# The Romanian 2009 presidential election featured one or more high Condorcet cycles

By Warren D. Smith, final draft 13 Dec 2009.

Abstract. With confidence≈99.7%, the Romanian 2009 presidential election included the "Condorcet cycle" Geoana⊃Oprescu⊃Basescu⊃Geoana where "⊃" means "majority-preferred." B (followed by G) was the official winner. Other cycles also may have occurred; in all there is 99.9% confidence at least one existed that contained both B & G. With "approval voting" (and probably also "range voting") Oprescu would have been the most likely winner, even though he officially finished 6th. This marks the first time a Condorcet Cycle has been detected high in a large national election.

## 1. Introduction

It has been well known since the days of the Marquis de Condorcet (1743-1794) – and previously clearly had been understood by Ramon Llull (1232-1315) – that Condorcet cycles can exist in elections. That is, the preferences of the voters can be such that every candidate has at least one rival preferred over him by a voter majority. For example, consider this 19-voter election:

Table 1. Example election with 19 voters and three candidates A,B,C, exhibiting a "Condorcet cycle."
#voters Their Preference Order
8 B>C>A
6 C>A>B
5 A>B>C

Should candidate B win? One could argue "No, since A is preferred over B by 11 out of 19 voters." So should A win? "No, since C is preferred over A by 14 out of 19 voters." So C must be the right winner? "Still no, since B is preferred over C by 13 out of 19 voters."

Condorcet cycles are tremendously important morally and philosophically since they show that "naive majoritarianism" is logically untenable, since self-contradictory. That is, it is untenable to claim "if there are a finite number of choices and choice A is preferred by a voter majority over choice B, then A is 'better' than B." The whole idea of "majority rule" therefore seems to need to be discarded as a basis for "democracy." Condorcet cycles also underlie, e.g. Arrow's impossibility theorem for rank-order-ballot voting systems.

This can be considered a good reason to prefer non-rank-order-ballot voting systems not afflicted by this kind of logical paradox, unaffected by Arrow's theorem, and obeying a different (and now logically consistent) notion of "majority rule," e.g. range voting.

All that was fine as a matter of abstract philosophy – but in "real world" elections Condorcet cycles seem to be quite rare.

Although they seem to arise considerably more frequently in legislative votes than in single-winner public elections, because legislators often strive to create them intentionally, e.g. they form the basis for the "poison pill" legislative tactic. Many examples have been documented in the writings of William H. Riker, and see also endnote 3 of Kurrild-Klitsgaard 2001.

Indeed, in the 750 years since Llull, apparently not a single Condorcet top-cycle has ever been clearly documented in any large governmental election! Which leads to the question: how frequent are Condorcet cycles? Some attempted answers are in table 2.

Table 2. The probability of a Condorcet top cycle (i.e the probability that no "Condorcet Winner" exists) in various probabilistic models of elections.
#Canddts Probabilistic model Prob(No Condorcet winner) Source
3 Random election model aka "Impartial Culture" (all 3!V rank-order-ballot elections with V voters equally likely, in limit V→∞) G = [3arcsec(3)-π]/[4π] = [3arctan(√8)-π]/[4π] ≈ 8.78% G.Gilbaud (stated in footnote without proof) 1952.
4 Random election model aka "Impartial Culture" (all 4!V elections with V voters equally likely, in limit V→∞) 2G ≈ 17.55% "R.May's theorem," reproven very simply by P.Fishburn 1973.
C Random election model aka "Impartial Culture" (all C!V elections with V voters and C candidates equally likely, in limit V→∞) Exact formula as 1D integral known, yields Prob(no CW)≈51.125% when C=10 and Prob→100% when C→∞, indeed Prob(Cond.Winner exists)<C-0.99 for all sufficiently large C. W.D.Smith Jan. 2009, see CRV Puzzle #102.
3 Dirichlet model [the 3! kinds of vote-total, using rank-order-ballots, are uniformly distributed on 5-simplex; V→∞ voters] 1/16 = 6.25% W.D.Smith, see CRV Puzzle #85. and #42.
4 Dirichlet model [the 4! kinds of vote-total are uniformly distributed on 23-simplex] 331/2048=16.162109375%
C Dirichlet model [the C! kinds of vote-total are uniformly distributed on (C!-1)-simplex] Goes to 100% when C→∞, indeed Prob(Cond.Winner exists)<C-0.99 for all sufficiently large C. A.Quas 2004.
3 Random election model aka "Impartial Culture" (all 12V elections with V voters equally likely, rank-orderings with equalities allowed as "votes" in limit V→∞) [3arccos(2/5)-π]/[4π] ≈ 2.68% W.D.Smith 2004, see CRV Puzzle #7.
C Candidates are distinct points on a real line and each voter's utility function for the candidates is an increasing-then-decreasing function of position on that line; voting honest & odd number of voters 0 D.Black's singlepeakedness theorem from 1948
C≥1 "Random tournament" model where every pairwise victory is decided by a fair coin toss (independent coins for all pairs A,B) 1-21-CC easy
3 Data-fit statistical model of "real world" (1.2±0.2)% T.N.Tideman 2006, recalculated by W.D.Smith 2009
4 Data-fit statistical model of "real world" (2.5±0.4)%
30 Data-fit statistical model of "real world" (25.1±1.7)%

This absence, however, does not necessarily prove cycles rare. They might have been quite common but merely escaped notice! [Actually, all it proves is that Condorcet cycles easily provable from the usually-small subset of the full preference dataset that manages to become publicly available, are rare.] Why? Because in order to prove a Condorcet cycle exists in some election, you need to have either

1. rank-order-ballots,
2. or extensive "A versus B" pairwise-preference polling.
In fact, in the last 750 years, it is doubtful that there have even been 100 major government elections where it even was possible to hope to detect a Condorcet cycle. In all the other major government elections, the necessary data simply was never collected, or never made public. To explain that:

1. Elections with rank-order ballots are rare since almost all countries employ "name one candidate" (plurality-voting-style) ballots. But in Ireland and Australia, the top two countries using "single transferable vote" rank-order-ballot schemes) their governments have refused to publish the ballot data and indeed refused even to publish the table of pairwise-preference counts. Instead, they merely publish certain incomplete summaries of the ballots inadequate to reconstruct these things. These countries also will not give this data to researchers who request it. There have been about 3 experiments in which Ireland or Australia did publish a full set of ballots, but in those few cases no Condorcet top-cycle arose.

2. In countries (and there are many) which conduct two-round "plurality+top-two-runoff" presidential elections, it has become common for pre-election and/or exit-polling agencies to ask pairwise questions of the form "if it comes down to a top-2-runoff between A and B, who would you vote for?" But it only becomes possible to see a Condorcet cycle if these questions are asked for enough candidate-pairs (A,B). And usually they are not.

Our purpose here is to report the first ever observation (via the second method) of a Condorcet cycle high in an important government election: namely in the 2009 Romanian presidential election.

I detected this cycle about 30 seconds after the official results were announced early in the morning of Monday 7 Dec 2009 Romanian time, and had been eagerly awaiting and expecting its appearance for about 5 hours.

## 2. The Romanian 2009 Presidential election – results

Table 3. Summary of 2009 Romanian presidential election (round 1 on 22 Nov; round 2 on 6 Dec) from Biroul Electoral Central (http://www.bec2009p.ro/rezultate.html)
Traian Basescu Democratic Liberal Party (PDL) 3153640=32.44% 5275808=50.33% (wins)
Mircea Geoana Social Democ (PSD) & Conservative Ptys (PC) 3027838=31.15% 5205760=49.66%
Crin Antonescu National Liberal Party (PNL) 1945831=20.02%
Corneliu V. Tudor Greater Romania Party 540380=5.56%
Hunor Kelemen Democ'c Union of Hungarians in Romania 372764=3.83%
Sorin Oprescu Independent (i.e, no party) 309764=3.18%
George Becali New Generation / Christian Democratic Ptys 186390=1.91%
Remus Cernea Green Party 60539=0.62%
Constantin Rotaru Socialist Alliance Party 43684=0.45%
Gheorghe-Eduard Manole Ecologist Party 22515=0.23%
Constantin-Ninel Potirca Independent 21306=0.21%
Totals 9946748=100% (54.4% turnout) 10620116=100% (58.0% turnout)

Comments: The "totals" given by the Biroul are not exactly equal to the column sums. Also 50.33+49.66=99.99≠100, which is another error made by the Biroul. Geoana contested his loss in constitutional court, accusing Basescu of electoral fraud. On 11 December the court decided to recount all 138476 invalid votes. This recount actually slightly increased Basescu's lead (it found 2247 ballots actually were valid, 1260 for Basescu and 987 for Geoana). On December 14 the court declared Basescu the victor, and Geoana conceded defeat.

There is evidence that some fraud occurred, although it does not amount to a large quantity; Geoana suggests those with more evidence send it to fraudarealegeri2009@gmail.com. I personally am convinced based on exit polls that the fraud, if any, was small in percentage terms – but unfortunately, only a small fraud was needed.

'Instant' runoff (IRV) would immediately have eliminated everybody except Basescu, Geoana, and Antonescu (since the sum of all their top-preference votes failed to exceed Antonescu's 20.02%). However, it is not obvious whom IRV would have eliminated next. Asset voting would presumably have elected Geoana since – somewhat unexpectedly – Antonescu, Marko, and Becali all endorsed Geoana in the G vs B runoff. (Antonescu called Geoana the "lesser evil" and denounced Basescu as a "demagogue and a populist." Oprescu also argued Geoana was the lesser evil, but refused to fall into the trap of [therefore] endorsing him, since in Oprescu's view both G and B were "false Romanians.") Evidently, though, Romania's voters were mostly unmoved by those endorsements.

## 3. Pairwise Polls

The principal pollsters and their pairwise-style pre-election polls are reported in the following tables.

Unfortunately, the media simply invented some fake "results" for "polls" which never actually happened. E.g. CURS and CCSB both posted warnings on their web pages about that. Multiple media coverage of the same "poll" does not necessarily prove that poll existed since different media outlets copy each other. It is preferable to check primary sources (i.e. the polling firms) in all cases and distrust media accounts – or at least, only rely on the media which actually sponsored that poll! Unfortunately, I have not always been able to do that, but I have done it for all the polls in the most important and critical tables 7, 9, and 11, the online polls, and the Gallup polls in §5.

Table 4. Important polling firms in Romania, and media or political entities that sponsored their efforts in 2009.
CURS: Center for Urban & Rural Sociology http://www.curs.ro/ Evenimentul Zilei & Televiziunea Romana
INCOR: Institutul pentru Cooperare Regionala si Prevenirea Conflictelor, http://www.incor.ro/PNL
CCSB: Compania de Cercetare Sociologica si Branding, http://www.ccsb.ro/ inpolitics.ro, Antena 3
CSOP: http://www.csop.ro/PDL
IOP: Institutul Operations Research, http://www.operationsresearch.ro/ ?? inpolitics.ro
INSOMAR: http://www.insomar.ro/ Realitatea TV
BCS Statele Unite

Comments: Sometimes pollsters conduct and publicize polls without being paid for it by anybody, for public relations, self-calibration, and/or advertising purposes.

