Nicholas R. Miller University of Maryland Baltimore County (UMBC)

1. THE ALTERNATIVE VOTE (AND COOMBS) VERSUS FIRST-PAST-THE-POST: A SOCIAL CHOICE ANALYSIS OF ENGLISH CONSTITUENCY ELECTIONS, 1992-2010 Nicholas R. Miller University of Maryland Baltimore County (UMBC) Second World Congress of the Public Choice Societies Miami, March 8-11, 2012 http://userpages.umbc.edu/~nmiller/NRMILLER.AVvsFPTP.PCS12.pdf

2. Overview: An Opportunistic Paper • Peter K-K’s invitation for paper on “empirical social choice” • I initially expected to do something relating to the U.S. Electoral College. • I was also working “Monotonicity Failure and IRV” and wanted to examine 2010 UK election data. • I found Pippa Norris’s Shared Datasets website, with 1992-2010 UK data. • The UK held a May 2011 referendum on electoral systems: AV vs. FPTP • The paper has no particular thesis or research hypothesis. • Rather it examines a sample of 2642 “real” ballot profiles, in terms of • the plurality (and anti-plurality) status of candidates; • the Condorcet status of, and relationships among, the candidates ; • plurality-Condorcet interrelationships; • winners under the three alternative electoral systems: FPTP, AV [IRV], Coombs; • the votes-seats proportionality of results under these electoral systems; • spoiler effects under all three systems; and monotonicity failure under AV and Coombs. • I also examine several more hypothetical sets of ballot profiles that “bracket” the primary set. • I do not here come to grips with “tactical voting” or a general assessments of the relative merits of the three systems. • FPTP won the referendum by a large margin. • The pre-referendum campaign did not stimulate an especially enlightening discussion of the properties and relative advantages and disadvantages of the two electoral systems.

4. Alternative Electoral Systems • On an FTPT (“First-Past-The Post”) ballot, • voters put an “X” beside the name of the candidate they wish to vote for, and • the candidate with the most votes is elected. • On an AV (“Alternative Vote”; known as Instant Runoff Voting [IRV] in US) ballot, voters rank the candidates in order of preference. • If one candidate has a majority of first preferences, that candidate is elected. • If no candidate is supported by a majority of first preferences, • the candidate with the fewest first preferences is eliminated and • his or her ballots are transferred to other candidates on the basis of the second preferences expressed on the ballots for the eliminated candidate. • This process is repeated until one candidate is supported by a majority of ballots and is elected. • “If you like the Alternative Vote, you ought to know about the Coombs rule” (Grofman and Feld, 2004): similar AV but with a different elimination rule: • the candidate with the most last preferences is eliminated. Bernard Grofman and Scott Feld, “If You Like the Alternative Vote (a.k.a. the Instant Runoff), Then You Ought to Know about the Coombs Rule,” Electoral Studies, 2004.

5. Political Context • Since they were displaced by Labour as the main opposition to the Conservatives in the inter-war period, the Liberals have joined the Electoral Reform Society in advocating the Single Transferable Vote for general elections. • STV is • the multi-member district generalization of AV and • (quasi-) proportional in nature. • More recently the Liberal Democrats have advocated AV as less radical but more acceptable alternative to FPTP. • Nick Clegg: “AV is a baby step in the right direction – only because nothing can be worse than the status quo.” • The Liberals demanded the AV referendum a condition for joining the Conservatives in a coalition government after the May 2010 general election.

6. Political Context (cont.) • As a generally centrist party, the Liberals have a lot of second preference support, though relatively little first preference support. • Since AV asks voters to rank candidates and often counts second preferences, at first blush it seems that AV would help the Liberals. • But while second-preference support helps a candidate if and when he gets into a runoffs, second-preference support doesn’t help a candidate get into a runoff in the first place. • As Aldrich et al. (2012) observe, if AV were applied at the national level (e.g., in a direct election of Prime Minister), the Liberals would still lose, since their candidate would not get into the runoff. • Aldrich et al. examine four alternatives to FPTP (Condorcet, AV, Coombs, and Borda) and conclude that the Liberals would have won such a national vote in 2010 under all four systems except the one they actually advocated. • A basic question is the extent to which individual constituencies are approximate replicas of the national electorate, implying that Liberal candidates would consistently fail to get into runoffs in which their second-preference support could elect them. • Coombs should be much more advantageous to Liberals because • second preferences not only help candidates win a runoff in the event they get in, but also • help them get into a runoff in the first place (by leaving them fewer last preferences). John Aldrich et al., “Strategic Voting in the 2010 U.K. Election,” APSA Meeting, 2012.

