Consider a polity which wishes to elect a congress (or parliament) with one delegate from each of \(N\) districts. Suppose there are \(M\) political parties competing for each district. It is well-known that the naive method of region-by-region plurality vote (sometimes called ‘first past the post’) can produce a congress where the allocations of seats to parties is wildly disproportionate to their share of the popular vote. If the political districts themselves are subject to redesign, then the system is also vulnerable to ‘gerrymandering’, where district boundaries are redrawn to maximize this disproportionality in favour of the currently ruling party.
This article proposes a new method, Fair Majority Voting (FMV) to solve these problems. FMV works as follows. Consider a matrix where the \((n,m)\) entry is the total number of votes for party \(m\) in district \(n\). Each row of this matrix corresponds to a party, and each column corresponds to a district. We first decide how many congressional seats party \(m\) should receive, so that its portion of the congress is as close as possible to its portion of the total popular vote (i.e. the sum of the entries in the row \(m\)). This can be done using a number of ‘apportionment methods’, but Balinksi favours the Jefferson/D’Hondt apportionment method for several reasons.
Suppose we decide a party \(m\) is to recive \(K_m\) out of the \(N\) congressional seats. The problem then is which \(K_m\) of the \(N\) districts should be represented by party \(m\). Balinski’s solution is to find scalars \(S_1,...,S_m>0\) such that, if we multiply the \(m\)th row of the matrix by \(S_m\) for all \(m\), then there are exactly \(K_m\) columns in the rescaled matrix where the \(m\)th entry is the largest; these are the \(K_m\) districts which go to party \(m\). Balinski’s Theorem 2 shows that there is always a collection of scalars \(S_1,...,S_M\) with this property. Furthermore, any such collection always produces the same allocation of districts to parties. Furthermore, instead of expressing the solution in terms of rescaling the rows (parties), we can equivalently express it in terms of rescaling the columns (districts) to arrive at the same solution.
If there are exactly 2 parties, then FMV has a very simple description: compute the percentage of the vote received by each party in each district, and make a matrix of these percentages. Then allocate to party \(m\) the \(K_m\) districts where it received its highest \(K_m\) percentage rankings (even if in some of these districts, this percentage was less than 50%).
Balinski offers several informal arguments for FMV. First, FMV clearly eliminates the possibility of gerrymandering district boundaries. Also, it is no longer necessary to ensure all districts have the same population (since the votes are rescaled anyways). Instead, districts can be separated along natural political, administrative, cultural, or geographical boundaries. FMV guarantees that the distribution of the congress reflects the distribution of popular vote. FMV makes every vote ‘count’ (because even if one candidate in your district has an overwhelming majority, your vote also helps your party win seats from other districts). Finally FMV forces the different candidates in each party to compete against each other as well as against the other parties; thus, they have an incentive to distinguish themselves as individuals and personally attract votes, rather than simply relying on their party affiliation.
Balinski also offers an interesting formal justification for FMV. For any allocation \(x\) of congressional seats, let \(\pi(x)\) be the product, over all districts, of the number of votes for the candidate chosen to represent that district. This can be seen as a ‘social welfare function’ measuring the ‘social welfare’ of allocation \(x\). Balinski’s Theorem 3 shows that FMV chooses the allocation of congressional seats which maximizes \(\pi(x)\). (This is intriguingly reminiscent of the Nash solution to a multilateral bargaining problem, which maximizes the product of the utilities of the bargainers).
Other voting systems have been proposed to reconcile regional representation with proportional representation, such as single transferable vote (STV) and mixed member proportional representation (MMPR). Unfortunately, nowhere in the article does Balinski compare FMV to these methods (or indeed, even mention them). This reviewer will offer some superficial comparisons. The ‘rescaling’ mechanism in FMV is similar to the ‘vote-transfer’ mechanism in STV, in that it allows ‘excess’ voter support for a party in one district to ‘flow’ into other districts in a very specific way; however, the actual flow mechanism is different (the STV flow mechanism is more nuanced). Indeed, FMV is much simpler than STV, and almost as simple as MMPR. However, like STV (and unlike MMPR) it keeps the total number of congressional delegates constant, and guarantees that all delegates represent specific districts. On the other hand, like MMPR, FMV has the disadvantage that it only looks at each voter’s top choice, and ignores the rest of her preference ordering (whereas STV uses the whole preference ordering).