Some Romanian conspiracy-theorist bloggers and media have insinuated that INSOMAR is supported by Sorin Ovidiu Vantu (the "media mogul" who owns Realitatea TV and who was linked with Geoana) while CCSB is 67% owned by Camelia Voiculescu, the daughter of another media "mogul" whom Basescu also accused (during a G-vs-B debate) of associations with Geoana and a founder of the PC – and therefore, their polls should not be trusted. (And indeed, the CCSB and INSOMAR polls comparing Geoana versus Basescu were the ones with the most outrageous pro-Geoana errors, while Gallup, CURS, and CSOP basically got correct answers.) Fortunately, the reader may verify that even if all CCSB and INSOMAR polls are discarded from our dataset, the main conclusions of this paper will not be greatly affected.

The problem in a nutshell is this: there are two different voter-populations (a) the voters in the official B-vs-G runoff and (b) the pollsters' attempts to get representative samples (with different pollsters using different techniques). Our statistical analyses will all be predicated on the assumption that (a) and (b) are the same. But at least in the case of the largest INSOMAR polls, table 5 indicates differences existed.

The usual patter about why such conspiracy theories are dubious is that pollsters are very motivated by market forces – especially in the case of publicly-released "showcase" polls like these – against delivering wrong/inaccurate answers; and further, there is little motivation to fake poll results anyhow (since, e.g, there is no logical reason why a poll wrongly predicting a Geoana victory, would motivate anybody to vote Geoana in a G-vs-B election). Indeed, I would expect that CCSB and INSOMAR will now be punished by having fewer market research customers.

The fact that, in some cases, different polls disagreed, means that anybody aware of only a subset of those polls, risks being deluded.

Table 5. Head-to-head pre-election polls (listed in chronological order): Basescu vs Geoana.
PollsterDate 2009 SampleBasGeo
INCOR19-28 June?52%48%
CCSB27 July?53%47%
CCSB2-3 Oct?47%53%
CSOP7 Oct?53%47%
INSOMAR8-11 Oct?49.4%50.6%
CSOP14-15 Oct513 telephone52%48%
IOP20-23 Oct1355 telephone & face to face45%55%
CURS15-22 Oct1110 face-to-face interviews in their homes, but only 73% responded50%50%
INSOMAR1 Nov1207 over age 18, but only 767 answered46.9%53.1%
IOP1-3 Nov120047%53%
Gallup Romania31Oct-2Nov101450%50%
INSOMAR6-9 Nov1206 people over 18, but only 807 answered46.8%53.2%
CURS7-12 Nov1500 voters49%51%
CCSB13-14 Nov119547%53%
INSOMAR13-15 Nov11973 interviews46%54%
IOP15-17 Nov125946%54%
CSOP15-18 Nov2064 over age 1851%49%
INSOMAR28-29 Nov11971 interviews46%54%
Official Runoff6 Dec1062011650.33%49.66%

Comments: The polls in table 5 are unnecessary for the purposes of this paper, since they all are trumped by the official runoff election count, which is allegedly exact. We give this table purely as a check on the non-fraudulence of the official count or on the abilities of the pollsters. The CURS, CSOP, and Gallup polls got the right answer. The two large INSOMAR polls at the end both were far outside their expected margins of error, suggesting either (a) election fraud by Basescu, or (b) some kind of sampling bias by the pollsters relative to the true voter distribution, or (c) the pollsters lied. Probably the reason is mostly (b) because 4 exit polls of the runoff (which presumably has greatly reduced biased-sampling problems) found: INSOMAR 48.8%, CSOP 50.4%, CURS 49.3%, CCSB 49% respectively for Basescu. These exit polls overall still wrongly predicted a Geoana victory, but much less dramatically than the pre-election polls, indeed their errors seem within the margin of error; and Basescu claims his victory was due to the 146876 valid votes from Romanian voters residing in foreign countries (who voted in B's favor about 78.9% and presumably were never polled). Note that Basescu's official winning margin over Geoana from table 3 was 70048 votes (0.66%); and without the foreign voters, B actually would have lost to Geoana by 14738 votes (0.14%).

Table 6. Head-to-head polls (listed in chronological order): Basescu vs Antonescu. Antonescu wins (or in one case ties) all polls after mid-october except for the final CSOP poll.
PollsterDate 2009 SampleBasAnt
INCOR19-28 June?53%47%
CCSB27 July?55%45%
CCSB2-3 Oct?46%54%
CSOP7 Oct?57%43%
CSOP14-15 Oct513 telephone52%48%
CURS15-22 Oct1110 face-to-face interviews in their homes, but only 73% responded 49%51%
IOP20-23 Oct1355 telephone & face to face44%56%
INSOMAR1 Nov1207 over 18, but only 741 answered49.4%50.6%
IOP1-3 Nov120047%53%
INSOMAR6-9 Nov1206 people over 18, but only 779 answered46.7%53.3%
CURS7-12 Nov1500 voters50%50%
INSOMAR13-15 Nov11973 interviews48%52%
CCSB13-14 Nov119548%52%
IOP15-17 Nov125944%56%
CSOP15-18 Nov2064 over age 1852%48%

Comments: If the samples were independent and from the correct voter distribution, and if we were allowed to ignore the annoying rounding of the pollsters' figures, then, based on the polls after mid-october, the confidence Antonescu was genuinely majority-preferred over Basescu would be over 5.5σ, that is over 99.999998% confidence. By the use of "optimally weighted stratified sampling" techniques (see CRV Puzzle #101; at least some pollsters, e.g. Gallup, have claimed they employ these techniques) pollsters should achieve error bars actually below the Bernoulli variance formula we are employing, therefore our conclusions ought to be valid.

