Automation and Remote Control

, Volume 80, Issue 6, pp 1058–1068 | Cite as

The Nontransitivity Problem for Three Continuous Random Variables

  • A. V. LebedevEmail author
Stochastic Systems


The nontransitivity problem of the stochastic precedence relation for three independent random variables with distributions from a given class of continuous distributions is studied. Originally, this issue was formulated in one problem of strength theory. In recent time, nontransitivity has become a popular topic of research for the so-called nontransitive dice. Some criteria are first developed and then applied for proving that nontransitivity may not hold for many classical continuous distributions (uniform, exponential, Gaussian, logistic, Laplace, Cauchy, Simpson, one-parameter Weibull and others). The case of all distributions with a polynomial density on the unit interval is considered separately. Some promising directions of further investigations on the subject are outlined.


nontransitivity nontransitive dice stochastic precedence continuous distributions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Steinhaus, H. and Trybula, S., On a Paradox in Applied Probabilities, Bull. Acad. Polon. Sci., 1959, vol. 7, pp. 67–69.zbMATHGoogle Scholar
  2. 2.
    Trybula, S., On the Paradox of Three Random Variables, Zastos. Mat., 1961, vol. 5, no. 4, pp. 321–332.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Trybula, S., On the Paradox of n Random Variables, Zastos. Mat. (Appl. Math.), 1965, vol. 8, no. 2, pp. 143–156.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Usyskin, Z., Max–min Probabilities in the Voting Paradox, Ann. Math. Stat., 1964, vol. 35, no. 2, pp. 857–862.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bogdanov, I.I., Nontransitive Roulette, Mat. Prosveshch., 2010, vol. 3, no. 14, pp. 240–255.Google Scholar
  6. 6.
    Podd'yakov, A.N., Nontransitivity of Dominance Relations and Decision-Making, Psikhol. Zh. Vyssh. Shkol. Ekon., 2006, no. 3, pp. 88–111.Google Scholar
  7. 7.
    Permogorskii, M.S. and Podd'yakov, A.N., Dominance Relation between Objects and the Nontransitivity of Human Preference for Them, Vopr. Psikhol., 2014, no. 2, pp. 3–14.Google Scholar
  8. 8.
    Podd'yakov, A.N., Nontrasitivity—A Mine for Inventors, Troitskii Variant, 2017, no. 242.Google Scholar
  9. 9.
    Podd'yakov, A., Intransitive Machines. (Accessed September 8, 2018).Google Scholar
  10. 10.
    Vasil'ev, V.A., On the k-accessibility of Cores of TU-cooperative Games, Autom. Remote Control, 2017, vol. 78, no. 12, pp. 2248–2264.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Larichev, O.I., Properties of the Decision Methods in the Multicriteria Problems of Individual Choice, Autom. Remote Control, 2002, vol. 63, no. 2, pp. 304–315.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gorokhovik, V.V. and Trofimovich, M.A., First- and Second-Order Optimality Conditions for Vector Optimization Problems with a Nontransitive Preference Relation, Tr. Inst. Mat. Mekh. Ur. Otd. Ross. Akad. Nauk, 2014, vol. 20, no. 4, pp. 81–96.Google Scholar
  13. 13.
    Arzhenenko, A.Yu., Kazakova, O.G., and Chugaev, B.N., Optimization of Binary Questionnaries, Autom. Remote Control, 1985, vol. 46, no. 11, pp. 1466–1472.zbMATHGoogle Scholar
  14. 14.
    Lepskiy, A.E., Stochastic and Fuzzy Ordering with the Method of Minimal Transformations, Autom. Remote Control, 2017, vol. 78, no. 1, pp. 50–66.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Boland, P.J., Singh, H., and Cukic, B., The Stochastic Precedence Ordering with Applications in Sampling and Testing, J. Appl. Probab., 2004, vol. 41, no 1, pp. 73–82.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Arcones, M.A., Kvam, P.H., and Samaniego, F.J., Nonparametric Estimation of a Distribution Subject to a Stochastic Precedence Constraint, J. Am. Stat. Assoc., 2002, vol. 97, no. 457, pp. 170–182.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shakhnov, I.F., A Problem of Ranking Interval Objects in a Multicriteria Analysis of Complex Systems, J. Comp. Syst. Sci. Int., 2008, vol. 47, no. 1, pp. 33–39.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Martinetti, D., Montes, I., Diaz, S., and Montes, S., A Study on the Transitivity of Probabilistic and Fuzzy Relations, Fuzzy Sets Syst., 2011, vol. 184, pp. 156–170.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sekei, G., Paradoksy v teorii veroyatnostei i matematicheskoi statistike (Paradoxes in Probability Theory and Mathematical Statistics), Moscow: Inst. Komp. Issled., 2003.Google Scholar
  20. 20.
    Gardner, M., The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference, Sci. Am., 1970, vol. 223, no. 6, pp. 110–114.CrossRefGoogle Scholar
  21. 21.
    Gardner, M., On the Paradoxical Situations That Arise from Nontransitive Relations, Sci. Am., 1974, vol. 231, no. 6, pp. 120–125.CrossRefGoogle Scholar
  22. 22.
    Savage, R., The Paradox of Nontransitive Dice, Am. Math. Monthly, 1994, vol. 101, no. 5, pp. 429–436.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Bozoki, S., Nontransitive Dice Sets Releazing the Paley Tournament for Solving Shütte's Tournament Problem, Miskolc Math. Notes, 2014, vol. 15, no. 1, pp. 39–50.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Conrey, B., Gabbard, J., Grant, K., Liu, A., and Morrison, K.E., Intransitive Dice, Math. Mag., 2016, vol. 89, pp. 133–143.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Buhler, I., Graham, R., and Hales, A., Maximally Nontransitive Dice, Am. Math. Monthly, 2018, vol. 125, no. 5, pp. 387–399.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Weibull, W., A Statistical Theory of the Strength of Materials, Stockholm: Generalstabens Litografiska Anstalts Förlag, 1939.Google Scholar
  27. 27.
    Bulinskaya, E.V., Teoriya riska i perestrakhovanie (Theory of Risk and Reinsurance), Moscow: OOO Meiler, 2008.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations