Computing the Margin of Victory in Preferential Parliamentary Elections

Abstract

We show how to use automated computation of election margins to assess the number of votes that would need to change in order to alter a parliamentary outcome for single-member preferential electorates. In the context of increasing automation of Australian electoral processes, and accusations of deliberate interference in elections in Europe and the USA, this work forms the basis of a rigorous statistical audit of the parliamentary election outcome. Our example is the New South Wales Legislative Council election of 2015, but the same process could be used for any similar parliament for which data was available, such as the Australian House of Representatives.

Appendices

A Full List of Margins for the NSW 2015 State Election

Table 5 records the last-round and true victory margins for each seat in the 2015 NSW lower house elections. In most seats, the last-round margin – the difference between the two last candidates in the elimination order – is the true margin. Exceptions to this rule are marked with an asterisk. The 8 Liberal/National coalition seats with the smallest margins are shown in bold. The total of the margins of these 8 seats gives the smallest number of vote changes required to produce a hung parliament, 10,398.

Table 5. LRM and MOV for each seat in the 2015 NSW lower house election.

Table 6 lists the number of vote changes (denoted $\varDelta$) necessary to elect an LAB or CLP candidate. This is at least the true margin (from the previous table), but may be strictly more, for example if an independent candidate was the runner-up. The rows inside the double lines are the 10 seats with the smallest changes necessary to give the labor parties 47 seats. The combined total number of votes needed to produce this is the sum of those rows: 22746.

Table 6. LRM, MOV, and the number of vote changes ($\varDelta$) required to elect an LAB or CLP candidate for each seat in the 2015 NSW lower house election.

Table 7 records the margins for a Labor-Green coalition. In this case the total number of vote changes required to produce this outcome is 16349.

Table 7. LRM, MOV, and the number of vote changes ($\varDelta$) required to elect a LAB, CLP, or GRN for each seat in the 2015 NSW lower house election.

B Modified margin-irv: Computing the MOVC

The margin-irv algorithm for computing the MOVC for an IRV election $\mathcal {B}$ given a set of alternate winners $\mathcal {A}$ is shown in Fig. 2. An initial upper bound on the MOVC is initialised to the last round margin ($LRM_\mathcal {B}$) in Step 2. For each candidate in $\mathcal {A}$, we add a partial elimination order to our frontier F. Each order $\pi '$ is assigned a lower bound (computed as described by Blom et al. [1]) on the degree of manipulation required to realise an elimination sequence ending in $\pi '$ – only orders with an estimated lower bound (l) that is less than the current MOVC upper bound (U) are added to the frontier (Steps 6 and 7). Steps 8 to 12 repeatedly select the partial order $\pi '$ in F with the smallest associated lower bound for expansion. To expand an order $\pi '$, we create a new order for each candidate c not already present in $\pi '$, appending c to the start of the sequence (Step 17). If the created sequence $\pi$ contains all candidates, it is a leaf node, and we evaluate the exact number of vote changes required to realise the sequence with a mixed integer linear program (MIP) denoted DistanceTo.

Section B.1 provides the formulation of the DistanceTo MIP, replicated from Blom et al. [1]. Otherwise, we compute a lower bound on the on the degree of manipulation required to realise an elimination sequence ending in $\pi$ ($l''$) and add $\pi$ to our frontier if this lower bound is less than our current upper bound on the MOVC (Steps 21 to 22). The algorithm terminates once there are no further partial orders to be expanded in our frontier, returning the current MOVC upper bound (U) as the computed MOVC.

Fig. 2.

MOVC computation for an IRV election $\mathcal {B}$ with candidates $\mathcal {C}$, winner $c_w \in \mathcal {C}$, and alternate winner set $\mathcal {A}$.

1.1 B.1 The DistanceTo MIP

The following MIP formulation, originally presented in the work of Magrino et al. [5], has been replicated as it appears in Blom et al. [1]. Let $\mathbf {R}$ denote the set of possible (partial and total) rankings R of candidates $\mathcal {C}$ that could appear on a vote, $N_{R}$ the number of votes cast in the election with ranking $R \in \mathbf {R}$, and N the total number of votes cast. For each $R \in \mathbf {R}$, we define variables:

$$\begin{aligned} q_{R}&\quad \text {integer number of votes to be changed into } R\text {;} \\ m_{R}&\quad \text {integer number of votes with ranking } R \text { in the unmodified} \\&\quad \text { election to be changed into something other than } R\text {; and} \nonumber \\ y_{R}&\quad \text {number of votes in the modified election with ranking } R. \end{aligned}$$

Given a partial or complete order $\pi$, the $\textsc {DistanceTo}$ MIP is:

$$\begin{aligned}&\min \sum _{R \in \mathbf {R}} q_{R}&\nonumber \\ N_{R} + q_{R} - m_{R}&= y_{R}&\forall R \in \mathbf {R} \end{aligned}$$

(3)

$$\begin{aligned} \sum _{R \in \mathbf {R}} q_{R}&= \sum _{R \in \mathbf {R}} m_{R}&\end{aligned}$$

(4)

$$\begin{aligned} \sum _{R \in \mathcal {R}_{i,i}} y_{R}&\le \sum _{R \in \mathcal {R}_{j,i}} y_{R}&\forall c_i, c_j \in \pi \,\,.\,\, i < j \end{aligned}$$

(5)

$$\begin{aligned} n \ge y_{R} \ge 0, \,\,\, N_{R}&\ge m_{R} \ge 0, \,\,\, q_{R} \ge 0&\forall R \in \mathbf {R} \end{aligned}$$

(6)

Constraint (3) states that the number of votes with ranking $R \in \mathbf {R}$ in the new election is equal to the sum of those with this ranking in the unmodified election and those whose ranking has changed to R, minus the number of votes whose ranking has been changed from R. Constraint (5) defines a set of special elimination constraints which force the candidates in $\pi$ to be eliminated in the stated order. $\mathcal {R}_{j,i}$ denotes the subset of rankings in $\mathbf {R}$ ($\mathcal {R}_{j,i} \subset \mathbf {R}$) in which $c_j$ is the most preferred candidate still standing (i.e., that will count toward $c_j$’s tally) at the start of round i (in which candidate $c_i$ is eliminated). Constraint (4) ensures that the total number of votes cast in the election does not change as a result of the manipulation.