According to Bernoulli, given N pollees a fraction p of whom answer "yes" and q=1-p of whom answer "no," the estimated mean-yes fraction in the larger universe is p and the estimated variance in that estimate is σ²=pq/N. Concerning the value of N, note that in 14 poll-questions mentioned in this paper whose response rates were provided by the pollster, the mean response rate was 64%. Therefore I employ N=0.64T where T is the total number of pollees, as my estimate of N for polls where the pollster did not inform us of the response rate.

However, the poor performance of some of the pollsters in table 5 at predicting the official runoff result contradicts that and casts some doubt on their expertise and on all our conclusions.

Indeed, the B-vs-G polls tabulated after mid-October would have yielded about 10σ "confidence" of a majority-preference for G over B. The official count actually found a slight G>B majority among the non-foreign (i.e the polled) Romanian voters – Basescu won due to foreign voters – but the polls found about 9.6σ "confidence" it should have been larger than it was.
Table 7. Head-to-head polls (listed in chronological order): Geoana vs Antonescu. Split results.
PollsterDate 2009 SampleAntGeo
CURS15-22 Oct1110 face-to-face interviews in their homes, but only 66% responded 52%48%
INSOMAR1 Nov1207 over age 18, but only 662 answered52.0%48.0%
INSOMAR6-9 Nov1206 people over 18, but only 685 answered47.6%52.4%

Comments: If the samples were independent and from the correct voter distribution, and if we were allowed to ignore the annoying rounding of the pollsters' figures, then, based on the above 3 polls, we get about 0.5σ confidence that Antonescu was majority-preferred over Geoana, which is only 70% confidence. On the other hand if we only employ the November polls (which both were available rounded to ±0.05% accuracy, unlike the CURS poll) then it is Geoana who wins, albeit with only 56% confidence. We shall do the latter. The habit of pollsters of rounding to the nearest integer percent may be naively appealing (if, say, their error bar is ±2.6%) but is a very bad practice since it severely handicaps efforts to compute confidence values or to combine data from different polls. CURS did not respond to requests for more accurate data.

Table 8. Head-to-head polls (listed in chronological order): Basescu vs Oprescu. Oprescu wins (or in one case ties) all polls after mid-October except for the INSOMAR poll.
PollsterDate 2009SampleBasOpr
CCSB2-3 Oct?43%57%
CSOP14-15 Oct513 telephone53%47%
CURS15-22 Oct1110 face-to-face interviews in their homes, but only 76% responded 48%52%
IOP20-23 Oct1355 telephone & face to face 47%53%
INSOMAR1 Nov1207 over age 18, but only 709 answered53.2%46.8%
Gallup Romania31Oct-2Nov101444%56%
IOP1-3 Nov120049%51%
CCSB7-11 Nov134849%51%
CCSB13-14 Nov119550%50%
IOP15-17 Nov125948%52%

Comments: If the samples were independent and from the correct voter distribution, and if we were allowed to ignore the annoying rounding of the pollsters' figures, then, based on the above after-mid-october polls, the confidence Oprescu was genuinely majority-preferred over Basescu would be 3.15σ, that is 99.92% confidence. Press coverage of the CURS poll commented that "paradoxically, Geoana has the best chances to get in the runoff but [once there] would have smaller chances than either Antonescu or Oprescu to defeat Basescu." This paradox is highly related to the Condorcet cycles.

Table 9. Head-to-head polls (listed in chronological order): Geoana vs Oprescu. All polls agree Geoana wins this matchup. There also was a press report of a "subjective estimate" by Gallup Romania that Geoana and Oprescu would each get 50% in a head-to-head runoff, but I could not find any actual Gallup poll backing that estimate with genuine data.
PollsterDate 2009 SampleGeoOpr
CURS15-22 Oct1110 face-to-face interviews in their homes, but only 66% responded 53%47%
INSOMAR1 Nov1207 over age 18, but only 653 answered54.8%45.2%

Comments: If the samples were independent and from the correct voter distribution, and if we were allowed to ignore the annoying rounding of the pollsters' figures, then, based on the above 2 polls, the confidence Geoana was genuinely majority-preferred over Oprescu would be about 2.86σ, that is 99.79% confidence.

Finally, there appears to have been no reputable polling organization that did a head-to-head comparison between Oprescu and Antonescu. But there were two online internet polls (anybody on the internet could participate) addressing this question.

```Results of online poll of 519 evidently extremely pro-Antonescu people,
(poll started "21/03/2009" but it remains possible to add your new vote, even now!)
http://www.petitieonline.ro/sondaj-s25710044.html

Q1. Daca maine ar fi alegeri, pe care dintre urmatorii potentiali candidati ati pune stampila?
Traian Basescu   10.6%     55 raspunsuri
Sorin Oprescu    10.98%    57 raspunsuri
Crin Antonescu   78.42%   407 raspunsuri
Q2. Daca in turul 2 ar ajunge Traian Basescu si Sorin Oprescu, pe cine ati vota?
Traian Basescu   15.03%    78 raspunsuri
Sorin Oprescu    84.97%   441 raspunsuri
Q3. Daca in turul 2 ar ajunge Traian Basescu si Crin Antonescu, pe cine ati vota?
Traian Basescu   11.37%    59 raspunsuri
Crin Antonescu   88.63%   460 raspunsuri
Q4. Daca in turul 2 ar ajunge Sorin Oprescu si Crin Antonescu, pe cine ati vota?
Sorin Oprescu    18.69%    97 raspunsuri
Crin Antonescu   81.31%   422 raspunsuri
```
```Results of online poll of 1291 evidently extremely pro-Antonescu people,
(poll started "27/10/2009" but it remains possible to add your new vote, even now!)
http://www.petitieonline.ro/sondaj-s64940048.html

Q1. Cu cine veti vota in TURUL 1?
Traian Basescu            14.64%   189 raspunsuri
Crin Antonescu            35.48%   458 raspunsuri
Mircea Geoana              4.18%    54 raspunsuri
Sorin Oprescu              1.16%    15 raspunsuri
Kelemen Hunor              1.01%    13 raspunsuri
Corneliu Vadim Tudor       3.02%    39 raspunsuri
Gheorge Becali             0.23%     3 raspunsuri
Remus Cernea              39.97%   516 raspunsuri
Constantin Ninel Potarca   0.08%     1 raspunsuri
Altcineva                  0.23%     3 raspunsuri
Q2. Cu cine veti vota in TURUL 2 dintre BASESCU si ANTONESCU?
Traian Basescu            17.66%   228 raspunsuri
Crin Antonescu            82.34%  1063 raspunsuri
Q3. Cu cine veti vota in TURUL 2 dintre BASESCU si GEOANA?
Traian Basescu            23.86%   308 raspunsuri
Mircea Geoana             76.14%   983 raspunsuri
Q4. Cu cine veti vota in TURUL 2 dintre BASESCU si OPRESCU?
Traian Basescu             21.3%   275 raspunsuri
Sorin Oprescu              78.7%  1016 raspunsuri
Q5. Cu cine veti vota in TURUL 2 dintre ANTONESCU si GEOANA?
Crin Antonescu            90.47%  1168 raspunsuri
Mircea Geoana              9.53%   123 raspunsuri
Q6. Cu cine veti vota in TURUL 2 dintre ANTONESCU si OPRESCU?
Crin Antonescu             84.9%  1096 raspunsuri
Sorin Oprescu              15.1%   195 raspunsuri
Q7. Cu cine veti vota in TURUL 2 dintre GEOANA si OPRESCU?
Mircea Geoana              20.06%  259 raspunsuri
Sorin Oprescu              79.94% 1032 raspunsuri
```

These polls obviously cannot be taken seriously because they both were from extremely unrepresentative, pro-Antonescu biased, samples which would have Antonescu defeating all rivals (except perhaps Cernea!) by huge margins, e.g. in the first poll defeating Basescu pairwise by 88.6% to 11.4%. However, it is perhaps of interest that Antonescu would according to the first poll defeat Oprescu by smaller margins than those by which either O or A defeated Basescu. In the second poll, Antonescu defeats Oprescu by a smaller margin than he defeats Geoana (albeit by a slightly larger margin than his defeat of Basescu). These facts suggest that in an unbiased sample, Oprescu might defeat Antonescu. This same speculation also is supported by Gallup's approval-voting-style poll discussed in our §5. Therefore we shall estimate (because this is the estimate arising from the Gallup approval poll) that Oprescu would win a matchup against Antonescu with probability 89%. Observe that in that poll (restricting attention to the 72-73% pollees who provided a response) at least 1.55% approved Oprescu but not Antonescu, and at least 1.55% disapproved A but not O. Hence in a pairwise A-vs-O contest, we suspect that O would win by at least (as our mean estimate of a lower bound) 51.55-to-48.44%.

## 4. The Condorcet Cycles and Pairwise Table

Table 10a. Pairwise-margins table (best estimates) for Antonescu, Basescu, Geoana, and Oprescu. Figures represent the estimated vote percentage for the candidate in that row, in a head-to-head matchup with the one in that column. They are based on polls after mid-october only, and in the Geo-vs-Ant and Geo-vs-Opr cases only on polls whose genuineness has been confirmed; the Bas-vs-Geo result is the exact official runoff result. Apparently there was no Ant-vs-Opr poll but indirect evidence suggests Opr would have had better chances (estimate 89%) in that matchup and won it by at least 51.55%. Error bars are 1 standard deviation.
AntBasGeoOpr
Ant ***52.07±0.3749.78±1.36≤48.44
Bas 47.93±0.37***50.33±048.24±0.56
Geo 50.22±1.3649.66±0***53.85±1.34
Opr ≥51.5551.76±0.5646.15±1.34***
Table 10b. Pairwise-defeat-confidence table for Antonescu, Basescu, Geoana, and Oprescu. Figures PRC represent the confidence the candidate R in that row would win a vote-majority versus the candidate C in that column, in a head-to-head matchup. From now onward (for better or worse) we shall regard these probabilities PRC as independent of each other.
AntBasGeoOpr
Ant ***99.999998%44%11%
Bas 0.000002%***100%0.08%
Geo 56%0%***99.79%
Opr 89%99.92%0.21%***

