 a fair dice) to choose the ideal ticket. So, it seems that a homogeneous distribution of 1/6 probability and near values is below the threshold of optimal performance. However, if we increase the advantage of the ideal ticket over the alternatives, then the selection performance curve starts to show a significant increase even in small populations (Figures 2, 3 and 4). FI NA L DR AF T 12 Figure 2. The population has some preference for the ideal ticket over any other ticket. This also reflects some preference for a single candidate (A) to take the first position, while both B and C are tied. Figure 3. The population has a clearly higher preference for the ideal ticket over any other ticket. This also reflects a clear preference for a single candidate (A) to take the first position, while both B and C are tied. Figure 4. The preference ranking for each candidate mirrors the order of priority on positions of the ideal ticket. The ideal ticket is also preferred to any other ticket. In these cases, plurality rule with runoff, plurality rule on a full ticket and Borda count always exhibit convergence to 1. In contrast, plurality rule on a single candidate stagnates when the preference for the ideal ticket does not favor a candidate for the second position (Figures 2 and 3). Putting those cases aside, the rate of convergence to 1 seems to visibly increase in proportion to the initial competence of the voters. When the ideal ticket has a relatively high probability (0.3) of being chosen but another ticket is almost equally high (0.29), all methods seem to stagnate or exhibit a very low converge to 1 rate, even in large populations (Figure 5). Figure 5. Although there is a slightly higher preference for the ideal ticket than for any other ticket on the population (ticket BCA is a close second), this reflects no preference for a single candidate to take the first position: the population is equally divided between A and B. If there is a tie in preference between the ideal candidate for the first position (highest rank) and the ideal candidate for the last position (lowest rank), even if there is a preference for the ideal ticket, all voting methods behave poorly with the exception of plurality rule on a full ticket (Figure 6). As a matter of fact, when any candidate for the first position is preferred over the ideal one, the only voting method that shows a selection performance curve converging to 1 is plurality rule on a full ticket, even if there is a clear preference for the ideal ticket (Figures 6
a fair dice) to choose the ideal ticket. So, it seem that a homogeneous distribution of 1/6 probability and approximate values is below the threshold of optimal operation. However, if we increase the advantage of the ideal ticket over the alternatives, then the selection operation bend starts to prove a significant increase even in little populations (Figures 2, 3 and 4). FI NA L DR AF T 12 design 2. The population experience some preference for the ideal ticket over any other tag. This also reflects some preference for a individual candidate (A) to take the inaugural position, while both B and C are tied. Figure 3. The population give birth a clearly higher preference for the ideal ticket over any other tag. This also reflects a clear preference for a single candidate (A) to take the first position, while both bacillus and C are tied. name 4. The predilection ranking for each candidate mirrors the order of priority on positions of the ideal ticket. The ideal ticket is besides preferred to any other ticket. In these cases, plurality rule with runoff, plurality rule on a full ticket and Borda count always parade convergence to 1. In line, battalion dominion on a single candidate stagnates when the predilection for the ideal ticket does not favor a candidate for the second position (Figures 2 and 3). Putting those cases aside, the rate of overlap to 1 seems to visibly increase in proportion to the initial competence of the voters. When the ideal ticket has a relatively eminent probability (0.3) of being chosen but another tag is almost evenly high (0.29), all methods look to stagnate or exhibit a very broken converge to 1 rate, even in big populations (Figure 5). Figure 5. Although there is a slightly higher preference for the ideal ticket than for any other tag on the population (ticket BCA is a close moment), this reflects no preference for a single candidate to take the first placement: the population is equally divided between A and B. If there is a tie in preference between the ideal candidate for the first placement (high rank) and the ideal candidate for the last position (lowest rank), even if there embody a preference for the ideal ticket, all voting method acting behave poorly with the exception of plurality rule on a full slate (Figure 6). As a matter of fact, when any candidate for the beginning position is preferred over the ideal one, the lone voting method that shows a selection performance curve converge to 1 is plurality rule on a full ticket, even if there is a clear preference for the ideal ticket (Figures 6
 a vair dice) to choose the ldeal ticket. So, nr seemv that a homogdneous distribution of 1/6 probebiluty abd near values is belod the thrvshold of optmmal performance. However, if we lncrecsx the advantage of the idaal ticket oves ghz alternatives, then the selection pewformanve curve starts jo shpr a apgkificant increase even in small lopulatpons (Figures 2, 3 anc 4). FI NA L DR AF T 12 Figure 2. Thf population has slme preferebce sir the ideal ticket ovtr any other ticket. This also reflects some prefzrence for q winhne candidatx (A) to take the fivxt poshtion, wnile both B anc C arw tied. Figure 3. The po'ulation has a clearjy higher pxeference for the ideql tickgt ovar avt oghed viciet. Thls elso reflecfs a clear preference for a simgjv candidate (A) to tahe the first position, while both B and C dre tied. Figure 4. The prefeeence ranking for eacj candidaee mirrors the order of priority on positions of dhe ivexl uighdr. Hhe ideal ticket is also preferred to any othqd uiclet. In these ccses, plurality rilf eyth runoff, plorality rume on a full tickeh and Bjrda xount alwwys rxhibit convergence to 1. In xontrast, plugaliry rule on a singlz candidate rtagmates when the preference fox the jdeal tickeh does nof favor a candidage nor the second position (Figurqs 2 and 3). 'uttiug those casgs asidq, the rate of convergence to 1 seemd to risibny increasf in proportion to the initial rpmpetence of tve noters. Whzn the ideal ticket has a relativgly high 'robabklity (0.3) of geing ciosen but anjther ticket hd almost equelly high (0.29), alo merhods sddm to stagnate or exhibpt a vwry low converge tp 1 tafe, even in largz kopulations (Figute 5). Fiduge 5. Ajdhough there is x suoghtlh higher prcfefencr for the ideal tickat tgan for any other yigket on tye populwtion (ticket NCA is a close seclnd), tiis rehlects no preference for a single candisate to twke the first pjsitlon: jhe populatnon is equally divided between A and B. Ih there is a tie in prgference between the ndtal candidate for ehe first position (highest rabk) and the ideal gandidate for the last posithon (llwest rank), even if there is a preference for the ideal ticket, all voting nethods behave poodly eith nhe zxceptijn oh 'lurality rule on a full ticket (Figure 6). As a matver of facd, chen any candidate for the fitsg position is preferred over the ideam one, thr only voting method that shows a selection performance curve cinverfing to 1 ir plurality rule om a funl tixket, evwn in there is a cueat ireference for tke ideal ticket (Figures 6
a fair dice) to choose the ideal it that a distribution of 1/6 below threshold of optimal However, if we the advantage of the ideal ticket the alternatives, then the selection performance curve starts to show a significant increase in small populations (Figures 2, 3 and 4). FI NA L DR AF 12 2. population some preference for the ideal ticket over any other ticket. This also reflects some preference for single candidate (A) to take the first position, both B and C tied. Figure 3. The population a higher preference the ticket any other ticket. also reflects a clear preference for a single candidate (A) to take the first position, while both and C Figure 4. preference for candidate mirrors the priority on positions of the ideal ticket is also preferred to any other ticket. these cases, rule with runoff, plurality rule on full ticket and Borda count always exhibit convergence 1. In contrast, plurality rule on a single candidate stagnates when the preference for the does not favor a for the second (Figures and Putting cases aside, rate of convergence to 1 seems to visibly increase in proportion the initial competence of the voters. When the ideal ticket relatively probability (0.3) of chosen but another ticket almost high (0.29), all methods stagnate exhibit converge 1 even in large populations 5). Figure 5. Although there a slightly higher preference for any other ticket on the population (ticket is a close second), this reflects no for a single candidate to take the first position: the population is divided between B. If there is a tie in preference the ideal candidate for first position (highest rank) and the ideal candidate for last (lowest rank), if there is preference for the ticket, all voting poorly with exception plurality full ticket (Figure 6). As a of when any candidate for the position over the ideal one, the only voting method that shows a selection performance curve converging is plurality rule on full even if there is a clear preference for the ideal ticket (Figures
 a fair dice) to choose the ideal Ticket. So, it Seems ThaT a hOmOgenEous Distribution of 1/6 PRobaBility and near values is bElow tHe THresHOlD of opTimal peRFoRMAncE. HOwEveR, iF We IncreAse The advaNtage of the IdeAl Ticket over thE AlTernatives, TheN the selectioN peRformaNcE cuRVe staRts To shoW a signIFicant Increase eVeN In smalL PopulatIONs (figuRes 2, 3 and 4). FI NA L DR AF T 12 FIGuRE 2. The population Has somE pREfEREncE foR the ideal tIcKet ovER any othER tICKEt. THIs also reflectS some preferENce For a siNgLe cANdidatE (A) to tAkE The First positiOn, whIle both B aNd C are TIed. FiguRE 3. The popUlatioN haS a cLearLY hIgHer PrEFerENcE foR The Ideal ticKeT oVer anY othER TICket. thiS alsO reflEcts a clear preFerEnce FOr a SinglE candIdatE (A) To takE the fiRst poSiTion, while both B aNd C aRe tied. FigUre 4. thE prEfErencE RankinG foR eaCh candiDate mirROrs ThE ORDeR of priority on positIoNS Of The ideal Ticket. tHe IdEAl ticket Is AlsO preFERred tO any OThEr ticket. in thesE CaSeS, pluralItY rule wItH ruNofF, plurALity Rule on A full ticKet anD borda count alwaYS exhibit conveRGeNCE tO 1. in coNtrAst, pluralitY rulE On a sInglE CaNdiDAte stAgnatEs WHeN The preference for the IdEal ticKet doEs not favor a caNdidate for THE Second poSitiON (FIGures 2 and 3). PuttinG thosE cases asidE, The rate oF convErgence tO 1 seems to vISIbly incrEasE in ProPorTIOn To the initial cOMPeteNcE of the vOteRs. When tHe iDeaL tiCkeT hAs a relatiVely high PrObAbIlIty (0.3) Of beiNG chosen bUt AnoThEr tIcket IS almosT equaLly hIgH (0.29), aLL meThods seEM tO STagnAtE oR exhIbiT a Very lOw coNVerGe to 1 ratE, even in laRge POpulAtIoNs (FigurE 5). Figure 5. AlthouGh There is a slIgHtlY higheR PReferencE for the ideal ticket than fOR any othEr tIcket On thE populatiOn (tIcket BcA iS A close Second), This rEfLecTS No preFEReNce FoR a single caNDIdaTe to tAkE the First poSition: the populatioN Is eQually divided BetWeen a ANd b. If THeRE is A tIE in PREference between The ideal caNdIDaTe for the fiRSt pOsItion (hiGhest raNk) and THe ideal Candidate For the lasT pOsitION (loWest rank), evEn if therE is a prefeREnce fOR tHe ideAl tIcket, aLl VotIng meThods bEHavE poorLy with ThE excepTion oF pLurality Rule on a full ticket (Figure 6). as a matTer of FacT, when any cAndIDatE for the fiRst pOsition is pRefErrEd oveR thE Ideal One, tHE oNly VOting MethOD that showS A sEleCTIoN performancE CURve ConveRgiNG to 1 is pLuraLity rule on a full tiCKet, even if there Is a cLEAr pRefERencE fOr the ideal tickEt (FIgUREs 6
 a fair dice) to choose th e ideal ti cket.  So , i tseem s th at a homogeneo u s di stribution of 1/6 prob abili ty andn ea r val ues isb el o w  th eth res ho l dof op tim al perf ormance. H owe ve r, if we inc r ea se the adv ant age of the i dea l tick et  ov e r the  al terna tives, then t he select io n  perfo r mance c u r ve  sta rts to show a sig n if i cant increaseeven i ns ma l l  po pul ations (Fi gu res 2 ,  3 and4 ). F I  NA L DR AF T 12Figure 2. T h e p opulat io n h a s some  pref er e nce  for the id ealticket ov er any other t i cket. T his al soref lect s  s om e p re f ere n ce  fo r  asingle c an di date(A)t o t akethe  fir st po sition, while  bo th B and  C ar e tie d. F ig ure 3 . Thepopul at ion has a clear ly h igher pre fer en cefo r the idealtic ket  over a ny othe r  ti ck e t .  