Marie-Edith Bissey, Mauro Carini and Guido Ortona (2004)
ALEX3: a Simulation Program to Compare Electoral Systems
Journal of Artificial Societies and Social Simulation
vol. 7, no. 3
<https://www.jasss.org/7/3/3.html>
To cite articles published in the Journal of Artificial Societies and Social Simulation, reference the above information and include paragraph numbers if necessary
Received: 05-Nov-2003 Accepted: 11-May-2004 Published: 30-Jun-2004
(1) |
where j refers to the electoral system, n is the number of parties, S_{j,i} is the number of seats obtained by party i under system j, S_{pp,i} is the number of seats obtained by party i under perfect proportionality rule (PPR), and S_{u,i} is the number of seats obtained by party i if all the seats go to the largest party in system j ^{[6]}.
g_{f} / [1/m - 1/(m+1)] = (f-t/2)/(t-t/2) |
which yields
g_{f} = [1/m - 1/(m+1)] (f-t/2)/(t/2) |
where t is the total number of seats in the Parliament.
g = g_{m} +g_{f} = 1/(m+1) + [1/m - 1/(m+1)] (f-t/2)/(t/2) |
where m is the number of crucial parties supporting the Government, f is the number of seats of the majority and t is the total number of seats. The value of g reaches its maximum, 1, when a party has all the seats, and decreases with the increase of m, thus justifying the claim that the range of g is the interval (0,1].
U_{x} > U_{y} iff Ag^{a}r^{b } > AG^{a}R^{b } | (1) |
where we write for simplicity the values of g and r for X in lower-case and those for Y in upper-case.
U_{x} > U_{y} iff (g/G)^{bp} > (R/r)^{b} |
Hence the condition may be written as
pLn(g/G) > ln(R/r) | (2) |
i.e.
p > ln(R/r)/Ln(g/G) if g > G or p < ln(R/r) / ln(g/G) | (3) |
provided that g, or G, or both are < 1^{[12]}.
Table 1: r and g for eleven electoral system in an Italy-like case | |||
System ^{[15]} | r | g | rg^{[16]} |
Borda | 0.303 | 0.687 | 0.208 |
Condorcet | 0.250 | 0.825 | 0.206 |
Plurality | 0.036 | 0.975 | 0.035 |
Run-off Plurality | 0.303 | 0.787 | 0.238 |
Mixed-Member 1 ° | 0.337 | 0.755 | 0.254 |
Mixed-Member 2 ° | 0.339 | 0.762 | 0.258 |
One-district Proportionality | 1 | 0.168 | 0.168 |
Multi-district Proportionality* | 0.643 | 0.367 | 0.236 |
Threshold proportionality+ | 0.821 | 0.256 | 0.210 |
VAP | 0.7 | 0.667 | 0.467 |
Single Transferable Vote* | 0.661 | 0.362 | 0.239 |
° | 25% of seats elected through one-district proportionality |
* | Hare quota (simple rounding was used for one-district proportionality) |
+ | 4% threshold (as for real) |
Table 2: r and g for eleven electoral system in an UK-like case | |||
System | r | g | rg |
Borda | 0.200 | 0.875 | 0.175 |
Condorcet | 0.240 | 0.850 | 0.204 |
Plurality | 0.280 | 0.800 | 0.224 |
Run-off Plurality | 0.280 | 0.825 | 0.231 |
Mixed-Member 1° | 0.440 | 0.700 | 0.308 |
Mixed-Member 2° | 0.540 | 0.650 | 0.351 |
One-district Proportionality | 1 | 0.337 | 0.337 |
Multi-district Proportionality | 0.800 | 0.350 | 0.280 |
Threshold proportionality | 0.920 | 0.346 | 0.318 |
VAP | 0.781 | 0.618 | 0.483 |
Single Transferable Vote | 0.840 | 0.346 | 0.291 |
Table 3: r and g in another Italy-like case | |||
System | r | g | rg |
Borda | 0.444 | 0.650 | 0.286 |
Condorcet | 0.381 | 0.625 | 0.238 |
Plurality | 0.413 | 0.346 | 0.143 |
Run-off Plurality | 0.333 | 0.637 | 0.212 |
Mixed-Member 1 | 0.619 | 0.254 | 0.157 |
Mixed-Member 2 | 0.619 | 0.254 | 0.157 |
One-district Proportionality | 1 | 0.205 | 0.205 |
Multi-district Proportionality | 0.730 | 0.265 | 0.193 |
Threshold proportionality | 0.921 | 0.252 | 0.232 |
VAP | 0.624 | 0.719 | 0.449 |
Single Transferable Vote | 0.841 | 0.254 | 0.214 |
Table 4: r and g in another UK-like case | |||
System | r | g | rg |
Borda | 0.545 | 0.700 | 0.381 |
Condorcet | 0.318 | 0.825 | 0.262 |
Plurality | 0.114 | 0.925 | 0.105 |
Run-off Plurality | 0.114 | 0.938 | 0.107 |
Mixed-Member 1 | 0.364 | 0.788 | 0.287 |
Mixed-Member 2 | 0.432 | 0.738 | 0.319 |
One-district Proportionality | 1 | 0.375 | 0.375 |
Multi-district Proportionality | 0.864 | 0.375 | 0.324 |
Threshold proportionality | 0.841 | 0.396 | 0.333 |
VAP | 0.880 | 0.667 | 0.587 |
Single Transferable Vote | 0.886 | 0.379 | 0.336 |
where X is the number of seats of the m largest (i.e. major) parties in the Government, and T is the total number of seats in the Parliament. This way, the Government keeps a majority of y if the small parties of the governing coalition defeat. In ALEX3 the value of y is fixed to 1, while m may be established by the experimenter through the assignment of the share of seats in the proportional system necessary to be considered major.
^{2} Approval voting has been excluded as previous experiments showed that it is inferior to Condorcet voting with reference both to G and R. See Ortona (1998).
^{3} For a review, see Lijphart (1994, p.67).
^{4} See Shugart and Wattenberg (2000), p.30; Shugart (2001).
^{5} For a more detailed discussion of the indices, see Ortona (2002a).
^{6}The value of S_{u,i} is the total number of seats for the largest party, and 0 for all the others. If several parties are the largest ones ex aequo, we take one at random.
^{7} ALEX3 is the third version of the program. The first one required a specific database and did not include plurinominal systems. The second is like this one, but without plurinominal systems.
^{8} An interesting discussion of power topics in is Mudambi et al. (2001).
^{9} In a previous paper, the Gini coefficient relative to the Banzhaf’s power indices was employed as an additional measure of governability; but the computing was performed outside the program. See Ortona (2002a).
^{10} A more detailed description of the procedure may be found in Fragnelli, Monella and Ortona (2002).
^{11} Here is the proof.
From U = Ag^{a}r^{b} and a =pb we get dU = dg(bpAg^{bp-1}r^{b}) + dr(bAg^{bp}r^{b-1})
If U does not change 0 = dg(bpAg^{bp-1}r^{b}) + dr(bAg^{bp}r^{b-1})
dg(bpAg^{bp-1}r^{b}) = - dr(bAg^{bp}r^{b-1})
dr/r = - p(dg/g)
^{12} Remember that g ≤ 1. If (trivially) the value of both g and G is 1, the value of r will establish the best system.
^{13} Lower Chamber only.
^{14} It may be of some interest to recall that in Italy some 1,500 respondents are commonly assumed to be sufficient to provide a reliable nation-wide survey.
^{15} See Appendix 1 for details.
^{16} Assuming p = 1, i.e. a = b, the selection criterion becomes "system X is preferred to system Y iff g_{x}^{a}r_{x}^{a} > g_{y}^{a}r_{y}^{a}, i.e. g_{x}r_{x} > g_{y}r_{y}. The rank of r times g provides the rank of the electoral systems.
^{17} See Appendix 1 for details.
^{19} When parties obtained (rearranged) 3.30, 45.31, 17.16, 28.78, 1.07, 3.30 and 1.07% of votes respectively.
^{20} Not necessarily complete.
^{21} Java Runtime 1.4 can be downloaded from http://java.sun.com.
^{22} To avoid the mushrooming of possible Governments, it is almost inevitable to adopt some rules concerning the coalition that will actually be chosen. A reasonable set is to suppose that (a) the Government must be supported by the majority of MPs and (b) the Government is made by a minimum winning coalition of parties adjacent on the left-right axis. The program could easily implement the conditions, thus producing automatically the government; however, we preferred to leave this task to the user, as condition (b) is not that frequent in real world, and it may be of interest to explore different possibilities.
^{23} This claim could be tested through ALEX3.
FRAGNELLI V, Monella G and Ortona G (2002) "Governability and Representativeness: Simulation of Concrete Voting Situations", paper presented to the Spanish Meeting on Game Theory and applications, Sevilla, July.
LIJPHART A (1994) Electoral Systems and Party Systems: a Study of Twenty-seven Democracies, 1945-1990. Oxford: Oxford University Press.
MUDAMBI R (1997) A Complete Information Index for Measuring the Proportionality of Electoral Systems. Applied Economics Letters, 4. pp. 101-104.
MUDAMBI R, Navarra P and Sobbrio G (2001) Rules, Choices and Strategy: The Political Economy of Italian Electoral Reform. Cheltenham: Edward Elgar.
ORTONA G (1998) Come funzionano i sistemi elettorali: un confronto sperimentale. Stato e mercato, 54. pp. 83-112.
ORTONA G (2000), "A weighted-voting electoral system that performs quite well", in Dardanoni V and Sobbrio G (eds.), Istituzioni politiche e finanza pubblica, Angeli, Milano; also as a working paper, Dep. of Public Choice, Università del Piemonte Orientale, 4 (1999).
ORTONA G (2002a) "Experimental Assessment of a suggested two-stage electoral system", Paper presented to the 2002 Conference of the European Public Choice Society, Belgirate (Italy), April.
ORTONA G (2002b) "Choosing the electoral system: why not simply the best one?" Working Paper, Department of Public Choice, Università del Piemonte Orientale, 32.
SHUGART M S (2001) Electoral "efficiency" and the move to mixed-member systems. Electoral Studies, 20. pp. 173-193.
SHUGART M S and Wattenberg M P Eds. (2000) Mixed-member Electoral Systems: the Best of Both Worlds? Oxford: Oxford University Press.
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