What about the foreign voters? Foreign voters boosted Basescu's margin over Geoana by 0.80% (from -0.14% to +0.66%). According to Alina Mungiu-Pippidi (Democracy Studies Chair at Hertie School of Governance in Berlin, and a Romanian) expatriate Romanians automatically vote heavily against parties linked to the old communist regime, which in this case means they automatically vote against Geoana. However, this presumably would not have motivated voting against either Antonescu or Oprescu. A different theory about the expatriates is that many live in Moldova, and since Basescu has loudly championed the right of Moldovan Romanians to get Romanian citizenship, they support Basescu. Presumably that motivation to vote for Basescu would still apply. If one guesses (which we shall not) that the same boost would have occurred for Basescu versus either Antonescu or Oprescu, then that would have diminished the Ant⊃Bas vote from 52.07% to 51.67% and decreased the confidence of the "⊃" to 4.5σ (99.9997%); and it would have diminished the Opr⊃Bas margin from 51.76% to 51.36% and diminished the confidence to 2.43σ (99.25%). This is not enough diminishment to hurt the main conclusion of this paper, i.e. that a B&G-containing cycle existed with confidence≈99.9%.

Cycle analysis: Two possible Condorcet 3-Cycles are

Cycle #1:    A⊃B⊃G⊃A
Cycle #2:    O⊃B⊃G⊃O

where "⊃" means "is preferred by a voter majority over." Let Pk for k=1,2 denote our statistical confidence levels that cycle #k genuinely existed. We claim

P1 ≈ 0.56 ≈ 0.99999998 × 1 × 0.56 ,     P2 ≈ 0.9971 ≈ 0.9992 × 1 × 0.9979.

From this we conclude with confidence 99.87% that at least one of these two cycles genuinely exists, i.e. with confidence 99.87% the official winner Basescu, and the official runner-up Geoana, both were members of at least one of these two cycles. Also there are additional possible cycles (such as B⊃G⊃O⊃A⊃B, which has estimated 89% chance of being genuine, almost independently of P1), which pushes the probability of at least one G&B-containing cycle really existing up over 99.9%.

To be precise about this, the exact formula (got by writing it as a sum of 64 monomial terms, each a product of 6 probabilities and expressing the probability of a directed-graph configuration among the 4 vertices {A,B,G,O}, where all configurations not including a cycle are multiplied by zero; then simplifying using PBG=1) for the probability that at least one of the three cycles O⊃B⊃G⊃O or A⊃B⊃G⊃A or A⊃B⊃G⊃O⊃A is present is

Prob(B&G-containing cycle)   =   [PGA + ([1+POBPGA-PGA-POB]POA - POBPGA)PGO]PAB + PGOPOB

With the PXY in table 10b this yields Prob(BGcycle)=99.904%; and even if we instead use the decreased values PAB=99.9997% and POB=99.25% due to a postulated foreign pro-Basescu vote, we still get Prob(BGcycle)=99.871%. It's important to realize that despite the unclarity about the pairs A-vs-O and A-vs-G, we still get high confidence of cycles, mainly because the leading cycle candidate O⊃B⊃G⊃O does not involve either unclear pair.

The confidence that no Condorcet winner existed is at least 56%. Almost all of the remaining 44% probability would represent the chance that Antonescu was really the Condorcet winner. However, that estimate of "44%" is ignoring the possibility that Antonescu would lose in a head-to-head comparison with Oprescu. Although I am unaware of any reputable polling directly addressing that question, from the online polls corrected to remove pro-Antonescu bias, and also from the Gallup approval-style poll, it appears that Oprescu would have 89% chances in an A-vs-O matchup. (There also is a small chance Antonescu might also lose pairwise versus any of the other candidates such as Tudor, Kelemen, etc.) In view of this, we reduce this 44% chance to 5%≈PAOPAGPAB. (More precisely, we find a 5.02% chance that a Condorcet winner exists, in which case there is 96% chance it is Antonescu, 4% chance it is Oprescu, and negligible chance it is anybody else. PAOPAGPAB is the exact probability it is Antonescu, POAPOGPOB is the exact probability it is Oprescu, [among [A,B,G,O}], etc. and these probabilities may be added since they represent disjoint events.) So our final estimate of the chance that no Condorcet winner existed is 95%.

What sort of Condorcet cycles were these? "Laatu types": Juho Laatu (after reading an earlier draft of this paper) proposed classifying Condorcet cycles in large elections into three types:

I. "Weak" cycle (aka random cycle or noise-generated cycle):
• the looped candidates are almost tied
• can be a result of some almost random variation in the votes
• one could say that this kind of a loop is one special version of a tie
• any of the looped candidates could be the winner (with no big violation of any of the majority opinions) against any
• the expected winner may change from day to day in the polls
II. "Strong" cycle (stable cycle, rational cycle, cycle with a stable identifiable reason):
• there is some specific, in principle describable, reason that has led to the formation of this loop (not random variation in the votes)
• the cycle / opinions are strong enough to that they are unaffected by daily/weekly random opinion fluctuations
III. "Strategic" cycle:
• artificially generated by strategies employed by some voters to "game the election"
• not based on sincere opinions

Which kind of cycle do we have here in Romania 2009? I do not believe the O⊃B⊃G⊃O cycle is "weak" and "noise-generated" because, if the votes were just the results of 107 voters tossing independent identical coins we would expect margins (and "noise levels") of order ±3162 votes, i.e. ±0.03%. The vote-margins in this cycle instead were about 3.5%, 0.66%, and 7.7% respectively – far larger. I believe this cycle was stable from day to day and there were some underlying rational reasons behind it. Indeed, any cycle detected via polls rather than exact vote counts will necessarily (if its detection is statistically significant) not be "weak" in Laatu's sense.

It also seems not "strategic" since its pairwise claims are based on 2-choice polls and elections, in which it is difficult or impossible to see any strategic reason for the pollees/voters to be dishonest.

On the other hand, obviously the huge distortions in the official round-1 plurality-counts away from the underlying truth, were mostly caused by strategic decisions by voters arising because of the well-known flaws in plurality as a voting method. With the plurality+top2 method employed, voting for anybody not perceived to be in the "leading three" in terms of election chances, was a "wasted vote." When the decision was made (almost certainly as part of a deal between the major political parties and media moguls) that only Antonescu, Basescu, and Geoana would be allowed to participate in a widely publicized and televised debate, they immediately became the leading three, regardless of whether anybody else was felt by Romania to be a superior candidate, and Oprescu's doom was sealed. Then: Because Basescu had a large lead in plurality-style polls going into round 1 (but clearly not large enough to get a majority and thus avoid the runoff) while (pairwise poll data showed) Antonescu was more likely than Geoana to defeat him in the runoff, it became strategically the smart move for many Basescu supporters to vote for Geoana in the first round, even if they honestly felt Geoana was the worst candidate. Meanwhile, anti-Basescu voters would have been motivated to vote Antonescu, even if they honestly preferred Geoana.

Therefore, the cycle was "strong" (Laatu type II). Nevertheless, it was not very "strong," since 0.66%, 3.5%, and 7.7% represent small to moderate margins.

All of Laatu's three types do occur: Bochsler 2008 found a preference cycle

```     {P or A}⊃S    by 109812 to 102796    (51.6% majority)
S⊃P    by 106832 to 104144    (50.6% majority)
P⊃A    by 106863 to 101586    (51.1% majority)
```
in a 3-choice referendum about changing a law about state employees held on 28 November 2004 in the Swiss Canton of Bern. (This election was held using unusual "decision tree" style ballots, which was the reason the cycle was detectable.) Bochsler in his paper explained the reasons he believed this cycle was Laatu type III, i.e. was mostly caused by "strategic voters" trying to "game" the decision-tree election.

Kurrild-Klitsgaard 2001 found a cycle

```     E⊃J   by 50.6% majority    among the 77% respondents among the 1169 pollees
J⊃R   by 51.1% majority    among the 92% respondents among the 1169 pollees
R⊃E   by 52.8% majority    among the 89% respondents among the 1169 pollees
```
among the three candidates {Hans Engell, Uffe Ellemann-Jensen, and Poul Nyrup Rasmussen} for Danish Prime Minister in mid-May 1994. Note that (a) the Danish PM is not elected by the people, hence this poll had nothing to do with any real election, (b) the ±1σ margin of error for this poll was ±(15-16) votes, i.e. ±(1.52% to 1.67%), so that the statistical confidences that these "majority" preferences genuinely existed, were respectively 64%, 76%, and 96%. This is of course laughable for the purpose of drawing any conclusion about the "reality" of this cycle in the Danish electorate. But it is legitimate to remark that among these 1169 pollees alone there was a cycle. But if so, there is no evidence it was not merely "noise-generated," i.e. of Laatu type I. Similarly, in the June 2008 election by the Wikimedia Foundation to fill a vacant seat on its Board of Trustees (Schulze's beatpath Condorcet method with rank-order ballots was employed), there were 15 candidates, about 26000 eligible voters, and 3019 valid ballots. Ting Chen was the clear Condorcet winner hence won the seat, but there was a cyclic tie for sixth to ninth position between J.Heiskanen, R.Postlethwaite, S.Smith, and R.Saintonge: JH⊃RP⊃SS⊃RS⊃JH. However reversing only 5 pair-preferences on ballots would have eliminated the cycle. Thus, again, there was no evidence this cycle was not "noise generated," i.e. Laatu type I.

What were the underlying rational reasons behind our cycles? That is very difficult to ascertain (or even merely to verify), especially for a foreigner like me unfamiliar with the Romanian language. But it is not hard to invent a plausible-sounding story "explaining" a cycle. There is a comparative sketch of the candidates here: /Romania2009MainCanddts.txt. Consider these three issue-sets (call them T1, T2, T3):

1. Foreign relations & diplomacy. Pensions and unemployment benefits.
2. Accusations of corruption and cozy relations with media moguls & wealthy businessmen; worries about trampling constitutional limits on power; best for hospitals & schools; feeling that Romania's current leaders doing poor job so need a change.
3. Desires to cut government size and budget; racist preferences for Christianity over Islam; Moldovans who want an easy path to Romanian citizenship; expatriate Romanians who automatically vote against parties linked to the old communist regime.

Voters caring mostly about T1 would have preference order G>B>O. Voters caring mostly about T2 would have preference order O>G>B. Finally, voters caring mostly about T3 might have preference order B>O>G and apparently B was felt to have won his final debate with G.

If these three kinds of voters were roughly equinumerous, that would produce an O⊃B⊃G⊃O cycle. I do not claim this is the right, true, or only explanation; I merely say it is one possible hypothesis.

## 5. Approval- and range-voting style polls

Gallup Romania during 5-8 November conducted both an Approval-voting-style poll and a Plurality-voting-style poll (1253 adults [age≥18] representative of the Romanian voter population; "stratified sampling" techniques used; face-to-face interviews at their homes), and the two graphics (provided by Gallup) give the results.

