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Voting Model for Demographic Decision Making
O.A. Malafeyev Saint-Petersburg State University malafeyevoa@mail.ru
A.A.Cherkasova Saint-Petersburg State University st087116@student.spbu.ru
V.F. Bogachov Institute of regional economics vic-bogachev@mail.ru
Elections, as an integral part of a democratic state, are held in many countries around the world. Each country has its own unique characteristics related to their experience, popular traditions, system of government and political regime. The question of whether elections are democratic and whether the results of elections are in accordance with the will of the people is a subject of debate among politicians, journalists, scholars and ordinary citizens.
Key words: relative majority rule, de Borda rule, Copeland rule, candidates, election, model.
The question of organizing a proper electoral system has been debated since the 18th century, when elections were just becoming a popular way of electing candidates. Mathematicians such as Jean Charles de Bordeaux, J.A. Condorcet, C. May and Kenneth J. Arrow have contributed to this problem. Kenneth J. Arrow was even awarded the Nobel Prize in economics. An electoral system is a set of legal rules that determine how voting results are translated into election results.
In this paper we will be looking at different voting rules such as: majority rule, de Borda rule, Copeland rule and problems based on these rules. This work makes use ofthe ideas ofthe articles [1-150].
Section 1. Application ofthe relative majority rule
Relative majority rule is the principle that the winner is defined as the candidate who receives the most votes, but not necessarily more than half of all votes. This means that a candidate can be elected even if he or she received less than 50% of the votes ifhis or her result exceeds that ofthe other candidates.
We should discuss the difficulty of collective choice, in line with Condorc and Borde's theories. Here we consider the problem of voting, where several electors have to choose one of the proposed candidates. The candidates may be different - people, versions of the law, and so on. When we have only two candidates, the usual majority rule is the simplest and most reasonable. However, difficulties arise when the number of candidates increases. For the purpose, the case of three candidates will be considered.
The relative majority rule, known as the "plurality rule", is the simplest and most popular extension of the majority rule for the case with multiple candidates. In this rule, each participant must cast a vote for only one candidate, and the candidate with the most votes is the winner. However, Condorcet and Borda realized the shortcomings of this rule. To illustrate them, consider the following preference table: a, b, c.
Table 1.
In this paper, the > sign will indicate the preference of one candidate over another. This table shows that 6 voters prefer a to candidate b, b is preferred to c for 5 voters, and 4 voters have c preferred to b and b preferred to a. According to the relative majority rule, a will be the chosen winner. Bord and Condorce notice that for the majority of voters (9 out of 15) a is the worst option, so the choice of a cannot be considered an adequate majority choice. Thus, the relative majority rule cannot be recognized as reasonable and fair.
The winner in this case must be candidate b, since he defeats both c (11 out of 15) and a (9 out of 15) in a pairwise vote based on majority rule.
Section 2. Example ofapplication ofthe de Borda rule
In the candidate evaluation process, each voter builds his or her own preference profile, putting the best candidate in order and moving to the worst, e.g., a > b > c when there are three candidates. Points are then assigned to each candidate: the worst gets 0 points, the next worst gets one point, and so on. De Borda score is then computed for each candidate, which is the sum of the points received from all agents. The candidate with the highest score is declared the winner.
Consider the same table (1), with the addition of points according to the de Borda rule.
Table 2.
In this way the candidates get the following number ofpoints:
a: 2*6 + 0*5 + 0*4= 12 b: 1*6 + 2*5 + 1*4 = 20 c: 0*4+ 1*5 + 2*4= 13 Whence it follows that candidate b wins by de Borda's rule.
Section 3. Example of application of Copeland's rule.
The Copeland score for each candidate a is calculated based on a pairwise comparison using the majority principle, where the difference between the number x of candidates who lost to candidate a and the number of candidates y who won against it is determined. The candidate with the highest Copeland score is considered the winner.
Let us consider the following preference table (2) and prove for it that de Borda's and Copeland's scores place the candidates in the opposite order.
Table 3.
Let's transform the table to make it convenient to calculate the De Borda estimate.
CONCORDE, 2024, N2 Table 4.
