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Public Management. Electronic journal Issue 4. 10 september 2004

Briedis V. Pareto Structures
A great Italian economist Vilfredo Pareto (1848 ­ 1923) in his unique economics and sociology studies [1, 2] continuously emphasised the systematic approach, reviewing the society development problems. Pareto, who himself had an excellent background in exact sciences and was a frequent user of mechanical terminology, analysed the balance and stability conditions in socio-economic systems in various contexts: during changes in «elite», in the circumstances of population and dynamic development, as well as upon changes in the general level of welfare or incomes. Pareto discovered the income distribution law strongly differing from the normal (Gaussian) distribution that was known to researchers from natural science applications. Nowadays, the 80/20 structure principle initiated by Pareto has become widely known and extensively used, however, often unreasonably advised as a panacea in the solution of inadequate problems. In the last century, the concept of a system has become a core of a lot of sciences; studies of system balance and stability spread over biology, natural sciences, linguistics, theory of communications, modern medical science and gene engineering (this listing could be continued long enough, from astronomy to zoology...). Such studies still remain urgent in economics and sociology. A wide range of various models and hypotheses is still lacking a really universal tool or formula that would identify the general description of a system by the analysis of its components or the structure of an object. The author proposes a new systematic formula featuring a rather universal theoretical structure that is generated in a natural way upon rather weak requirements to the nature of system components (objects). I have named the discovered structures «Pareto Structures» (PS) in honour of the Italian genius. Using the probability theory and statistical ideas with inductive heuristic approach, the discovered formula cardinally differs from those resulting from classical techniques, such as thermodynamical (Boltzmann), informational (Claude Shennon) or chaos theory (Mandelbraught). Stressing the influence of V.Pareto's ideas, I would like to cite widely here Albert Lasszlo Barabasi, a US researcher of Hungarian origin, whose book «Linked. The New Science of Networks» [3] inspired me to recommence my previous studies of system structures, since nothing seems featuring better this classic. Formal model of Pareto Structure First of all, provide a formal mathematical model adequately featuring the Pareto Structure. Identify a system as a totality of n elements (hereinafter referred to as «the object»), which is integrated, governed by a certain criterion or a combination thereof. Use the following denotations: iPn(s) = iZn(s)/Zn(s-1); i-th member in Pareto Structure (1) Use the Euler-Reaman denotations for the known zeta-function (the power exponent s will be hereinafter referred to as «the characteristic») Zn(s)=1+1/2s+1/3s+..... +1/ns (2) (3) iZn(s)=Zn(s)-Zi-1(s); corresponding Zn(s) is the i-th residue, or remainder. The totality {Pn(s)} (*) is the Pareto Structure for an n-object system at s characteristic; At s = 1, the structure becomes easily interpreted; considering that Zn(1) is simply a sum of inversed natural numbers up to n, we obtain Zn(1)=Hn=1/1+1/2+1/3+.......+1/n-1+1/n (4) where Hn is a generally accepted denotation of a harmonic progression;
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iHn=Hn-Hi=1/i+1/i+1+............1/n-1+1/n (5) Hn is the i-th residue, or remainder iPn=iHn/n (6) accordingly, let us call the i-th member as the i-th member of the Pareto Harmonic Structure (PHS); The totality {Pn(s)} (**) is the Pareto Harmonic Structure (PHS) at given n. A priori hypotheses and opinions. The author puts forward a hypothesis that the Pareto Structure reflects the degree or condition of balance, natural non-uniformity or nonhomogeneity of theoretically any universal system made up by n elements or objects. A system of n objects or also with the possibility of its unbiased splitting into n objects, where each object can be comprehensively featured by an index generally acceptable for the system, is arranged according to the descending sequence of this index of the objects. Balance, equilibrium, stability are the terms that should be understood in the interpretation of neither classical mechanics nor vulgar socio-economic inequality. I daresay to assert that if a universal law of diversity, heterogeneity or non-uniformity exists in the material reality and society, the Pareto Structures are the form in which such law is expressed. The