7. Social Choice Analysis • The analysis is based on three simplifying conditions: • there are only three candidates, • so there is at most only one “instant runoff.” • voters ranks all three candidates. • In practice, AV may allow “truncated” ballots, • but Coombs really cannot do so. • In practice Coombs (even more than AV) may be vulnerable to “donkey voting.” • there are no plurality (or anti-plurality) or pairwise ties. • A ballot profile is a set of n rankings of three candidates X, Y, and Z, where n is the number of voters.

8. Plurality Status • Given a particular profileB: • the candidate with the most first preferences is the Plurality Winner, • the candidate with the second most first preferences is the Plurality Runner-Up, and • the candidate with the fewest first preferences is the Plurality Loser. • Let n(PW), n(P2), and n(PL) be the number of ballots that rank the Plurality Winner, the Plurality Runner-Up, and Plurality Loser first. • Given three candidates, n(PW) > n/3 > n(PL). • If n(PW) > n/2, the Plurality Winner is also a Majority Winner. • The AV winner is • the Majority Winner if one exists and • otherwise either the Plurality Winner or the Plurality Runner-Up, depending on the outcome of the instant runoff between them. • Note that all these definitions depend on the distribution of first preferences only.

9. Condorcet Relationships • Condorcet relationships take account of second (and lower) preferences. • Given ballot profile B, • candidates X, Y, and Z have x, y, and z first preferences respectively, where x + y + z = n. • Likewise xyis the number of voters who have a first preference for X and second preference for Y (and therefore a third preference for Z),xz is the number who have a first preference for X and a second preference for Z, so xy + xz = x; and likewise for other candidates. • If under election profile B a majority of voters rank X over Y, i.e., if x + zx > y + zy, we say that ‘X beats Y’ in a “straight fight” (to use an appropriately British phrase). • This is the basic Condorcet relationship.

10. Condorcet Relationships (cont.) • Given three candidates and a particular profileB: • aCondorcet Winner is a candidate who beats both other candidates. • a Condorcet Loser is a candidate who is beaten by both other candidates; and • Condorcet Carrier is a candidate who beats one candidate and is beaten by the other. • In the theory of directed graphs, a carrier is a point with ‘in-degree’ and ‘out-degree’ both equal to 1. • If X beats Y, Y beats Z, and Z beats X or if Y beats X, X beats Z, and Z beats Y, • there is a Condorcet cycle, and • all three candidates are Condorcet carriers.

11. Anti-Plurality Status • Given ballot profileB: • the candidate with the fewest last preferences is the Anti-Plurality Winner, • the candidate with the second fewest last preferences is the Anti-Plurality Runner-Up, and • the candidate with the most last preferences is the Anti-Plurality Loser. • Let n(APW), n(AP2), and n(APL) be the number of ballots that rank the Anti-Plurality Winner, the Anti-Plurality Runner-Up, and Anti-Plurality Loser last. • Given three candidates, it follows that n(APW) < n/3 < n(APL). • The Coombs winner • is the Majority Winner if one exists and • otherwise is either the Anti-Plurality Winner or the Anti-Plurality Runner-Up, depending on the outcome of the instant runoff (based on the Condorcet relationship) between them.

12. Plurality--Condorcet Interrelationships • Some elementary social choice propositionsfor the case of three candidates: PROPOSITION 1. A Majority Winner is always a Condorcet Winner. PROPOSITION 2. The Plurality Winner always wins under FPTP (by definition). PROPOSITION 3. Plurality and Condorcet status are completely independent, in particular, • the Plurality Winner may be a Condorcet Loser, and • the Plurality Loser may be a Condorcet Winner, so • a Condorcet Winner may fail to be elected under FPTP; and • a Condorcet Loser may be elected under FPTP. PROPOSITION 4. Either the Plurality Winner or Plurality Runner-Up may be elected under AV, • so FPTP and AV may produce different winners. PROPOSITION 5. The Plurality Loser cannot be elected under AV, • because it is eliminated from the instant runoff. PROPOSITION 6. A Condorcet Loser cannot be elected under AV, • because, even if it gets into the runoff, it will be beaten in the runoff.