4 3 2 10
1 a > b > c > d > e
4 c > d > b > e > a
1 e> a>d>b >c
3 e> a>b>d >c
Let's calculate the candidates' points: a:4*l + 0*4 + 3*1 + 3*3 = 4 + 3 + 9=16 b:3*l+ 2*4 +1*1 + 2*3 = 3 + 8+1 + 6=18 c:2*l + 4*4 + 0*1 + 0*1=2+ 16= 18 d: 1*1 + 3*4 + 2*1 + 1*3 = 1 + 12 + 2 + 3 = 18 e: 0*1 +1*4 + 4*1 + 4*3 =4 + 4 +12 = 20
Thus the ranking ofthe candidates will be as follows:
Table 5.
e 20
b 18
c 18
d 18
a 16
By de Borda's rule, candidate e wins.
Now consider Copeland's rule. We start with candidate a. Candidate a is preferred to candidate b in 5 out of 9 cases. This is more than half, hence in a pairwise comparison, candidate a wins.
In a pairwise comparison with candidate c, candidate a will be favored by 5 voters. It turns out that in this case candidate a is preferred.
Reasoning similarly we get that a wins over d (5 out of 9) and loses to e (1 out of 9). Thus the score will be: 3 - 1 = 2
For candidate b the score will be 0, similarly for c and d, also 0. For candidate e, the score will be-1.
The ranking ofthe candidates will look as follows:
Table 6.
a 2
b 0
c 0
d 0
e -1
We can notice that by Copeland's rule candidate a wins and candidate e loses. It is interesting that these results are opposite to the results obtained by de Borda's rule.
Section 4. Election problem.
Consider the following model. There are three candidates: a, b, c. Each of them has a certain amount of resource - millions, which candidates can spend on their promotion - advertising to increase the final number of points. Candidate a has 2 millions, b and c have 4 millions each. Voting will be held in two stages. Points will be calculated according to the de Borda rule. Pre-known limits are set - how much each candidate can spend at the first and second stages: candidate a -1 million rubles, b and c-no more than 2 millions. Initially, the table ofpreferences has the following form:
Table 7.
№1 a > b>c
№2 b>a>c
№3 c >a> b
Candidates have the following scores on this table: a-4,b- 3,c - 2.
Each candidate can change the opinion of only one voter, and they must not repeat themselves. If a candidate wants to promote himself in one position, he invests 1 million on his advertising, if in two positions, then 2 millions.
So, for example, on the first move candidate a can change the opinion of voter №3 by spending 1 million on advertising, then his position will change as follows: a > c > b. If candidate c spends 2 millions on his promotion, then he will change the opinion of voter №1 as follows: c > a > b. The order in which the candidates spend resources is known: a - b - c.
The decision tree is given below. It is a dynamic game with complete information. A voter who changes his mind is indicated on the left (the number of millions spent on this move is indicated in parentheses iftwo choices are possible).
Since this is a game with complete information, it is possible to find the optimal path. These moves are highlighted in green. The numbers in the diagram indicate the number of choices in which the candidate choosing this move can win.
Thus, the optimal path is the following one.
First stage: candidate a spends 1 million on advertising, opinion changes voter №2, candidate b spends 2 millions on voter №3 and c spends 1 millions on advertising voter №1.
The second stage: a -1 million rubles, voter №3, b -1 million rubles, voter №2, and candidate c isina deadlock and can no longer spend the resource on promotion.
According to the voting results, candidate b wins in the optimal strategy.
It should be noted that in this case an example for two stages is considered, since in another case it would be quite difficult to display the scheme. In general, it is possible to consider this problem for a large number of stages.
Figure 1. Decision Trees
Fig.l
Figure 2. Decision Trees
Figure 3. Decision Trees
Conclusion. Thus, different voting rules have been analyzed in this article. It can be seen that different approaches give different results and depending on the situation it is worth applying one or another method. That is, it is impossible to choose one effective method for all cases, an individual approach is required.
The de Borda voting model was considered for the case of three voters and three candidates, because in other cases it would be difficult to represent the tree of possible solutions on the diagram. This problem can be considered for a larger number ofparticipants.
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98. Malafeyev O.A., Stohasticheskoe ravnovesie v obshchej modeli konkurentnoj dinamiki// V sbornike: Matematicheskoe modelirovanie i prognoz social'no-ekonomicheskoj dinamiki v usloviyah konkurencii i neopredelennosti - 2004 - S. 143-154.
99. Malafeyev O.A., Eremin D.S. Konkurentnaya linejnaya model' ekonomiki // V sbornike: Processy upravleniya i ustojehivost' - 2008 - S. 425-435.
100. Malafeyev O.A., Redinskikh N.D. Quality estimation of the geopolitical actor development strategy// CNSA 2017 Proceedings - 2017-http://dx.doi.org/10.1109/CNSA.2017.7973986
101. Ivashov L.G., Kefeli I.F., Malafeyev O.A. Global'naya arkticheskaya igra i ee uchastniki// Geopolitika i bezopasnost' -2014 - № 1 (25) - S. 34-49.
