Hostname: page-component-cb9f654ff-p5m67 Total loading time: 0 Render date: 2025-08-23T05:29:13.521Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction of aggregation paradoxes through load-sharing models

Part of: Special processes Distribution theory - Probability Mathematical economics

Published online by Cambridge University Press:  08 August 2022

Emilio De Santis  and
Fabio Spizzichino
Emilio De Santis*
Affiliation:
University of Rome La Sapienza
Fabio Spizzichino*
Affiliation:
University of Rome La Sapienza
*
*Postal address: University of Rome La Sapienza, Department of Mathematics, Piazzale Aldo Moro, 5, 00185, Rome, Italy.
*Postal address: University of Rome La Sapienza, Department of Mathematics, Piazzale Aldo Moro, 5, 00185, Rome, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that load-sharing models (a very special class of multivariate probability models for nonnegative random variables) can be used to obtain basic results about a multivariate extension of stochastic precedence and related paradoxes. Such results can be applied in several different fields. In particular, applications of them can be developed in the context of paradoxes which arise in voting theory. Also, an application to the notion of probability signature may be of interest, in the field of systems reliability.

Keywords

Minima among random variablesorder-dependent load-sharingmajority graphsranking patternsvoting theoryaggregation paradoxes

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60E15: Inequalities; stochastic orderings 91B06: Decision theory

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 223 - 244
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alon, N. (2002). Voting paradoxes and digraphs realizations. Adv. Appl. Math. 29, 126–135.CrossRefGoogle Scholar
Arcones, M. A. and Samaniego, F. J. (2000). On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint. Ann. Statist. 28, 116–150.CrossRefGoogle Scholar
Bachmeier, G. et al. (2019). k-Majority digraphs and the hardness of voting with a constant number of voters. J. Comput. System Sci. 105, 130–157.CrossRefGoogle Scholar
Blyth, C. R. (1972). Some probability paradoxes in choice from among random alternatives. J. Amer. Statist. Assoc. 67, 366–373.CrossRefGoogle Scholar
Blom, G. and Thorburn, D. (1982). How many random digits are required until given sequences are obtained? J. Appl. Prob. 19, 518–531.CrossRefGoogle Scholar
Boland, P. J., Singh, H. and Cukic, B. (2004). The stochastic precedence ordering with applications in sampling and testing. J. Appl. Prob. 41, 73–82.CrossRefGoogle Scholar
De Santis, E. (2021). Ranking graphs through hitting times of Markov chains. Random Structures Algorithms 59, 189–203.CrossRefGoogle Scholar
De Santis, E., Fantozzi, F. and Spizzichino, F. (2015). Relations between stochastic orderings and generalized stochastic precedence. Prob. Eng. Inf. Sci. 29, 329–343.CrossRefGoogle Scholar
De Santis, E., Malinovsky, Y. and Spizzichino, F. (2021). Stochastic precedence and minima among dependent variables. Methodology Comput. Appl. Prob. 23, 187–205.CrossRefGoogle Scholar
De Santis, E. and Spizzichino, F. (2012). First occurrence of a word among the elements of a finite dictionary in random sequences of letters. Electron. J. Prob. 17, 1–9.CrossRefGoogle Scholar
De Santis, E. and Spizzichino, F. (2016). Some sufficient conditions for stochastic comparisons between hitting times for skip-free Markov chains. Methodology Comput. Appl. Prob. 18, 1021–1034.CrossRefGoogle Scholar
Erdős, P. and Moser, L. (1964). On the representation of directed graphs as unions of orderings. Publ. Math. Inst. Hung. Acad. Sci. A 9, 125–132.Google Scholar
Finkelstein, M. and Hazra, N. K. (2021). Generalization of the pairwise stochastic precedence order to the sequence of random variables. Prob. Eng. Inform. Sci. 35, 699–707.CrossRefGoogle Scholar
Fishburn, P. C. (1981). Inverted orders for monotone scoring rules. Discrete Appl. Math. 3, 27–36.CrossRefGoogle Scholar
Foschi, R. Nappo, G. and Spizzichino, F. (2021). Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes. Dependence Modeling 9, 394–423.CrossRefGoogle Scholar
Gehrlein, W. V. and Lepelley, D. (2017). Elections, Voting Rules and Paradoxical Outcomes. Springer, Cham.CrossRefGoogle Scholar
Guibas, L. J. and Odlyzko, A. M. (1981). String overlaps, pattern matching, and nontransitive games. J. Combinatorial Theory A 30, 183–208.CrossRefGoogle Scholar
Hazla, J., Mossel, E., Ross, N. and Zheng, G. (2020). The probability of intransitivity in dice and close elections. Prob. Theory Relat. Fields 178, 951–1009.CrossRefGoogle Scholar
Li, S.-Y. R. (1980). A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann. Prob. 8, 1171–1176.CrossRefGoogle Scholar
Marichal, J.-L. and Mathonet, P. (2011). Extensions of system signatures to dependent lifetimes: explicit expressions and interpretations. J. Multivariate Anal. 102, 931–936.CrossRefGoogle Scholar
McGarvey, D. C. (1953). A theorem on the construction of voting paradoxes. Econometrica 21, 608–610.CrossRefGoogle Scholar
Montes, I., Rademaker, M., Perez-Fernandez, R. and De Baets, B. (2020). A correspondence between voting procedures and stochastic orderings. Europ. J. Operat. Res. 285, 977–987.CrossRefGoogle Scholar
Navarro, J. and Rubio, R. (2010). Comparisons of coherent systems using stochastic precedence. Test 19, 469–486.CrossRefGoogle Scholar
Navarro, J., Spizzichino, F. and Balakrishnan, A.N. (2010). The role of average and projected systems in the study of coherent systems. J. Multivariate Anal. 101, 1471–1482.CrossRefGoogle Scholar
Nurmi, H. (1999). Voting Paradoxes and How to Deal with Them. Springer, Berlin.CrossRefGoogle Scholar
Saari, D. G. (1989). A dictionary for voting paradoxes. J. Econom. Theory 48, 443–475.CrossRefGoogle Scholar
Saari, D. G. (1990). The Borda dictionary. Social Choice Welfare 7, 279–317.CrossRefGoogle Scholar
Saari, D. G. (1995). A chaotic exploration of aggregation paradoxes. SIAM Rev. 37, 37–52.CrossRefGoogle Scholar
Saari, D. G. (2018). Discovering aggregation properties via voting. In New Handbook of Mathematical Psychology, Vol. 2, Cambridge University Press, pp. 271–321.Google Scholar
Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York.CrossRefGoogle Scholar
Savage, R. P., Jr. (1994). The paradox of nontransitive dice. Amer. Math. Monthly 101, 429–436.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1990). Dynamic construction and simulation of random vectors. In Topics in Statistical Dependence (IMS Lecture Notes—Monogr. Ser. 16), Institute of Mathematical Statistics, Hayward, CA, pp. 415–433.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2015). Multivariate conditional hazard rate functions—an overview. Appl. Stoch. Models Business Industry 31, 285–296.CrossRefGoogle Scholar
Shelah, S. (2009). What majority decisions are possible. Discrete Math. 309, 2349–2364.CrossRefGoogle Scholar
Spizzichino, F. (2019). Reliability, signature, and relative quality functions of systems under time-homogeneous load-sharing models. Appl. Stoch. Models Business Industry 35, 158–176.CrossRefGoogle Scholar
Steinhaus, H. and Trybula, S. (1959). On a paradox in applied probabilities. Bull. Acad. Pol. Sci. Ser. Math. 7, 67–69.Google Scholar
Trybula, S. (1969). Cyclic random inequalities. Zastos. Mat. 10, 123–127.Google Scholar