 A,??) = #(??, A, B) + #(??, B, A) = #(A,??, B) + #(B,??, A) = #(A, B,??) + #(B, A,??) = n ; once a candidate is fixed in a position, the votes are distributed in the only two possible options to occupy the remaining positions. • A winning assignment will be represented in the form [α, β, γ] and it will indicate that α, β, γ were assigned to positions 1, 2, and 3, respectively. We can now describe the selection mechanisms (voting methods) on which the computational simulations are based. 2.2 Voting methods for a problem involving priority on positions (I) Plurality rule on a single candidate with priority. [α, β, γ] is the winning assignment if and only if #(α) > #(β) > #(γ). In this case, the positions are being assigned based on which candidate has a greater number of votes. (II) Plurality rule with runoff with priority. Let α ∈ T be such that #(α) > #(β) and #(α) > #(γ). Let β be such that #(??, β, γ) > #(??, γ, β). Then [α, β, γ] is the winning assignment. In the first part of this process, the voter is asked to choose a candidate for the first position. The candidate with the greater number of votes takes the first position. In the second stage (runoff), the occupant of the second position is chosen among the two remaining candidates that were not winners on the first stage. (III) Plurality rule on a full ticket. In this process, each voter is asked to choose one of the six possible full tickets (as it turns out, for FI NA L DR AF T 6 this method it is not significant whether there is a priority or not on the positions): If (α, β, γ) ∈ S3 = {(A, B, C), (A, C, B), (B, A, C), (B, C, A), (C, A, B), (C, B, A)} and #(α, β, γ) > #(x, y, z) for every (x, y, z) ∈ S3\{(α, β
A, ? ?) = # (?? , A, B) + # (?? , B, A) = # (A, ?? , B) + # (B, ?? , A) = # (A, B, ? ?) + # (boron, A, ? ?) = n; once a campaigner is fixed in a position, the right to vote are spread in the only two possible option to occupy the remaining position. • A winning assignment will be represent in the form [ α, β, γ ] and it will indicate that α, β, γ were arrogate to position 1, 2, and 3, respectively. We can now describe the selection mechanisms (voting method) on which the computational simulations are based. 2.2 Voting method acting for a problem involving priority on positions (iodine) Plurality rule on a single candidate with priority. [ α, β, γ ] is the winning grant if and only if # (α) > # (β) > # (γ). In this case, the positions are being assigned based on which candidate has a greater number of votes. (II) Plurality rule with runoff with priority. Let α ∈ T be such that # (α) > # (β) and # (α) > # (γ). Let β be such that # (?? , β, γ) > # (?? , γ, β). Then [ α, β, γ ] is the winning grant. In the first part of this summons, the voter is ask to choose a candidate for the first position. The campaigner with the greater numeral of votes takes the inaugural position. In the second stage (runoff), the occupant of the second position is chosen among the two stay candidates that were not winners on the first stage. (III) Plurality dominion on a full tag. In this process, each voter is asked to choose one of the six possible full tickets (as it turn out, for FI NA L DR AF T 6 this method it is not significant whether there is a precedence or not on the positions ): If (α, β, γ) ∈ S3 = { (A, B, C), (A, C, B), (B, A, C), (B, C, A), (C, A, B), (C, B, A) } and # (α, β, γ) > # (x, y, z) for every (x, y, z) ∈ S3\{(α, β
