In L. Sandy Maisel (ed.), Political Pareties and Elections in the United States: An Encyclopedia, vol. 1. New York: Garland, 1991, pp. 23-31.


Steven J. Brams
Department of Politics
New York University
New York, NY l0003

Peter C. Fishburn
AT&T Bell Laboratories
Murray Hill, NJ 07974


Several alternatives to plurality voting, with and without a runoff, are compared: the Hare system of single transferable vote (STV), the Borda count, cumulative voting, additional-member systems, and approval voting. The Hare system and the Borda count allow voters to rank candidates, and cumulative voting allows voters to allocate a fixed number of votes among candidates. These systems are designed to ensure proportional representation of different groups in the electorate, as are additional-member systems, in which extra seats can be given to parties underrepresented in a legislature. Approval voting is a nonranked voting system that tends to help majority candidates.

All these voting systems are vulnerable to strategic manipulation, with the Borda count being the most manipulable. STV is nonmonotonic, which means that a candidate can be hurt when raised in the rankings of some voters, and additional-member systems may make it advantageous for parties to throw elections in districts without a strategyproofness constraint, whose satisfaction impedes proportional representation. Proportional representation under cumulative voting requires careful calculations by the parties and disciplined supporters. Strategic calculations may also be required under approval voting, though it is more sincere than other nonranked voting systems. Examples are given to illustrate these properties of alternative voting systems, and places where they have been used are indicated.

l. Introduction

The cornerstone of a democracy is fair and periodic elections (Riker, l982). The voting procedures used to elect candidates determine to a crucial degree whether elections are considered fair and their outcomes legitimate. By procedures we mean the rules that govern how votes in an election are aggregated and how a winner or winners are determined.

Of course, the disembodied study of procedures qua procedures misses the human element that so often enlivens elections, from the struggle by candidates to capitalize on an opponent's weaknesses to the often difficult choice by voters of a preferred candidate who, in addition, can win. Yet this human side of campaigns and elections is inevitably played out against a background of rules--in large part structured by voting procedures--that give election contests their shape and form if not their substance. While our focus will be on the political-strategic calculations that different procedures engender, the term "voting system" also encompasses the cultural and sociological forces that impinge on and shape election competition.

The two best-known and most commonly used voting procedures in the United States restrict voters to voting for only one candidate, regardless of how many run. They are plurality voting (the candidate with the most votes wins) and plurality voting with a runoff (the two candidates with the most votes are paired against each other in a second, or runoff, election; the candidate with the most votes in the runoff election wins). Runoff elections are held only if the winner in the first election does not receive a majority--or some other designated percentage, such as 40 percent--of the votes.

The ostensible purpose of a runoff election is to prevent a strong minority candidate, who may be preferred by, say, 35 percent of the voters, from defeating majority candidates who might split the remaining 65 percent of the vote. By pairing one of the majority candidates against the minority candidate, runoff elections help to ensure that one of the majority candidate gets elected, provided that most of the supporters of the defeated majority candidate vote for the majority candidate in the runoff.

In this chapter, we shall exclude from consideration plurality voting, with or without a runoff. We also exclude voting procedures commonly used in legislatures, councils, and other voting bodies, wherein the alternatives that are voted on are not candidates or parties but bills and resolutions. Like the procedures we discuss, these other procedures are vulnerable to strategic voting, whereby rational voters vote insincerely (not in accordance with their preferences) to try to obtain a preferred outcome.

If two majority candidates are pitted against a minority candidate, under plurality voting (without a runoff), for example, a voter might vote for the majority candidate who seems to have the better chance of winning --even though this voter might prefer the other majority candidate--to try to prevent the minority candidate from winning. On the other hand, if there were a runoff, supporters of either of the majority candidates would presumably vote sincerely in the first election, because at least one of the majority candidates would make the runoff and defeat the minority candidate if that candidate were also in the runoff.

We have used the words "minority" and "majority" rather loosely here. In our subsequent analysis, we shall be more precise about strategic calculations and their effects on outcomes. Our purpose in analyzing better and worse strategies under alternative procedures is not only to illuminate their strategic properties but also to illustrate paradoxes that can arise under them.

We shall concentrate on practical voting procedures that have actually been used in elections, including the most prominent ranking procedures and methods of proportional representation. We then discuss an alternative nonranking procedure--approval voting--that is presently used in many universities and several professional societies and has been proposed for public elections, such as party primaries and nonpartisan elections in the United States, that often draw more than two candidates. We conclude with a brief normative assessment of the alternative voting procedures assayed.

2. The Hare System of Single Transferable Vote (STV)

First proposed by Thomas Hare in England and Carl George Andrae in Denmark in the l850s, STV has been adopted throughout the world. It is used to elect public officials in such countries as Australia (where it is called the "alternative vote"), Malta, the Republic of Ireland, and Northern Ireland; in local elections in Cambridge, MA, and in local school board elections in New York City; and in numerous private organizations. John Stuart Mill (l862) placed it "among the greatest improvements yet made in the theory and practice of government."

Although STV is known to violate a number of properties of voting systems discussed in the literature on social choice theory (Kelly, l987), it has a number of strengths as a system of proportional representation. Minorities, in particular, can get a number of candidates, roughly proportional to their numbers in the electorate, elected if they rank these candidates at the tops of their lists. Also, if a person's vote does not help elect his or her first choice, it can still be counted toward lower choices.

To describe how STV works and also illustrate two properties that it fails to satisfy, consider the following examples (Brams, l982; Brams and Fishburn, l984c). The first shows that STV is vulnerable to "truncation of preferences" when two out of four candidates are to be elected, the second that it is also vulnerable to "nonmonotonicity" when there is one candidate to be elected and there is no transfer of so-called surplus votes.

Example l. Assume that there are three classes of voters who rank the set of four candidates {x,a,b,c} as follows:

I. 6 voters: xabc

II. 6 voters: xbca

III. 5 voters: xcab

Assume also that two of the four candidates are to be elected, and a candidate must receive a quota of 6 votes to be elected on any round. A "quota" is defined as (n/(m+l)) + l, where n is the number of voters and m is the number of candidates to be elected.

It is standard procedure to drop any fraction that results from the calculation of the quota, so the quota actually used is q = [(n/(m+l)) + l], the integer part of the number in brackets. The integer quota is the smallest number that makes it impossible to elect more than m candidates by first-place votes on the first round. Since q = 6 and there are l7 voters in our example, at most two candidates can attain the quota on the first round (l8 voters would be required for three candidates to get 6 first-place votes each). In fact, what happens is as follows:

First round: x receives l7 out of l7 first-place votes and is elected.

Second round: There is a "surplus" of ll votes (above q = 6) that are transferred in the proportions 6:6:5 to the second choices (a, b, and c, respectively) of the three classes of voters. Since these transfers do not result in at least q = 6 for any of the remaining candidates (3.9, 3.9, and 3.2 for a, b, and c, respectively), the candidate with the fewest (transferred) votes (i.e., c) is eliminated under the rules of STV. The supporters of c (class III) transfer their 3.2 votes to their next-highest choice (i.e., a), giving a more than a quota of 7.l. Thus, a is the second candidate elected. Hence, the set of winners is {x,a}.

Now assume that 2 of the 6 class II voters indicate x is their first choice, but they do not indicate a second or third choice. The new results are:

First round: Same as earlier.

Second round: There is a surplus of ll votes (above q = 6) that are transferred in the proportions 6:4:2:5 to the second choices, if any (a, b, no second choice, and c, respectively) of the voters. (The 2 class II voters do not have their votes transferred to any of the remaining candidates because they indicated no second choice.) Since these transfers to do not result in at least q = 6 for any of the remaining candidates (3.9, 2.6, and 3.2 for a, b, and c, respectively), the candidate with the fewest transferred votes (i.e., b) is eliminated. The supporters of b (4 voters in class II) transfer their 2.6 votes to their next-highest choice (i.e., c), giving c 5.8, less than the quota of 6. Because a has fewer transferred votes (3.9), a is eliminated, and c is the second candidate elected. Hence, the set of winners is {x,c}.

Observe that the 2 class II voters who ranked only x first induced a better social choice for themselves by truncating their ballot ranking of candidates. Thus, it may be advantageous not to rank all candidates in order of preference on one's ballot, contrary to a claim made by a mathematical society that "there is no tactical advantage to be gained by marking few candidates" (Brams, l982). Put another way, one may do better under the STV preferential system by not expressing preferences--at least beyond first choices.