Pareto ideas may be to a certain extent compared to Darwin's ideas of general natural selection theory; however, the problem of system structuring may spread far beyond the limits of an economic system. Analysing the PS behaviour at various characteristics s, I have received rather surprising results. Indeed, I was primarily interested in the classical (harmonic structure) model at s = 1. I was slightly disappointed discovering that at big n values the distribution curve was far enough from the Pareto point. I identify the Pareto point as a point of the 80/20 distribution with the same coordinates. At s = 1, the increase in n entailed a relatively fast convergence of the curve into the border curve that roughly corresponded to the 70/30 distribution. Studying the asymptotic variations in specific weight, or «contribution», of a particular system element, my moderate knowledge of the number theory had to be refreshed, and therefore I was happy getting in my hands a wonderful book «Gamma. Exploring Euler's Constant» by Julian Havil [5]. Naturally, I was soon enlightened that the famous zeta-function of Euler-Reaman should be taken instead of harmonic progression, and s-parameter appeared: it was one of the happiest days in my life and the rest was, as the saying is, the matter of technique. Further discoveries followed each other. The trajectory of distribution corresponding to the Pareto point was formed at s = 1.9 (but only approximately, since n modifies the distribution with approaching the border curve). A cursory analysis of various structures led to the conclusion that the explored structures are incorporated into real life within a comparatively small interval 1 < s < 3 (a very subjective evaluation for the time being). Having become more or less familiarised with the probable range of s values in actually existing processes and events, I was indeed interested at what s values comes the thermodynamic «heat death», or the absolute equality; I do not know why it first seemed that at s = 0. However, I was surprised with the absolute equality only appearing at s = -. The other extremity with the total domination of sole element and 0 contribution by other elements evidently comes up only at s = +. I came to a probably incorrect (in political aspect) conclusion that from the standpoint of the Pareto Structure analysis the slave-owning system appears to be a little more probable (still, the number of slaveowners exceeded 1, and s, respectively, would be nevertheless the final number) than the utopian communism and absolute equality, when s is infinite. So easy: if physicists can reach the maximum entropy in the utopian space, then the followers of equality, strongly criticised by Pareto, may as well create the absolute equality in the utopian environment! Who will surrender first ... Thank God, neither the Universe nor the mankind is endangered by total homogeneity!
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Fig. 1 illustrates the Pareto Structures at various s. The proposed structures (the idealised standards of theoretical models) may serve as a wide basis for studies of system evolution from the standpoint of dynamic equilibrium description. For the purposes of this publication, the systems are interpreted as social and economic systems, providing that the investigation of PS will yield considerably greater opportunities compared to the well-known Genie ratio and Lorenz curve that are used in socio-economic studies. However, the potential applications could be actually unlimited: scientometry (research science), mathematical linguistics, communications network science, biometry, various statistical studies, all «systematic» sciences... Which science does not use it? Lots of studies, where distribution of non-Gaussian or hyperbolic (power law) type appears, may use the PS as an efficient tool for the evaluation of stability and equilibrium of a system. Indeed, it is hard to predict s values most widely spread in actual systems, and still I can put forward a hypothesis (as I mentioned earlier) that in 90% and even more of cases s value could fall within 1 and 3 (1 < s < 3). Surprisingly enough in examples studied by me I never came upon a structure that would be featured by s < 1; however, I think that a structure like that (and probably more than one) exists. The other problem is whether such structures are durable and long-term. At the same time I would like to assume a priori that the lower s is (of course, at s < 1), the less stable the system will be. In the next section I would like to consider more in detail the case when s = 1, (identified by me as the Pareto Harmonic Structures) since it was just the historical origin. Three options of the Pareto Harmonic Structures genesis.
