ApplicationBordaFPtoDutchElections006.nb Application of the Borda Fixed Point voting rule to the Dutch Parliamentary elections 006 Thomas Colignatus November 3 & 8 006 The Borda Fixed Point voting rule can be found in the book "Voting Theory for Democracy", http://www.dataweb.nl/~cool/papers/vtfd/index.html See also http://www.dataweb.nl/~cool/papers/socialwelfare/withouttimenomorality.html For Dutch readers there is also http://www.dataweb.nl/~cool/svhg/dok/dok-aankondiging.html Start Needs["Economics`Pack`"] ResetAll Economics[Voting] Data Final results November 7 006 Parties = CDA, },, 6}, D66, 3}, GL, 7}, PvdA, 33}, PvdD, }, PvdV, 9}, SGP, }, SP, }, VVD, }} i CDA y 6 D66 3 GL 7 PvdA 33 PvdD PvdV 9 SGP SP k VVD Items = First ê@ Parties NumberOfItems = Length@ItemsD 8CDA,, D66, GL, PvdA, PvdD, PvdV, SGP, SP, VVD< 0
ApplicationBordaFPtoDutchElections006.nb vlis = Last ê@ Parties; NumberOfVoters = Length@vlisD; Votes = vlisêadd@vlisd : ÅÅ ÅÅÅÅÅ, ÅÅ, ÅÅ, 7 ÅÅÅÅÅ, ÅÅ, ÅÅ, 3 ÅÅ, ÅÅ, ÅÅÅÅÅ 6, ÅÅ > StatusQuo@D CDA Hypothesis These routines require party preferences on the selection of a Prime Minister. Each party can present a candidate PM and then the Members of Parliament enter their orders of preference on the candidates. These preferences should best expressed not by the parties but by the individual Members of Parliament. Parties might increase their chances by proposing candidates that are well received by other parties. Perhaps it is simplest though to presume that their candidates will be the leaders at the elections. Lacking those data we must enter an educated guess, and it is useful to assume some party homogeneity. Pref@CDAD = 8CDA > > VVD > PvdD > GL > SP > SGP > PvdA > D66 > PvdV<; Pref@D = 8 > CDA > SGP > PvdA > GL > SP > VVD > PvdD > D66 > PvdV<; Pref@D66D = 8D66 > PvdA > GL > VVD > PvdD > > SP >CDA > SGP > PvdV<; Pref@GLD = 8GL > SP > PvdA > PvdD > D66 > > CDA > VVD > SGP > PvdV<; Pref@PvdAD = 8PvdA > GL > D66 > SP > PvdD > > CDA > VVD > SGP > PvdV<; Pref@PvdDD = 8PvdD > D66 > GL > > SP > PvdA > CDA > VVD > SGP > PvdV<; Pref@PvdVD = 8PvdV > VVD > > CDA > PvdD > SGP > SP > PvdA > D66 > GL <; Pref@SGPD = 8SGP > > CDA > PvdD > VVD > PvdV > SP > PvdA > GL > D66 <; Pref@SPD = 8SP > GL > PvdA > D66 > PvdD > > CDA > VVD > SGP > PvdV<; Pref@VVDD = 8VVD > CDA > > D66 > PvdD > PvdV > GL > PvdA > SP > SGP<; These preference patterns can be translated in Borda ordinal preference scores. Preferences = PrefToList@ToPref @@ Pref@#DD& ê@ Items i 0 9 6 3 7 5 8 y 9 0 6 7 3 8 5 3 5 0 8 9 6 7 5 6 0 8 7 9 3 5 8 9 0 6 7 3 7 9 8 5 0 6 3 7 8 3 6 0 5 9 8 9 3 7 5 0 6 5 7 9 8 6 0 3 k 9 8 7 3 6 5 0
ApplicationBordaFPtoDutchElections006.nb 3 The Borda Fixed Point selection Given the above data and assumptions the Borda Fixed Point algorithm determines that fixed point, i.e. the winner who also wins from the runner up (the alternative winner if the overall winner would not partake). BordaFP@D BordaAnalysis@D êê N :Select Ø, BordaFPQ Ø 8True<, WeightTotal Ø 86.78667, 6.99333, 5.3333, 6.76667, 5.9333, 6.6667,.8,.9667, 5.9, 5.9333<, i.8 PvdVy.9667 SGP 5.3333 D66 5.9333 PvdA Position ØH.L, Ordering Ø 5.9333 VVD 5.9 SP > 6.6667 PvdD 6.76667 GL 6.78667 CDA k 6.99333 Alternative: Pairwise voting It appears that the is also the Condorcet winner - i.e. wins from all pairwise votes. This criterion however is not a strong one since there can be elections where there is no such winner or there can be elections where that winner loses in a Borda approach.
