Elections R. Ramanujam, The Institute of Mathematical Sciences, Chennai India is getting ready to elect a new parliament. In April and May, 2009, more than sixty crore Indians will be exercising their democratic right. They will decide which candidate, among all those contesting the election, will get to represent their constituency. All these elected representatives will together form the next Indian Parliament. The political party, or coalition of parties, that has won a majority of seats will get to form the Government, electing the next Indian Prime Minister. Democracy is supposed to be Government by the people themselves. Since it is not possible for crores of Indians to directly participate in decision making, they elect persons to represent them. The idea is that these representatives act in the interests of those whom they represent, and through them, the people participate in Government indirectly. This is the central principle of modern democracies. Thus elections form the most critical tool by which democracy is implemented. A citizen gets to vote only once every five years, so the tool has to be used intelligently, effectively. Therefore, elections should make sure that the final outcomes indeed reflect the will of the people. Do they? Do results of elections really reflect what people want? But then the real question is: {\sf what} do people want, and how can we ever find out what they want? ***** In the Thulirpatti constituency, 4 lakh voters participate in the election, with 4 candidates Anita, Balu, Chandra and Dilip contesting. Anita gets 1.3 lakh votes, Balu gets 1.2 lakh votes, Chandra gets 1.1 lakh votes and Dilip gets 40,000 votes. Anita wins the election, but is she the candidate whom {\em the people} of Thulirpatti want as their representative? Suppose that all those who voted for any of the other three actually oppose Anita. Then more than two-thirds of the people oppose their own representative! This is a problem typical of {\bf First Past the Post} system of elections. We have had a Member of Parliament (MP) who won even though he had only 22% of the votes cast. Once we had a Parliament in which less than 5% of the MPs had got more than 40% of the votes in their constituencies. Contrary to what many of us believe, this is not the system of elections practised all over the world. In fact, it is prevalent mainly in the countries that were once British Colonies: apart from the United Kingdom, this includes India, Canada and the United States of America. There are many advantages to this system: it is very simple for voters to understand and votes are easily counted. It makes it easy for the voter to express her views clearly. But many political scientists consider that these advantages are little compared to the many disadvantages of the system. Apart from the problem pointed to above, that elections can be won on small support base, it wastes a large number of votes. In Thulirpatti, votes of nearly two-thirds of the people do not matter in the final outcome in any way. In such a system, if a group is absolutely sure of the votes from an organized minority, that group can easily swing the result, making all issue based promises by others irrelevant. (This is often called {\em electoral arithmetic}, in which addition has very funny properties!) Because of these problems, even a minor redrawing of the geographical boundaries of a constituency can cause a major change in electin outcome. Another problem is that of {\sf tactical voting}: many voters, rather than expressing their actual preference which may simply get wasted, evaluate which candidates are the "two most likely winners" and among them choose the one they prefer, or very often, vote against the one they oppose most. This way of voting effectively restricts voter choice and hence goes against democratic principles. However, majority based decisions are very common in our lives, and we grow up thinking of majority opinion as democratic decision. In class, when a decision is to be made on whether to do a revision or go on to next lesson, the teacher asks, ``How many people want a revision?'', counts the hands that go up, and uses majority to decide. This is clearly democratic. In fact, in any situation, when the choice is between two possibilities, simple majority works very well. The trouble is when we have more than two choices, or as in the case of elections, more than two candidates. **** If you prefer strawberry icecream over butterscotch, and if you like butterscotch more than vanilla, it would be extremely surprising to hear that you would choose vanilla when asked to choose between strawberry and vanilla. In general, if a voter prefers Anita over Balu, and Balu over Chandra, then she prefers Anita over Chandra. This is a reasonable conclusion. Unfortunately what holds good for an individual does not hold good for a majority. Consider a town with 100 voters selecting among 3 candidates Anita, Balu and Chandra. Suppose the voters' preferences are as in the table below: C > A > B: 34 B > C > A: 33 A > C > B: 11 A > B > C: 11 B > A > C: 11 C > B > A: 0 Then we see that 56 of the 100, which is a majority, prefer Anita to Balu; 55, again a majority, prefer Balu to Chandra; but only one third of the population prefer Anita to Chandra. In the scenario above, the candidate with maximum number of first place votes is Balu, with 44 votes. But then, this has to be seen alongside the fact that even more voters (45) consider Balu to be the worst possible choice. So it is hard to justify the choice of Balu as their representative. **** Why should each voter vote only for one candidate? Why can't we have a system where each voter votes for as many candidates as she favours? This is called {\sf Approval Voting}, and is practised, for example, in the election of the Secretary General to the United Nations. Societies like the American Mathematical Association use this system, which prefers the choice of the most acceptable candidate rather than the most popular candidate. In particular, while a voter can exclude those whom she does not favour, she has no way of expressing relative preferences among those she consider acceptable. In many countries, rather than one vote as in the Indian system, or in many votes as in Approval Voting, each voter gives two preferences. This is the basis of the {\sf Single Transferable Vote} (STV) system. In its simplest version, the first preferences are all counted. If the maximum of first place votes is also a majority, that candidate is declared winner. Otherwise the candidate with the lowest first place votes is removed from the election. All