7#(P+۱ 7JJKXLD@LLMvMvMvMjMjN^N^NnxMvN O3O*O[LOO9mOOP+OOOOOO.c1.Comparison of Voting Systems; .c2.Definitions; The oldest and most often used voting system is called single-vote plurality. Each voter gets one vote which he can give to one candidate. The candidate who gets the most votes, a plurality, wins. In multi-candidate races the winner often gets less than a majority, less than 50% of the votes. Plurality has a very low Condorcet efficiency: it often picks a winner who is not at the center of the electorate and who would be beaten by one or more other candidates in one-on-one contests. In this sense it leads to the election of candidates who are disliked by a majority of the voters. It is very easy to manipulate too. The runoff system starts with a single-vote plurality election. The top two finishers of that election go are put on to a new campaign and a new one-on-one election. Briefly,Approval voting was first promoted in the 1970s. It has recently been put into use by several professional societies in the United States. footnote The American Psychology Association in 1986? and the American Society of Mechanical Engineers in 1988? But should I mention that APA dropped Hare? It lets a voter give one vote to each candidate. Steven or Stephen? Brams (1979) suggests to each voter that cast an approval for one and only one of the top two candidates and as many minor candidates as he rates above that one. Note that in the common sense of the word he might not approve of that one leading candidate. But if she is the lesser of two evils, one of whom is likely to win, then it is in his interest to cast an approval vote for her. The candidate with the most approvals wins. Note that a majority is not required. The system of counts created by Jean-Charles de Borda  in 1781 gives a candidate points for each rank voted. A first-rank vote gives [that candidate] points equal to the number of candidates minus one. A second rank gets vote gives points equal to the number of candidates minus two and so on. XL use this method to get Borda scores for any # of candidates The candidate who gets the most points wins. Thus voters can hurt the leading candidate simply by ranking her last. That is bad for most voters because it can often defeat the moderate, central candidates. Several polls for ranking sports teams use Borda voting. The next 6 rules are Condorcet completion systems. Duncan Blacks 1958 rule elects the Condorcet winner if one exists. Otherwise it elects the Borda winner. Clyde Coombs 1954 alternative vote, like Hares, eliminates candidates until one gets a majority. But it eliminates the candidate with the most last-place votes. Voters who move a major rival to the bottom of their ballot help eliminate her. A. H. Copelands 1950 rule gives a candidate 1 point for winning a pairwise contest against another candidate and -1 for losing. (In voting cycles, Copeland often produces ties so it does not complete Condorcet.) A voter who ranks the leader last might increase the number of candidates who beat the leader. Charles Ludwidge Dodgson (author Lewis Carrol) proposed in 1876 to elect the Condorcet winner or, in the event of a cycle, the candidate who needs to change the fewest ballots to become the Condorcet winner. John Kemenys 1959 system determines how many rank pairs must be exchanged (or flipped) on voters ballots to make a candidate win by Condorcets rule. The candidate who requires the fewest changes wins. A voter who ranks the leader last increases the number of rank pairs which must be changed to give his ballot to the leader. Given sincere ballots it may measure how close other Condorcet-completion methods come to picking a centrist candidate. The 1950s _____s max-min system elects the candidate with the smallest pairwise loss. (It is not the same as Dodgson. A candidate may lose pairwise elections to two rivals by 5% each. Her max-min score would be -5%. But she might have to change 10% of the ballots to become Dodgsons winner.) In Samuel Merrills 1988 standard-score voting system, voters rate candidates on a fixed scale, say 0 to 100. It then makes each voters ratings average zero (some ratings become negative). It also normalizes the variation within a voters ballot. This keeps any one voter from spreading out his ratings to influence the election more than others voters. If voters know each candidates chances and vote carefully, this system can select the candidate with the maximum utility to the electorate. Merrill and Straffin more fully explain most of the decision rules mentioned in this article. I shall not describe in detail other utility voting systems such as Clarks incentive revealing device or Hylland and Zeckhausers point voting. The tables I have borrowed from other authors do not include any. Most are quite complicated. They often fail to elect a Condorcet winner (because they elect the utility maximizer), are easy to manipulate (punishing and coalitions defeat most such systems), and their ballots have high information costs may confuse and burden voters(see footnote ). They do not fit majority rules one person / one vote. Hopefully they can be adapted for groups such as legislatures seeking proportional outcomes for all parties. A group might adapt a utility voting system to get proportional outcomes for all members but not to meet the Supreme Courts standard of one person/one vote. Manipulation Statistical and psychological proofs or arguments here? communications theory for quantifying the difficulty of a conspiracy Define manipulated results: sincere Condorcet winner (SCW) = candidate A, the runner-up = B, other candidates are C and D. Voting patterns may be limited to manipulation by one party usually the runner-up, two parties the runner-up and the SCW, all parties voting undominated strategies. We could symulate assymetries in the percentage of each partys voters who attempt manipulation. .2C.Condorcet efficiency; Even though Condorcet winners can beat each of the other candidates in one-on-one elections, most voting rules do not always elect them. MSTV failed to do so in figure 2 and examples 2 and 5. Given 100 elections with no voting cycles, what percentage of the 100 Condorcet winners will each voting system elect? This number is a voting systems Condorcet efficiency. To estimate the efficiency of each voting system, several political scientists have used computers to simulate groups of voters. I suspect Tidemans Condorcet efficiencies from ISR thermometer surveys would make Hare look better. .c6.Tables 7 and 8. Condorcet Efficiencies; Table 7. Condorcet Efficiencies in computer simulated elections with 4 candidates and 4 issues data from Chamberlin & Cohen (1978) 21 Voters 1000 Voters Voting Impartial Candidate Dispersion Impartial Candidate Dispersion system culture Low Medium High culture Low Medium High Coombs 93 96 98 99 91 81 99 99 Borda 86 83 83 92 89 85 86 97 Hare 92 72 75 90 92 32 60 84 Plurality 69 59 53 77 69 27 33 70 Table 8. Condorcet Efficiencies in computer simulated elections with 5 candidates and 1000 voters from Merrill, page 24 Spatial model Dispersion = 1.0 Dispersion = 0.5 . Voting Random C = 0.5 C = 0.0 C = 0.5 C = 0.0 . system society D = 2 D = 4 D = 2 D = 4 D = 2 D = 4 D = 2 D = 4 Plurality 60 57 67 61 81 21 28 27 42 Runoff 82 80 87 79 96 31 44 39 62 Hare [MSTV] 88 78 86 83 97 34 50 38 72 Approval 67 74 78 81 84 73 76 75 82 Borda 85 86 89 89 92 84 87 86 88 Coombs 90 97 97 95 97 90 91 90 94 Black (Con) 100 100 100 100 100 100 100 100 100 Utility maximizer 78 83 88 88 90 80 85 83 86 % [of elections with] 76 99+ 99 99+ 99+ 98 98 98 99 Condorcet winners Spatial model refers to simulations with a bell-curved distribution of voters on each issue. A dispersion of 1.0 (or medium) means the average distance between candidates opinions is as wide as the average distance between voters opinions; 0.5means the candidates tend to be more moderate than the voters. (See figures 3 and 4 on page 24.) The latter corresponds to the assumption that most candidates seek the large group of voters in the middle of the bell curve (as shown inSee figure 1). Low dispersion=0.4 and high=1.5. C=0.5 means there is some relationship between a voters position on one issue and his position on others; C=0.0means there is no correspondence between issues. Dis the number of issues simulated. Plurality has the worst scores. Runoff and MSTV also do poorly in some situations. Often MSTVs flaw results from the squeeze effect, that was shown in Figure2. In computer simulations, MSTV drops from finding about 80% of the Condorcet winners in contests with 3 candidates to less than 50% with 5 candidates.(Merrill:24) MSTV suffers from a mediocreThe Concorcet-completion rules by Black, Coomb?, Copeland, Dodgson, and Kemeny, Max-min have Condorcet efficiencies of 100% like CSTV. But manipulation of those rules can hide Condorcet winners, as we saw in Example6 and even promote false ones. CSTVs resistance to manipulation maintains is key to its high Condorcet efficiency in real life. simple declarative sentence at the end Merrill explores Condorcet efficiencies in more complex situations too. CSTVs Condorcet efficiencies in complex environments remain 100% by definition. Because CSTV is not subject to the squeeze effect. It cannot drop in a polarized society as MSTVs does. It will tend to pick the most central candidate. (Please see figure 8 on page 27.) Nor could CSTVs efficiency rise (above 100%) with rising voter perceptual uncertainty about candidates positions on issues. MSTVs efficiency does rise as voter perceptual uncertainty rises, (Merrill, page 39) but it remains lower than CSTVs. Condorcet-completion rules pick the majority winner no matter the number of canidates. The chance of a voting cycle [increases sightly?] as the number of candidates increases. The Condorcet efficiency of M-STV and other non-Condorcet rules drops as the number of candidates increases. Obviously, the elections in which MSTV picks the Condorcet winner are a subset of those in which CSTV does. Surveys and actual elections reveal some randomness, some clusters of like-minded voters and some agreement on the candidates relative positions left to right. A mixture of random and a spatial models roughly resembles these actual patterns. But just as random and spatial models lead to different results, so the actual data differs from both of them. Tideman reportedly foundthat even plurality picked the Condorcet winner in 95% of three-candidate elections. He used survey data to simulate rank-order ballots. (Merrill, page 70) This does not recommend plurality since its efficiency drops as the number of contestants rises and all other systems scored higher. Chamberlin and Featherston simulated ballots to resemble the clustering and distribution they found in the APA electorate. The simulated ballots [see Table 9] So the pattern of opinion dispersion and clusters effects Condorcet efficiencies. But the relative standing of the voting systems does not change. After the Condorcet completion rules Coombs is best; the worst is always plurality. * Condorcet efficiency, the ability to select pick-out choose the Condorcet winners in elections which have them, has great importance because they tend to be the median candidates and a happy result for the greatest number of voters. This is not necessarily the greatest total happiness as utility voting systems attempt to define it. [Merrill used computer simulations to compare voting systems. The percentages cited resulted from simulations which assumed: two issues, a non-random, normal or bell-curved distribution of voters opinions on each issue, no zero correlation of opinions on the two issues, candidates with a range of opinions half as dispersed as the voters, and finally no voter uncertainty about the candidates positions on the issues. In the these simulations, 200 voters chose among 5 candidates in each of 1,000 elections. Merrill also included looked at the effects of voter uncertainty, pre-election polls and strategic voting in which each person uses polling information, optimizing his ballot to elect candidates he likes and block those he dislikes. Chamberlin, Cohen, and Coombs (1984) found similar Condorcet efficiencies from slightly different simulation assumptions. Using ballots data simulated to resemble the APA data, Chamberlin and Featherston (1986) found Hares Condorcet efficiency about 85%.] .c3.Utility efficiency; The major competitor to Condorcet efficiency is utility efficiency. It attempts to measure how likely a voting system is to elect the candidate with supporters who feel strongly and opponents who don't much care. Many theorists people are skeptical about trying to compare utilities inter-personally; so Condorcet efficiency remains the most widely accepted measure. .c6.Table 9. Utility Efficiencies; Table 18. Utility Efficiencies in computer simulated elections with 5 candidates and 1000 voters. from Merrill, page 35 Spatial model . Dispersion = 1.0 Dispersion = 0.5 . Voting Random C = 0.5 C = 0.0 C = 0.5 C = 0.0 . system society D = 2 D = 4 D = 2 D = 4 D = 2 D = 4 D = 2 D = 4 Plurality 70 64 75 74 93 -1 0 22 52 Runoff 81 86 92 88 98 28 47 48 75 Hare (MSTV) 82 88 92 91 98 40 59 52 82 Approval 90 96 96 97 98 96 96 95 98 Borda 95 98 98 98 99 97 97 96 99 Coombs 87 96 96 96 98 92 92 92 94 Black (Con) 93 97 98 98 99 96 97 96 98 Spatial model refers to simulations with a bell-curved distribution of voters on each issue. A dispersion of 1.0 means the candidates range of opinions is as wide as the voters range of opinions; 0.5 means the candidates tend to be more moderate than the voters. The latter corresponds to the assumption that most candidates seek the large group of voters in the middle of the bell curve. C = 0.5 means there is some relationship between a voters position on one issue and his position on others; 0 means there is no correspondence between issues. D is the number of issues simulated. I think the pattern that simulates reality best is 50/50 random/spatial with C=0.5 and D=4 Simulation suggestion from Don Heyrich: for each voter, vary the number of issues and their weight. Many cast their vote based on a single issue. .c3.Utility efficiency estimate for CSTV; Merrill concludes his chapter on utility efficiency saying that : The candidate with the maximum social utility is no more likely to be the Condorcet candidate than is the candidate selected by many if not most of the systems studied. That is to say, the Condorcet criterion and the criterion of maximizing social utility are in fact very different. [Please see figure 7 on page 25.] Looked at from the other side of the coin, however, one sees that the Condorcet candidate generally has high social utility, although she may not have the highest of all candidates. This can be seen by comparing the social-utility efficiencies of the Black and Borda systems. The two systems differ only when there is a Condorcet candidate; [Black chooses the Condorcet candidate when there is one] the fact that the former has almost as high an efficiency as the latter indicates that the Condorcet candidate has relatively high social utility, although not as high as the Borda winner even when a Condorcet candidate exists. (Merrill, page 37) Whenever the two criteria indicate different winners, the Condorcet winner would get more votes than beat the utility winner would in a one on one election according to all of these voting rules. The problem with all utility voting systems is that a minority of rabid voters can claim on their ballots that their candidate has a much higher utility value for them than any other candidate. With this claim / ballots they may be able to steal the election from acomplacent majority. For spatial model simulations I estimate CSTVs social utility efficiency will be between 95% and 97%. In the case where CSTV does worst, a random society,with Condorcet winners in only 76% of the elections, times Blacks 93% efficiency in a random society; plus 24% non-Condorcet winners times Hares 82%, I estimate CSTVs social utility efficiency at 90%. This estimate is as high as Merrill's simulations of utility efficiency for approval voting a voting system based on measuring social utilities. Bordleys graphs, from simulations which were based on somewhat different assumptions than Merrills, show Copelands Condorcet-completion method is usually a little less efficient than Borda but better than approval voting. Table 9 shows Merrill found the same relationship between Borda, Black and approval. Thus the Condorcet-completion rules all have very high utility efficiencies. It is likely, although not certain, that the elections in which MSTV picks the central candidate will be a subset of those in which CSTV does. Certainly CSTV will have a higher utility efficiency than MSTV . .C3.Distribution of Winners; Chamberlin and Cohens 1978 spatial-model simulations showed Condorcet picked the candidate nearest the center of the electorate 87% of the time. So its utility efficiency equaled 87% plus some utility points for the second and third nearest candidates. Borda was the second best rule with 81%. This is from a spatial-model simulation with 4 candidates less dispersed than the 1000 voters. It included only elections with Condorcet winners, voting cycles were thrown out I think this suggests a political measure of political outcomes in contrast to the economic measure of utility. To measure the dispersions of voters and candidates and the distributions of winners and budget allocations (or other laws) assumes that each citizen has an equal right not only to vote but to be represented and to live under government programs compatible with the citizens philosophy. A system that produced proportional outcomes would reduce majority domination of minorites and so make empire building unattractive. The majority would lose some of its autonomy for every increase in territory. .c6.Table 10. Nearness to the Center of the Theoretical Electorate; Table 10. Nearness to the Center of the Theoretical Electorate 4 candidates with low dispersion relative to 1000 voters from Chamberlin and Cohen (1978) [Voting Nearest Furthest system] [candidate] [candidate] Condorcet .87 .11 .02 .00 Borda .81 .17 .02 .00 Coombs .75 .20 .05 .00 Hare [MSTV] .33 .33 .29 .05 Plurality .23 .27 .12 .38 Condorcet has the narrowest distribution. Hare has the second widest. CSTVs distribution of winners will depend on the percentage of elections with natural or manipulated voting cycles. We know that natural cycles are rare. Chamberlins 1986 article on the ratio of of natural to manipulated voting cycles. * Perhaps Condorcet tends to elect the centrist high utility candidates because it directly compares every candidate with each of the others. Simulations by Bordley and Merrill found Condorcets rule sometimes picked winners a bit farther from the utility center lower in social utility than Borda which uses all information in one step. Condorcet certainly beats Hare which uses only first-choice information at each of several steps. Notice that plurality tends to elect the least-favorite candidate, the one toward one edge out on a wing on a bell curve or scattergram. Thats because she has no competition for the first-place votes of voters in that area of the electorate. Meanwhile other candidates split-up the first-choice votes from the electorates center. of the... All voting systems, even plurality, always pick the centrist when there are only 2 effective partives with a chance of winning. But 2 party systems lead to low voter turnout which produces low information which may result in less than optimal social choices. .c2.Manipulation; Transition from efficiency to manipulation: Researchers have analized andsimulated voting systems with greater realism by considering investigating the posiblities for and likely effects of various types of manipulations. Any voting system is manipulable, sometimes. That is, all decisive, non-dictatorial voting systems in this article can be manipulated by introducing irrelevant alternatives. Clarks Tax and Hylland-Zeckhausers point voting might not be manipulable by introducing irrelevant alternatives, but parties or factions can avoid the tax and steal a bills outcome. The operant questions are How often is each voting system manipulable in a realistic electorate, how easy is the manipulation, and how damaging is its effect? The empirical evidence to date suggests CSTV resists manipulation best. .c3.Punishing the leading candidate with last-place votes; : a one-sided manipulation Most of these voting rules occasionally reward some opposition opposition voters for punishing give each voter a strong incentive to punish the leading candidate with last-place votes. That usually hurts the leaders score, which helps the oppositions favorite candidate to win toward victory. Example 6 demonstrates this. In contrast, punishing the leading candidate, A, with a last-place vote cannot help the voters first choice, B, to win under Condorcets rule. The voter already ranks his favorite as number 1. So an insincere ballot cannot increase the number or percentage of voters who rank his favorite, B, ahead of the main rival, A. But the punishing vote might decrease the chance that A could win by Condorcets rule, because the insincere voter might be helping another popular candidate, C, (whom he would rank below both A and B on a sincere ballot) to beat the original leader. This may make C win by Condorcets rule or it may create a voting cycle which moves Condorcet-completion rules the election from Condorcets rule to their completion rules. i.e. Black switches from Condorcets to Bordas rule. In fact, even if most voters although 8 of 9 would honestly rank C last, insincere ballots can sometimes many voters do this, they can even make her look like a Condorcet winner. - an outcome they would like even less than B Systems which reward punishing votes are unlikely to find true Condorcet winners. and may indicate false ones as shown in Example 5 b. Such systems may (destroy) true Condorcet candidates. Bold? To block B without electing C, the insincere voter(s) must manage to purposefully create a voting paradox a cycle in which C beats B who beats A who beats C. (See Example 1.) In theory this is possible but practical opportunities are rare and difficult to manipulate. It requires a fairly large conspiracy, accurate knowledge of other voters preferences, and it cannot work unless the candidates are close in popularity to begin with. Condorcet offers voters no strategy which produces more favorable results than sincere ballots do. The most they can do to hurt the major rival is to create a voting cycle - in effect a tie - with an even less desirable candidate. And a voters insincere ballot is likely to help elect the candidate he really detests [but insincerely ranked ahead of the leading candidate]. I have adapted Example 6 altered from one Merrill (on page 66) used to prove that Condorcet-completion rules do not necessarily elect true Condorcet winners when voters have polling information and then vote strategically. Blacks, Coombs?, Copelands, Dodgsons, and Kemenys Condorcet-completion rules all fail this real-world test. - this test of hard-ball politics. CSTV succeeds. .c5.Example 6. Punishing Vote Strategy; Example 6. Punishing Vote Strategy a) Sincere Voting Interest groups ballots . Pairwise comparisons .Ballot441 A gets 5 votes to 4 against B etc.ranksvotersvotersvoterABC1st choiceABCA wins5:48:12ndBAAB4:58:13rdCCBC1:81:8 From this pre- election poll the major voting systems produce a unanimous result: Here are the results of this poll as calculated by several voting rules: Candidates A B C Agenda Plurality 4 4 tie Runoff Approval 1 4 4 1tie - indecisive tie Approval 2 9 8 1 Black (Con) Borda 13 12 2 Chamberlin (Con) Coombs X Copeland (Con) 2 0 -2 CSTV (Con) Dodgson (Con) 0 -1 -4 MSTV (Hare) X Kemeny (Con) 0 -1 -8 Max-Min (Con) +11 -11 -78 Nanson Net % Std-score (1,0,-1) 4 3 -7 (Con) = a Condorcet-completion system. X = an eliminated candidate.  