102. Kolokoltsov V.N., Malafeyev O.A., Mean field game model of corruption // arxiv.org https://arxiv.org/abs/1507.03240
103. Zaitseva I.V., Malafeyev O.A., Zakharov V.V., Zakharova N.I., Orlova A.Yu. Dynamic distribution of labour resources by region of investment // Journal of Physics: Conference Series. - 2020 - S. 012073.
104. Malafeyev O.A. Sushchestvovanie situacij ravnovesiya v differencial'nyh beskoalicionnyh igrah so mnogimi uchastnikami // Vestnik Leningradskogo universiteta. Seriya 1: Matematika, mekhanika, astronomiya. - 1982 -№13- S. 40- 46.
105. Malafeyev O.A., Petrosyan L.A. Igra prostogo presledovaniya na ploskosti s prepyatstviem //Upravlyaemye sistemy - 1971 -№9-S. 31-42.
106. Malafeyev O. A, Galtsov M., Zaitseva I., Sakhnyuk P., Zakharov V., Kron R. Analysis of trading algorithms on the platform QIUK // Proceedings - 2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency - 2020 - S. 305-311.
107. Malafeyev O.A. Ustojchivye beskoalicionnye igry N lie // Vestnik Leningradskogo universiteta. Seriya 1: Matematika, mekhanika, astronomiya. - 1978 - № 4 -S. 55-58.
108. Malafeyev O.A., CHernyh K.S. Prognosticheskaya model' zhiznennogo cikla firmy v konkurentnoj srede // V sbornike: Matematicheskoe modelirovanie i prognoz social'noekonomicheskoj dinamiki v usloviyah konkurencii i neopredelennosti. - 2004 - S. 239-255.
109. Drozdov G.D., Malafeyev O.A. Modelirovanie tamozhennogo dela - SPb: Izd-vo SPbGUSE -2013 -255 s.
110. Pichugin YU.A., Malafeyev O.A., Alferov G.V. Ocenivanie parametrov v zadachah konstruirovaniya mekhanizmov robotov-manipulyatorov // V sbornike: Ustojchivost' i processy upravleniya. Materialy III mezhdunarodnoj konferencii - 2015 - S. 141-142.
111. Malafeyev O.A., Zajceva I.V., Komarov A.A., SHvedkova T.YU., Model' korrupcionnogo vzaimodejstviya mezhdu kommercheskoj organizaciej i otdelom po bor'be s korrupciej // V knige: Linejnaya algebra s prilozheniyami k modelirovaniyu
korrupcionnyh sistem i processov - 2016 - S. 342-351.
112. Malafeyev O.A., Strekopytova O.S. Teoretiko-igrovaya model' vzaimodejstviya korrumpirovannogo chinovnika s klientom: psihologicheskie aspekty // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov. Malafeyev O.A., i dr. Kollektivnaya monografiya. Pod obshchej redakciej d.f.-m.n., professora O. A. Malafeyeva - 2016. S. 134151.
113. Zaitseva I.V., Malafeyev O.A., Zakharov V.V., Smirnova T.E., Novozhilova L.M. Mathematical model of network flow control // IOP Conference Series: Materials Science and Engineering - 2020 -S.012036.
114. Malafeyev O.A., Petrosyan L.A. Differencial'nye mnogokriterial'nye igry so mnogimi uchastnikami // Vestnik Leningradskogo universiteta. Seriya 1: Matematika, mekhanika, astronomiya - 1989 -№3-S. 27-31.
115. Malafeyev O.A., Kefeli I.F. Nekotorye zadachi obespecheniya oboronnoj bezopasnosti// Geopolitika i bezopasnost' - 2013 - № 3 (23) - S. 84-92.
116. Malafeyev O.A., Redinskih N.D. Stohasticheskij analiz dinamiki korrupcionnyh gibridnyh setej // V sbornike: Ustojchivost' i kolebaniya nelinejnyh sistem upravleniya (konferenciya Pyatnickogo) -2016-S. 249-251.
117. Malafeyev O.A., Farvazov K.M. Statisticheskij analiz indikatorov korrupcionnoj deyatel'nosti v sisteme gosudarstvennyh i municipal'nyh zakupok // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 209-217.
