Borda s Paradox. Theodoros Levantakis

Transcription

1 orda s Paradox Theodoros Levantakis

2 Jean-harles de orda Jean-harles hevalier de orda (May 4, 1733 February 19, 1799), was a French mathematician, physicist, political scientist, and sailor. In 1770, orda formulated a ranked preferential voting system that is referred to as the orda count. The French cademy of Sciences used orda's method to elect its members for about two decades until it was changed by Napoleon onaparte the year following orda's death. The orda count is in use today in some academic institutions, competitions and several political jurisdictions. The orda count has also served as a basis for other methods such as the Quota orda system and Nanson's method.

3 Plurality voting System orda had serious doubts about the plurality voting system So shortly before the outbreak of the French Revolution, orda presented a paper to the Royal cademy of Sciences in Paris which called attention to a problem associated with the plurality voting system and suggested a completely new electoral system

4 orda s Paradox The problem concerning the plurality voting system was stated by orda as: When the alternative ranked first by more voters than any other alternative is defeated by every other alternative in pairwise contests by the majority of votes, then we speak of orda's paradox.

5 Plurality rule/pairwise Majority rule Plurality rule: If the number of ballots ranking as the first preference is greater than that of every other candidate then candidate is the winner (voting only for top choice candidate). Pairwise Majority rule: Every candidate is matched against every other candidate and the winner of each contest is decided by majority rule

6 orda s Paradox (example) The original example that orda considered has a profile for 21 voters with complete preferences on three candidates: 1 Voter 7 Voters 7 Voters 6 Voters

7 orda s Paradox (example) The concern expressed by orda is related to the outcome of the election when plurality rule is used, versus the outcome when PMR (pairwise majority rule) If we use plurality rule for the example in table 1 then we can see that: > (8 7) > (8 6) > (7 6) So a complete and transitive rank according to plurality rule would be >>. So would be the winner. The voters would think that the decision was a fair and a democratic one since a majority of the voters have preferred to the others. orda showed that this belief is ill-founded.

8 orda s Paradox (example) very different result will occur if we use the PMR rule (Let M denote the situation in which more voters have than have in their preference rankings, regardless of the relative position of andidate in the preference ranking.) M (13 8) M (13 8) M (13 8) a complete and transitive PMR rank, would be MM and would be the winner using PRM rule.

9 orda s Paradox (example) With this particular profile, plurality rule and PMR reverse the ranks on the three candidates. This phenomenon is considered orda s Paradox. orda s major concern in this example seemed to be that the loser by PMR is elected by plurality rule orda strongly endorsed the notion of selecting the PMRW, and proposed a voting rule to be assured of selecting that candidate. Thus he introduced the orda ount.

10 orda s ount The orda count is a single winner election method in which voters rank candidates in order of preference. The orda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all votes have been counted the candidate with the most points is the winner. ecause it sometimes elects broadly acceptable candidates, rather than those preferred by the majority, the orda count is often described as a consensus-based electoral system, rather than a majoritarian one.

11 orda s ount (continue) Under the orda count the voter ranks the list of candidates in order of preference. So, for example, the voter gives a '1' to their first preference, a '2' to their second preference, and so on. In this respect a orda count election is the same as elections under other preferential voting systems, such as instantrunoff voting, the Single Transferable Vote or ondorcet's method. The number of points given to candidates for each ranking is determined by the number of candidates standing in the election. Thus, under the simplest form of the orda count, if there are five candidates in an election then a candidate will receive five points each time they are ranked first, four for being ranked second, and so on, with a candidate receiving 1 point for being ranked last. In other words, where there are n candidates a candidate will receive n points for a first preference, n-1 points for a second preference, n-2 for a third, and so on. lternatively votes can be counted by giving each candidate a number points equal to the number of candidates ranked lower than them, so that a candidate receives n-1 points for a first preference, n-2 for a second, and so on, with zero points for being ranked last. nother way to express this is that a candidate ranked in ith place receives n-i points. For example, in a five candidate election, the number of points assigned for the preferences expressed by a voter on a single ballot paper might be:

12 orda s ount (continue) Lets get back to our example 1 Voter 7 Voters 7 Voters 6 Voters Formula Points n 3 n-1 2 n-2 1

13 orda s ount (example) If we use the orda count we will have the following results: andidates Points candidate is the winner like it was when we used the PMR rule, so the paradox that we faced using plurality rule, that the least preferred candidate ( candidate) was selected, is avoided by using orda s count Winner

14 Statistical Significant? It is important to know whether the orda effect is likely to occur in committee decisions and elections or whether the distribution of preferences necessary to produce the effect is statistically so improbable as to rob it of any practical or political significance. The first attempts to illustrate that orda effect is indeed probable were made by Paris (1975) and Gillet (1976) but they only came close to doing this. olman and Pountney in their orda s Voting Paradox: Theoretical Likelihood and Electoral Occurrences article in 1978 were the first to prove that orda s effect had a significant probability to occur. In their study out of 261 constituencies in the 1966 ritish General Election contested by a onservative, a Labor and a Liberal candidate only, 246 failed to exhibit a orda effect and 15 exhibited a orda effect This represents a frequency of occurrence in excess of.057.

