Evaluating the likelihood of the referendum paradox
for mixed voting systems
Michel Le Breton ∗, Dominique Lepelley †, Vincent Merlin
‡
Abstract
A referendum paradox (Nurmi, 1999) occurs, in a two party competition, each time a party gets a
majority of the seats in the parliament while it did not obtain a majority of votes nationwide. The
election of George W. Bush in 2000 (with 271 electoral vote against 267, but less votes than Albert
Gore) can be viewed as an occurrence of the paradox. In fact, any two tiers voting systems can be
plagued by this flaw. This paradox can be also viewed as an instance of the Borda paradox, as the
voting rules fails to select the Condorcet winner. Feix, Lepelley, Merlin and Rouet (2004), Wilson MC,
G Pritchard (2007) and Lepelley, Merlin and Rouet (2011) computed the probability of the referendum
paradox under the IC and IAC assumptions when two parties compete in A equal sized districts. A
priori models for voting are extensively described in Gerhlein (2006).
The same paradox may occur for mixed electoral systems. On the top of electing A representatives
in districts, the voters also elect D members of the parliament at large. Hence, the parliament is of size
A+D. Blais and Massicote (2009) propose an extensive survey of all the mixed electoral systems that
are used worldwide. In this paper, assuming that A representative are elected in equal size jurisdictions,
we estimate the probability of the referendum paradox for different mixed systems : 1) when the D at
large seats are apportioned according to the proportional rule and 2) when the all the D at large seats
are attributed to the party which obtained a majority of votes nationwide
Keywords : referendum paradox, mixed voting rules, probability calculations, IAC hypothesis, IC hypothesis.
JEL Classification : C9, D72
∗ Institut
Universitaire de France and Toulouse School of Economics, France. E-mail: [email protected]
CEMOI, Faculté de Droit et d’Economie, Université de La Réunion, 97715 Saint-Denis cedex 9 France. E-mail:
[email protected]
‡ Normandie Université, UCBN, CREM CNRS UMR6211. UFR des sciences économiques et de gestion, 19 rue Claude
Bloch, 14032 Caen cedex, France. E-mail: [email protected]
†
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References
[1] Blais B and L Massicote (2009) Mixed electoral systems: a conceptual and empirical survey.
Electoral Studies 18: 341-366.
[2] Feix M, D Lepelley, V Merlin and JL Rouet (2004) The probability of paradox in a U.S. presidential
type election. Economic Theory 23: 227-257.
[3] Gehrlein W, Condorcet’s Paradox. Springer Publishing, 2006,
[4] Lepelley D, Merlin V and JL Rouet (2011) Three ways to compute accurately the probability of
the referendum paradox, Mathematical Social Sciences 62-1: 28-33.
[5] Nurmi H (1999). Voting Paradoxes, and how to deal with them. Springer.
[6] Wilson MC, G Pritchard (2007) Probability calculations under the IAC hypothesis. Mathematical
Social Sciences 54: 244-256.
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