Fundraising and vote distribution: A non-equilibrium statistical approach

Autoři: Hygor P. M. Melo aff001;  Nuno A. M. Araújo aff001;  José S. Andrade, Jr. aff004
Působiště autorů: Centro de Física Teórica e Computacional, Universidade de Lisboa, Lisboa, Portugal aff001;  Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Avenida Des. Armando de Sales Louzada, Acaraú, Ceará, Brazil aff002;  Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal aff003;  Departamento de Física, Universidade Federal do Ceará, Fortaleza, Ceará, Brazil aff004
Vyšlo v časopise: PLoS ONE 14(10)
Kategorie: Research Article
doi: 10.1371/journal.pone.0223059


The number of votes correlates strongly with the money spent in a campaign, but the relation between the two is not straightforward. Among other factors, the output of a ballot depends on the number of candidates, voters, and available resources. Here, we develop a conceptual framework based on Shannon entropy maximization and Superstatistics to establish a relation between the distributions of money spent by candidates and their votes. By establishing such a relation, we provide a tool to predict the outcome of a ballot and to alert for possible misconduct either in the report of fundraising and spending of campaigns or on vote counting. As an example, we consider real data from two proportional elections with more than 6000 candidates each, where a detailed data verification is virtually impossible, and show that the number of potential misconducting candidates to audit can be reduced to less than ten.

Klíčová slova:

Brazil – Decision making – Elections – Entropy – Fats – Finance – Probability distribution – Statistical distributions


1. UN Secretary General. Guidance Note of the Secretary-General on Democracy. 2009.

2. International Institute for Democracy and Electoral Assistance. International Electoral Standards: Guidelines for Reviewing the Legal Framework of Elections. 2002.

3. Melo HPM, Reis SD, Moreira AA, Makse HA, Andrade JS. The price of a vote: Diseconomy in proportional elections. PloS One. 2018;13(8):e0201654. doi: 10.1371/journal.pone.0201654 30133469

4. Jacobson GC. The effects of campaign spending in congressional elections. American Political Science Review. 1978;72:469–491. doi: 10.2307/1954105

5. Morton R, Cameron C. Elections and the theory of campaign contributions: A survey and critical analysis. Economics & Politics. 1992;4:79–108. doi: 10.1111/j.1468-0343.1992.tb00056.x

6. Gerber AS. Does campaign spending work? Field experiments provide evidence and suggest new theory. American Behavioral Scientist. 2004;47:541–574. doi: 10.1177/0002764203260415

7. Gordon SC, Hafer C, Landa D. Consumption or investment? On motivations for political giving. The Journal of Politics. 2007;69(4):1057–1072. doi: 10.1111/j.1468-2508.2007.00607.x

8. Costa Filho RN, Almeida MP, Andrade JS, Moreira JE. Scaling behavior in a proportional voting process. Physical Review E. 1999;60:1067. doi: 10.1103/PhysRevE.60.1067

9. Costa Filho RN, Almeida MP, Moreira JE, Andrade JS. Brazilian elections: voting for a scaling democracy. Physica A: Statistical Mechanics and its Applications. 2003;322:698–700. doi: 10.1016/S0378-4371(02)01823-X

10. Castellano C, Fortunato S, Loreto V. Statistical physics of social dynamics. Reviews of Modern Physics. 2009;81(2):591. doi: 10.1103/RevModPhys.81.591

11. Mantovani MC, Ribeiro HV, Moro MV, Picoli S Jr, Mendes RS. Scaling laws and universality in the choice of election candidates. EPL (Europhysics Letters). 2011;96:48001. doi: 10.1209/0295-5075/96/48001

12. Mantovani MC, Ribeiro HV, Lenzi EK, Picoli S Jr, Mendes RS. Engagement in the electoral processes: scaling laws and the role of political positions. Physical Review E. 2013;88:024802. doi: 10.1103/PhysRevE.88.024802

13. Bokányi E, Szállási Z, Vattay G. Universal scaling laws in metro area election results. PloS One. 2018;13:e0192913. doi: 10.1371/journal.pone.0192913 29470518

14. Moreira AA, Paula DR, Costa Filho RN, Andrade JS. Competitive cluster growth in complex networks. Physical Review E. 2006;73:065101. doi: 10.1103/PhysRevE.73.065101

15. Araújo NAM, Andrade JS, Herrmann HJ. Tactical voting in plurality elections. PloS One. 2010;5:e12446. doi: 10.1371/journal.pone.0012446 20856800

16. Fernández-Gracia J, Suchecki K, Ramasco JJ, San Miguel M, Eguíluz VM. Is the voter model a model of voters? Physical Review Letters. 2014;112:089903. doi: 10.1103/PhysRevLett.113.089903

17. Calvão AM, Crokidakis N, Anteneodo C. Stylized facts in brazilian vote distributions. PloS One. 2015;10:e0137732. doi: 10.1371/journal.pone.0137732 26418863

18. Borghesi C, Raynal JC, Bouchaud JP. Election Turnout Statistics in Many Countries: Similarities, Differences, and a Diffusive Field Model for Decision-Making. Plos One. 2012;7:e36289. doi: 10.1371/journal.pone.0036289 22615762

19. Fortunato S, Castellano C. Scaling and universality in proportional elections. Physical Review Letters. 2007;99:138701. doi: 10.1103/PhysRevLett.99.138701 17930647

20. Lehoucq F. Electoral fraud: Causes, types, and consequences. Annual Review of Political Science. 2003;6:233–256. doi: 10.1146/annurev.polisci.6.121901.085655

21. Alvarez RM, Hall TE, Hyde SD. Election fraud: detecting and deterring electoral manipulation. Brookings Institution Press; 2009.

22. Deckert J, Myagkov M, Ordeshook PC. Benford’s Law and the detection of election fraud. Political Analysis. 2011;19:245–268. doi: 10.1093/pan/mpr014

23. Klimek P, Yegorov Y, Hanel R, Thurner S. Statistical detection of systematic election irregularities. Proceedings of the National Academy of Sciences. 2012;109(41):16469–16473. doi: 10.1073/pnas.1210722109

24. Beber B, Scacco A. What the numbers say: A digit-based test for election fraud. Political Analysis. 2012;20:211–234. doi: 10.1093/pan/mps003

25. Enikolopov R, Korovkin V, Petrova M, Sonin K, Zakharov A. Field experiment estimate of electoral fraud in Russian parliamentary elections. Proceedings of the National Academy of Sciences. 2013;110:448–452. doi: 10.1073/pnas.1206770110

26. Beck C, Cohen EGD. Superstatistics. Physica A: Statistical Mechanics and its Applications. 2003;322:267–275. doi: 10.1016/S0378-4371(03)00019-0

27. Jaynes ET. Information theory and statistical mechanics. II. Physical review. 1957;108:171. doi: 10.1103/PhysRev.108.171

28. Dataset for the 2014 election for federal deputies in Brazil, from

29. Dataset for the 2018 election for federal deputies in Brazil, from

30. Chatterjee A, Mitrović M, Fortunato S. Universality in voting behavior: an empirical analysis. Scientific Reports. 2013;3:1049. doi: 10.1038/srep01049

31. Brent RP. Algorithms for minimization without derivatives. Courier Corporation; 2013.

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2019 Číslo 10