For calculation of Kemeny as distance from majority ordering see end note. Now all voters know that A leads the race is the leader. Voters opposed to A can punish her with last-place votes to decrease her score relative to the other candidates. In Example 5 b, supporters of B decide to vote strategically under most systems. To help their favorites, voters hurt most those candidates who score highest in the polls. b) Strategic Voting by Bs party Interest groups ballots . Pairwise comparisons .Ballot441 A gets 5 votes to 4 against B etc.ranksvotersvotersvoterABC1st choiceABCA5:44:52ndBCAB4:58:13rdCABC5:41:8 A bests B who bests C who bests A. This voting cycle makes the Condorcet-completion rules use a second rule to decide their winners. Whether or not they are based on Condorcet, almost all rules are easily defeated by punishing votes. Our major rules produce these results for the final election: Here are the results of the final election: ` Candidates A B C Agenda tie Plurality 4 4 tie Runoff Approval 1 4 4 1 tie Approval 2 5 8 5 Black (Con) Borda 9 12 6 Chamberlin tie? Coombs X X Copeland (Con) 0 0 0 tie CSTV (Con) X Dodgson (Con) -1 -1 -4 tie MSTV (Hare) X Kemeny (Con) -2 -2 -5 tie Max-Min % (Con) -11 -11 -77 tie Nanson X drops lowest Borda scorer: C Net percentage leads to rankings = Borda (Young 1974) Std-score (1,0,-1) 0 3 -3 For calculation of Kemeny as distance from majority ordering see end note. By voting strategically, Bs supporters would win or tie the election according to most voting rules. Notice that strategic voting by Bs supporters gives a less favorable result not only for voters who ranked Afirst but also for those who honestly favored C. So strategic voting hurt a majortity of the voters in this example. Cs supporters also can vote insincerely against A. But they would only help B not C. As supporters may try to counter Bs strategy by punishing B. In that case all of these rules would choose C, the least-liked candidate. The important point is that A would not need to counter Bs strategy in this case under CSTV, or MSTV Chamberlin . If Bs supporters have the will to manipulate [and to risk electing C if As party counters] then only CSTV, and MSTV and Chamberlin are truly Condorcet efficient in this example. Only these two rules are truly Condorcet efficient in this example. Chamberlin s rule might say A wins because B beats C by more than A does. To avoid this B could ask some voters to give C first place but that is as difficult as the squeeze to manipulate STV. And like STV the window of opportunity is small. To reach the majority ordering B A C requires fewer changes than the ordering A B C does. I only achieved ties on the Condorcet-completion rules. How does one manipulate Dodgson. xc3.Punishing and inclusion: a voters point of view Standard-score, like Borda, uses all information in one step. Both lead to voters punishing the leading rival even if she is the voters second favorite candidate. If voters of both major parties (center-left and center-right) do this they negate each other and produce net scores near zero for each major candidate Herbert What do Merrill, Neimi, and Brams call this? Why do they discount it? Let us consider the most common approval voting strategies, inclusion and exclusion or punishing, from the perspectives of two voters. Wolfgang likes the Greens. He does not like the Social Democratic Party nor the Christian Democratic Union. Should he cast as approval for the SPD one of the two leading parties? If he shows his honest disdain for SDP-CDU, then he does not approve of either and he has no effect on the election. He will be, in effect, a non-voter, disregarded, disengaged, unrepresented. This is so to varying degrees for all voters outside the two major parties. This forces him to so corrupt and dissemble my political values in order to influence government. He might even convince himself that he had won. With CSTV he would see clearly that he had won (only) his second choice. He does not rank the SDP equal to the greens. Herbert, the centrist voter, will often feel forced to punish one of the (usually two) major candidates. If he votes for both he will have no effect on the election result. If he votes for only one he lumps the other together with the Communists and far right New Republicans. Analyses of manipulation under approval have failed to define this dissembling as a significant strategic manipulation. So estimates of the incentive to vote strategically have been low. Political designers must consider honesty from the various voters perspectives. (compare Chamberlin, Cohen , and Coombs 1984:490) This is more than a semantic problem with the name approval voting. The psychological quandary would be the some no matter what name was used for n votes per voter. MSTV or CSTV also transfer Wolfegangs vote to the SDP. So it is no more sincere in this sense than approval is. The only difference here is that approval requires voters to think-out their best strategy, where as CSTV does it without the voters effort - or consent. .c3.Frequency of manipulable elections; Punishing is simply one of the easiest ways for voters to manipulate an election. Others include decapitation (not voting for ones sincere first choice). How difficult is it to manipulate an election? How often can voters and their party leaders manipulate an election? The first question requires a framework based in psychology and information science (degrees of insincerity, degrees of risk, amont or type of information needed, communication needed). The second question needs mathimatical proofs or statistical profiles. Each analysis must rest on observations and data from actual elections. Randomly generated ballots and purely mathematical analysis do not resemble accurately actual ballots and human psychology/game playing. Chamberlin, Cohen, and Coombs assessed the minimum numbers of voters needed to change the winners of actual elections. They did not include possible manipulations by politicians such as the introduction or deletion of irrelevant alternatives. They used the ballots from 5 previous elections for the presidency of the American Psychological Association. .c6.Tables 2 and 3. Manipulability; Tables 2 and 3. Manipulability data from Chamberlin, Cohen, and Coombs (1984) The total number of voters tended to increased over the years. Each year, about 6,000 voters ranked all 5 candidates. Year 1976 1978 1979 1980 1981 # of voters 11560 15285 13535 15449 14223 Table 2. Minimum Coalition Sizes Necessary for Manipulations Voting 1976 1978 1979 1980 1981 system U P U P U P U P U P Plurality 500 500 552 552 551 551 778 778 1 1 Borda 444 964 72 476 591 842 158 849 28 104 Hare * * 35 * * * * * * * Coombs 834 1430 468 26 63 64 36 524 254 517 Approve 2 375 662 99 379 293 406 32 428 286 307 Approve 3 714 1199 454 740 373 705 868 1277 20 156 Kemeny (Con) 1312 1822 572 819 821 971 240 957 467 566 Max-min (Con) 1410 2110 575 801 783 1240 242 1006 467 566 Black (Con) 1200 1588 531 649 616 971 231 616 321 410 * = Manipulation not possible. (Con) = a Condorcet completion system. Approve 2 = Approval votes for the voters top 2 choices. Approve 3 = Approval votes for his top 3 choices. U = Uniform majority ordering P = Proportional majority ordering Uniform and proportional majority orderings were used to fill the empty ranks of ballots on which the voters marked only their first few choices. Uniform ordering filled-up ballots randomly so as to give no net advantage to any remaining candidate. The researchers state, This corresponds to the assumption that voters are indifferent to candidates whom they do not rank. Proportional ordering made the artificially-completed ballots resemble voter-completed ballots with the same top preferences. This method corresponds to an assumption that voters omitted candidates because they lacked sufficient knowledge, and that if these voters had the knowledge necessary to complete their ballots they would have done so with the same preferences as those on the similar but complete ballots. More people must conspire to manipulate a proportionally-filled ballot set than [ are needed ] to manipulate the same ballots filled uniformly. It rewards the voters who complete their ballots by giving them influence over how candidates get ranked on the incomplete ballots. {So it gives the consiecious voter a ballot and a half.} About 6,000 voters ranked all 5 candidates each year. Table 3. Minimum Coalition Sizes as a Percentage of Voters with the Incentive and Ability to Aid in Manipulation Voting 1976 1978 1979 1980 1981 system U P U P U P U P U P Plurality 16.0 14.6 13.3 12.6 16.0 6.2 18.6 18.2 0.1 0.1 Borda 8.6 19.2 1.0 7.0 10.2 14.5 2.2 12.4 0.5 0.5 Hare * * 0.7 * * * * * * * Coombs 16.2 28.2 6.8 0.4 1.1 1.1 0.5 7.9 4.0 8.2 Approve 2 12.6 21.3 2.7 9.5 14.2 21.2 0.8 10.5 11.0 13.1 Approve 3 21.2 33.4 11.6 23.1 14.1 26.5 27.3 39.4 0.6 4.5 Kemeny (Con) 27.7 46.4 8.1 12.1 14.1 16.7 3.5 15.0 18.9 22.4 Max-min (Con) 63.1 91.5 23.2 32.8 45.4 74.7 10.3 32.1 18.9 22.4 Black (Con) 25.4 35.2 7.6 9.6 10.6 16.7 3.3 9.6 13.0 16.2 The researchers reported: The most striking result is the difference between the manipulability of the Hare system and the other systems. Because the Hare system considers only current first preferences, it appears to be extremely difficult to manipulate. To be successful, a coalition must usually throw enough support to losing candidates to eliminate the sincere winner (the winner when no preferences are misrepresented) at an early stage, but still leave an agreed upon candidate with sufficient first-place strength to win. This turns out to be quite difficult to do. One other factor also distinguishes the Hare system from the other[s]. The strategy by which Hare can be manipulated, on the occasions when this is possible, is quite complicated in comparison with the strategies for the other methods. (Chamberlin, Cohen, and Coombs) The authors contrast those strategies for 2 pages. { Also / Add to this As they and Merrill imply, the first preference is the rank most likely to be sincere on each ballot. Voters often must change that preferrence to manipulate STV and this probably extracts a high psychological cost more than many voters would feel comfortable with. is be { It seems to be the item to focus on to discourage manipulationif combined with an elimination process. Together they create the need to know other voters complete preference orders. Merrill states Indeed, since the Hare system appears very difficult to manipulate, strategic voting tends to be identical with sincere voting... (on page 66) The manipulability of the three Condorcet-completion rules (Kemeny, max-min, and Black) proves that in each election a group of voters could create a voting cycle and also change a count such as the Borda used by Blacks rule or the percentage used by the max-min rule. That will be possible for at least one party in any election.? Still, page 6 shows the need to create a cycle makes CSTV even harder to manipulate than MSTV because it increases the number of voters who must be organized into a conspiracy. Tidemans findings reportedly agree with these.(Merrill, page 70) He used data from thermometer polls surveys of voter opinions about the candidates for the 1972 and 1976 presidential nominations. It is worth noting that he found Dodgsons Condorcet-completion rule to be about as resistant to manipulation as Hares (MSTV) rule. But to manipulate Dodgsons rule needs less information than STV requires about other voters preference lists. So those who want to manipulate Dodgson can plan and coax voters into a simple strategy. For now I must base this point on psychological factors. Perhaps someone will quantify it in information units to be gathered, estimated, calculated, and communicated. It may be more difficult to quantify degrees of psychological uneasiness and risk typically required for manipulating each rule. Also, as shown above, CSTV resists manipulation even more than MSTV does. Alternative voting is hard to manipulate because it looks at only (recalculated) first preferences. The first-place is the rank most likely to be sincere on voters ballots. Moving a rival up or down the ranks below has no immediate effect. Manipulation of Dodgson, CSTV, and other Condorcet-completion elections requires creating a voting cycle; unless a cycle would have occured anyway. Chamberlin has shown that the occuracne of a cycle often may inidcate an attempted manipulation. If the cycle was not manipulated, it indicates a decission with rather weak support compared with one or more alternatives. So a second vote might be a good policy. But with complete election results manipulations would be easier the second time around. Therefore the election comission must not announce any results: who tied nor even who lost. Why then use STV? To limit the process to two elections. > I dont want to even suggest this mass of confusion. .c3.Irrelevant alternatives; The winner under Condorcets criterion cannot does not be changed if we by removing any other candidate(s), nor if we by introducing any less popular candidate(s). (Merrill, page 98) Political scientists would say no one can manipulate it by through the introducing of irrelevant alternatives. Politicians rather easily can manipulate many almost all? elections under other voting systems by using this strategy. That means in most systems politicians can make the winner become a loser by introducing a candidate who is less popular than the former winner. Introducing irrelevant alternatives includes related to the strategy by which single-vote plurality defineparties help start-up candidates on the opposite political wing to divide the opposition. Example ii in the introduction showed how an irrelevant alternative could create a voting cycle and change CSTVs loser into its winner. Niemi and Riker cite an example in which the entry of a new last-place finisher reverses the order of finish in a Borda election. (See Example 2 in the introduction.) Hylland-Zeckhausers Point voting (and Merrill's Standard-score?) may be manipulated by introducing irrelevant alternatives to divide the supporters of similar alternatives. [This paragraph is too short.] For approval voting, voters are usually told to vote for just under half of the candidates - Brams rule. A gaggle of leftists, perhaps even 2 strong ones, against a single