13. Plurality---Anti-Plurality—Condorcet Interrelationships • For the case of three candidates: PROPOSITION 7. Plurality and anti-plurality status are completely independent; in particular, • the Anti-Plurality Winner may be the Plurality Loser, and • the Anti-Plurality Loser may be the Plurality Winner. PROPOSITION 8. Condorcet and anti-plurality status are in the general case completely independent; in particular, • the Anti-Plurality Winner may be a Condorcet Loser, and • the Anti-Plurality Loser may be a Condorcet Winner. PROPOSITION 9. However, in the special case of single-peaked preferences, • the Condorcet winner has no last preferences and therefore is always the Anti-Plurality winner, so • Coombs always elects the Condorcet winner. • In fact, this is true regardless of the number of candidates.

14. Empirical Data • The empirical analysis that follows is based on constituency-level data from U.K. general elections from 1992 through 2010. • However, I use data from English constituencies only, • because virtually all constituency elections in England are essentially three-party (Labour, Liberal Democrat, and Conservative) affairs, • while those in Wales, Scotland, and Northern Ireland almost always include strong (and often winning) candidates of ‘nationalist’ parties as well. • A handful of English constituencies that do not fit the three-party pattern are also excluded. • Over the five general elections, this gives us a sample of 2642 three-candidate elections (527 to 531 per year). This data comes from Pippa Norris’s Shared Datasets website (http://www.hks.harvard.edu/fs/pnorris/ Data/Data.htm). I am extremely grateful to Professor Norris for making this valuable data readily available.

15. The Problem of Second Preferences • An obvious problem is that these elections were conducted under FPTP and therefore the election data provides us only (what we take to be) the first-preferences of voters, • while analysis of AV (and Coombs) elections requires voters’ full preference rankings of the three major-party candidates. • I have addressed this problem in a way that is fairly standard among those British psephologists, which is to allocate second preferences (and by default third preferences) in each district in proportion to second preferences nationwide, as determined by surveys that produce individual level data about second preferences. • Data for 1992 through 2005 comes from Curtice (2009), which in turn comes from the British Election Study (post-election) for 1992 and 1997 and from ICM/BBC (pre-election) for 2001 and 2005. Data for 2010 comes from Ritchie and Gardini (2012), which in turn was taken a (pre-election) poll conducted for ITV News and The Independent newspaper. • Survey respondents who gave a ‘nationalist’ or other fourth-party second preference, or who did not give a second-preference were excluded in these calculations, and proportions were calculated on the basis of Labour plus Liberal plus Conservative second preferences only. John Curtice, “Recent History of Second Preferences” (http://news.bbc.co.uk/ nol/shared/ spl/hi/uk_politics/10/ alternative _vote/alternative_vote_june_09_notes.pdf) Ken Ritchie and Alessandro Gardini, “Putting Paradoxes into Perspective — In Defence of the Alternative Vote,” in Dan S. Felsenthal and MoshéMachover, eds., Electoral Systems: Paradoxes, Assumptions, and Procedures, 2012

16. Survey-Based Second Preferences • As we would expect, that British voter preferences are “partially single-peaked,” • that is, most but not all Labour(presumptively ‘left-of-center’) voters have the Liberals (the presumptively ‘centrist’ party) as their second preference, and • most but not all Conservative (presumptively ‘right-of-center’) voters likewise have the Liberal as their second preference, while • Liberal voters have second preference more equally divided between the two other parties (though generally leaning in the Labour direction but with the proportion varying considerably from election to election).

