The third-place trap: how vote fragmentation keeps deciding Colombian elections

Colombia is about to hold its presidential first round on May 31. The polls, as of today, look like this: Iván Cepeda 38.6%, Abelardo De la Espriella 23.2%, Paloma Valencia 18.4%, the minor center-right candidates around 9% in total, undecideds about 11%. If those numbers hold, De la Espriella faces Cepeda in the runoff and Valencia goes home.

That outcome is mathematically avoidable. It is also exactly what happened in 2018 and in 2022. This is the third-place trap, and it has decided three Colombian elections in a row.

The arithmetic is the same every cycle. The third-place candidate sits a few points behind the second. The fourth and fifth candidates, all from the same political family as the third, hold enough votes between them that 3 + 4 + 5 exceeds the second-place share. In every Colombian first round since 2018, that simple sum has been positive — and in every cycle, the alliance it implies has failed to form.

Three elections, one pattern

Fig 1. First-round shares for 2018, 2022, and current 2026 polls. Dashed line: runoff cutoff (2nd-place share). Dotted green line: alliance arithmetic (3rd + 4th + 5th).

In 2018, Sergio Fajardo finished third with 23.7%, just 1.35 points behind Petro. Vargas Lleras and Humberto de la Calle, both natural center-coalition partners for Fajardo, took 7.3% and 2.1%. Together that's 33.07% — eight points above Petro's 25%. Fajardo went home. Petro faced Duque in the runoff and lost. Colombia got four years of Duque.

In 2022, Federico Gutiérrez ("Fico") finished third with 23.9%, four points behind Rodolfo Hernández. Sergio Fajardo (now as 4th) and John Milton Rodríguez (5th) held 4.2% and 1.25%. Together: 29.36% — one point above Hernández's 28.15%. Fico went home. Hernández faced Petro in the runoff and lost. Colombia got Petro.

In 2026 the math is even cleaner. Paloma Valencia sits at 18.4%, with the minor center-right candidates together around 9%. 18.4 + 9.0 = 27.4%. De la Espriella is at 23.2%. The alliance arithmetic is +4.2 percentage points. The polls have been roughly stable for two months. Valencia is currently scheduled, on present trajectories, to go home.

What 2022 looked like in real time

Fig 2. The 2022 polling trajectory. Petro stable around 40%. Fico stuck at 24–27% for nearly three months. Fajardo collapsed from 17% to 5%. Hernández surged 16% → 28% in the final three weeks.

The 2022 case is the cleanest demonstration of the trap because the dynamic played out month by month in published polls. Petro was the polarized leader at 40% throughout. Fico arrived at 24–27% after winning the right-bloc primary in March and stayed there for nearly three months. Fajardo collapsed from 17% in January to under 5% by May. And then Rodolfo Hernández, an anti-system populist polling at 14% in late April, surged through 16, 18, 21, 22, 23, 26, 27% in successive weekly polls — and finished at 28.15% on election night.

Fico had been the runoff candidate of the right for two months. His arithmetic showed Fico + Fajardo + Rodríguez beating Hernández in essentially every poll from January through mid-May. The signal was visible. The coordination wasn't done. Hernández did the consolidation for the right — on his own terms, from outside the establishment, and against the candidate that the establishment's own polls had been telling them was vulnerable. Fico spent the campaign attacking Petro. Hernández spent it attacking Fico.

What Colombia looks like under topological data analysis — and what it doesn't look like

There is a separate way to measure how fragmented a political conversation is, independent of the vote shares: build the network of Twitter mentions during the campaign, treat that network as a simplicial complex, and read off its topological invariants — connected components, loops, cavities, alphabet letter. Colombia 2018 sits at the high end of this measurement too. The figure below, from our companion methodological paper on topological data analysis of evolutionary and social systems, shows the discourse signature for four interpretively distinctive elections.

Fig 4. Each row is one election. Columns are (i) the mention graph at month −1, (ii) the top-25-node 2-skeleton with triangles and red-outlined tetrahedra, (iii) the persistence barcode (H₀ blue, H₁ red, H₂ orange), (iv) the alphabet trajectory of β₁ and β₂ over the five months before the vote.

The differences are large and interpretable.

Colombia 2018 (top row) has |V|=225, |E|=1251, 62 triangles, 36 tetrahedra, β₁ = 8, β₂ = 28, and an alphabet trajectory that oscillates around high β₁ across the five months before the vote. Many edges, many higher-order simplices, a stable cycle count, a non-trivial β₂. The discourse is highly reticulate without being cleanly partitioned. This is the topological signature of a system in which third-place candidates have allies who are not coordinating with them.

Canada 2021 (second row) has β₁ = 41, β₂ = 9. More 1-cycles than Colombia, but with a markedly cleaner partition into communities — the 2-skeleton is sparser, β₂ is much smaller. The Canadian Westminster system produces a multi-party conversation that organizes cleanly into blocs; Colombia's runoff system produces one that doesn't.

Belarus 2020 (third row) has |V|=91, |E|=133, 5 triangles, no tetrahedra, β₂ = 0. The 1-skeleton is tiny; the 2-skeleton is almost empty. The Belarusian Twitter conversation in 2020 was an asymmetric structure dominated by Lukashenko's machine and a few opposition handles, with no reticulate community structure at all. The topological signature of a closed political system: there is no fragmentation because there is no contest.