solid right-wing candidate, would split the lefts votes - and hand victory to the right. [Politicians have and will exploit this divide-and-conquer formula.] Approval does poorly in Table 4, Violations of Subset Rationality. Chamberlin and Cohen suggest their models strategy of casting approvals for a number of candidates makes approval look worse than it is. Yet it is certainly more vulnerable to this common political trick than Condorcet is and probably more so than Hare is. Standard-score is probably subject to this flaw as much as Approval or Borda. Copy the Cal Tech Borda example refd in Sci Am 6/76 This political trick is fairly simple and common. .c6.Table 4. Violations of Independence of Irrelevant Alternatives; Table 4. Violations of Independence of Irrelevant Alternatives Spatial model of 200 voters and 5 candidates repeated in 1,000 elections from Merrill, page 98 Voting system Violation % Plurality 19 Runoff 10 Approval 9 Borda 7 Hare (MSTV) 6 Coombs 1 Black (Condorcet/Borda) 0.1 [ CSTV (Condorcet/Hare) 0.1 estimated] Approval 9 Black (Condorcet/Borda) 0.1 Borda 7 Case Study. Republicans Split Northern and Southern Democrats In the late 1950s the US House of Representatives considered a bill to increase federal funds for local schools. The Democratic Party favored the bill and had enough votes to pass it. Republicans, opposed to the whole bill, reasoned that if they proposed an amendment to block the funding of segregated schools, districts, the Northern Democrats would be compelled by constituents to support it. The Southern Democrats then would have no political choice but to join the Republicans in voting against the amended bill. The Northern and Southern Democrats behaved predictably and the Republicans succeeded in killing the school-funding bill. Lets see what would have happened under different voting procedures. Here are the approximate sizes and preferences of the three voting blocks. .c5.Example 7. Republicans Split Democrats North & South (3 Interest groups) Example 7. Republicans, Northern & Southern Democrats Ballot 161 Northern 80 Southern 160 ranks Democrats Democrats Republicans Pairwise comparisons 1st Amended Bill No bill Amended Bill . 2nd Bill No bill Amended Bill 80:321 3rd No bill Amended Bill No bill 240:161 160:241 . This is a voting cycle. The amended bill beats the plain bill by 321 votes to 80 votes. The plain bill beats no bill by 241 votes to 160. And no bill beats the amended bill 240 to 161. A > B > N > A. This is the paradox of voting noted by Merrill in footnote 1 and the Glossary. Should they pass a bill to increase funding and fight segregation? No voting system picks the plain bill. It gets the fewest first-choice votes. If the House votes first on funding then on desegregation both would pass; if the Republicans vote for the desegregation amendment they proposed. But most parliamentary procedures require voting on the amendment before the bill. So the House would pass the amendment and then defeat the bill as actually happened. This case follows Duncan Blacks rule of thumb as cited by Straffin, ...the later you bring up your favored alternative, the better chance it has of winning(page 20) Here Bill which could beat No bill was itself beaten in the previous round by Amended bill. They get the same result, nothing, from the CSTV and MSTV voting systems as noted below. Keep in mind that without the amendment, the plain Bill would have passed by 241 Democratic votes to 160 Republican votes. Options . Amended Bill No bill Agenda Plurality 161 80 160 Runoff Approve of 1 161 80 160 Approve of 2 321 241 240 Black (Con) Borda 482 321 400 Chamberlin (Con) X Coombs X Copeland (Con) 0 0 0 tie CSTV (Con) X Dodgson (Con) -40 -121 -41 MSTV (Hare) X Kemeny (Con) -40 -121 -41 Max-Min % (Con) -19.7 -60 -20.2  Nanson 161 X 240 drops lowest Borda scorer: B Net percentage leads to rankings = Borda (Young 1974) Std-score (1,0,-1) drops fewest 1st place votes: B drops most last-place votes: N tie - indecisive after eliminating B Borda 482 321 400 Kemeny -40 -121 -41 Max-Min % -79 -240 -80 If the Northern Democrats out-number the Republicans then Amended bill would win by most rules. Under Copeland all options would tie. Under agenda, MSTV, and CSTV No bill would win. We would say CSTV was manipulated by an irrelevant alternative because the new alternative did not win, yet reversed the order of the original two options. The Republicans might argue that they exposed the fact that some of the school funds would have gone to support racist school districts which most voters did not approve of and did not want to pay taxes for. Because of this new issue dimension, previously unconsidered, CSTV reverses its result. If the Republicans had added an amendment to set funding higher or lower than the Democrats bill, then the CSTV result would not be reversed. No bill would still be defeated by either the original Bill or the Amended bills funding amount. The amendment certainly was not an irrelevant issue; but strictly speaking it was an irrelevant alternative. If parties propose several funding levels then the Republicans could not manipulate CSTV to avoid all funding. Could they manipulate voting rules subject to the punishing vote strategy ? I think they could play the agenda game. $ 2 Issue Dimensions 161 N. Dems. 80 S. Dems. Funding $ Desegregation Amended - O O -Bill O -no amend Yes 241 321 (maybe) Desegregation Segregation No 160 80 O -No bill 160 Republicans A similar campaign trick to divide and conquer the opposition for primary or general elections is discussed on page 4. I will discuss the most important social and normative differences affected by voting systems and CSTV in particular at the end of this article. If the Republicans outnumber the Northern Democrats, that switch of one vote changes the result to No bill under most voting systems. If the Republicans rank the desegregation Amended bill last, and raise the plain Bill to second place, then the plain Bill would beat each of the other options in one-on-one contests and win under most voting systems. No bill could still win only under agenda and Hare. The Republicans in this case used several manipulation techniques. First they introduced an amendment that some theorists might consider an irrelevant alternative. It created a voting cycle. Then they probably voted insincerely to punish the leading option. No one can prove insincere votes but many of these same Republicans often voted against desegregation so I doubt they sincerely preferred the Amended bill over the plain Bill. I give this negative example of CSTV last to impress upon readers that no decisive, non-dictatorial voting system can guarantee complete resistance to manipulation in all situations. CSTV is most subject to manipulation in committee voting. Dennis Mueller writes in a section titled Cycling, Thus it would seem that when committees are free to amend the issues proposed, cycles must be an ever present danger. (page64) If the amendments create a cycle, then CSTV starts to eliminate proposals. It is hard to manipulate that process, but it is possible. A committee might adopt procedures that allow the chairperson to put aside for a time such a cyclical vote and issue or to continue debate and hold another election, perhaps one with new proposals. This is a procedural approach to Condorcet completion. If debate cannot resolve the voting cycle then the procedure may eventually require a completion rule. Particularly in elections with few voters or few voting blocks. Neimi and Riker cite a case in which the entry of a new last-place loser reverses the results of a Borda count. Deleting less popular candidates can change the winner