The 2006 election in Peru:

Thwarted-majority, non-monotonicity, and "no-show" pathologies

(Executive summary)

This election is an apparent real-world example of the failure of the plurality+runoff system and presumably also of the IRV (instant runoff) system. The actual election was plurality with a second runoff election among the top two if nobody got a majority (over 50%) in the first election. [Voting is mandatory in Peru; you are fined if you don't.] But the delayed and instant runoff systems happen to be mathematically equivalent if the voters are consistent between rounds and if there are ≤3 candidates.

Actually there were 20 candidates but it is approximately true there were only three since candidates 4-20 got a lot fewer votes than the top three. Instant and delayed runoff, in this case, therefore were approximately equivalent (and exactly equivalent if the voters for candidates 4-20 felt the same about candidates 1-3 relative to each other, as the 1-3 voters felt).

Pre-election polls consistently showed that one candidate (Flores) was pairwise-preferred over each of the other two frontrunners. Specifically, pollees asked about a head-to-head contest (hypothetical second round) indicated they preferred Flores>Humala by about a 55-to-45 margin, and Humala>Garcia, by a slight margin (or some sources instead had Garcia>Humala by a slight margin, e.g. from the Miami Herald of 10 Apr 2006: "Pre-election polls had shown that Flores would be favored over Humala to replace outgoing President Alejandro Toledo, who cannot succeed himself. But Garcia had little or no advantage against Humala in the pre-election polls.") The last polls before the election, by Compania Peruana de Estudios de Mercado y Opinion Publica S.A.C. (CPI) on 27-31 March and independently by Apoyo (see El Comercio 2 de abril del 2006) both showed that Flores would win a runoff against either Humala (55.1%-44.9%) or Garcia by a whopping 58.5%-41.5% based on 3896 people polled by CPI, and (55-45 and 58-42) based on those polled by Apoyo. Also Conecta, UL, PUCP, UAP, and UNI also, in their last polls before the election, all found Lourdes Flores would have beaten or tied both of the others head-to-head, making in total 7 independently-done polls all reaching this same conclusion during March 2006.
(Wikipedia article surveying such polls, see table about "Hypothetical Runoff") (local copy)

And this would be compatible with the official vote table below if, e.g, those voters who voted for candidates 4-20 tended to prefer Lourdes Flores; it also would be compatible with it provided that, to quote the analysis, "in a Flores-Humala contest, Flores would likely win a larger share of Garcia's supporters than Garcia would gain among Flores' supporters."

But, Flores was eliminated in the first round!


In the 9 April election, no single presidential ticket obtained more than half of the total valid votes, causing a runoff election (held 4 June 2006) between the two candidates with the most votes – Ollanta Humala and Alan Garcia. Garcia grabbed the second runoff spot by only a thin margin over Lourdes Flores, causing her to concede defeat only reluctantly. (E.g, her margin of defeat was 8 times smaller than the number of "invalid ballots.") Then in the runoff, Humala conceded defeat to Garcia when, with 77% of votes counted, he found himself behind 45.5% to 55.5%. (Although mathematically, victory was still possible at that point, Humala evidently felt his chances were microscopic. In my opinion, however, this concession decision by Humala was extremely unwise, because as the table of final results shows, the final score moved very substantially away from 55.5-45.5, i.e. a far greater motion than any statistical analysis that assumed the initial count was a random subsample of the final count, could consider probable. Even though by the official tally Humala did indeed lose, the probability of such a huge motion under the random subsample assumption is well below 0.0000000000000001.)

Candidates - Parties Votes 1st round % of valid votes Votes 2nd round % of valid votes
1 Ollanta Humala Tasso - Union for Peru 3758258 30.616 6270080 47.375
2 Alan Garcia Perez - Peruvian Aprista Party 2985858 24.324 6965017 52.625
3 Lourdes Flores Nano - National Unity 2923280 23.814 - -
4 Martha Chavez Cossio - Alliance for the Future 912420 7.433 - -
5 Valentin Paniagua Corazao - Center Front 706156 5.753 - -
6 Humberto Lay Sun - National Restoration 537564 4.379 - -
7 Susana Villaran - Decentralist Concertation 76105 0.620 - -
8 Jaime Salinas - National Justice 65636 0.535 - -
9 Javier Diez Canseco - Socialist Party 60955 0.497 - -
10 Natale Amprimo - Alliance for Progress 49332 0.402 - -
11 Pedro Koechlin Von Stein - With Force Peru 38212 0.311 - -
12 Alberto Moreno - New Left Movement 33918 0.276 - -
13 Alberto Borea - Democratic Force 24584 0.200 - -
14 Ulises Humala - Go On Country 24518 0.200 - -
15 Ciro Galvez - Andean Renaissance 22892 0.186 - -
16 Javier Espinoza - Let's Make Progress Peru 13965 0.114 - -
17 Jose Cardo Guarderas - Democ'c Reconstruction 11925 0.097 - -
18 Antero Asto - Peruvian Resurgence 10857 0.088 - -
19 Ricardo Wong - And It's Called Peru 10539 0.086 - -
20 Luis Guerrero - Peru Now 8410 0.069 - -
Total valid (Turnout: 1st Round 88.7% Runoff 87.7%) 12275384 100 13235097 100
Blank 1737045 11.872 157863 1.091
Invalid 619574 4.234 1075089 7.431
2006 Peru Election.   Source: Peru National Office of Electoral Processes.