17. Other Second-Preference Assumptions • Strictly Single-Peaked Preferences: all Labour and Conservative voters have Liberal as their second preferences. • No ideological structuring of preferences: • Random second preferences: on average, second preferences are equally split, and • Impartial second preferences: on each ballot profile, the second preference of each voter is determined by a flip of a coin. • SSP and Random/Impartial in effect “bracket” survey-based data. • In addition, we occasionally compare the English data with • wholly random ballot profiles and • wholly impartial (culture) ballot profiles., • in which first as well as second preferences are determined “randomly” or “impartially.”

18. Plurality Status of Party Candidates(All Second-Preference Assumptions)

19. Plurality Rankings of Party Candidates (All Second-Preference Assumptions)

20. Condorcet Relationships(All Second-Preference Assumptions Plus Random Profiles and Impartial Culture)

21. Plurality-Condorcet Interrelationships

22. AV vs. FTPT Winners • Given a switch from FPTP to AV, an average 7.5% constituencies change party hands. • Liberals are consistently helped by AV, losing not a single seat, gaining on average 4.3% of all seats (a 65% gain in seats). • Conservatives are almost as consistently hurt by AV, gaining a handful of seats (from Labour in 1992) but losing on average 6.3% of all seats (a 14.5% loss in seats). • Labour is inconsistently affected by AV, gaining seats in good Labour years, losing seats in bad Labour years, and gaining on average 1.9% of all seats (a 3.8% gain in seats.

23. Coombs vs. FPTP Winners • Given a switch from FPTP to Coombs, an average 16% constituencies change party hands. • Liberals are greatly and consistently helped by Coombs, losing not a single seat, gaining on average 11.2% of all seats (a 170% gain in seats), and placing second in three years. • Conservatives are almost as consistently hurt by Coombs, gaining seats from Labour in 1992 but losing on average 11.4% of all seats (a 26% loss in seats). • Labour is inconsistently affected by Coombs, gaining seats in good Labour years, losing seats in bad Labour years, and gaining/losing nothing on average.

24. Coombs vs. AV Winners • Given switch from AV to Coombs, an average 9.5% constituencies change party hands. • Liberals are consistently helped by AV, losing not a single seat, gaining on average 6.9% of all seats (a 63% gain in seats). • Conservatives are almost as consistently hurt by AV, gaining a few seats (from Labour in 1992) but losing on average 5.1% of all seats (a 13.8% loss in seats). • Labour is inconsistently impacted by AV, gaining seats in good Labour years, losing seats in bad Labour years, and losing on average 1.8% of all seats (a 3.5% loss in seats.

25. AV and FPTP Winners with DifferentSecond Preferences • The Liberals gain slightly fewer seats under AV if preferences are strictly single peaked. • With random second preferences, there is very little difference between AV and FPTP (random “erosion” of the FPTP distribution). • With impartial second preferences, there is essentially no difference between AV and FPTP.

26. Coombs and FPTP Winners with DifferentSecond Preferences • The Liberals gain vastly more seats under Coombs if preferences are strictly single-peaked, winning more seats than either Labour or the Conservatives, • exactly paralleling Condorcet Winner status (from previous table). • With random second preferences, there is almost no difference between AV and FPTP (random “erosion” of the FPTP distribution). • With impartial second preferences, there is essentially no difference between AV and FPTP.

27. Coombs and AV Winners with DifferentSecond Preferences • The Liberals gain vastly more seats under Coombs if preferences are strictly single-peaked, winning more seats than either Labour or the Conservatives, • With random second preferences, there is almost no difference between AV and FPTP (random “erosion” of the FPTP distribution). • With impartial second preferences, there is essentially no difference between AV and FPTP.

28. Disproportionality(Survey Second Preferences) • Gallagher Index of Disproportionality = • Coombs delivers quite proportional results in 2010 but highly disproportional results in 1997 and 2001. • But overall all three system are substantially and approximately equally disproportional. Michael Gallagher, “Proportionality, Dispropor-tionalityand Electoral Systems,” Electoral Studies, 1991

29. Disproportionality(Strictly Single-Peaked) • In this data, Coombs vastly overcorrects the penalty imposed on the Liberals by FPTP and generally is less proportional than FPTP and AV. • Overall all three system are substantially disproportional, FPTP and AV being about equal disproportional as under the previous data.