 A,??) = #(??, A, B) + #(??, B, A) = #(A,??, B) + #(B,??, A) = #(A, B,??) + #(B, A,??) = n ; oncg q candmdate ia fixed kn a position, the votes are visteibuttb in the only two porsible opnions to iccuky the remaining 'ksitions. • A wjkning essignment will be represanted in the fmro [α, β, γ] and it will indicate that α, β, γ werq assigmef to positions 1, 2, amq 3, rsspectively. We can now describe ths selecuion mechanisms (voying methods) on which the fompktational simulatilns are basgs. 2.2 Diting methodr for a problem involvjng priority on positions (I) Plufalitv rule on a sungpg candidate xith pgiority. [α, β, γ] is the winting asxignment if anc oily uf #(α) > #(β) > #(γ). In this case, vhe positions are beyng assigteb based on which candudqte hds a grexrer nujbxr kf votfs. (MI) Pluralitg rule with runoff with prioriuy. Jvy α ∈ T be sucg that #(α) > #(β) and #(α) > #(γ). Let β be such that #(??, β, γ) > #(??, γ, β). Then [α, β, γ] js the winning assignmebt. In the first part lf this pwocess, the voter is asked to choose a candidate fmr thx wirwt oisltion. The candidate with the greater number os vptvs takes the firsb position. In the xefomq stage (runofw), the occhpant of the seconf posityon iw chosen wmonb the two remaining candidares that werv nor winners on the fnrst stage. (INI) Plutality rule on a full ticket. Nn thia process, ewch voter ks asked to choore pna of the six possible full tickets (es it turns uut, gor FI NA L DR AV T 6 bvis method it is nlt siynifiwant whethfr there is a priority or not oi the positioms): If (α, β, γ) ∈ S3 = {(C, B, C), (S, C, B), (B, A, C), (B, C, A), (C, A, B), (C, B, A)} and #(α, β, γ) > #(x, y, z) for enery (x, y, v) ∈ S3\{(α, β
A,??) = #(??, A, B) + #(??, = B) + A) = #(A, n once a candidate fixed in a the votes are distributed in the two possible options to occupy the remaining positions. • A winning assignment will represented in the form [α, β, γ] and it will indicate that α, γ assigned positions 2, and 3, respectively. We can now describe the selection mechanisms (voting methods) on which the simulations are based. 2.2 Voting methods for a involving priority on positions Plurality rule on a single with [α, β, is winning if and only #(α) > #(β) > #(γ). In this case, the positions are being assigned based on which candidate a greater votes. (II) rule runoff priority. Let α be such that #(α) > #(β) #(γ). Let β be such that #(??, β, > #(??, β). Then [α, β, γ] is winning assignment. In the first part of this the voter is asked to choose a candidate for the first position. The candidate with number of votes takes first position. In second (runoff), occupant the second is chosen among the two remaining candidates that were not winners the first stage. (III) Plurality rule on a full ticket. process, voter is asked choose one of the possible tickets (as it turns FI L 6 method is not significant whether is a priority or not the positions): If (α, {(A, B, C), (A, C, B), (B, A, (B, C, A), (C, A, B), (C, A)} and #(α, β, γ) > #(x, y, z) for every (x, z) ∈
 A,??) = #(??, A, B) + #(??, B, A) = #(A,??, B) + #(B,??, A) = #(A, B,??) + #(B, A,??) = n ; once a candidAte is fixed In a poSitIon, ThE votEs arE distributed in THe onLy two possible options to OccupY tHE remAInIng poSitions. • a WiNNIng AsSiGnmEnT WiLl be rEprEsented In the form [α, Β, γ] aNd It will indicaTE tHat α, β, γ were AssIgned to positIonS 1, 2, and 3, reSpEctIVely. WE caN now dEscribE The selEction mecHaNIsms (voTIng methODS) oN whiCh the computationaL SiMUlations are basEd. 2.2 VotiNg MEtHODs fOr a Problem invOlVing pRIority oN PoSITIonS (i) Plurality rulE on a single cANdiDate wiTh PriORity. [α, β, Γ] is thE wINniNg assignmenT if aNd only if #(α) > #(Β) > #(γ). In thIS case, thE PositioNs are bEinG asSignED bAsEd oN wHIch CAnDidATe hAs a greatEr NuMber oF votES. (ii) plurAliTy ruLe witH runoff with prIorIty. LET α ∈ T Be sucH that #(Α) > #(β) anD #(α) > #(Γ). Let β Be such That #(??, β, Γ) > #(??, γ, Β). Then [α, β, γ] is the wiNninG assignmeNt. IN tHe fIrSt parT Of this ProCesS, the votEr is askED to ChOOSE a Candidate for the firSt POSiTion. The cAndidaTE wItH The greatEr NumBer oF VOtes tAkes THe First posItion. IN ThE sEcond stAgE (runofF), tHe oCcuPant oF The sEcond pOsition iS chosEN among the two reMAining candidaTEs THAt WEre nOt wInners on the FirsT StagE. (III) pLuRalITy rulE on a fUlL TiCKet. In this process, eacH vOter is Asked To choose one of The six possIBLE full ticKets (AS iT Turns out, for FI Na L DR Af T 6 this methOD it is not SigniFicant whEther therE IS a prioriTy oR noT on The POSiTions): If (α, β, γ) ∈ S3 = {(A, B, c), (a, c, B), (B, A, c), (B, c, A), (C, A, B), (C, B, a)} anD #(α, β, γ) > #(x, y, z) For EveRy (x, Y, z) ∈ S3\{(Α, β