15 1860 US Elections Tabbarok and Spector (1999) presented a paper concerning the election of This election was one of the most important and contentious elections in US history. Four candidates from three parties battled for the presidency The main issue concerning the paper was if Lincoln s victory was sound or would a different voting system have represented the preferences of the voters more accurately

16 1860 US Elections ased on the data of the paper I constructed the following table andidate First Second Third Fourth Lincoln 39.8% 9.73% 20.11% 30.36% ouglas 29.5% 36.64% 21.87% 11.99% ell 12.6% 46.12% 40.95% 0.33% reckinridge 18.1% 7.51% 17.07% 42.68%

17 1860 US Elections Using the orda rule we would have the following results: andidate Lincoln ouglas ell reckinridge First 39.8 * *3 12.6*3 18.1*3 Second 9.73* * *2 7.51*2 Third 20.11* * * *1 Fourth 30.36* *0 0.33* *0 So the new winner would be ouglas and Lincoln would have dropped down to third place!!! Points

18 Unlike most other voting systems, in the orda count it is possible for a candidate who is the first preference of an absolute majority of voters to fail to be elected. This is because the orda count affords greater importance to a voter's lower preferences than most other systems, including other preferential methods such as instant-runoff voting and ondorcet's method. The orda count satisfies the monotonicity criterion, the summability criterion, the consistency criterion, the participation criterion, the Plurality criterion (trivially), Reversal symmetry, Intensity of inary Independence and the ondorcet loser criterion (excludes the possibility that a ondorcet loser will win) It does not satisfy the ondorcet criterion, (not always a ondorcet winner is elected) the Independence of irrelevant alternatives criterion, the Independence of lones criterion.

19 Paradox in orda s ount Using orda s count sometimes leads to another paradox: Withdrawal of a loser may influence the outcome

20 Example For example seven voters trying to choose among,,, candidates have expressed the following preferences: 1 n-3 2 n-2 3 n-1 4 n Points Formula

21 Example (continue) pplying the orda rule we get: andidates Points Winner

22 Unfortunately due to a scandal candidate had to withdraw his candidacy Lets see what happens 1 n-2 2 n-1 3 n Points Formula

23 pplying once again the orda rule we have: andidates Points new winner

24 Potential for tactical manipulation the orda count is vulnerable to tactical voting. In particular, it is vulnerable to the tactics of 'compromising' that is, voters can help avoid the election of a less preferred candidate by insincerely raising the position of a more preferred candidate on their ballot and 'burying' where voters can help a more preferred candidate by insincerely lowering the position of a less preferred candidate on their ballot.

25 Not strategic proof n effective tactic is to combine these two strategies. For example, if there are two candidates whom a voter considers to be the most likely to win, the voter can maximize his impact on the contest between these front runners by ranking the candidate whom he likes more in first place, and ranking the candidate whom he likes less in last place. If neither front runner is his sincere first or last choice, the voter is employing both the compromising and burying tactics at once. If many voters employ such strategies, then the result will no longer reflect the sincere preferences of the electorate. For example in the example of the 1860 Elections voters of ell with the preference : ell>ouglas>lincoln>reckinridge that believe Lincoln and ouglas are the front runners could switch to the insincere ballot ouglas>ell> reckinridge>lincoln in order to help ouglas win who is more desirable than Lincoln.

26 Relation to other voting systems The orda count is classified as a positional voting system because each rank on the ballot is worth a certain number of points. Other positional methods include first-pastthe-post (plurality) voting, and minor methods such as "vote for any two" or "vote for any three". The orda count has also served as a basis for other methods such as the Quota orda system and Nanson's method

27 Quota orda system The Quota orda System or Quota Preference Score is a voting system that was devised by the ritish philosopher Michael ummett and first published in 1984 in his book, Voting Procedures, and again in his Principles of Electoral Reform. If proportionality is required in a orda count election, a quota element should be included into the counting procedure, which works best in multi-member constituencies of either 4 or 6 members. The quota is calculated accordingly; in a single-seat constituency, the quota would be an absolute majority, i.e., (50% + 1) of the valid vote; in a 2-seat constituency, it is (33% + 1); in a 3-seat, it's (25% + 1); and in a 4-seat, it is (20% + 1) of the valid vote. The four-seat selection goes as follows; Stage i) ny candidate gaining a quota of 1st preferences is elected. Stage ii) ny pair of candidates gaining 2 quotas is elected. ( pair of candidates, Ms J and Mr M, say, gains 2 quotas when that number of voters vote either 'J-1, M-2' or 'M-1, J-2'.) If seats still remain to be filled, then, ignoring all those candidates who have already been elected; Stage iii) ny pair of candidates gaining 1 quota gains 1 seat, and the seat is given to the candidate of that pair who has the higher M score (modified orda count). Stage iv) ny seats still remaining are given to those candidates with the highest M scores.

28 Nanson's method The Nanson method is based on the original work of the mathematician Edward J. Nanson. Nanson's method eliminates those choices from a orda count tally that are at or below the average orda count score, then the ballots are retallied as if the remaining candidates were exclusively on the ballot. This process is repeated if necessary until a single winner remains.

29 Present The orda count is used for certain political elections in at least three countries, Slovenia and the tiny Micronesian nations of Kiribati and Nauru is a popular method for granting awards for sports in the United States It is also used in a number of educational institutions in the United States to elect Student Governments and Unions nd in several small committees and boards

30 QUESTIONS?