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Now review three completely different approaches, all of them leading to the Pareto Harmonic Structure (PHS) model. First: Sports competition model. Although a model like that was probably invented and even utilised long ago, I would like to identify possibly the simplest and most obvious way of telling the others about the Pareto Harmonic Structure idea. Assume that n competitors distribute a monetary fund that is split into n units, competing in n-1 orders /tours?/ stages?. In each order, the participants split one unit into equal shares of the monetary fund while those remained in the last place shall be expelled from the further competition that will go on until two strongest participants remain at n-1 stage: the final players. The loser shall receive a half of penultimate monetary unit while the winner shall receive its second half and the whole last monetary unit. It is obvious that the most unsuccessful participant shall only receive 1/n-th part of the first unit of the monetary fund, or 1/n*1/n, while the winner shall receive the lion's share, that is 1/n(1/n+1/n-1+.....+1/2+1), and thus each of the participants shall receive a part of the monetary fund in compliance with the Pareto Harmonic Structure (PHS) model. This model very obviously demonstrates realisation of a really fair («natural») principle of inequality: equal opportunities at the start and further success only depending on fair play. I do not know whether such award granting algorithm exists in sports world since usually the winner (or 3 best participants) gets everything; however, it would be interesting if, for example, California Governor Arnold Schwarzenegger arranged an exclusive tournament of strong men according to such model of awarding; and I think it would be even more challenging to arrange a contest of world wisemen under such rules (but it would not get TV interested in the entailed financial consequences...) Second: Amortisation model. Assume that a property to be amortised is totally depreciated within n years and choose the following writing-off strategy: in the first year write off 1/n-th part of the value and besides write off also 1/2nd part of the value that is totally depreciated within 2 years, 1/3rd part of value that is totally depreciated within 3 years, and so on; respectively, already in the first year we will actually amortise 1/n(1+1/2+... +1/n1++1/n), in the second year ­ 1/n(1/2+1/3... +1/n-1++1/n), and in the last year ­ 1/n*1/n, and thus we again obtain the familiar PHS! This model was thoroughly analysed by Kristina Meistare (my worker for doctor's degree) by its comparison with traditional amortisation algorithms that were reviewed in detail by A. Buhvalov, V. Buhvalova and A. Idelson in their book «Financial Estimates for Professionals» [7]. In more aspects the PHS algorithm was found not only being competitive but also providing certain advantages when a businessman wishes to use the accelerated amortisation; the government loses nothing in the aspect of taxation while the businessman is always happy to enjoy tax holidays. The Schedule (refer to Example No.5) evidently shows the comparison with four algorithms most frequently used in practice. It is interesting to note that the dependence of prices for used (therefore, partly depreciated) cars on their age generally complies with the Pareto Structure (lack of time barred me from more detailed analysis of this example). It is natural that n is higher for better maker names (brands) and, indeed, the prices also drop in a different manner; I was surprised much with the opinion of authoritative experts: the most rational way is buying a 2.5- to 3year old car since it nearly coincides with the Pareto point: thus, for 30% of the price you get 70% of total service life (PHS!!). Not a bad offer... Third: Stochastic model. Review an abstract mathematical model where upon realisation of the uniform distribution law (respectively, at weakest conditions: remember the Laplas principle) casual events occur with constant intensity (the simplest flow is known to follow the exponential distribution law). It could be obviously illustrated by a straight line of unit length, above which a «guillotine» moves to and fro and divides the line into n sections each time upon occurrence of n-1 «chopping events». Repeating such an experiment a number of times long enough and each time assorting (ranking) the chopped sections by their length,