118. Malafeyev O.A., Zajceva I.V., Zenovich O.S., Rumyancev N.N., Grigor'eva K.V., Ermakova A.N., Rezen'kov D.N., SHlaev D.V. Model' raspredeleniya resursov pri vzaimodejstvii korrumpirovannoj struktury s antikorrupcionnym agentstvom // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 102-106.
119. Malafeyev O., Parfenov A., Smirnova T., Zubov A., Bondarenko L., Ugegov N., Strekopytova M., Strekopytov S.,Zaitseva I. Game-theoretical model of cooperation between producers // V sbornike: AIP Conference Proceedings - 2019 - S. 45-59.
120. Neverova E.G., Malafeyev O.A., Alferov G.V. Nelinejnaya model' upravleniya antikorrupcionnymi meropriyatiyami // V sbornike: Ustojchivost' i processy upravleniya. Materialy III mezhdunarodnoj konferencii - 2015 - S. 445-446.
121. Malafeyev O.A., Marahov V.G. Evolyucionnyj mekhanizm dejstviya istochnikov i dvizhushchih sil grazhdanskogo obshchestva v sfere finansovoj i ekonomicheskoj komponenty XXI veka // V sbornike: K. Marks i budushchee filosofii Rossii.
Busov S.V., Dudnik S.I. i dr. - 2016. S. 112-135.
122. Malafeyev O.A., Borodina T.S., Kvasnoj M.A., Novozhilova L.M., Smirnov I.A. Teoretikoigrovaya zadacha o vliyanii konkurencii v korrupcionnoj srede // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016-S. 88-96.
123. Malafeyev O.A., Zajceva I.V., Koroleva O.A., Strekopytova O.S. Model' zaklyucheniya kontraktov s vozmozhno korrumpirovannym chinovnikom-principalom // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 114-124.
124. Malafeyev O.A., Sajfullina D.A. Mnogoagentnoe vzaimodejstvie v transportnoj zadache s korrupcionnoj komponentoj // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 218-224.
125. Malafeyev O.A., Neverova E.G., Smirnova T.E., Miroshnichenko A.N. Matematicheskaya model' processa vyyavleniya korrupcionnyh elementov v gosudarstvennoj sisteme upravleniya // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov -2016 - S. 152-179.
126. Malafeyev O.A., Koroleva O.A., Neverova E.G. Model' aukciona pervoj ceny s vozmozhnoj korrupciej // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 96-102.
127. Malafeyev O.A., Neverova E.G., Petrov A.N. Model' processa vyyavleniya korrupcionnyh epizodov posredstvom inspektirovaniya otdelom po bor'be s korrupciej // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov -2016 - S. 106-114.
128. Malafeyev O.A., Parfenov A.P. Dinamicheskaya model' mnogoagentnogo vzaimodejstviya mezhdu agentami korrupcionnoj seti // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 55-63.
129. Malafeyev O.A., Andreeva M.A., Gus'kova YU.YU., Mal'ceva A.S. Poisk podvizhnogo ob"ekta pri razlichnyh informacionnyh usloviyah // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 63-88.
130. Zajceva I.V., Popova M.V., Malafeyev O.A. Postanovka zadachi optimal'nogo raspredeleniya trudovyh resursov po predpriyatiyam s uchetom izmenyayushchihsya uslovij // V knige: Innovacionnaya ekonomika i promyshlennaya politika regiona (EKOPROM-2016) -2016 - S. 439-443.
131. Malafeyev O.A., Novozhilova L.M., Redinskih N.D., Rylov D.S., Gus'kova YU.YU. Teoretiko-igrovaya model' raspredeleniya korrupcionnogo dohoda // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 125-134.
132. Malafeyev O.A., Salimov V.A., SHarlaj A.S. Algoritm ocenki bankom kreditosposobnosti klientov pri nalichii korrupcionnoj sostavlyayushchej // Vestnik Permskogo universiteta. Seriya: Matematika. Mekhanika. Informatika - 2015 - № 1 (28) - S. 35-38.
133. Kefeli I.E, Malafeyev O.A. Problemy ob"edineniya interesov gosudarstv EAES, SHOS i BRIKS v kontekste teorii kooperativnyh igr // Geopolitika i bezopasnost' - 2015 - № 3 (31) - S. 33-41.
134. Kefeli I.F., Malafeyev O.A., Marahov V.G. i dr. Filosofskie strategii social'nyh preobrazovanij XXI veka - SPb: Izd-vo SPbGU -2014- 144 s.
