]$, $[0,7]$, ⋯, $[0,20]$.

Notice that it could be of interest to consider the mean and standard deviation of the data distribution on each axis, such that one can obtain a possible “universal comparison” of final rankings and hierarchies. If the statistical characteristics make sense, the measured values could be normalised or, at least, constrained to vary in an identical range. One could assume that all the values involved in the decision process belong to $[0,1]$. The value 0 is achieved in the worst case for the considered parameter (bad realisation) and, accordingly, 1 is taken in the “best choice” case. Nevertheless, such a scaling is not a fundamental request. Although of interest, this normalisation aspect is not discussed further here and is left for further investigation.

Then, one reports the values on equidistant axes. These are disposed in a regular star-shaped network with a central node, a common starting point and identical angles between consecutive axes; the angle value actually depends on the number of axes (criteria). The non-common nodes of the consecutive segments are connected. The resulting polygon has a number of sides identical to the number of evaluation parameters; the polygon is not necessarily regular and can be considered as being made of triangles. If N is the number of axes, the largest possible number of different polygons is $(N-1)!/2$. The number of different triangles is N!.

In this paper, let us focus on hexagons, since evaluation processes are quite often associated with six different parameters.

In Figure 2, we present an example for two hexagons out of sixty possible cases; let $a,b,c,d,e,f$ (e.g., $\in [0,1]$) be the values of the six parameters. The surface of the resulting hexagon $\mathcal{S}$ can be straightforwardly computed. The evaluation score coincides with the value of $\mathcal{S}$, so that a high $\mathcal{S}$ means a high score. However, notice that $\mathcal{S}$ belongs to the range associated to the limiting cases $a=b=c=d=e=f=0$ (worst result of the evaluation process, with $\mathcal{S}=0$) and $a=b=c=d=e=f=1$ (best result of the evaluation exercise, with $\mathcal{S}=3\sqrt{3}/2$).

Certainly, the surface of the hexagons shown in Figure 2 depends on how the segments are disposed: the values of the six measures are always $a,b,c,d,e$ and f, but the placement of c and f have been reverted in both displays.

Let ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ be the surface of the hexagons in Figure 2, left, and Figure 2, right, respectively. One has

$${\mathcal{S}}_1=\frac{sin(\pi /3)}{2}\left(ab+bc+cd+de+ef+fa\right)=\frac{\sqrt{3}}{4}\left[(ab+bc+cd+de+ef+fa\right)],$$

$${\mathcal{S}}_2=\frac{sin(\pi /3)}{2}\left(ab+bf+fd+de+ec+ca\right)=\frac{\sqrt{3}}{4}\left[ab+bf+fd+de+ec+ca\right];$$

In the case $b+d-e-a\ne 0$, one has that ${\mathcal{S}}_1={\mathcal{S}}_2$, if and only if $c=f$.

Such a remark leads to the questionability of the evaluation exercise through polygons. Indeed, the resulting polygons, differently shaped according to the disposition of the axes, have different surfaces. This means that the evaluation is strongly dependent on how selection parameters are included in the graphical representation of the problem, as we again stress.

To overcome such an inconsistency, and in order to obtain a more indubitable scoring, we propose taking the average $\overline{\mathcal{S}}(a,b,c,d,e,f)$ of all the possible polygon surfaces, thus over all the possible dispositions of the axes, here corresponding to six evaluation criteria:

$$\overline{\mathcal{S}}(a,b,c,d,e,f)=\frac{sin(\pi /3)}{5!}\sum _{\mathbf{H}(a,b,c,d,e,f)}\left(x_1{x}_2+x_2{x}_3+x_3{x}_4+x_4{x}_5+x_5{x}_6+x_6{x}_1\right),$$

(5)

where

$$\mathbf{H}(a,b,c,d,e,f)\equiv (x_1,\cdots ,x_6)\in \{a,b,c,d,e,f\}$$

(6)

with $x_i\ne x_j\mathrm{for}\mathrm{each}i\ne j$.

By construction, $0\le \overline{\mathcal{S}}(a,b,c,d,e,f)\le 3\sqrt{3}/2$. Such a mean surface measure achieves the intended job effectively, i.e., providing a consistent score of the evaluated object whose six parameters (criteria) take values $a,b,c,d,e,f$.

