Recently the city of madrid consulted with us about the possibility of running an election with two ballot types, an unusual but interesting circumstance. The election was to produce a ranking over a large number of options (you can imagine these to be candidates, or policy decisions or whatever, it’s not important for the discussion) The two ballot types being considering were: I) a standard preferential or range voting ballot, and II) a pairwise ballot. Voters would have a choice as to what ballot type to cast. I won’t go into the details as to whether this kind of set up is desirable or not. Rather what I will briefly discuss is, given these requirements, how can one perform a tally that represents voter intent and does not obviously penalize a choice of ballot.

The election process would go as follows. Voters first choose what ballot type is more convenient for them, for whatever reason. They fill in the ballot and cast it. At the end of the election we have a set of ballots of type I (preferential or range) and a set of ballots of type II (pairwise). The tally needs to determine a global ranking over the options.

Without going into technical details, there are four possibilities, depending on what operations we perform and when. The two operations we can perform are ballot conversion (C), and ballot aggregation (A). Ballot conversion refers to converting a ballot of one type into a ballot of the other. Ballot aggregation is simply the application of a standard tally for the given ballot type.

Method C-A

This method converts ballots of type I into ballots of type II, and then runs the aggregation for the complete set of type II ballots. This aggregation corresponds to the global tally. The conversion, which occurs at the ballot level, is the complicated part.

The first step is to realize that a particular preferential/range ballot can be mapped to a set of pairwise ballots with which it is consistent. This set does not necessarily contain all the information in the converted ballot, but does contain all the information that it is consistent with. The mapping may be lossy, but not logically contradictory. In these terms there are two main approaches:

  • Consistent set sampling conversion

This method samples from the full consistent set, which allows easier size matching with the target ballot.

  • Full consistent set conversion

This method uses the full consistent set. Size matching requires scaling, possibly using a method based on LCM’s. The method is in principle lossless when using preferential in type I. It is in principle lossy when using ranged in type I.

Note that the hypothetical symmetric A-C method is not present because a conversion from type II ballots (pairwise) to type I is not possible, it would require adding information that we simply do not have at the ballot level. However, as we will see below (A-AC) inference allows this in a pairwise tally, at the aggregate level of all ballots.

Method AC-A

This method performs aggregation for type I, converts the results into type II ballots, and aggregates at II for the final result. Because aggregation for type I ballots results in information similar to a type I individual ballot, the conversion procedure above can be reused. The difference is that this conversion will not be one-ballot one ballot, but one aggregation to n ballots (or 1 weighted ballot if using total set conversion). The same problems of information loss and size matching must be addressed.

Method A-AC

This method performs aggregation for type II, converts the result into a weighted type I ballot, and aggregates at I for the final result. Conversion is easier since aggregation results are isomorphic to type I ballots: a type II result can be interpreted as a range ballot as it contains numerical strenghts for each option.

Method AAA

This method performs aggregation for type I and type II, applies weights to the results, and then aggregates at the global level using ranged voting or preferential voting tallying. This is possible because both type I and type II aggregation produce results isomorphic to preferential or ranged ballots.


We have left most of the technical details out above in order to list the four main methods as an overview of how multi-tally elections can be handled. One conclusion we do offer is that methods that make conversions from type I to type II are probably inferior as they discard information: type I ballots are richer than type II, which is precisely why type II tallies must perform inference (for example using pairwise-beta or pairwise-bradleyterry)

Fairness and expected loss

At the beginning we stated that one objective is to not penalize voters for their choice of ballot. Although the four methods above take measures to ensure this, these measures are informal and their performance not demonstrated. This issue can be investigated with simulations that calculate the expected loss with respect to choice of ballot.