How could it be that the candidate pairwise-preferred over all others could lose – indeed lose in the first IRV voting round?!? This can easily happen (and in Peru 2006, apparently did) as follows.

#voters their vote
35 Left>Center>Right
14 Center>Left>Right
17 Center>Right>Left
34 Right>Center>Left

In this 3-candidate 100-voter example IRV election by Jan Kok, "Center" is eliminated in the first round, and then, under the rules of the Instant (or delayed) Runoff System, Right ends up with 34+17=51 ballots (a majority) to beat Left and win. This happens even though the electorate prefers Center>Left by 65-to-35 and prefers Center>Right by 66-to-34, i.e. in both cases by nearly a 2-to-1 margin!

So don't believe IRV advocates who claim that they have never heard of any real IRV election in which pathologies occur, because that sort of thing is "incredibly rare." Nonsense. Caveat: Greg Dennis points out (correctly) that we do not really know IRV would not have elected Flores in Peru, because the 21% of the voters for candidates 4-20 might conceivably have voted differently about the top 3 than the other 79% of the voters – and differently-enough that they would have caused Flores to be the IRV winner.

The simplest scenario of this type is

#voters their vote
2 C>A>B
2 B>A>C
1 A>B>C

in which A is the Condorcet winner but is eliminated in the first Instant Runoff Voting (IRV) round.

Did this affect the candidates' strategies?

You bet it did! As the election approached, Garcia and Humala both realized Flores was the one Peru wanted, and both realized the only way for them to win was to eliminate her in the first round – precisely the way that it happened. They then both ran anti-Flores negative ads designed to try to make that happen.

IRV advocates sometimes claim that IRV will reduce "negative campaigning." We do not know if that claim is true. But we know it wasn't true in Peru 2006!

Would this affect voters' strategies?

Yes! In this kind of scenario, if the Humala voters knew what was going to happen (and preferred Flores as the lesser evil over Garcia) then they would have been better off dishonestly ranking Flores top and betraying their true favorite Humala. That betrayal (in the runoff or IRV systems) would then cause Flores to win over the hated Garcia.

Further, if we were (and Peru was) using two round runoff not "instant" runoff, then this works even better: the Humala voters could have just ranked Flores dishonestly top in the first round. That way, they are totally protected: if Humala makes it to the second round, they can always vote Humala in that round and still have their chance to elect him, whereas if not, then they are happy because Flores will not allow Garcia to win.

So don't ever believe IRV advocates when they tell you that IRV solves the "spoiler" problem and allows every voter to vote their honest favorite top without fear this will "waste their vote" causing the "greater evil" to win. Range voting solves that problem; IRV doesn't. Somebody ought to tell David Cobb that.

Peru 2006 also exhibited "nonmonotonicity"

"Humala>Flores>Garcia" voters would have been better off dishonestly voting their favorite Humala "worst" (i.e. "Flores>Garcia>Humala") because that would have improved the election result for them, making Flores win.

But it gets crazier: there is in fact a severe form of "nonmonotonicity." Suppose they'd actually not only voted their true-favorite as "worst," but also they'd voted their truly-most-hated Garcia artificially "best"! I.e: "Garcia>Flores>Humala." If just 63000 had done that (apparently trying to help Garcia) that act of voting for Garcia would have stopped Garcia from winning and made Flores win.

This maximally-dishonest vote ought to (if things acted logically) be a suicidal move influencing things in the worst possible way. But it isn't – it's a good strategy which improves the election result, while voting honestly is a bad strategy.

Peru 2006 also exhibited "participation failure," a.k.a. "no-show paradox"

"Humala>Flores>Garcia" voters would have been better off if about 60000 did not vote at all.

I thank Jan Kok and Rob LeGrand for pointing this all out; and Rob LeGrand's brother who was living in Peru at the time.

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