NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR THE CASE AGAINST BULLET. APPROVAL AND PLURALITY VOTING Donald G. Saari Jill Van Newenhizen Collaborative Papers report work which has not been performed solely at the International Institute for Applied Systems Analysis and which has received only limited review. Views or opinions expressed herein do not necessarily represent those of the Insti- tute, its National Member Organizations, or other organizations supporting the work. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria PREFACE In this paper Don Saari pursues his inquiry into the paradoxes of vot- ing systems and shows that bullet, approval and plurality voting can have most unlooked-for consequences. This research was carried out in collaboration with the System and Decision Sciences Program. ALEXANDER KURZHANSKI Chairman System and Decision Sciences Program THE CASE AGAINST BULLET. APPROVAL AND PLURALtTY VOTING Donald G. Saari Jill Van Newenhizen It is well Known that plurality voting need not reflect the true sentiments of the uoters. When there are more than two candidates running for the same office, it is probable that the candidate approved by most of the voters won't win. Examples abound; perhaps the best Known one is the Senatorial election in the state of New YorK during 1970. Conservative James Buckler benefited fram a split uote for his two liberal .opponents; he was elected with 3YL of the vote even though 61% of the electorate preferred a liberal. In the 1983 Democratic Party primary campaign for the mayor of Chicago, the black candidate, Harold Washington, was elected because of a split vote for his two white opponents, Jane Bryne and Richard Daley. Indeed, examples can be found in many closely contested elections among three or more candidates. To illustrate the problem, consider the following example of fifteen voters and three candidates A, B, and C. Suppose the uoters' preferences are split in the following way: Six of the voters have the ranking A)C)B; five haue the ranking B)C)A, and four haue the ranking C)B)A. The result of a plurality election is A)B)C with a tally 6:5:4. But, although A is elected, a majority (60%) of these voters prefer B to A. More seriously, a majority (60%) of these voters prefer C, the last place candidate, to 4, and 213 of them prefer C to B! So, C is the preferred candidate, but this fact isn't reflected in the election results. This is because with a plurality vote the voters can vote for only their top ranked candidate. Voters are aware of this phenanena and react by using strategic voting behavior. This is manifested by the cmon refrain near election time of 'Don't waste your uote, mark your ballot for ---.' To remain viable, candidates must devote valuable campaign time to counter this effect. For instance, during the 1984 presidential primaries, Jesse Jackson urged his followers to vote for him rather than strategically voting for Walter Mondale. There is little question that this instrument of democracy is in need of reform. But, what should be its replacement? One reform measure, which is intended to capture how strongly a voter prefers his favorite candidate, is bullet votinq. Here a voter has two votes; he can vote for his top ranked candidate, his top two candidates, or he can cast both votes for his top ranked candidate. For instance, in the above example, if all of the voters had sufficient regard for candidate C, then she would be elected. Bullet voting was used in Illinois for certain legislative off ices. A second proposed reform method is a~~rovalvotinq. This is where a voter votes for of the candidates he approves of. As such, when there are N candidates, the voter has N choices; he can vote approval for his top i candidates, i=l,..,N. Again for the above example, depending upon the degree which the voters favor candidate C, she may, or mar not, emerge victorious. Approval voting enjoys the support of several experts in this field. It was employed for a straw ballot during the Pennsylvania Democratic party conference in December, 1983, C11, and it was used to select faculty members to the Northwestern University Presidential search cmittee in November, 1983. In addition, it '..is now used in academic societies such as the Econanetric Society, in the selection of members of the National Academy of Science during final balloting, and by the United Nations Security Council in the election of a Secretary General. Bills to enact this reform are now before the state legislatures of New York and Vermont .' t21. Much of this support is a consequence of the analysis of its properties by two of its foremost advocates, Steven Brams and Peter Fishburn. Most of their conclusions, which high1 ight several of the desirable properties of this system, are summarized in their book 'Approval Voting'C31. To demonstrate the strength of approval voting, often they canpared it with those cmonly used systems which distinguish between two sets of candidates - the top k and the rest. Plural i ty voting is the special case where k=l; it distinguishes between the top ranked candidate and all others. Because of the technical difficulties involved, not all of the properties of 'approval uoting' were found, and in 131 it wasn't compared with a1 1 other voting systems. At a conference in July, 1984, one of us (DGS) presented several negative results concerning the behavior of approval voting (see 14,531 as part of a lecture discribing what can occur with voting methods. Later, S. Brams privately asked whether the techniques developed to obtain these results could expose other properties of approval uoting. We started this project with the expectation that approval voting is, in sane sense, better than most systems. But, we found that it has several disturbing features which makes it worse than even the plurality voting system. Indeed, these properties appear to be sufficiently bad disaualif~ a~provalvotina as a viable reform alternative. In this paper, we report on sane of these negative features which are shared by voting systems, such as approval uoting, bullet uoting, cardinal uoting, etc., where there is more than one way to tally each voter's ranking of the candidates. The feature emphasized here is that the election outcome can be 'random" in nature rather than being decisiue. More precisely, if there are N candidates, then there are N! possible ways to rank them without ties. The purpose of an election is to determine which one of them is the group's choice. If a voting system is decisive, a given set of voters' profiles uniquely determines one of these rankings. However, for approval and bullet uoting, there are a large number of examples where N! outcanes occur for the same profile! That is, each voter votes honestly according to his fixed ranking of the candidates. But, as the voters vary their choice of how their ballots will be tallied, each of the possible N! rankings emerge! This phenomenon can be illustrated with the above example. Let w,y,z denote, respectively, the number of voters from the three types of voters who vote approval for their top two candidates rather than just their top ranked candidate. Then, O(w(6, O(y(5, O(z(4, and the tally for A:B:C is 6:5tz:4twty. It follows immediately that 9 elect ion outcome is attainable from these uoters. For instance the result B>A>C occurs when 212 (at least two from the last set of voters vote for their top two ranked candidates) and wty(1. Even ties are possible. A deadlocked election of &B=C results from z=1 and w+y=2, while the result B=C>A results from w+y-l=z>2.- This example and the general result indicates that systems such as approval and bullet voting possess features which are more undesirable than even those of the plurality voting system! Just this stochastic, random nature of the election outcome raises serious questions whether approval voting, and related methods, truly offer any reform. The proposed cure seems to be much worse than the disease. This result doesn't mean we are doomed to accept and to live with the failings of the plural i t~ voting. There are other ways to tally a ballot which would reflect a voter's first, second, .., last ranked candidates. For instance, a Borda Count is where when there are N candidates, N points are tallied for a voter's top ranked candidate, (N-1) for his second ranked candidate, ..., (N-k) for his kTH ranked candidate, .., and 1 point for his last ranked candidate. (The Borda ranking for the above example is the desired one of C>B>A with a tally of 34:29:27.) So, out of all possible ways there are to tally ballots, the problem is to isolate those ways which best capture the wishes of the electorate. It turns out that the unique solution for this problem is the Borda Count.161 2. The main results. Assume there are N13 candidates denoted by (a~,..,aNl. Let W = (WI, ...,WN) be a votin~vector where its components satisfy the inequalities WJ~WK if and only if j<k, and WI>WN. Furthermore, we require all of the weights to be rational numbers. (The only purpose of the last requirement is to simplify the proofs. Clearly it doesn't impose any practical limitations because for all commonly used methods the weights are fractions and/or integers.) Such a vector defines the tallying process for an elect ion -- WJ points are tall ied for a voter's jTH ranked candidate.