Strategic voting in multi-winner elections with approval balloting: a theory for large electorates

So far, the literature on elections with approval balloting has mainly focused on the different ways approval-type ballots can be counted for electing a committee (of fixed size or not). Electing the candidates with the largest approval scores is the simplest but not the only idea one can have (Fishburn 1981; Aziz et al. 2015). Kilgour (2010) surveys the many proposals which have been made, and Laffond and Lainé (2010) survey the representativeness issue under an assumption of separable preferences. This issue is often tackled in the theoretical literature under the assumption that the committee size is not fixed, which makes the problem similar to a multiple referendum problem. In this vein, see Gehrlein (1985), Bock et al. (1998), Brams et al. (1997, 1998, 2007). We here focus on the case—often met in practice—of a fixed-size committee.

This is an important assumption. The models in Myerson (2002), De Sinopoli et al. (2006) or Nuñez (2010b) do not meet this assumption. This may lead to equilibria (with possible ties in the expected scores) in which a Condorcet winner is not elected. As will be seen this is impossible in our model.

The additional assumption that the expected vote difference between any two candidates in \(\widehat{s}_{-i}\) is at least 3 guarantees that the expected winners and losers in the election remain the same whatever the ballot chosen by voter i.

Lemma 2 takes into account the votes of all voters, including voter i. To derive the best response of voter i, the argument has to be adjusted to take into account the fact that voter i takes the votes of other voters as given, but not her. These adjustments are made in the proof in the appendix, but the intuition about the orders of magnitude of the different pivot events remains similar.

Lemma 2 deals with exact ties for election. A voter can also be pivotal in case of a near tie (one vote margin) for election between two candidates. Noting that a requirement of a near tie is no larger than the requirement of an exact tie plus one, the arguments carry through when explicitly taking into account the possibility of near ties (which is done in the proof).

To be precise, the voting rule studied by Cox was slightly different from the one considered here, since voters are only allowed to cast up to two votes (“Restricted Approval Voting”). Yet, it is straightforward to check that strategic voting implies never voting for one’s least prefer candidate, therefore, when there are only three candidates, a best response entails casting at most two votes. The two rules are therefore equivalent from a strategic point of view, for three candidates.

Remember that we assume strict preferences over the set of candidates.

The survey was conducted on line by Harris International, relying on a panel from the Swiss polling firm Link. The sample was representative as to age, gender, and education level. A more detailed analysis of this election can be found in Lachat et al. (2014).

Because this may happen several times, and for the sake of simplicity, what we did is to replicate each participant 100 times, breaking all ties randomly.

This kind of variation in the preference profile has a long history inChoice Theory; see the “Axiom 2.6” in Milnor (1951), the “Independence of Clones” in Tideman (1987), the “Composition-Consistency” in Laffond et al. (1996).

Note nevertheless that there are events with requirement \(\widehat{s}(c_{k})-\widehat{s}(c_{M+1})\) where candidate \(c_{k}\) is caught in a tie for election with candidate \(c_{M+1}\) but also with another candidate. Indeed, consider an event where \(\widehat{s}(c_{k})-\widehat{s}(c_{M})\) votes for \(c_{k}\) are not recorded, and where \(\widehat{s}(c_{M})-\widehat{s}(c_{M+1})\) NO votes for \(c_{M+1}\) are wrongly recorded as YES votes for \(c_{M+1}\), no other mistake being made. The requirement of this event is \(\widehat{s}(c_{k})-\widehat{s}(c_{M+1})\) and it involves a three-way tie for election between \(c_{M}\), \(c_{M+1}\) and \(c_{k}\). As mentionned in the description of the model (Sect. 2), we assume that the voter neglects this type of events involving three-way ties.

Here again, note that there exists an event with requirement \(\widehat{s}(c_{M})-\widehat{s}(c_{k})\) involving a three-way tie for election between \(c_{M}\), \(c_{M+1}\) and \(c_{k}\). We assumed that the voter neglects this type of events involving three-way ties.