 A,??) = #(??, A, B) + #(? ?, B, A) =  #(A, ??,  B)  +  #(B ,??,  A) = #(A, B,? ? ) +#(B, A,??) = n ; oncea can di d atei sfixed  in a p o si t i on,  t he  vo te s  a re di str ibutedin the onl y t wo  possible op t io ns to occu pythe remainin g p ositio ns . • A win nin g ass ignmen t  willbe repres en t ed int he form [ α,  β,γ] and it will in d ic a te that α, β,γ were  a s si g n edtopositions1,  2, a n d 3, re s pe c t i vel y . We can nowdescribe th e  se lectio nmec h anisms  (vot in g  me thods) on w hich  the comp utatio n al simu l ationsare ba sed . 2 .2 V o ti ng  me th o dsf or  ap rob lem invo lv in g pri orit y o n  pos iti ons(I) P lurality rule  on  a s i ngl e can didat e wi th  prio rity.[α, β ,γ] is the winni ng a ssignment  if  a ndon ly if #(α) >  #( β)> #(γ).  In thi s  ca se , t he  positions are bei ng a ss igned ba sed on wh ic h  candida te  ha s ag r eater  num b er  of vote s. (II )  P lu ralityru le wit hrun off  with prio rity.Let α ∈T bes uch that #(α)>  #(β) and #(α )  > # (γ ) . Le t β  be such th at # ( ??,β, γ )  >  #( ? ?, γ,  β).Th e n[ α, β, γ] is the win ni ng ass ignme nt. In the fi rst part o f t his proc ess, th e  voter is aske d tochoose a c a ndidatefor t he first  position . The cand ida tewit h t h e  g reater number o f vo te s takes  th e first  po sit ion . I nthe secon d stage(r un of f) , t he oc c upant of  t hese con d pos i tion i s cho senam on g  th e two r e ma i n ingca nd idat esth at we re n o t w innerson the fi rst stag e.  ( III) Pl urality ruleon  a full ti ck et.  In th i s  process , each voter is asked t o  choose  on e ofthesix possi ble  fulltic k ets (a s it t urnsou t,f o r FIN A  L  DR  A F T 6 this m eth od it  i s no t signi ficant whether the r e i s a priorityornoto n  t hep os i tio ns ) : I f (α, β, γ) ∈ S3= {(A, B,C) ,  ( A, C, B),( B,A,  C), (B , C, A) , (C, A, B),(C, B, A) } and #(α ,β, γ ) > # (x, y, z)for ever y (x, y,z ) ∈ S 3 \{ (α, β 
 A,??)_= #(??,_A, B) + #(??,_B, A)_=_#(A,??, B)_+_#(B,??, A) =_#(A, B,??) +_#(B, A,??) = n_; once a_candidate_is fixed in a position, the votes are distributed in the only two possible_options_to occupy_the_remaining_positions. • A winning assignment_will be represented in the_form [α,_β, γ] and it will indicate that α,_β,_γ were assigned_to positions 1, 2, and 3, respectively. We can_now describe the selection mechanisms (voting_methods) on which_the_computational_simulations are based. 2.2_Voting methods for a problem involving_priority on positions (I) Plurality rule_on a single candidate with priority. [α,_β, γ] is the winning assignment_if and only if #(α)_> #(β)_> #(γ). In this case,_the positions are_being assigned_based on which_candidate has a greater number of_votes. (II) Plurality_rule with runoff with priority. Let_α_∈ T be_such_that_#(α) >_#(β) and #(α)_>_#(γ). Let_β_be such that #(??, β, γ)_>_#(??, γ, β). Then [α, β, γ]_is the winning assignment._In_the first part of_this process, the voter is_asked to choose a candidate for_the first_position. The_candidate with the greater number of votes takes the first position._In the second stage (runoff), the_occupant of the second_position is_chosen_among the two_remaining_candidates that_were not winners on the first stage._(III) Plurality_rule on a full ticket. In_this process, each voter_is_asked to choose one of the_six possible full tickets (as it_turns out, for FI NA_L_DR_AF T 6 this method_it is not significant whether there_is a priority_or not on the positions): If (α,_β,_γ) ∈ S3 = {(A, B,_C),_(A, C, B), (B, A, C),_(B,_C,_A), (C, A, B), (C,_B, A)} and #(α, β, γ)_> #(x, y, z) for every (x, y, z)_∈ S3\{(α, β