The reason for this in the example is that the 2 class II voters, by not ranking bca after x, prevent b's being paired against a (their last choice) on the second round, wherein a beats b. Instead, c (their next-last choice) is paired against a and beats him or her, which is better for the class II voters.

Lest one think that an advantage gained by truncation requires the allocation of surplus votes, we next give an example in which only one candidate is to be elected, so the election procedure progressively eliminates candidates until one remaining candidate has a simple majority. This example illustrates a new and potentially more serious problem with STV than its manipulability due to preference truncation.

Example 2. Assume that there are four candidates, with 21 voters in the following four ranking groups:

I. 7 voters: abcd

II. 6 voters: bacd

III. 5 voters: cbad

IV. 3 voters: dcba

Because no candidate has a simple majority of q = ll first-place votes, the lowest first-choice candidate, d, is eliminated on the first round, and class IV's 3 second-place votes go to c, giving c 8 votes. Because none of the remaining candidates has a majority at this point, b, with the new lowest total of 6 votes, is eliminated next, and b's second-place votes go to a, who is elected with a total of l3 votes.

Next assume that the 3 class IV voters indicate only d as their first choice. Then d is still eliminated on the first round, but since the class IV voters did not indicate a second choice, no votes are transferred. Now, however, c is the new lowest candidate, with 5 votes; c's elimination results in the transfer of his or her supporters' votes to b, who is elected with ll votes. Because the class IV voters prefer b to a, it is in their interest not to rank candidates below d to induce a better outcome for themselves, again illustrating the truncation problem.

It is true that under STV a first choice can never be hurt by ranking a second choice, a second choice by ranking a third choice, etc., because the higher choices are eliminated before the lower choices can affect them. However, lower choices can affect the order of elimination, and hence the transfer of votes. Consequently, a higher choice (e.g., second) can influence whether a lower choice (e.g., third or fourth) is elected.

We wish to make clear that we are not suggesting that voters would routinely make the strategic calculations implicit in these examples. These calculations are not only complex but also might be neutralized by counterstrategic calculations of other voters. Rather, we are saying that to rank all candidates for whom one has preferences is not always rational under STV, despite the fact that it is a preferential voting procedure. Interestingly, STV's manipulability in this regard bears on its ability to elect so-called Condorcet candidates (Fishburn and Brams, l984). We say more about this shortly.

Example 2 illustrates another paradoxical aspect of STV: raising a candidate in one's preference order can actually hurt that candididate. This is referred to as nonmonotonicity (Smith, l973; Doron and Kronick, l977; Fishburn, l982; Bolger, l985). Thus, if the three class IV voters raise a from fourth to first place in their rankings--without changing the ordering of the other three candidates--b is elected rather than a. This is indeed perverse: a loses when he or she moves up in the rankings of some voters and thereby receives more first-place votes. Equally strange, candidates may be helped under STV if voters do not show up to vote for them at all, which has been called the "no-show paradox" (Fishburn and Brams, l983; Moulin, l986; Ray, l986; Holzman, l987).

The fact that more first-place votes or even no votes can hurt rather than help a candidate violates what arguably is a fundamental democratic ethic. STV also does not guarantee the election of Condorcet candidates (Condorcet, l785)--those who can defeat all other candidates in separate pairwise contests. Thus in Example 2, b is the Condorcet candidate: b is preferred to a by l4 voters (class II, III, and IV voters), whereas a is preferred to b by only 7 voters (class I); similarly, b is preferred to c, 13-8, and to d, l8-3. However, a is elected under STV.

3. The Borda Count

Under a system proposed over two hundred years ago (Borda, l78l), points are assigned to candidates so that the lowest-ranked candidate of each voter receives 0 points, the next-lowest l point, and so on up to the highest-ranked candidate, who receives m-l votes if there are m candidates. Points for each candidate are summed across all voters, and the candidate with the most points wins. To the best of our knowledge, the Borda count and similar scoring methods (Young, l975) are not used to elect candidates in any public elections, but they are used by many private organizations.

Like STV, the Borda count need not elect the Condorcet candidate. This is illustrated by the case of three voters with preference order abc and two voters with preference order bca. Under the Borda count, a receives 6 points, b 7 points, and c 2 points, making b the Borda winner; yet a is the Condorcet candidate.