135. Malafeyev O.A., Andreeva M.A., Alferov G.V., Teoretiko-igrovaya model' poiska i perekhvata v N-sektornom regione ploskosti // Processy upravleniya i ustojchivost' - 2015 -Vol. 2.№1- S. 652-658.
136. Zajceva I.V., Malafeyev O.A. Issledovanie korrupcionnyh processov i sistem matematicheskimi metodami //Innovacionnye tekhnologii v mashinostroenii, obrazovanii i ekonomike - 2017 - Vol. 3. № 1-1 (3) - S. 7-13.
137. Kolchedancev L.M., Legalov I.N., Bad'in G.M., Malafeyev O.A., Aleksandrov E.E., Gerchiu A.L., Vasil'ev YU.G.Stroitel'stvo i ekspluataciya energoeffektivnyh zdanij (teoriya i praktika s uchetom korrupcionnogo faktora) (Passivehouse) - Borovichi: NP "NTO strojindustrii Sankt-Peterburga" -2015 - 170 S.
138. Malafeyev O.A., Novozhilova L.M., Kvasnoj M.A., Legalov I.N., Primenenie metodov setevogo analiza pri proizvodstve energoeffektivnyh zdanij s uchetom korrupcionnogo faktora // V knige: Stroitel'stvo i ekspluataciya energoeffektivnyh zdanij (teoriya i praktika s uchetom korrupcionnogo faktora) (Passivehouse) -2015-S. 146-161.
139. Malafeyev O., Awasthi A., Zaitseva I., Rezenkov D., Bogdanova S. A dynamic model of functioning of a bank // AIP Conference Proceedings. International Conference on Electrical, Electronics, Materials and Applied Science - 2018 - S. 020042.
140. Malafeyev O.A. Obzor literatury po modelirovaniyu korrupcionnyh sistem i processov, ch.I // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov -2016 - S. 9-17.
141. Asaul A.N., Lyulin P.B., Malafeyev O.A. Matematicheskoe modelirovanie vzaimodejstvij organizacii kak zhivoj sistemy // Vestnik Hmel'nickogo nacional'nogo universiteta Ekonomicheskie nauki - 2013 - № 6-2 (206) - S. 215-220.
142. Awasthi A., Malafeyev O.A. Is the indian dtock market efficient - a comprehensive study of Bombay stock exchange indices // arxiv.org -https://arxiv. org/abs/1510.03704
143. Malafeyev O.A., Demidova D.A. Modelirovanie processa vzaimodejstviya korrumpirovannogo predpriyatiya federal'nogo otdela po bor'be s korrupciej // V knige: Vvedenie v modelirovanie korrupcionnyh sistem i processov - 2016 - S. 140-152.
144. Malafeyev O.A., Chernyh K.S. Matematicheskoe modelirovanie razvitiya kompanii // Ekonomicheskoe vozrozhdenie Rossii - 2005 -№2-S.23.
145. Kulakov F.M., Alferov G.V., Malafeyev O.A. Kinematicheskij analiz ispolnitel'noj sistemy manipulyacionnyh robotov // Problemy mekhaniki i upravleniya: Nelinejnye dinamicheskie sistemy - 2014 -№46- S. 31-38.
146. Kulakov F.M., Alferov G.V., Malafeyev O.A. Dinamicheskij analiz ispolnitel'noj sistemy manipulyacionnyh robotov // Problemy mekhaniki i upravleniya: Nelinejnye dinamicheskie sistemy - 2014 -№46- S. 39-46.
147. Zajceva I.V., Malafeyev O.A., Stepkin A.V., CHernousov M.V., Kosoblik E.V. Modelirovanie ciklichnosti razvitiya v sisteme ekonomik // Perspektivy nauki - 2020 - № 10 (133) - S. 173-176.
148. Bure V.M., Malafeyev O.A., Some game-theoretical models of conflict in finance //Nova Journal ofMathematics, Game Theory, and Algebra. - 1996 - T.6.№1- S. 7-14.
149. Malafeyev O.A., Redinskih N.D., Parfenov A.P, Smirnova T.E. Korrupciya v modelyah aukciona pervoj ceny // V sbornike: Instituty i mekhanizmy innovacionnogo razvitiya: mirovoj opty i rossijskaya praktika. - 2014 - S. 250-253.
150. Marahov V.G., Malafeyev O.A. Dialog filosofa i matematika: «0 filosofskih aspektah matematicheskogo modelirovaniya social'nyh preobrazovanij XXI veka» // V sbornike: Filosofiya poznaniya i tvorchestvo zhizni. Sbornik statej - 2014 - S. 279-292