ApplicationBordaFPtoDutchElections006.nb PairwiseMaority@D VoteMarginToPref::cyc : Cycle 8PvdD, D66, GL, PvdD< :VoteMargin Ø VoteMargin ii 0 - kk 0 - - 5 5 5 5 0-3 5 5 7 5 5 - - 6 8 5-3 0-5 5-7 6-0 - 5 - - 8 0 5 5 - - 6 - - - - 73-7 - 58-7 - 67 - ÅÅÅÅ - 3-5 5 5 - - - 5 - - - - 73 - - 6 73 73 7 5 5 - ÅÅÅÅ - 5 5 58 67 0-3 7 - - - - - 67 0-58 58 0-67 0 yy, Ø 8StatusQuo Ø CDA, Sum Ø88, 9, 3, 6,, 5, 0,,, 5<, Max Ø 9, Condorcet winner Ø, Pref Ø PrefHPvdV, SGP, 8D66, GL, PvdA, PvdD, SP, VVD<, CDA, L, Find Ø, LastCycleTest Ø False, Select Ø <,NØ:Sum Ø: ÅÅ 93 ÅÅÅÅÅ, ÅÅÅÅÅ, - ÅÅ 8, 38 ÅÅ 5, 6 ÅÅ, 3 ÅÅ 5, - ÅÅ 66 ÅÅÅÅÅ, - ÅÅ 386 ÅÅÅÅÅ, ÅÅ, 6 ÅÅ >, Pref Ø PrefHPvdV, SGP, D66, 8PvdA, VVD<, SP, PvdD, GL, CDA, L, Select Ø >, All Ø > Alternative: Plurality voting Plurality selects the person with the highest vote - that might be less than %. All parties vote for their own candidate and there is no clear winner. Plurality@D :Sum Ø i CDA D66 SP k ÅÅ 7 GL ÅÅ PvdA PvdD PvdV 3 SGP ÅÅÅÅ 6 VVD y i PvdDy SGP D66 7 ÅÅ GL, Ordering Ø, Max Ø :CDA, ÅÅÅÅÅ>, Select Ø 8<> 3 PvdV VVD ÅÅÅÅ 6 SP PvdA k ÅÅ CDA
ApplicationBordaFPtoDutchElections006.nb 5 % êê N i CDA 0.73333 y i 0.033333 PvdDy 0.0 0.033333 SGP D66 0.0 0.0 D66 GL 0.066667 0.0 :Sum Ø PvdA 0. 0.066667 GL, Ordering Ø PvdD 0.033333 0.06 PvdV, Max Ø 8CDA, 0.73333<, Select Ø8<> PvdV 0.06 0.6667 VVD SGP 0.033333 0.66667 SP SP 0.66667 0. PvdA k VVD 0.6667 k 0.73333 CDA An example pairwise vote The following example shows that the candidate of the would win from the candidate of the CDA in a pairwise vote. There are however 5 of such pairwise votes and thus it is simplest if all Members of Parliament would enter a single preference list whereafter the algorithm determines the overall result. SelectPreferences@8CDA, <D CheckVote::ad : NumberOfItems adusted to :Number of Voters Ø 0, Number of items Ø, Votes are nonnegative and add up to Ø True, Preferences fit the numbers of Voters and Items Ø True, Type of scale Ø Ordinal, Preferences give a proper ordering Ø True, Preferences add up to Ø 83<, Items Ø 8CDA, <, Votes Ø : ÅÅ ÅÅÅÅÅ, ÅÅ, ÅÅ, 7 ÅÅÅÅÅ, ÅÅ, ÅÅ, 3 ÅÅ, ÅÅ, ÅÅÅÅÅ 6, ÅÅ >> Plurality@D :Sum Ø i k CDA 9 y, Ordering Øi CDAy 9 k, Max Ø :, ÅÅ 9 >, Select Ø > Conclusion The simplest scheme is where parties vote for their own candidate. Then the CDA will get the highest score, which is still only 7.3% of the vote. Thus, "simplest" doesn't seem to be too useful. In pairwise voting it so happens that the is the Condorcet winner. However, that kind of voting is notoriously unstable. In many elections there is no such winner, leaving one with the question what to do next. The overall best approach is the Borda Fixed Point. In this case this coincides with the Condorcet winner since the apparently is rather high on the preference lists anyway.