On the other hand, the Borda count would elect the Condorcet candidate (b) in Example 2 of the preceding section. This is because b occupies the highest position on the average in the rankings of the four sets of voters. Specifically, b ranks second in the preference order of l8 voters and third in the order of 3 voters, giving b an average ranking of 2.14, which is higher (i.e., closer to l) than a's average ranking of 2.19 as well as the rankings of c and d. Having the highest average position is indicative of being broadly acceptable to voters, unlike Condorcet candidate a in the preceding paragraph, who is the last choice of two of the five voters.

Unfortunately, the Borda count is readily subject to manipulation. Consider again the example in which three voters have preference order abc and two voters have order bca. Recognizing the vulnerability of their first choice, a, under the Borda count, the three abc votes might insincerely rank the candidates acb, maximizing the difference between their first choice (a) and a's closest competitor (b). This would make a the winner.

In general, voters can gain under the Borda count by ranking the most serious rival of their favorite candidate last in order to lower his or her point total (Ludwin, l978). This strategy is relatively easy to effectuate, unlike a manipululative strategy under STV that requires estimating who is likely to be eliminated, and in what order, so as to be able to exploit STV's dependence on sequential eliminations and transfers.

The vulnerability of the Borda count to manipulation led Borda to exclaim, "My scheme is intended only for honest men" (Black, l958, p. 238). Nurmi (l984) has shown that the Borda count, like STV, is also vulnerable to preference truncation, giving voters an incentive not to rank all candidates in certain situations. However, Chamberlin and Courant (l983) contend that the Borda count would give effective voice to different interests in a representative assembly, if not always ensure their proportional representation.

Another type of paradox that afflicts the Borda count and related point-assignment systems involves manipulability by changing the agenda. For example, the introduction of a new candidate, who cannot win--and, consequently, would appear irrelevant--can completely reverse the point-total order of the old candidates, even though there are no changes in the voter's rankings of these candidates (Fishburn, l974). Thus, in the example below, the last-place finisher among three candidates (a, with 6 votes) jumps to first place (with l3 votes) when "irrelevant" candidate x is introduced, illustrating the extreme sensitivity of the Borda count to apparently irrelevant alternatives:

3: cba c = 8 3: cbax a = 13

2: acb b = 7 2: axcb b = l2

2: bac a = 6 2: baxc c = ll

x = 6

Clearly, it would be in the interest of a's supporters to encourage x to enter simply to reverse the order of finish.

4. Cumulative Voting

Cumulative voting is a voting system in which each voter is given a fixed number of votes to distribute among one or more candidates. This allows voters to express their intensities of preference rather than simply to rank candidates, as under STV and the Borda count. It is a system of proportional representation in which minorities can ensure their approximate proportional representation by concentrating their votes on a subset of candidates commensurate with their size in the electorate.

To illustrate this system and the calculation of optimal strategies under it, assume that there is a single minority position favored by one-third of the electorate and a majority position favored by the remaining two-thirds. Assume further that the electorate comprises 300 voters, who are required to elect a six-member governing body, and that the six candidates with the most votes win.

If each voter has six votes to cast for as many as six candidates, and if each of the l00 voters in the minority casts three votes each for only two candidates, these voters can ensure the election of these two candidates no matter what the 200 voters in the majority do. For each of these two minority candidates will get a total of 300 (l00 x 3) votes, whereas the two-thirds majority, with a total of 1,200 (200 x 6) votes to allocate, can at best match this sum for its four candidates (l,200/4 = 300).

If the two-thirds majority instructs its supporters to distribute their votes equally among five candidates (l,200/5 = 240), it will not match the vote totals of the two minority candidates (300) but can still ensure the election of four (of its five) candidates--and possibly get its fifth candidate elected if the minority (mistakenly) puts up three candidates and instructs its supporters to distribute their votes equally among the three (giving each 600/3 = 200 votes).

Against these strategies of either the majority (support five candidates) or the minority (support two candidates), it is easy to show that neither side can improve its position. To elect five (instead of four) candidates with 30l votes each, the majority would need l,505 instead of l,200 votes, holding constant the 600 votes of the minority; similarly, for the minority to elect three (instead of two) candidates with 241 votes each, it would need 723 instead of 600 votes, holding constant the l,200 votes of the majority.

