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On Ehrhart polynomials and probability calculations in voting theory

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  • Dominique Lepelley

    ()

  • Ahmed Louichi

    ()

  • Hatem Smaoui

    ()

Abstract

In voting theory, analyzing how frequent is an event (e.g. a voting paradox) is, under some specific but widely used assumptions, equivalent to computing the exact number of integer solutions in a system of linear constraints. Recently, some algorithms for computing this number have been proposed in social choice literature by Huang and Chua [17] and by Gehrlein ([12, 14]). The purpose of this paper is threefold. Firstly, we want to do justice to Eug`ene Ehrhart, who, more than forty years ago, discovered the theoretical foundations of the above mentioned algorithms. Secondly, we present some efficient algorithms that have been recently developed by computer scientists, independently from voting theorists. Thirdly, we illustrate the use of these algorithms by providing some original results in voting theory.

(This abstract was borrowed from another version of this item.)

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Bibliographic Info

Article provided by Springer in its journal Social Choice and Welfare.

Volume (Year): 30 (2008)
Issue (Month): 3 (April)
Pages: 363-383

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Handle: RePEc:spr:sochwe:v:30:y:2008:i:3:p:363-383

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References

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  1. William Gehrlein, 2004. "Consistency in Measures of Social Homogeneity: A Connection with Proximity to Single Peaked Preferences," Quality & Quantity: International Journal of Methodology, Springer, vol. 38(2), pages 147-171, April.
  2. Pierre Favardin & Dominique Lepelley & Jérôme Serais, 2002. "original papers : Borda rule, Copeland method and strategic manipulation," Review of Economic Design, Springer, vol. 7(2), pages 213-228.
  3. William V. Gehrlein, 2002. "Obtaining representations for probabilities of voting outcomes with effectively unlimited precision integer arithmetic," Social Choice and Welfare, Springer, vol. 19(3), pages 503-512.
  4. Pierre Favardin & Dominique Lepelley, 2006. "Some Further Results on the Manipulability of Social Choice Rules," Social Choice and Welfare, Springer, vol. 26(3), pages 485-509, June.
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Citations

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Cited by:
  1. Wilson, Mark C. & Pritchard, Geoffrey, 2007. "Probability calculations under the IAC hypothesis," Mathematical Social Sciences, Elsevier, vol. 54(3), pages 244-256, December.
  2. Fabrice Barthélémy & Dominique Lepelley & Mathieu Martin, 2013. "On the likelihood of dummy players in weighted majority games," Social Choice and Welfare, Springer, vol. 41(2), pages 263-279, July.
  3. Le Breton, Michel & Lepelley, Dominique & Smaoui, Hatem, 2012. "The Probability of Casting a Decisive Vote: From IC to IAC trhough Ehrhart's Polynomials and Strong Mixing," TSE Working Papers 12-313, Toulouse School of Economics (TSE).
  4. Sebastien Courtin & Mathieu Martin & Issofa Moyouwou, 2013. "The q-Condorcet efficiency of positional rules," Working Papers hal-00914907, HAL.
  5. Mostapha Diss, 2013. "Strategic manipulability of self-­selective social choice rules," Working Papers halshs-00785366, HAL.
  6. Achill Schürmann, 2013. "Exploiting polyhedral symmetries in social choice," Social Choice and Welfare, Springer, vol. 40(4), pages 1097-1110, April.
  7. William Gehrlein & Dominique Lepelley, 2010. "On the probability of observing Borda’s paradox," Social Choice and Welfare, Springer, vol. 35(1), pages 1-23, June.
  8. William v. Gehrlein & Dominique Lepelley, 2009. "A note on Condorcet's other paradox," Economics Bulletin, AccessEcon, vol. 29(3), pages 2000-2007.
  9. Lepelley, Dominique & Merlin, Vincent & Rouet, Jean-Louis, 2011. "Three ways to compute accurately the probability of the referendum paradox," Mathematical Social Sciences, Elsevier, vol. 62(1), pages 28-33, July.
  10. Sébastien Courtin & Boniface Mbih & Issofa Moyouwou, 2012. "Are Condorcet procedures so bad according to the reinforcement axiom?," THEMA Working Papers 2012-37, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
  11. Gehrlein, William V. & Moyouwou, Issofa & Lepelley, Dominique, 2013. "The impact of voters’ preference diversity on the probability of some electoral outcomes," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 352-365.
  12. Sébastien Courtin & Mathieu Martin & Issofa Moyouwou, 2013. "The q-Condorcet efficiency of positional rules," THEMA Working Papers 2013-29, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
  13. Diss, Mostapha & Louichi, Ahmed & Merlin, Vincent & Smaoui, Hatem, 2012. "An example of probability computations under the IAC assumption: The stability of scoring rules," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 57-66.
  14. William Gehrlein & Dominique Lepelley, 2009. "The Unexpected Behavior of Plurality Rule," Theory and Decision, Springer, vol. 67(3), pages 267-293, September.
  15. Cervone, Davide P. & Dai, Ronghua & Gnoutcheff, Daniel & Lanterman, Grant & Mackenzie, Andrew & Morse, Ari & Srivastava, Nikhil & Zwicker, William S., 2012. "Voting with rubber bands, weights, and strings," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 11-27.
  16. Mostapha Diss, 2013. "Strategic manipulability of self-selective social choice rules," Working Papers 1302, Groupe d'Analyse et de Théorie Economique (GATE), Centre national de la recherche scientifique (CNRS), Université Lyon 2, Ecole Normale Supérieure.
  17. Maurice Salles, 2014. "‘Social choice and welfare’ at 30: its role in the development of social choice theory and welfare economics," Social Choice and Welfare, Springer, vol. 42(1), pages 1-16, January.

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