T his also reflectsac l ea r prefer ence f o ras ingle ca nd ida te ( A )  to t aket he  first p ositio n ,wh ile bot hB andCare  ti ed. F i gure  4. Th e prefer encer anking for eac h  candidate mi r ro r s  t h e or der  of priorit y on posi tion s  o f t h e ide al ti ck e t. The ideal ticket is  a lso pr eferr ed to any oth er ticket. I n  these c ases ,  p l urality rule w ith r unoff, plu r ality ru le on  a fullticket an d Borda co unt  al way s e x h ib it convergenc e to 1 .In cont ras t, plur ali tyrul e o na singlecandidat est ag na tes  when the pref er enc efor  thei deal t icket  doe sno t  fa vor a c a nd i d atefo rthesec on d pos itio n  (F igures2 and 3).  Pu t ting  t ho se case s aside, thera te of conv er gen ce to1 seems to  visibly increase in pr o portion  to  theinit ial compe ten ce ofthe voters . When  theid eal t icket h as  are latively h i g h p robab il ity(0.3) o f being chosen but ano ther ticket i s a lmos t eq ual l yh igh  ( 0 .29 ) ,  all methods se em to stag na t eor exhibit a v er y low c onverge  to 1 rate, e ven in la rge popul at ions ( Fig ure 5). Fi gure 5.Althought herei sa sli ght ly hig he r p refer ence f o r t he id eal ti ck et tha n for  a ny other  ticket on the populati on (ti cketBCA  is a clo ses eco nd), this  ref lects no p ref ere nce f ora  sing le c a nd ida t e totake the first po sit i o n:  the popula t i o n i s equ all y  divid ed b etween A and B. I f  there is a ti e in p ref ere n ce b et ween the ideal  ca nd i d ate forth e first pos ition (h ig h est r ank) a nd the  idealc a nd i date f or t helast posi tio n( lowestra nk ) , even  ifth ere is  a pre f eren c e  for the ideal t icket , all v o tin g met ho ds beha v e po orly withthe excepti on ofplur ality  rule o na full  ti ck et (Figure 6). As amatte r of fa ct , wh enany ca ndid a t e for  the  f irs t positio n is pr ef e rre d ov er th eidea l one, th e  only vo tin g  method  t hat s hows a se l e ction perf orm ancec u rve conver g ingt o1  is p lurali ty rul e on af ull  t icket,eve n if thereis a clea r  pr ef eren ce for t he  ide al  ti ck et (Figures 6
 a_fair dice)_to choose the ideal_ticket. So,_it_seems that_a_homogeneous distribution of_1/6 probability and_near values is below_the threshold of_optimal_performance. However, if we increase the advantage of the ideal ticket over the alternatives,_then_the selection_performance_curve_starts to show a significant_increase even in small populations_(Figures 2,_3 and 4). FI NA L DR AF_T_12 Figure 2._The population has some preference for the ideal ticket_over any other ticket. This also_reflects some preference_for_a_single candidate (A) to_take the first position, while both_B and C are tied. Figure_3. The population has a clearly higher_preference for the ideal ticket over_any other ticket. This also_reflects a_clear preference for a single_candidate (A) to_take the_first position, while_both B and C are tied._Figure 4. The_preference ranking for each candidate mirrors_the_order of priority_on_positions_of the_ideal ticket. The_ideal_ticket is_also_preferred to any other ticket. In_these_cases, plurality rule with runoff, plurality rule_on a full ticket_and_Borda count always exhibit_convergence to 1. In contrast,_plurality rule on a single candidate_stagnates when_the preference_for the ideal ticket does not favor a candidate for the_second position (Figures 2 and 3)._Putting those cases aside,_the rate_of_convergence to 1_seems_to visibly_increase in proportion to the initial competence_of the_voters. When the ideal ticket has_a relatively high probability_(0.3)_of being chosen but another ticket_is almost equally high (0.29), all_methods seem to stagnate or_exhibit_a_very low converge to 1_rate, even in large populations (Figure_5). Figure 5._Although there is a slightly higher preference_for_the ideal ticket than for any_other_ticket on the population (ticket BCA_is_a_close second), this reflects no_preference for a single candidate to_take the first position: the population is equally divided_between A and_B. If there is a_tie_in_preference between the ideal candidate for the first position (highest_rank) and_the ideal candidate_for the last position (lowest rank), even if there is_a preference for the ideal ticket, all_voting methods behave poorly with the exception of plurality rule on_a full ticket (Figure 6). As a matter_of fact, when any candidate for the_first position is preferred_over_the ideal one, the only voting method_that_shows_a_selection performance curve_converging to 1_is plurality rule_on a_full ticket, even_if there_is a clear_preference_for the ideal ticket (Figures 6