her first place votes are {\em transferred} to the candidates second placed in those votes. In the new tally, whichever candidate gets the maximum is the winner. (In some countries, this again has got to be the majority; otherwise the process is repeated.) With some variations, an STV like system is in place in the electoral system of Ireland, Scotland, Germany, New Zealand, Mexico, Bolivia and so on. Often, the second choice is for a party rather than a candidate. In that case, the first placed votes directly elect representatives, but the second choice votes for parties are used to add numbers to parties so that the number of seats a party gets is proportional to its vote share. Germans follow this interesting method. In the Irish Presidential election of 1990, the world saw how STV could make a big difference. The three candidates were Brian Lenihan of the traditionally strong Fianna Fail party, Austin Currie of the nation's second largest party, Fine Gael, and Mary Robinson of the Labour Party. After the first round, Lenihan had the largest tally of first preference votes. But no candidate attained the required majority. Table: Round 1 Brian Lenihan 694,484 44.1% Mary Robinson 612,265 38.9% Austin Currie 267,902 17.0% Invalid Ballots 9,444 0.6% Last placed, Currie was eliminated, and among the votes of those who gave him first place, the ballots that did not give a second preference were eliminated, and the second preference votes were reassigned to the other two. In this process, Robinson received over 80% of Currie's votes, being the second preference of the majority of his supporters, thereby overtaking Lenihan and becoming the seventh President of Ireland. Table: Round 2 Mary Robinson 817,830 51.6% Brian Lenihan 731,273 46.2% Invalid Ballots 34,992 2.2% French elections do not have vote transfer, but can go to multiple rounds, and second round voting can show patterns like the one above. **** Over many centuries, political scientists have studied many voting systems in an attempt to find one that is fair. One such system is the {\sf Borda Count}. It is followed in Slovenia, and in contests such as the one for the Most Valuable Player Award of the USA's National Basketball Association (with millions of voters). The idea of the Borda Count is simple. Voters give their complete preferences of candidates. For every vote which places a candidate last, she gets zero points. For every vote that places her second last, she gets 1 point, and so on. The candidate who gets maximum number of points is declared winner. As an example, consider a population of 100 voters with preferences among 3 candidates as follows. There is a majority group consisting of two-thirds of the population, from which there is one candidate A, whereas there are two candidates B and C from the minority. Among these two, B is more respected than C. A > B > C: 63 B > C > A: 37 A gets 126, C gets 37 and B gets 137, with B emerging winner. However, to solve this problem, the majority group only has to field one more candidate, D, known to be less popular than A: A > D > B > C: 63 B > C > A > D: 37 But the Borda counts now are: A - 226, B - 174, C - 74 and D - 126, with A winning the election. Borda counts nicely illustrate how groups can anticipate how others might vote and decide on their own voting patterns, effectively change the outcome. Suppose there are three parties A, B and C, with A and C representing two extreme viewpoints and members of B equally divided in sympathy between these two. Consider a scenario where A has 45% support, B has 40% support and C has only 15% support. The preferences are: A > B > C: 45 B > C > A: 20 B > A > C: 20 C > B > A: 15 The Borda counts are: A - 110, B - 140 and C - 50, thus the centrist party B wins the election. But then, since the party A can anticipate this, they can decide, as a group to express their preference as A > > C > B. In this case, the Borda counts would be A - 110, B - 95 and C - 95, and A would indeed emerge winner. However, if party B gets wind of such a plan, they would persuade their half who would vote as B > A > C to instead vote as B > C > A. In such a situation, the Borda count would read: A - 90, B - 95, C - 115. Thus, C, with only 15% of actual support can well win the election! **** There are no easy simplistic solutions to ensuring democracy. Elections are very important for democracies, and we should find the most suitable one for our society. From among the existing systems of the world, the Single Transferable Vote system seems to be the most representative one. In elections of the future, simplicity may not be much of an issue. With the advent of electronic voting machines, counting is very easy these days. Perhaps, with more computerized elections, voters will also be able to rank candidates easily. **** Figures: Make a generic ballot with the 4 names: Anita, Balu, Chandra, Dilip. Fig 1: Vote for one option: one box has an X. Fig 2: Vote for any number of candidates: two boxes have X. Fig 3: Rank candidates in order of preference: three of the boxes have entries (1, 2, 3), the fourth is blank. Fig 4: Rate each candidate between -10 and 10: boxes contain values (-7, 10, -3, 0). Fig 5: Distribute 10 votes among candidates: one box is blank, the other three have (6, 3, 1). Fig 6: Runoff voting: Two ballots, Round 1 and Round 2: the first as above, the second has only two names. ------------------ Fig 7: STV Box Children's party, 5 candidates for what to serve, 3 to be chosen. Candidates are: Oranges, Pears, Chocolate, Apples and Sweets. 20 guests are asked. To be elected any candidate should have at least 6 votes. Pref 1 Pref 2 Number O -- 4 P O 2 C A 8 C S 4 A -- 1 S -- 1 Round 1: O has 4, P has 2, C has 12, A has 1, S has 1. Chocolate is declared elected, since Chocolate has more votes than the quota. Round 2: Chocolate's surplus votes are transferred proportionately to Apple and Sweets according to the Chocolate voters' second choice preferences. Now, O has 4, P has 2, C has 6, A has 5, S has 3. However, even with the transfer of this surplus no candidate has reached the quota. Therefore Pear, who has the fewest votes, is eliminated. Round 3: Pear's votes transfer to their second preference, Oranges, causing Orange to reach the quota and be elected. Orange barely meets the quota, and therefore has no surplus to transfer. O has 6, C has 6, A has 5, S has 3. Round 4: Neither of the remaining candidates meets the quota, so Sweets are eliminated. Apple is the only remaining candidate and so wins the final seat. Result: The winners are Chocolate, Oranges and Apples. Thanks: Electoral Reforms Society, UK.