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we obtain the average lengths from the longest to the shortest one. It turns out that a fragment asymptotically ranked as the i-th by the length is 1/n*(1/i+1/i+1+1/i+2+...+1/n-1+1/n) unit long, i.e. the i-th member of the PHS! Unfortunately, I do not know who is the first author of this solution (within the framework of well-known Feller's theory of probability this problem is only mentioned incidentally in another context); however, the knowledge of the formula allows its elementary proof by means of the mathematical induction method. A peculiar recurrence formula is evidently valid: (n+1)iHn+1=niHn+1/n+1 (*), according to our denotations. It should be recognised that I saw this solution for the first time 35 (!) years ago in the small brochure «50 Binding Tasks in the Probability Theory» by a well-known mathematician Mosteller and already then the solution left an indelible impression on me. However, the author admitted he did not know a simple proof of the revealed formula; probably, he did not recognise the mathematical induction method as a correct one in that case or maybe simply did not think about it... In general, these three radically different methods, all of them leading to the Pareto Harmonic Structure, are the evidence that we deal with rather a universal system featuring structure. The reader most probably will mention other cases when the PHS is generated. Further, review several specific examples illustrating the compliance of the Pareto Structures with various actual realities. Examples of actual systems making up Pareto Structures Example 1. While until May 1, 2004 the European Union (EU) had existed as an association of 15 countries, it currently includes 25 members. Although it is easy to put forward a priori a hypothesis that the heterogeneity in the EU has increased, how can it be proved? Evidently, by studying variations in the Pareto s-characteristic. Graphic illustration (Fig. 1) demonstrates that the old structure (n = 15) was featured by the characteristic of heterogeneity s =1. ** while in the new structure the heterogeneity has grown up to s = 1. ***, that is s has increased by 0.xx.. Indeed, the importance of s should not be overestimated; however, in the integral review of the problem we can operate with an objective index that helps understand deeper the dynamic development of the process. It should be still pointed out that in no case the question is discussed whether the EU25 is better or worse than the EU15; simply, the situation is different and, from the abstract point of view, the EU25 possesses a greater growth potential (EU25>EU15) and, therefore, a greater vitality (remember the utopian events), in contrast with the opinion recently expressed by a Nobel Prize winner Milton Freedman.

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Figure 2. Pareto structures for EU (GDP 2002)

Figure 2a. Pareto structures for EU 25 countries (GDP 2002) Example 2. Competition among beer brewers in Latvia A student Didzis Persevics has studied the distribution on beer market in Latvia where rather a strong leading position is currently held by Aldaris Company. Indeed, for that reason
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the actual structure of market players is far from s = 1 pattern (respectively, the harmonic structure). Didzis Persevics analysed the structure variations within the last three years and found that notwithstanding a strong competition Aldaris successfully retained their market share and variations of the total structure were minimal (refer to Table 4); the diagram shows the obtained results. Example 3. USA.

Figure 3. Pareto structures for the USA(GDP 2002) Example 4. Distribution on fuel market in Latvia. Example 5. Amortisation estimate. Using the calculations presented in the book, skilfully (in pedagogic aspect) written by A. Bahvalov et al [7], for 4 different amortisation models, we added the fifth model that is based on the Pareto Harmonic Structure (that means s = 1, of course) at n = 8. The comparative results are shown in Figure 4 and Table 1. Upon consultations with professional accountants, the results were evaluated as rather positive, though accompanied by objections that manual calculations were hard to perform. Besides, each country has tough regulations of procedure under which businessmen shall report the depreciated values, thereby strongly restricting the incentives. And still, it is worth trying.

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Table 1. Comparison of fixed assets residual value

References: 1.V.Pareto. Selected works. Introduction and editorial matter, S.E.Finer.Translation, Basil Blackwell, Pall Mall Press Ltd., 1966. 2.V.Pareto. Manual of Political Economy. New York, 1971. 3.A.L.Barabasi. Linked. The New Science of Networks. Perseus Publishing. 2002. 4.Steven Strogatz. SYNC. The Emerging Science of Spontaneous Order. Theica, New York, 2003. 5. Julian Havil. Gamma. Exploring Eulers Constant. Princeton University Press. 2003. 6. F.Mostellers. 50 .1964. , 1971. 7. A.Bahvalov, V.Bahvalov, A.Idelson. . BHV-Saint-Petersburg. 2001.
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