30. Spoiler Effects • Given a potential three-candidate (X, Y, and Z) election, the entry of candidate X into what would otherwise be a two-candidate contest (or X’s exit from a three-candidate contest) can have three effects: (1) No Spoiler Effect I: the same candidate (Y or Z) wins in either event; (2) No Spoiler Effect II: X wins if X enters, Y or Z wins otherwise; or (3) Spoiler Effect: X does cannot win even if X enters, but Y wins if X is out and Z wins if X is in, i.e., X’s entry “spoils” Y’s election (or X’s exit “spoils” Z’s election). • Two types of spoiler effects: • unilateral spoiler: X’s entry spoils Y’s election but Y’s entry does not spoil X’s election; • mutual spoilers: X’s entry spoils Y’s election and Y’s entry spoils X’s election (e.g., X and Y are “clones”); • In the following discussion, majority, plurality, anti-plurality, and Condorcet status refers to (potential) the three-candidate ballot profile.

31. Spoiler Effects (cont.) • PROPOSITION 10. If there is a Majority Winner, no spoiler effects occur under FTPT, AV, or Coombs. • Otherwise we examine eight possible election configurations with respect to • the plurality status of the candidates and the Condorcet relationships among them (for FPTP and AV); and • the anti-plurality status of the candidates and the Condorcet relationships among them (for Coombs). • There are two configurations of each following type: • PW (or APW) is CW [(1a) and (1b)]; • P2 (or AP2) is CW [(2a) and (2b)]; • PL (or APL) is CW [(3a) and (3b)]; • there is no CW (cyclical) [(4a) and (4b)]. • The outcome of every straight fight is determined by the Condorcet (“beats”) relationship.

32. Spoilers under FPTP • Under FPTP, the winner of the three-candidate contest is (by definition) always PW. • The number by each candidate shows the spoiler effect that results when that candidate enters/exists the election.

33. Spoiler Effects under FPTP • PROPOSITION 11. The Plurality Winner can never be a spoiler under FPTP. • PW wins whenever it enters, so always No Spoiler Effect II. • PROPOSITION 12. FPTP is invulnerable to spoiler effects if and only if PW = CW [(1a) and (1b)]. • PROPOSITION 13. FPTP is vulnerable to mutual spoiler effects if and only if PW = CL [(2b) and (3b)]. • PROPOSITION 14. Otherwise FPTP is always vulnerable to unilateral spoiler effects.

34. Spoilers under AV • Under AV, the runoff (if any) is between PW and P2. • So the winner of a three-candidate election is • PW if PW  P2, and • P2 if P2  PW.

35. Spoiler Effects Under AV • PROPOSITION 15. The Plurality Loser can never be a spoiler under AV. • PL never makes the runoff and, in the runoff, second preferences are distributed just as they would be if PL never entered. • PROPOSITION 16. The Condorcet Winner is never a spoiler under AV • PROPOSITION 17. AV is vulnerable to spoiler effects if and only if PL = CW or the profile is cyclical [(3a), (3b), (4a), (4b)]. • PROPOSITION 18. AV is invulnerable to mutual spoiler effects. • PROPOSITION 19. AV does not [logically] “dominate” FPTP with respect to resisting spoiler effects; in particular, PW can be a spoiler under AV (if it is not the AV winner).

36. Spoilers under Coombs • Under Coombs, the runoff (if any) is between APW and AP2. • So the winner of a three-candidate election is • APW if APW  AP2, and • AP2 if AP2  APW.

37. Spoiler Effects under Coombs • PROPOSITION 20. The Anti-Plurality Loser can never be a spoiler under Coombs. [APL never makes the runoff, and second preferences are distributed just as they would be if APL did not enter.] • PROPOSITION 21. The Condorcet Winner is never a spoiler under AV. • PROPOSITION 22. Coombs is vulnerable to spoiler effects if and only if APL = CW or in the event of a cycle [(3a), (3b), (4a), (4b)]. • PROPOSITION 23. Coombs is invulnerable to mutual spoiler effects. • PROPOSITION 24. Coombs does not [logically] “dominate” FPTP with respect to resisting spoiler effects. • If preferences are single-peaked, • the Condorcet Winner has no last preferences, and • the Anti-Plurality Winner is therefore always the Condorcet Winner, • so only configurations (1a) and (1b) can occur. • PROPOSITION 25. If preferences are single-peaked, Coombs is invulnerable to spoiler effects.