also. This test uses real-world ballots to measure a voting systems vulnerability to changes in the slates minor candidates. They found Hare second only to Coombs in resistance to this strategy. .c6.Table 5. Violations of Subset Rationality; Table 5. Violations of Subset Rationality data from Chamberlin, Cohen, and Coombs (1984) Number of violations when X is reduced from 5 candidates to Voting 2 3 4 system Candidates Candidates Candidates Total Plurality 5 7 2 14 Borda 2 2 1 5 Hare (MSTV) 2 2 0 4 Coombs 0 0 1 1 Approve 2 1 17 1 19 Approve 3 3 6 5 14 [ Condorcet 0 0 0 0 R.L.] The authors note that Violations of this subset rationality condition when a single candidate is omitted seem most serious... [emphasis added] Hare was the only system with no violations when a single candidate was omitted. There were no voting cycles in the large-scale real-life elections and polls studied by these authors and Tideman. So CSTV would have found Condorcet winners and each eletions winner would not have been changed due to the withdrawal of any lesser candidate(s). .c2.Sensitivity to incomplete ballots; We may need or want to use incomplete ballots ones with some candidates not ranked. In this situation no voting system can guarantee electing the true Condorcet candidate. That is because there might not be enough information on the ballots to tell who would be the Condorcet winner if all ballots were complete. Voting systems may indicate a Condorcet winner, but that winner might change with more information. Unfortunately all vote-counting rules will miss lose some of their ability to find the most central candidate more often when voters cast incomplete ballots. Such ballots can perturb the results of all voting systems - but to different degrees. Some systems will stumble due to a small percentage of bad ballots (or non-voters?) while other systems probably tolerate this problem better. To better understand the effects of incomplete ballots, we need a study similar Chamberlin and Cohens on the deletion of candidates, shown in Table 5. For now I shall re-use some of their published results to estimate the sensitivity to incomplete ballots for five voting systems. Tables 2a and 2b in/from their article showed/shows how five election rules ranked all candidates, from winner to last-place loser. Table 2a showed/shows their results when they filled the incomplete ballots uniformly giving no favor to any candidate. The researchers state: This corresponds to the assumption that voters are indifferent to candidates whom they do not rank. Table 2b gave/gives their results when they filled the ballots proportionally making the artificially-completed ballots resemble voter-completed ballots with the same initial preferences. This method corresponds to an assumption that voters omitted candidates because they lacked sufficient knowledge, and that if these voters had the knowledge necessary to complete their ballots they would have done so with the same preferences as those on the similar but completed ballots. Uniformly filling ballots comes closer conceptually to blanks (all zeros) and so perhaps voters and legislators would accept it more easily than proportional completion. It is not certain that we need to complete ballots by any means. This is repitious If a voting system showed/shows many differences between those/these two tables, then it is very sensitive to how the incomplete ballots are filled and probably sensitive to non-completion or the use or deletion of incomplete ballots. If how the ballots are filled-up makes a difference, then whether voters completed them probably does also. Table 6 shows the number of differences, in winners and complete social rankings, between Chamberlin, and Cohens Tables 2a and 2b. .c6.Table 6. Sensitivity to Methods of Filling Incomplete Ballots; Table 6. Sensitivity to Methods of Filling Incomplete Ballots from data of Chamberlin and Cohen (1978) Ordering Generated by . Plurality Borda Hare (MSTV) Coombs Approve 2 Approve 3 Winners changed 0 1 1 1 0 0 Other positions " 0 1 1 6 1 4 Most of the systems tested by Chamberlin and Cohen sometimes picked a different winner depending on which completion method they used. A change of winners more seriously effects us than a change further down the collective ordering. So I tentatively rank the voting systems sensitivity to incomplete ballots as: plurality, approve 2, (approve 3, Borda, Hare), and Coombs. This list seems reasonable based on how the systems select winners. Plurality always picked the same winner, runner-up and so on, no matter which completion method the researchers used. It uses only the first choice; so whatever they filled in below made no difference. Coombs eliminates the candidate with the most last-place votes; so whether or how they filled the bottom of the ballots made a big difference. Approval appears more stable than it is. We simply cant tell if an approval ballot is complete or not: if the voter had enough information on each of the candidates to vote yes or no on each. But these were approval votes calculated from complete and incomplete rank-order ballots. Incomplete ballots cause no greater problem for CSTV than for most multi-candidate systems. except plurality, and approval? In the next chapter/ In a later section I will argue that attempt to show on page 21 faulty ballots are least likely to occur under CSTV. In the case of approval this is not provable. Incomplete ballots are not detectable in approval voting, but many voters may leave a candidate unapproved just because they are undecided or havent heard enough about her. In a large electorate with deleted or incomplete ballots randomly distributed, a change? would make no difference in the outcome of the election. But if one class of voters, let us say the less educated, fails to vote or complete their ballots, then they will have less influence on the election than their numbers entitle them to. This is so for all voting systems except single-vote plurality. Really? Approval ? What do the Australians do? Aspects of the problem question 1) What is the sensitivity of each voting rule to 2) What should we do with these ballots? 3) What is the frequency of incomplete ballots (for each voting system?) 4) The effectiveness of voting has been shown to encourage people to vote. Under proportional representation each vote adds to a parties number of seats in the legislature. Delete a set % of ballots or a set % of the incomplete ballots To sum-up this section comparing CSTV with the other multi-candidate voting rules: 1)CSTV probably is no more sensitive to incomplete ballots than the other rules. The next topic will show CSTV strains voters less: it induces much less worry about voting strategies. 2) CSTV has the highest possible efficiency at picking the canidate with broad support and it has a very high social utility efficiency. 3) Most importantly, CSTV resists manipulation very well and always elects a candidate close to the center. So in competitive political situations its winners probably will have higher Condorcet and utility efficiencies than any other voting systems. It induces the sincere ballots needed by any voting system for electing utility maximizing and Condorcet candidates and finding. It may promise the greatest happiness for the greatest number of voters. Kemeny as distance from majority ordering: Sincere voting: OrderingUnanimityMajorityOrderingUnanimityMajorityABC102ACB17BAC115BCA62CBA175CAB168 Strategic voting: OrderingUnanimityMajorityOrderingUnanimityMajorityABC102ACB17BAC115BCA62CBA175CAB168  Niemi shows that in some cases In the absence of dichotomous preferences, sincere approval voting need not select a [the] candidate who has a majority of first place preferences. and ...approval voting may select a candidate whom the majority of voters prefers least. These faults are impossible under CSTV. Note that the name approval does not always carry the common meaning of the word approval.  