It is evident that the optimal strategy for the leaders of both the majority and minority is to instruct their members to to allocate their votes as equally as possible among a certain number of candidates. The number of candidates they should support for the elected body should be proportionally about equal to the number of their supporters in the electorate (if known).

Any deviation from this strategy--for example, by putting up a full slate of candidates and not instructing supporters to vote for only some on this slate--offers the other side the opportunity to capture more than its proportional "share" of the seats. Clearly, good planning and disciplined supporters are required to be effective under this system.

A systematic analysis of optimal strategies under cumulative voting is given in Brams (l975). These strategies are compared with strategies actually adopted by the Democratic and Republican parties in elections for the Illinois General Assembly, where cumulative voting was used until l982. This system has been used in elections for some corporate boards of directors. In l987 cumulative voting was adopted by two cities in the United States (Alamogordo, NM, and Peoria, IL) to satisfy court requirements of minority representation in municipal elections.

5. Additional-Member Systems

In most parliamentary democracies, it is not candidates who run for office but political parties that put up lists of candidates. Under party-list voting, voters vote for the parties, which receive representation in a parliament proportional to the total numbers of votes that they receive. Usually there is a threshold, such as 5 percent, which a party must exceed in order to gain any seats in the parliament.

This is a rather straightforward means of ensuring the proportional representation (PR) of parties that surpass the threshold. More interesting are systems in which some legislators are elected from districts, but new members may be added to the legislature to ensure, insofar as possible, that the parties underrepresented on the basis of their national-vote proportions gain additional seats.

Denmark and Sweden, for example, use total votes, summed over each party's district candidates, as the basis for allocating additional seats. In elections to the Federal Republic of Germany's Bundestag and Iceland's Parliament, voters vote twice, once for district representatives and once for a party. Half of the Bundestag is chosen from party lists, on the basis of the national party vote, with adjustments made to the district results so as to ensure the approximate proportional representation of parties.

In Puerto Rico, no fixed number of seats is added unless the largest party in a house wins more than two-thirds of the seats in district elections. When this happens, that house can be increased by as much as one-third to ameliorate the underrepresentation of minority parties.

To offer some insight into an important strategtic feature of additional-member systems, assume, as in Puerto Rico, that additional members can be added to a legislature to adjust for underrepresentation, but this number is variable. More specifically, assume a voting system, called adjusted district voting, or ADR (Brams and Fishburn, l984a, l984b), that is characterized by the following simplifying assumptions:

l. There is a jurisdiction divided into equal-size districts, each of which elects a single representative to a legislature.

2. There are two main factions in the jurisdiction, one majority and one minority, whose size can be determined. For example, if the factions are represented by political parties, their respective sizes can be determined by the votes that each party's candidates, summed across all districts, receive in the jurisdiction.

3. The legislature consists of all representatives who win in the districts plus the largest vote-getters among the losers, necessary to achieve PR, if it is not realized in the district elections. Typically, this adjustment would involve adding minority-faction candidates, who lose in the district races, to the legislature, so that it mirrors the majority-minority breakdown in the electorate as closely as possible.

4. The size of the legislature is variable, with a lower bound equal to the number of districts (if no adjustment is necessary to achieve PR), and an upper bound equal to twice the number of districts (if a nearly 50-percent minority wins no district seats).

As an example of ADV, suppose that there are eight districts in a jurisdiction. If there is an 80-percent majority and a 20-percent minority, the majority is likely to win all the seats unless there is an extreme concentration of the minority in one or two districts.

Suppose the minority wins no seats. Then its two biggest vote-getters could be given two "extra" seats to provide it with representation of 20 percent in a body of ten members, exactly its proportion in the electorate.

Now suppose that the minority wins one seat, which would provide it with representation of l/8 Å l3 percent. If it were given an extra seat, its representation would rise to 2/9 Å 22 percent, which would be closer to its 20-percent proportion in the electorate. However, assume that the addition of extra seats can never make the minority's proportion in the legislature exceed its proportion in the electorate.