38. Frequency of Spoilers in England under FPT, AV, and Coombs • Overall, in this data AV and Coombs are considerably less vulnerable to spoilers effects than FPTP. • The incidence of spoilers is slightly greater under Coombs than AV, even though • the data is partially single-peaked, and • Coombs in invulnerable to spoilers given strictly single-peaked profiles. • It is notable than the 2010 election (close to a three-way tie with respect to first preferences) was highly vulnerable to spoiler effects under both FPTP and AV but not Coombs.

39. The “Nader Effect” • A fundamental virtue of AV (and Coombs) is that it precludes the “Nader effect,” i.e., • the fact that a minor candidate (i.e., one who wins only a small percent of the vote) can be a spoiler under FPTP (in the manner of Ralph Nader in FL [and therefore the US as a whole] in 2000). • Under AV (and Coombs), (most) Nader voters could have (and presumably would have) ranked Gore second, so • their votes would have transferred to Gore in the runoff (as if Nader were not on the ballot or all Nader voters had voted “tactically” under FPTP). • However, if Nader and been a third “major” candidate, rather than a “minor” candidate, he might still have been a spoiler under AV. • Suppose that Nader had been more “centrist,” • so Gore had been “squeezed” between Bush and Nader, • rendering Gore the Plurality Loser, while still the Condorcet winner. • The runoff then would have been Bush and Nader, rather than Bush and Gore, and Bush would (as the near Majority Winner, even if most [but not quite all] Gore voters ranked Nader over Bush) have won the runoff.

40. The “Nader Effect” (cont.) • We have seen that under AV, only the Plurality Runner-Up or Plurality Winner can be a spoiler. • Since spoiler effects can occur only in the absence of a Majority Winner, the smallest vote the Plurality Runner-Up can receive and possibly be a spoiler is slightly more than 25%. • Thus a candidate who wins less than 25% of the (first-preference) vote cannot be a spoiler under AV. • Note that this does not imply that that AV is invulnerable to spoiler effects if the Plurality Loser in a prospective three-way contest wins less than 25% of the vote. • Such a prospective Plurality Loser may prevent the Plurality Winner from being a Majority Winner, and thereby allow the Plurality Runner-Up to be a spoiler, i.e., • the Plurality Loser (e.g., Gore “squeezed between Bush and Nader) wins a straight fight with the Plurality Winner (e.g., Bush) but Plurality Winner wins if the Plurality Runner-Up (e.g., Nader) entersand displace the Plurality Loser from the runoff.

41. Propensity to Spoiler Effects by Support for the (Prospective) Plurality Loser and Plurality Runner-Up

42. Are The Same Elections Vulnerable to Spoiler Effects Under the Different Electoral Systems? • In this data, elections subject to spoiler effects under AV are a subset of those likewise subject under FPTP. • In contract, elections subject to spoiler effects under Coombs and FPTP and under Coombs and AV are almost disjoint.

43. Frequency of Spoilers by Party • Liberal candidate are never spoilers under AV, • evidently because they are typically Plurality Losers and/or Condorcet Winners. • As we would expect, mutual spoilers are always ideologically adjacent, i.e., “near clones.”

44. Monotonicity Failure • A disconcerting feature of AV (and Coombs) is that • getting more (first preference) votes can cause a candidate to lose an election, and • getting fewer votes can cause a candidate to win. • While FPTP is monotonic, electoral systems that incorporate (actual or ‘instant’) runoffs are subject to (upward and/or downward) monotonicity failure. • Though AV (and Coombs) are unarguably non-monotonic, the question arises how often instances of monotonicity failure arise in practice. • The English data provide a good opportunity to address this question.