ApplicationBordaFPtoDutchElections006.nb 6 Of course, voting would be conditional on agreements on policy and coalition forming. However, in "Voting Theory for Democracy" it appears that a Cabinet "mirrorring" Parliament would tend to be best, so that the issue on policy making could still be rather distinct from the selection of the Prime Minister. Appendix: Strategic voting Strategic voting can never be fully avoided. CDA might give its competitor much less weight and then itself becomes the Borda Fixed Point. Pref@CDAD = 8CDA > VVD > PvdD > GL > SP > SGP > PvdA > D66 > PvdV > <; Preferences = PrefToList@ToPref @@ Pref@#DD& ê@ Items i 0 3 7 8 5 6 9 y 9 0 6 7 3 8 5 3 5 0 8 9 6 7 5 6 0 8 7 9 3 5 8 9 0 6 7 3 7 9 8 5 0 6 3 7 8 3 6 0 5 9 8 9 3 7 5 0 6 5 7 9 8 6 0 3 k 9 8 7 3 6 5 0 BordaFP@D BordaFP::chg : Borda gave 8GL<, the selected Fixed Point is 8CDA< CDA BordaAnalysis@D êê N :Select Ø GL, BordaFPQ Ø 8False<, WeightTotal Ø 86.78667,.80667, 5.58667, 7.0, 6.8667, 6.5,.5333, 3., 6.333, 6.8667<, i.5333 PvdVy 3. SGP.80667 5.58667 D66 Position ØH.L, Ordering Ø 6.8667 PvdA 6.8667 VVD > 6.333 SP 6.5 PvdD 6.78667 CDA k 7.0 GL However, as other parties might anticipate such CDA strategic voting behaviour they might respond by entering much higher in their preferences.
ApplicationBordaFPtoDutchElections006.nb 7 Pref@CDAD = 8CDA > VVD > PvdD > GL > SP > SGP > PvdA > D66 > PvdV > <; Pref@D = 8 > CDA > SGP > PvdA > GL > SP > VVD > PvdD > D66 > PvdV<; Pref@D66D = 8D66 > > PvdA > GL > VVD > PvdD > SP >CDA > SGP > PvdV<; Pref@GLD = 8GL > > SP > PvdA > PvdD > D66 > CDA > VVD > SGP > PvdV<; Pref@PvdAD = 8PvdA > > GL > D66 > SP > PvdD > CDA > VVD > SGP > PvdV<; Pref@PvdDD = 8PvdD > > D66 > GL > SP > PvdA > CDA > VVD > SGP > PvdV<; Pref@PvdVD = 8PvdV > > VVD > CDA > PvdD > SGP > SP > PvdA > D66 > GL <; Pref@SGPD = 8SGP > > CDA > PvdD > VVD > PvdV > SP > PvdA > GL > D66 <; Pref@SPD = 8SP > > GL > PvdA > D66 > PvdD > CDA > VVD > SGP > PvdV<; Pref@VVDD = 8VVD > > CDA > D66 > PvdD > PvdV > GL > PvdA > SP > SGP<; Preferences = PrefToList@ToPref @@ Pref@#DD& ê@ Items i 0 3 7 8 5 6 9 y 9 0 6 7 3 8 5 3 9 0 7 8 5 6 9 5 0 7 6 8 3 9 7 8 0 5 6 3 9 8 7 5 0 6 3 7 9 3 6 0 5 8 8 9 3 7 5 0 6 9 6 8 7 5 0 3 k 8 9 7 3 6 5 0 BordaFP@D BordaFP::set : Local set found: 8CDA, < BordaFP::chg : Borda gave 8<, the selected Fixed Point is BordaAnalysis@D êê N :Select Ø, BordaFPQ Ø 8True<, WeightTotal Ø 86.6, 6.85333, 5., 6.6, 5.95333, 6.08667,.5333, 3., 5.9667, 6.0667<, i.5333 PvdVy 3. SGP 5. D66 5.9667 SP Position ØH.L, Ordering Ø 5.95333 PvdA 6.08667 PvdD > 6.0667 VVD 6.6 GL 6.6 CDA k 6.85333