The following is a guest post from University of Chicago political scientist Alberto Simpser. The first chapter of his forthcoming book, Why Governments and Parties Manipulate Elections: Theory, Practice, and Implications, can be found here.

The PRI won the presidency in the July 1st Mexican election, but the dust has not settled yet (see Marco A. Morales’s excellent post on this). Given the safeguards in place and the electoral commission’s (IFE) stewardship, election fraud almost certainly did not occur on a large scale. Vote buying, however, apparently did, as numerous media reports and YouTube videos suggest. A serious allegation is circulating: that the PRI in fact would have lost a clean election to the runner up, the PRD’s Andrés Manuel López Obrador (AMLO), and that its electoral result stemmed from massive vote buying. I have no evidence about whether this allegation is in fact true. Rather, in this note I explore the plausibility of the accusation by estimating how much money it would have cost the PRI to obtain the result it did. The credibility of the ensuing estimates, of course, depends on the solidity of the underlying assumptions. By clarifying my assumptions, I hope to spark informed debate on the question.

I use three pieces of information: the number of votes that the PRI would have needed to buy in order to obtain the result it did, the effectiveness of vote buying (i.e. the rate at which vote buying attempts translate into actual votes), and the cost per vote. I leave out other potentially important factors, such as payments to vote brokers and embezzlement of funds by these brokers.

The PRI obtained about 3.2 million votes more than the PRD (equivalent to about 6.5% of cast votes). Because the PRD could also have bought votes, as some have alleged it did, this figure is a lower bound on the amount of votes that the PRI would have needed to buy in order to upend the election outcome (assuming, for simplicity, that no other forms of vote manipulation were used).

The effectiveness of vote buying depends on a variety of factors. For one thing, ballot secrecy means that many citizens can take the money and vote their conscience. The PRI is alleged to have violated ballot secrecy through the use of cell phone pictures, as well as children who allegedly accompanied voters into the booth; but it is unlikely to have been able to verify every single vote it bought. Effectiveness also depends on the counterfactual scenario: whether and how the bribe recipient would have voted in the absence of a bribe. Even with perfect observability and enforcement, it is possible to waste a bribe on a PRI loyalist who planned to vote anyway. I do not have data on effectiveness for Mexico, so I use an estimate from the literature, based on a survey of Argentinean citizens in 2000-2001 (Brusco, Nazareno and Stokes 2004). Only 16% of surveyed citizens who received a bribe said that it influenced their vote. There exist additional factors that could impact effectiveness, such as embezzlement by brokers. I ignore those in my calculations here (Wang and Kurzman 2007 estimate money “leakage” in a 1993 Taiwan election at upwards of 45%).

Estimates of the cost per vote bought range from 100 to 1,800 pesos. A YouTube video suggests that 100 was low enough to insult the recipient, while 1,800 (a figure implied in recent allegations by AMLO about the amount of money spent and the number of votes bought by the PRI in the state of Mexico) was substantially more than at least some brokers appear to have received for their services. I use 700 pesos, reported on some internet videos, as my best guess (as a point of comparison, a study of the 2000 Mexican election suggests a price per vote of 250 to 500 pesos).

Based on these assumptions, the estimated cost to the PRI of stealing the election through vote buying is 3.2 million actual votes * 700 pesos per vote bought / 16% actual votes per vote bought = 14,000 million pesos approximately (roughly equivalent to USD $1 billion). This is an underestimate insofar as it leaves out brokerage costs and assumes that the PRD bought zero votes. It is an overestimate insofar as vote-buying effectiveness was greater than I have assumed, or if the price per vote figure is too high. At 100% effectiveness, the estimate is 3.2 * 700 = 2,240 million pesos.

What do costs of this magnitude imply? As a point of comparison, the total campaign funding of the PRI-PVEM coalition was about 1,390 million pesos (according to Alianza Cívica’s July 3 press bulletin), far less than even the smaller figure estimated here. This suggests that, if the PRI bought its electoral result, it must have done so with unaccounted-for funds. It is not inconceivable that the PRI should have access to such extra-official sums (the former PRI governor of Coahuila, for example, is involved in a debt and fraud scandal involving thousands of millions of pesos). Nevertheless, the magnitude of the requisite sums suggests that these should be difficult to conceal from authorities, should these wish to investigate the matter.

This analysis suggests a dilemma for the parties accusing the PRI of vote buying: the graver the accusation, the more scandalous, but also the less feasible it would appear to be for the PRI to have pulled it off. The runner-up AMLO today said that 5 million votes were stolen (presumably by the PRI). As a matter of strategy, he might have been better off leveling the more modest accusation that the PRI bought just enough votes to overturn a PRD victory. Doing so, however, would have amounted to an admission that the result of a clean election – which the PRD claims to have won – was close. I emphasize that I have no information about whether the PRI, the PRD, or other parties did in fact buy votes, with what money, and on what scale.

As an aside, from a theoretical perspective, it is interesting to note that, based on the internet videos, some voters appear to have been offended when discovering that they were offered too low a bribe in exchange for their vote. This is consistent with the idea that voter compliance in vote buying transactions might stem, at least in part, from normative or expressive behavior. Also, it suggests that a marginal change in the size of the individual bribe could have a non-linear effect on the effectiveness of vote buying efforts: a zero bribe could obtain more votes than a low, offensive one, which in turn could obtain fewer votes than a higher, non-offensive bribe.