On October 2nd, 2016, I watched in awe as Colombia's national plebiscite for its just-signed peace accord narrowly failed. For the following week, I brooded over the result: the disinformation campaign, Uribe's antics, and just how good the deal really seemed to be. Two days ago, I chanced upon this post, which reminds us that the razor-thin margin - 6,431,376 "No" vs. 6,377,482 "Yes" - is not particularly convincing, nor, as it happens, immune to human error.

And as with all manual voting systems, one cannot rule out at least some degree of misclassification of papers on some scale, no matter how small. We know of no evidence of cheating, and Colombia is to be lauded for the seriousness of its referendum process, but the distinction between intentional and unintentional misclassification by individual counters can occasionally become blurred in practice.

In other words, it was humans - tired humans - counting ballots by hand.

The technology of tired humans sorting pieces of paper into four stacks is, at best, crude. As a large research literature has made clear, we can reasonably assume that even well-rested people would have made mistakes with between 0.5% and 1% of the ballots. On this estimate, about 65,000-130,000 votes would have been unintentionally misclassified. It means the number of innocent counting errors could easily be substantially larger than the 53,894 yes-no difference.

Is it possible that the majority wanted "Yes" and still happened to lose?

plebiscite vote

To answer this question, we can frame the vote as a simple statistical process and ask: "if we were to re-hold the vote many more times, how often would the 'Yes' vote actually win?"

Should we choose, we could pursue this result analytically, i.e. solve the problem with a pencil and paper. This get messy quickly. Instead, we'll disregard closed-form theory and run a basic simulation; "if you can write a for-loop, you can do statistics."

We'll frame our problem as follows:

  1. \(V_t=13,066,047\) voters arrive to the polls.
  2. \(p_{\text{yes}}\%\) of them intend to vote "Yes", \((1-p_{\text{yes}})\%\) of them intend to vote "No."
  3. Each voter casts an invalid (unmarked or void) ballot with probability \(p_{\text{invalid}}\%\).
  4. Of the valid ballots, the poll workers misclassify the vote with probability \(p_{\text{misclassification}}\%\).
  5. Majority vote wins.
YES_BALLOTS = 6377482
NO_BALLOTS = 6431376

N_TRIALS = 100000

In each trial, we assume a true, underlying \(p_{\text{yes}}\%\) for the voting populace. For example, if \(p_{\text{yes}}\) is .48, we will have \(V_t * p_{\text{yes}}\) individuals intending to vote "Yes," and \(V_t * (1-p_{\text{yes}})\) voters intending to vote "No." We assume these values to be static: they are not generated by a random process.

Next, each voter casts an invalid ballot with probability \(p_{\text{invalid}}\), which we model as a Binomial random variable. Each remaining, valid ballot is then misclassified with probability \(p_{\text{misclassification}}\). Finally, the tallies of "Yes" and "No" votes are counted, and the percentage of "Yes" votes is returned.

def simulate_vote(probability_yes):
    yes_votes = int(TOTAL_VOTES * probability_yes)
    no_votes = TOTAL_VOTES - yes_votes

    yes_votes_samples = N_TRIALS * [yes_votes]
    no_votes_samples = N_TRIALS * [no_votes]

    invalid_ballots_yes = np.random.binomial(n=yes_votes_samples, p=P_INVALID)
    invalid_ballots_no = np.random.binomial(n=no_votes_samples, p=P_INVALID)

    valid_yes_votes = yes_votes - invalid_ballots_yes
    valid_no_votes = no_votes - invalid_ballots_no

    yes_votes_from_yes_voters = np.random.binomial(n=valid_yes_votes, p=1-P_MISCLASSIFICATION)
    no_votes_from_yes_voters = valid_yes_votes - yes_votes_from_yes_voters

    no_votes_from_no_voters = np.random.binomial(n=valid_no_votes, p=1-P_MISCLASSIFICATION)
    yes_votes_from_no_voters = valid_no_votes - no_votes_from_no_voters

    tallied_yes_votes = yes_votes_from_yes_voters + yes_votes_from_no_voters
    tallied_no_votes = no_votes_from_no_voters + no_votes_from_yes_voters

    return tallied_yes_votes / (tallied_yes_votes + tallied_no_votes)

Let's try this out for varying values of \(p_{\text{yes}}\). To start, if the true, underlying percentage of "Yes" voters were 51%, how often would the "No" vote still win?

In [16]:

percentage_of_tallied_votes_that_were_yes = simulate_vote(.51)
(percentage_of_tallied_votes_that_were_yes < .5).mean()


That's comforting. Given our assumptions, if 51% of the Colombian people arrived at the polls intending to vote "Yes," the "No" vote would have nonetheless won in 0 of 100,000 trials. So, how close can we get before we start seeing backwards results?

for epsilon in [1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6, 1e-7]:
    probability_yes = .5 + epsilon
    percentage_of_tallied_votes_that_were_yes = simulate_vote(probability_yes)
    proportion_of_trials_won_by_no = (percentage_of_tallied_votes_that_were_yes < .5).mean()

    results = "p_yes: {:1.6f}% | no_win_percentage: {:1.3f}%"
    print(results.format(100*probability_yes, 100*proportion_of_trials_won_by_no))

p_yes: 60.000000% | no_win_percentage: 0.000%
p_yes: 51.000000% | no_win_percentage: 0.000%
p_yes: 50.100000% | no_win_percentage: 0.000%
p_yes: 50.010000% | no_win_percentage: 0.191%
p_yes: 50.001000% | no_win_percentage: 38.688%
p_yes: 50.000100% | no_win_percentage: 48.791%
p_yes: 50.000010% | no_win_percentage: 50.063%

Our first frustration comes at \(p_{\text{yes}} = .5001\): if \(V_t * p_{\text{yes}} = 13,066,047 * .5001 \approx 6,534,330\) voters wanted "Yes" vs. \(\approx 6,531,716\) who wanted "No," the "No" vote would have still won \(0.191\%\) of the time. Again, this reversal derives from human error: both on the part of the voter in casting an invalid ballot, and on the part of the the poll-worker incorrectly classifying that ballot by hand.

As we move further down, the results get tighter. At \(p_{\text{yes}} = .50001\), the "Yes" vote can only be expected to have won \(1 - .38688 = 61.312\%\) of the time. Finally, at \(p_{\text{yes}} = .5000001\) (which, keep in mind, implies an "I intend to vote 'Yes'" vs. "I intend to vote 'No'" differential of just \(13,066,047 * (p_{\text{yes}} - (1 - p_{\text{yes}})) \approx 3\) voters), the "No" vote actually wins the majority of the 100,000 hypothetical trials. At that point, we're really just flipping coins.

In summary, as the authors of the above post suggest, it would be statistically irresponsible to claim a definitive win for the "No." Conversely, the true, underlying margin does prove to be extremely tight: maybe a majority vote just isn't the best way to handle these issues after all.


The notebook and repo for the analysis can be found here.

Key references include: