@ Copyright 2001 by Anatole Beck, Professor of Mathematics, university of Wisconsin,

480 Lincoln Drive, Madison WI 54706. E-mail beck@math.wisc.edu, a.beck@lse.ac.uk.

A More Proportional Representation

Much has been written in the past 250 years about the manner in which we elect representatives. The system which the British misname as "first past the post" (FPP), in which each constituency elects one representative by plurality of those voting, results in the United States in a substantial minority (and occasionally a majority) of the voters being "represented" in Congress or State legislatures by people whose policies they hate and against whom they voted, often bitterly. In the United Kingdom, the majority are so represented. For a long time, including all of the 20th century, the principal alternative has been Proportional Representation, either in pure form as in Israel or modified as in Germany. This takes as its basis the idea that it matters little which individuals are elected, since power lies in the hands of the Executive chosen and sustained by the parliamentary majority, so that the important thing is whether the parties have power proportional to their support in the electorate. (Clearly, in a government of separated powers like the United States, this view is less compelling.) Both these systems have difficulties, which appear below. More recently, we have Approval Voting (AV), wherein each voter marks the ballot against the name of each candidate whom he is willing to have as his representative. Then the candidate who is denoted as acceptable to the largest number of voters is the winner. (Cf. Science, May 25 2001). This system also has difficulties, notably its insensitivity to the strength of the voters' preferences among the candidates they find acceptable. All three rest on the sort of technology available fifty or one hundred years ago.

The advances of our technological time, notably the rather ordinary ability to add several hundred six-digit integers within a fraction of a second, make it possible to construct a system which apportions power very precisely and gives (nearly) every voter a representative to whom he is willing to give his proxy for the duration of the legislative term.

In the system which I have called Proxy Representation (PxR), we would replace the grand national tickets seen in the Israeli and German variants of PR with districts which would elect a number of representatives, the number to be a little less than twice the number of principal parties in the country. In the United States, I would suggest three, in the United Kingdom, five. Then, as in AV, each voter would decide which of the candidates he would be willing to empower with his proxy in the legislature, and vote for them all in a preferential ballot. For almost all voters, one of those whom he chose would in fact eventually have his proxy. Each of the elected candidates would then have a mandate which was the number of proxies he ended up with and, in the legislature, each of the elected candidates would be able to cast as many votes on each issue as there were proxies in his mandate. The machinery would add the votes, record each legislator's choice, and declare the result, all in less time than is ordinarily taken by a voice vote. Of course, the legislators would have different mandates, effectively voting directly on behalf of the electors who had given them their proxies. It would be a very democratic system, and would give almost all the people representatives whom they were willing to give their vote of confidence for the term of the legislature..

In FPP and PR, each individual vote usually counts for nearly nothing, as the election is almost never decided by a handful of votes. In stark contrast, when the contest is close, each vote counts for too much, making the issue of fraud or exclusion very critical. Indeed, PR is even worse than FPP in this regard. To see why, consider a nation in which the number of members of the parliament is nearly one for every 50,000.voters. Only when the vote in a constituency is near 25,000 for each of (say) two candidates will individual votes have great importance; otherwise many would feel that it was not worth their time to go to the polls. In PR, if the quota for each seat is 50,000 votes, there would rarely be slates with mandates within a few dozen votes of an integral multiple of 50,000, and only in that case would individual votes matter a great deal. And, of course, they would then matter even more. In PxR, every vote which ends up as a proxy for a successful candidate would be a vote that counted, and even if a particular candidate were winnowed out in the selection process, it is overwhelmingly likely that almost all the proxies would be passed to others among those whom the voters had selected as acceptable.

In PR, on each party list there is at most one candidate whose election could depend on a few hundred votes, and that only in the case that the number of votes for that party is almost exactly a multiple of the quota. Of course, no one knows for sure who that candidate is, but if it were known to just one voter, his ballot might reflect his preference among the vulnerable candidates rather than the party. The fact that this is not known means that for those who care about individual candidates, they might adjust their votes to what they imagine as the likely outcome of the polling. In America, we have the traditional party loyalist who "would vote for a mangy yellow dog if he were running on the Democratic ticket", but many voters reject such a narrow approach to elections. These are prominent among those who reject PR. Indeed, in the United Kingdom, where constituencies are smaller than in the United States, many people are swayed from normal their party attachments by personable candidates who appear on their doorsteps to seek their votes.

If the number of candidates were precisely equal to the number of seats at issue (an artificial situation which would almost never happen but which is chosen for its didactic value), then everyone who voted would give his proxy to one of these, and the representation would be precisely proportional. However, the system does not anticipate that. Instead, there would probably be a larger number of candidates. I would expect about three times the number of seats, including multiple candidates from the larger parties. In that case, some system of winnowing would be necessary to decide who would actually be elected. (I will suggest one method below; there might be others.) For those whose only interest is in voting for their party, they would probably choose that party's candidates for their highest choices, in some order. Assuming that party commanded substantial adherence, at least one, and maybe more, of that party's candidates would be elected, sharing the party's mandate among them, probably unequally. Those who adhered to a minor party, who expected that all their party's candidates might be eliminated, would have to decide whether they would prefer to empower someone from another party or to deny that power to all the other candidates. Note that they would not have to decide between voting for their own little party or supporting one with a broader mandate, as in both FPP and PR. Also, they would not be concerned that their adherence to the fall-back would help disqualify their favorite candidate, as in AV. They could merely add those fall-back choices as lower-ranking options. But they would have the option of not empowering anyone other than those they find acceptable.

One simple way to winnow the weaker candidates is to start by assigning to each candidate as a provisional proxy each ballot on which he is the first choice. Then the first stage of the winnowing is to find the candidate with the fewest proxies and eliminate him from the election, redistributing his proxies. Then in each subsequent stage, the candidate with the fewest proxies is again eliminated, and his name likewise removed from consideration, with his proxies redistributed, including those he might have received in an earlier elimination. The process would continue until the required number of candidates were left, together with the proxies which contained the name of at least one of them.

PxR is like AV in many respects, but its precision prevents some hard choices which the voter might have to make under AV. Under that system, if there is a candidate whom he really detests strongly, the voter might feel the necessity of voting for all the candidates except that one, including some whom he does not genuinely find acceptable. In PxR, he would not find the same pressure, or at most very rarely. Thus, while AV requires the voter to manipulate his vote less often than FPP or PR, in this respect PxR is even more accommodating to the voter who does not want to have to tailor his vote to the projected outcome of the election.

In an American election, districts with three seats might routinely elect two members from one of the major parties and one from the other. In a district with an overwhelming preference for one party (It would take more than 75 %.), it might be all three from that party, but at least the adherents of the other could decide whether there was any of these they might find acceptable. If not, they would at least have the option of denying their "legitimacy" to any of those elected. This last consideration might result in having at least one "moderate" candidate from the dominant party, even in areas where their margin was overwhelming. In some areas, there might even be a member from each major party and some third person: an independent or from a minor party. In the UK, a five-seat constituency might elect two each from two parties and one from a third, those being distributed in any one of many ways. Some highly fractionated areas might elect from four parties, or even five.

Those voters who vote exactly by party would be represented in the legislature even more precisely than with PR. But also the voter who wanted to vote for "the nice young woman who came to my door" or the off-beat candidate not from his usual party would have the option of doing that. In these ways, PxR is simpler and more acceptable than PR. To illustrate how rarely a voter allied to a major party would be denied a representative under PxR, let us consider the US model suggested above, where districts would elect three, and one party (say the Republicans) would hold a large majority. In the winnowing, we could assume that the Democratic candidates would be eliminated first, until only one remained. He would then provisionally hold all the Democratic proxies. In order for him to be eliminated, he would then have to hold fewer proxies than each of the remaining candidates, of whom there would be at least three, so that all the Republican votes together would have to exceed three times as many as his. Furthermore, the weakest remaining Republican would have to hold more votes than he, so that even 76 % would not suffice to eliminate him unless they were extremely evenly divided, a highly unlikely event. Indeed, even if the Democrat were to hold 20 % of the vote, he would likely be elected, with the top Republicans sharing the remaining proxies. The reckoning in the UK would be similar, but a little more complicated.

The business of having unequal voting power for representatives has an extensive history. It is commonplace in corporations, where we always see proxy voting, especially by the management. It is also frequent in some forms of voting affecting real property, (as in annexations), where each owner votes the assessed value of his holding. Union conventions often feature delegates voting the membership of their locals, and in the Council of Ministers of the European Union (EU), there is a system of qualified majority voting, in which each nation has an agreed voting power reflecting a complicated mixture of its size and importance. In PxR, the total power of any party or faction accurately reflects the voters who have given their vote of confidence to particular candidates.

There is another substantial benefit compared to PR. There, the placement of candidates on the party list is the work of the party machinery. It could be that a particular person, strongly favored by the party moguls, is given a place which is the equivalent of a safe seat. Regardless of his own personal mandate, the party's vote will bring him in. With PxR, he would have to obtain some sort of individual victory. Other members of the same party could compete for the votes of the faithful. If the party has factions, then the polling would serve the needs of a primary election at the same time. In a district where it is favored, the two seats it could expect might well be divided between representatives of two of the factions, without the voters having to weaken the total party vote by the fractionization. It transfers power away from the moguls to the people. Some would find that unwelcome, but I call it greater democracy.

As in the case of PR, the departure of a representative due to death or resignation would not necessitate a special election. In PR, the party owns the seat, and would fill it with the next candidate on its list. In PxR, the ballots could be recounted with the one name excised. (It might not be necessary to keep the actual ballots; the tabulations of the few dozen variants of voting orders would generally suffice.) This would then create the practice of adding to the ballot names of candidates in excess of the number of seats which a party might expect to win, with the unelected ones effectively holding a status equivalent to Vice-representative.

In both FPP and PR, much of the resolution of differences within the parties takes place behind closed doors. Especially in the former, a faction of the party can command who the candidates will be, excluding the dissenters. The bargaining is something like the coalition-formation process which is described as the most unpleasant feature of parliaments under PR, but further polluted by the fact that it happens outside the formal legislative process. The assignment of places on the party lists in PR suffers from this same defect, though less so. Under PxR, there would be a fractionated parliament, as with PR, but the factions would be in a position to negotiate based on an accurate apportionment of power, rather than the usual system of push-and-shove which characterizes intra-party decisions in either of the other options. And the participants would mostly be seasoned politicians, who would be better able to reach a reasonable accommodation even than voters in primary elections. In that sense, it serves the democratic purpose better.

For some reason, PxR has not had a prominent place in the discussion of elections. Perhaps it is because some aspects (such as the reckoning of how many voters are needed to constitute an overwhelming majority) require active mathematical thinking. But as long as PR is being considered, this variant, with its greater accuracy of reflection of public opinion and desire, should also be in the running.