45. Conditions for Monotonicity Failure under AV • Given that Z is the Plurality Loser, two separate conditions must hold for an AV ballot profile to be vulnerable to (Upward or Downward) Monotonicity Failure in the event that candidate X is moved up or down in some ballot orderings. • Condition 1 pertains to X’srunoff opponent;it requires that the ballot changes • deprive Y of enough first preferences (for Upward Monotoncity Failure), • or give Z enough additional first preferences (for Downward Monotoncity Failure), • to convert Z into the Plurality Runner-Up, • so the runoff that had been between X and Y is now between X and Z. • Condition 2 pertains to the runoff outcome; it requires that • Xmust lose (for Upward Monotoncity Failure) or win (for Downward Monotonicity Failure) the new runoff with Z. • The conjunction of the two conditions is necessary and sufficient to make profile B vulnerable to (Upward or Downward) Monotonicity Failure. • Note that Condition 1 depends on the distribution of first preferences only, while Condition 2 depends on second preferences as well.

46. Upward Monotonicity Failure under AV • If ballot profile B is vulnerable to Upward Monotonicity failure, • Condition 1 requires that X can gain enough first preference ballots at Y’s expense that two things are simultaneously true in the resulting profile Bʹ: • Xis still not a Majority Winner, and • Y becomes the Plurality Loser instead of Z. • This requires that n/2 − x > y − z. • Condition 2 requires that Z beat X under Bʹ, i.e., that zʹ + yzʹ > xʹ + yxʹ. • It turns out that Condition 1 can be simplified and Condition 2 can be restated in terms of the original ballot profile B, as follows: PROPOSITION 26. A ballot profile B in which X is the IRV winner and Z is the Plurality Loser is vulnerable to Upward Monotonicity Failure if and only if: (1) z > n/4; and (2) z + yz > x + yx. Nicholas R. Miller, “Montonicity Failure in IRV Elections with Three Candidates,” PCS, 2012 Dominique Lapelley et al., “The Likelihood of Monotonicity Paradoxes in Run-Off Elections,” Mathematical Social Sciences, 1996

47. Downward Monotonicity Failure under AV • If ballot profile B is vulnerable to Downward Monotonicity failure, • Condition 1 requires that it is possible for X to lose enough first preference ballots in favor of Z that two things are simultaneously true in the resulting companion profile Bʹ: • Zis no longer the Plurality Loser, and • Y, rather than X, becomes the Plurality Loser. • Thus it must be that x −  y > y − z. • Furthermore, in order for Z to gain these first preferences rather than Y, these (y − z) new first preference ballots for Z must all come from the xz ballots that initially ranked Z rather than Y second. • Condition 2 stipulates that X beats Z under Bʹ, i.e., xʹ + yxʹ > zʹ + yzʹ. • Again these conditions can be simplified and restated in terms of the original ballot profile Bonly: PROPOSITION 27. A ballot profile B in which Y is the IRV winner and Z is the Plurality Loser is vulnerable to Downwards Monotonicity Failure if and only if: (1) y < n/3 and xz> y − z; and (2) y + yz < n/2. Nicholas R. Miller, “Montonicity Failure in IRV Elections with Three Candidates,” PCS, 2012 Dominique Lapelley et al., “The Likelihood of Monotonicity Paradoxes in Run-Off Elections,” Mathematical Social Sciences, 1996

48. Monotonicity Failure in English Data vs. Random Profiles

49. Contrast Between English and Random Data • The English data contains considerably fewer profiles vulnerable to monotonicity failure (1.7%) than the random data (14.1%). • This might suggest that simulated data is irrelevant and misleading: once we look at “real” electoral data, the problem of monotonicity failure under AV almost disappears. • However, this low incidence reflects particular features of the English election data, and does not demonstrate that IRV’s non-monotonicity problem is practically irrelevant. • The primary determinant of vulnerability to monotonicity failure is election closeness, but very few of these English elections represented closely contested three-candidate contests (in part because they were actually conducted under FPTP, not AV). • In all English ballot profiles • 60% had a Majority Winner, and • in only 4.2% did the Plurality Loser get as much as 25% of first-preference support (and 39.2% of these profile were vulnerable to Monotonicity Failure). • Controlling for elections closeness, vulnerability to Monotonicity Failure looks very similar in the two data sets.

50. Monotonicity Failure by Closeness in English vs. Random Data