This is consistent with the measure concept, resulting in calculating the correlation between fluctuations in statistical physics, as can be found in linear response theory: see Appendix A. Examples of applications are given in Appendix B.

4. Conclusions

To summarize, the present study is devoted to resolve the contradictory situation when searching for an objective ranking procedure and subsequently observing some hierarchy through a set of criteria for a given population. This sort of scientometric practice is tied to considerations found in geometry and statistical physics, complementing aspects of research on opinion dynamics found in Galam’s numerous studies.

Observe the technical and somewhat philosophical frames in the present ranking study. Decisions often concern rather small sub-systems, like the funding of a project, the evaluation of a research group, or that of an individual at his or her hiring or promotion. Statistical laws cannot be immediately applied for such cases in which, due to the (very finite) size of the system, fluctuations which are not necessarily due to stochastic causes play a crucial role. Indeed, recall that scientific work is part of a social system; its actors are human beings. Moreover, statistics are like snapshots and do not often allow for predicting the future sufficiently well. In practice, indeed, any newly introduced criterion or “measure” will be met by the capacity of humans to interact and make decisions.

Without going back to the Bible and raising the question about who is the just and who will be saved (certainly, according to a single judge), one may, within modern scientific reasoning on choice, admit that when several partners and judges are implied, the resulting choice (solution) ends up into a difficulty pointed out by Arrow [37]. He considered the preference aggregation problem, that is the problem of passing from a set of known individual preferences to a pattern of social decision making. Arrow’s today oft-quoted theorem, while being variously worded, has nevertheless shown that the difficulties met in the building process of preference aggregation are highly general. The theorem implies that no rank order voting system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking, besides also meeting a specific set of natural criteria.

Why indeed is it impossible to reach a choice? Commonly, one constructs several filters and a priori decides on the order of their applications, like in the case of a decision tree (DT). That leads to a discussion on ranking the filters rather than the candidates.

Indeed, in the DT scheme, due to the order of filtering criteria, the final choice is significantly biased. The order of filters is often adapted a priori in order to select the final choice, e.g., maintaining a candidate, or candidates, in competition with others during the selection process, for hypocritical, political or other reasons.

In opinionology, the selection process consists in projecting from a multidimensional space onto various planes, and finally finding the intersection of the distribution. Recall that in physics, it is like projecting on various “external field axes” or making a scalar product. There might not be any “solution”. This may further mean that the set (of “candidates”, “events”, etc.) to be ordered is either not to be ranked or that the filters are not appropriate.

Thus, we propose, starting from Galam’s and others’ considerations on the dynamics of choices, a procedure for obtaining the “best choice” through an objective statistical physics method.

We would like make a remark at this stage. One may say that our method is not better or worse than another, but is merely another type of classification; for example, it will not be applied by the Hollywood Academy to award Oscar Prizes. Let us notice that we are not using the word “better”. We emphasise that our method is more rigorous, contains less arbitrariness and is based (through an analogy) on major statistical physics concepts. Certainly, we might regret that our method might not be applied in Los Angeles. Actually, we admit that we are not aware of all the criteria used by the Academy in order to award Oscars. The same holds true in other branches of opinion formation. In a decision process, the final ranking might be due to hidden or specifically weighted criteria; idem for the final choice. Our method does not apply outside the objective world. We do not consider subjective criteria.

Surely, the situation might not be closed, since the choice of criteria is left for many discussions between agents, be they surveyees, surveyors or other stakeholders.

Nevertheless, our argument, after outlining elements of reflexion from information theory, geometry and surely complex systems, as found in sociophysics considerations, proposes a strong argument for calculating choice values. It is grounded in linear response theory. In the latter, as it is recalled in Appendix A, the coefficients, that are measured in laboratories and estimated theoretically, are defined through the correlations between the dependent variables; thus, we suggest carrying out the same in sociophysics, i.e., calculating all correlation functions implying the relevant variables, thereby avoiding an arbitrarily ordered set of criteria. In particular, this could be useful and meaningful in the modelling of the opinion formation processes due to Galam. This methodology is missing in previous sociophysics studies. Thus, on a LRT basis, which is rather easily implemented, this should lead to a more objective hierarchy of values ahead of selection processes.