Mexico 2018 (fourth row) has |V|=226, |E|=1336, 85 triangles, 64 tetrahedra, β₂ = 15. AMLO's coalition produced a dense, highly tessellated 2-skeleton (the red-filled regions in the network drawing) but a low β₁ = 3 — the system was reticulate at the close-similarity scale but consolidated topologically because one candidate captured the runoff slot outright. The alphabet trajectory collapses to near-zero in the final months. The signature of an outright-winner election: the discourse loops eventually close into solid simplicial structure rather than persisting as cycles.

The implication for the 2026 Colombian race is that the topological signature predicts the third-place trap before the vote shares arrive. Colombia 2018 shows the K₄ alphabet signature — many independent loops, real 2-cavities, no clean partition — that corresponds to the alliance arithmetic being positive without the alliance forming. The same Twitter pipeline applied weekly to 2026 mention data would tell us in real time whether Valencia is in the same structural position Fico was in May 2022, or whether the right has consolidated. So far, the polling-trajectory evidence above suggests she is in the same position.

2026: the same trap, two weeks out

Fig 3. The 2026 race today. Valencia + minor center-right = 27.4% vs. De la Espriella's 23.2%, a +4.2 pp alliance margin. Sixteen days to the first round.

Today's polling places Paloma Valencia in the same structural position Fico held in May 2022 and Fajardo held in May 2018. She is the establishment-right candidate. She is third. The second-place candidate — De la Espriella — is an outsider with a base that explicitly rejects the political family Valencia represents. The minor center-right candidates, together polling around 9%, exist precisely in the space Valencia would need to absorb. None has dropped out. None has been formally offered a place. None has signaled they intend to.

If the pattern holds, De la Espriella reaches the runoff with 23–25% of the vote, Valencia goes home with 18–22%, and the Colombian right spends another four years trying to consolidate after the fact. The arithmetic that would change that outcome exists right now and is visible in every published poll. What's missing is the coordination.

Why this keeps happening

Three reasons, in order of how depressing they are.

First, fragmentation has been growing, not shrinking. The effective number of candidates (N_eff = 1 / Σ_i v_i²) was 3.59 in 2018, 3.32 in 2022, and 3.90 in current 2026 polls. The leading bloc fields one disciplined candidate every cycle. The opposing bloc fields three or four.

Second, the candidates who should ally find each other unacceptable. In 2018, Fajardo couldn't credibly absorb Vargas Lleras's right-leaning electorate or De la Calle's Santos-era peace baggage. In 2022, Fico and Fajardo had been mutually hostile for an entire campaign. In 2026, Valencia and the minor center-right candidates are bargaining as peers; no one has the institutional power to compel withdrawal, and they each have a different theory of why the others are unacceptable.

Third — and this is the structural part — the third-place candidate has a real reason to stay in until the end. Third place is one runoff away from the presidency. Fourth place isn't. Dropping out transfers maybe 60–70% of one's vote and delivers a cabinet position that may or may not actually materialize. Running through election day costs nothing and preserves the small but real chance that polls are off and one's actual vote will be higher than expected. Ex ante, both candidates have rational reasons to refuse the alliance. Ex post, the math says they both lose.

What it would take

A credible voto útil — a coordination signal trustworthy enough for voters to act on, plus willingness from the lagging center-right candidates to actively redirect their own voters in the final week — is the standard French solution to the same structural failure. It works there. It has never been seriously attempted in Colombia.

A coordinated late dropout, the way Vargas Lleras stepped aside for Pastrana in 1998, requires the dropping candidate to have a clear political future after the election. In 1998 Vargas got it: he was made part of the eventual government and went on to his own presidential run. Today, neither Valencia nor De la Espriella has offered anyone in the minor right such a deal. The minor right candidates haven't asked.

There are sixteen days left.

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Methods and sources

Vote shares for 2018 and 2022 are official first-round results from the Registraduría Nacional del Estado Civil. The 2022 polling trajectory is compiled from 20 polls by Invamer, Yanhaas, Guarumo, Datexco, and CNC, published between January 21 and May 27, 2022. The 2026 poll average is from Invamer (April 26, 2026), Fundación Génesis Crea (May 4–11, 2026), and the five-pollster aggregate maintained by La Silla Vacía.

Effective number of candidates uses the Laakso–Taagepera index N_eff = 1 / Σ_i v_i². Alliance arithmetic is the sum of the third, fourth, and fifth candidates' shares minus the second candidate's share. Voter-transfer rates between allied candidates in Colombian elections are estimated at 60–70%; applying that correction still flips the 2018 and 2026 cases, while 2022 becomes a tight call.

The TDA panel (Colombia 2018 / Canada 2021 / Belarus 2020 / Mexico 2018) is the four-row validation panel from the companion methodological paper. Each row shows, for one election: (i) the 1-skeleton of the Twitter mention network at month −1, (ii) the top-25-node 2-skeleton with triangles and red-outlined tetrahedra, (iii) the persistence barcode showing H₀, H₁, H₂ across the filtration, and (iv) the alphabet trajectory of β₁ and β₂ across the five months before the vote.

The argument here is structural, not a prediction. The first-round result on May 31 will speak for itself.

Comments, corrections, and counter-arguments welcome. The polls cited will be replaced with actual results on May 31; this post will be updated.