Each voting system traditionally is identified by its inventors surname.  ...for each voter separately, replace her ratings ri by their statistical standard scores, i.e., S ? Zi=(ri-ʵ)/s where and s are the mean and standard deviation of the voters ratings. The standard-score system, because of the complexity of its decision rule, should be recommended only for a mathematically knowledgeable electorate. (Merrill pages 101 and 103) Each voter needs to calculate each candidates probability of tieing for the win to make his vote tip the balance.  Merrill coined this term and defined it. The Condorcet efficiency of a voting procedure is the proportion or percentage of a class of elections (for which a Condorcet candidate exists) in which the voting system chooses the Condorcet candidate as winner. (Merrill : Glossary)  In simulated elections, Merrill found the frequency of cycles ranged from 47.5% for elections with 10 candidates and 25 voters randomly distributed on issues, to less than 1% with 5 candidates and 200 voters normally distributed in a bell-shaped curve. This distribution simply means most voters are moderates. (pages 20, 24) Chamberlin and found similar percentages for their simulation assumptions. On ballots from actual elections of the American Psychology Association, Chamberlin, Cohen, and Coombs found Condorcet winners, therefore no cycles, in 5 out of 5 elections. Those were five-candidate races using rank-order ballots. Did Tideman report the frequency of cycles? Hide this in the text ?  Researchers attempt to make utility measure the distance between a candidate and a voter on an issue. They average the scores for all issues to determine the expected utility value of the candidate for that voter. The candidates averaged utility score for all voters is said to be her social utility to the electorate. The highest candidate scores from from each election in a series of elections are averaged to find the highest average possible[ for the social utilities from those elections]. Then the social utility scores of winners under a voting rule are averaged and compared with the highest possible to give theorists a number for the utility of the rules utility winners efficiency as a percentage of the highest utility possible. Following R.J. Weber (1977), most Researchers subtract a large number of utility points, equal to the score of a randomly selected candidate, from both the utility maximizer and the voting systems winners. The size of each score is reduced. But the difference between their scores remains the same. So the difference is now a larger percentage of a score. This exagerates the differences between voting systems on utility efficiency. You must decide whether such exageration helps you see the differences or misleads your understanding of these differences. Get definitions from Merrill, Bordley, and Mueller. There are several different conceptions of distance, linear, logarithmic and so on (Bordley, ; Merrill page 42), and no standard unit to measure interpersonal utility for all types of issues. For these reasons, many people are skeptical about the meaning, comparison, and statistical manipulation of interpersonal utilities. Voting system designers [and economists in general] have not yet found a unit to measure interpersonal utilities that can overcome or bypass the inequalities and diversity of our society. Money, which is used in the Clark Tax voting system, certainly does not work equitably. Legislative power points have many other problems with coalitions, agenda changes. Most utility voting systems use a dimensionless number with and no grounded zero point, and no grounded meaning for the voters.  If Blacks 97% (97.25%, rounded to significant figures) average utility efficiency results from 1 or 2% of elections (those with voting cycles) at (multiplied by) Bordas 98 (97.75)% utility efficiency, plus 98 or 99% of elections at Condorcets (?)% utility efficiency , then (?) % = 96 to 98%. Then CSTVs utility efficiency = 98 or 99% of elections times 96 to 98% efficiency, plus 1 or 2% of elections at times the (40 to 98 %) efficiency of Hare = a 95 to 98% utility efficiency for CSTV. 97% is the top I calculated  If Blacks 93% efficiency results from 24% of elections (those with voting cycles) at Bordas 95% utility efficiency, plus 76% of elections at Condorcets (?)% utility efficiency , then (?) = 92%. CSTVs utility efficiency = 76% of elections at 92% efficiency, + 24% at 82% = 90% utility efficiency. (I have assumed that the presence of a voting cycle does not effect Bordas efficiency in random society.) I have had to assume that in a random society Bordas ability to elect the candidate with the highest social utility is as strong in elections with voting cycles as it is in elections with Condorcet winners. The calculations use 98% in both situations. I have made the same assumption for MSTV.  ...all voting systems permit manipulation, as was shown by Gibbard (1973) and Satterthwaite (1975). Thus, the practical questions for social choice theory to answer are the extent to which different systems encourage strategic calculations in voting, their effects on the nature and perceived legitimacy of the outcome, and their implications for political stability. (Merrill, page xvii)  2) Standard scores of -1, 0, and 1 are used for simplicity. just as approval voting made all voters cast the same number of votes. 1) Of course, the standard-score system allows any value between the extremes on a continuous scale.  Merrill states of the Hare system, There is also no incentive, as there is under the Borda count, for a voter to move the chief rival of his favorite to the bottom of his preference order. As long as his favorite remains in the race, lower preferences are not counted. If his favorite is eliminated, there is no motivation for the voter to try to punish his former chief rival. [Indeed, since the Hare system appears very difficult to manipulate, strategic voting tends to be identical with sincere voting, so that Condorcet efficiency is little affected.] (page 65) Move into text?  Other manipulations include changing the sequence of preferences as the third voter did on page vii. That particular change, not voting for his first choice, is called decapitation and is most common under single-vote plurality, Borda and standard score. (Please see figure 6 on page 25.) Many other rules reward bullet voting or plunking: voting only for ones first choice such as approval.  This concerns the frequency of manipulable elections as found in simulations and practise. It doesnt not contradict the theoretical proofs by Gibbard and Satterthwaite that any possible voting system rules is manipulable to some degree.  In 98% of Merrill's spatial-model elections there was no voting cycle, so Black's rule used the Condorcet criterion whose results did not change due to irrelevant alternatives. In the remaining 2% of elections Black it used Bordas rule which was vulnerable to irrelevant alternatives in 7% of these elections. 2% x Bordas 7% manipulability = 0.14% which rounds to 0.1%. This was the number Merrill reported for Black, but his number came from simulation of Black. He did not inferre it from simulations of Condorcet and Borda. CSTV's score also would equal be zero for 98% of the simulations elections. Add 2% of Hare's score for a total of 0.12%. 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