Comments: Oprescu and Antonescu are clearly ahead of everybody else in the approval-style poll. Oprescu had more approvals than any rival despite having the fewest (only 72%) pollees giving an opinion about him; and Oprescu also had the fewest disapprovals even after rescaling to discard all no-opinion votes. Antonescu is second, and Geoana third; all others are statistically-significantly behind, i.e. they would all lose with confidence exceeding 99.9%. To assess this more precisely, I did a Monte Carlo calculation. It indicated that the Gallup data predicts that, with confidence≈89.5%, Oprescu would defeat Antonescu in an approval-voting election; and with confidence≈99.4% Oprescu would defeat Geoana.

But with plurality voting the poll predicts Basescu wins (also with confidence>99.9%), which, of course, indeed happened in round 1. Observe that switching from approval to plurality voting swaps the fourth and first place finishers Oprescu and Basescu. I.e, Oprescu comes top and Basescu bottom among {A,B,G,O} in the approval-voting poll, exactly the opposite of the official results using plurality + top-2-runoff voting.

I am unaware of any range-voting-style polls. However, a poll by IOP (1355 telephone & face to face interviews, 20-23 October) asked a range-voting-like question "How much do you trust politician X?" ("Cata incredere aveti in X?") for many different X on an 0-100 trust scale. The most-trusted among the presidential candidates again was found to be Oprescu (37%). Oprescu had also been the most-trusted (48%) in an earlier (18-25 Sept) IOP poll asking the same question.

Table 11. IOP's "trust level" range-voting-like polls. (Also involved in these polls were the presidential non-candidates Emil Boc, Elena Udrea, and Klaus Johannis.)
CandidateTrust% 20-23 OctTrust% 18-25 Sept
Oprescu3748
Antonescu3535
Geoana3535
Basescu3436
Tudor2025
Becali1925
Hunor75

Other remarks: The polls I examined revealed no sign of a "gender gap." But there was an "age gap" with older voters favoring Geoana and younger ones Basescu.

## 6. Conclusions

1. The 2009 Romanian Presidential Election probably exhibited two or more different Condorcet cycles. I find about 99.9% confidence (assuming valid polls) that both of the top two official finishers Geoana & Basescu were members of at least one cycle.
2. There either was no Condorcet winner (95% confidence) or it was the official third-placer Antonescu (4.8%) or 6th-placer Oprescu (0.2%).
3. The official plurality+top2runoff (2 round) method used, elected Basescu who would have lost head-to-head versus either O or A by comparatively clear margins; B did beat G pairwise but only by an 0.66% margin. So, based on the pairwise table alone, B overall was probably the worst choice among {A,B,G,O}, but was elected; while A or O were probably the best choices, but finished third and sixth, respectively, in the official count.
4. The truth was this was a close 4-way race between {O, A, G, B} but the election method distorted that hugely and made it look like (and effectively be) a 2-way race between G & B, with O & A far behind.
5. The only approval-voting-style poll I have found says if it had been approval voting then O, A, and G all would have placed clearly ahead of B. So again, the official winner B appears to have been the worst choice. The approval poll, and also two range-like "trust level" polls, all gave the top spot to O despite the fact that Oprescu officially finished last among these four (and 6th overall) with only 3% of the vote. So approval and the officially-employed plurality+top2runoff method featured reversed results for the top and bottom candidates among these 4.
6. With instant runoff voting (IRV), one of {A,B,G} would have won, but it is unclear who. With asset voting, presumably Geoana would have won. I thank Ronald Rivest for informing me that (with the central estimates of the pairwise margins in table 10a) under both Borda Count and Schulze's "beatpath" Condorcet method, Geoana would have won.

## 7. Acknowledgments

I thank Daniel Bochsler, Juho Laatu, Alina Mungiu-Pippidi, Diana Viasiu, as well as some other Romanians who wish to remain anonymous, for useful comments.

## 8. Literature References

Daniel Bochsler: The Marquis de Condorcet goes to Bern (2008), Public Choice (to appear).

Steven J. Brams & Peter C. Fishburn: Approval Voting, Birkhauser, Boston 1983.

Peter C. Fishburn: A proof of May's theorem, Behav. Sci. 18 (1973) 212. Also explained in CRV Puzzle #42.

Wm.V.Gehrlein: Condorcet's paradox, Springer Theory and Decision Library 2006. [See also his paper Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences, Theory & Decision 52,2 (March 2002) 171-199.]

G.Th.Gilbaud: Economie Apliquee 5 (1952) 501-584. Also explained in CRV Puzzle #7 and by Gehrlein.

Peter Kurrild-Klitgaard: An empirical example of the Condorcet paradox of voting in a large electorate, Public Choice 107,1-2 (2001) 135-145.

Wm.H.Riker: The paradox of voting and congressional rules for voting on amendments, American Political Science Review 52 (1958) 349-366.

Anthony Quas: Anomalous outcomes in preferential voting (pdf), Stochastics & Dynamics 4,1 (2004) 95-105. See also CRV Puzzle #102.

Warren D. Smith: Explanation of "optimally weighted stratified sampling," CRV Puzzle #101.

Warren D. Smith: Condorcet cycle probability (Dirichlet model), CRV Puzzle #85.

Warren D. Smith: Exact formula for Probability Condorcet Winner exists in C-candidate random election model, CRV Puzzle #102.

Warren D. Smith et al: "Center for Range Voting" website .

Nicolaus Tideman: Collective Decisions and Voting: The Potential for Public Choice, Ashgate 2006. For my recalculations of his cycle-probability estimates in chapter 9, see /TidemanElModel.html.