Paradoxically, the minority would benefit by winning no seats and then being granted two extra seats to bring its proportion up to exactly 20 percent. To prevent a minority from benefitting by losing in district elections, assume the following no-benefit constraint: the allocation of extra seats to the minority can never give it a greater proportion in the legislature than it would obtain had it won more district elections.

How would this constraint work in the example? If the minority won no seats in the district elections, then the addition of two extra seats would give it 2/l0 representation in the legislature, exactly its proportion in the electorate. But we just showed that if the minority had won exactly one seat, it would not be entitled to an extra seat--and 2/9 representation in the legislature--because this proportion exceeds its 20-percent proportion in the electorate. Hence, its representation would remain at l/8 if it won in exactly one district.

Because 2/l0 > l/8, the no-benefit constraint prevents the minority from gaining two extra seats if it wins no district seats initially. Instead, it would be entitled in this case to only one extra seat, because the next-highest ratio below 2/l0 is l/9; since l/9 < l/8, the no-benefit constraint is satisfied.

But l/9 Å ll percent is only about half of the minority's 20-percent proportion in the electorate. In fact, one can prove in the general case that the no-benefit constraint may prevent a minority from receiving up to about half of the extra seats it would be entitled to--on the basis of its national vote total--were the no-benefit constraint not operative and it could therefore get up to this proportion (e.g., 2 out of l0 seats in the example) in the legislature (Brams and Fishburn, l984b).

This constraint may be interpreted as a kind of "strategyproofness" feature of ADV: it makes it unprofitable for a minority party deliberately to lose in a district election in order to do better after the adjustment that gives it extra seats. But strategyproofness, in precluding any possible advantage that might accrue to the minority from throwing a district election, has a price. As the example demonstrates, it may severely restrict the ability of ADV to satisfy PR, giving the following impossibility result: under ADV, one cannot guarantee a close correspondence between a party's proportion in the electorate and its representation in the legislature if one insists on the no-benefit constraint. Dropping it allows one to approximate PR, but this may give an incentive to the minority party to lose in certain district contests in order to do better after the adjustment.

It is worth pointing out that the "second chance" for minority candidates afforded by ADV would encourage them to run in the first place, because even if most or all of them are defeated in the district races, their biggest vote-getters would still get a second chance at the (possibly) extra seats in the second stage. But these extra seats might be cut by up to a factor of two from the minority's proportion in the electorate should one want to motivate district races with the no-benefit constraint. Indeed, Spafford (l980, p. 393), anticipating this impossibility result, recommended that only an (unspecified) fraction of the seats that the minority is entitled to be allotted to it in the adjustment phase to give it "some incentive to take the single-members contests seriously, . . . though that of course would be giving up strict PR."

6. Approval Voting

We indicated in section l that a minority candidate, with support from a relatively small percentage of the electorate, can either win a plurality election outright or qualify for a runoff. In the example given in that section, the runoff would deny the election of the minority candidate. On the other hand, a potential defect of runoffs is that a Condorcet candidate may not even make the runoff.

For example, if there are strong minority candidates on both the left and the right, a moderate candidate in the middle may receive the smallest percentage of the vote. Yet this candidate may be in fact be able to defeat each of the minority candidates in separate pairwise contests. Despite being the Condorcet candidate, however, his or her election would be obviated by plurality voting, with or without a runoff.

Approval voting, proposed independently by several analysts in the l970s (Brams and Fishburn, l983), is a voting procedure that is designed in part to prevent the election of minority candidates in multicandidate contests (i.e., those with three or more candidates). Under approval voting, voters can vote for, or approve of, as many candidates as they wish. Each candidate approved of receives one vote, and the candidate with the most votes wins.

Advantages of approval voting include the following:

l. It gives voters more flexible options. They can do everything they can under the plurality system--vote for a single favorite--but if they have no strong preference for one candidate, they can express this by voting for all candidates they find acceptable. For instance, if a voter's most preferred candidate has little chance of winning, that voter could vote for both a first choice and a more viable candidate without worrying about wasting his or her vote on the less popular candidate.

2. It would increase voter turnout. By being better able to express their preferences, voters would more likely go to the polls in the first place. Voters who think they might be wasting their votes, or who cannot decide which of several candidates best represents their views, would not have to despair about making a choice. By not being forced to make a single --perhaps arbitratry--choice, they would feel that the election system allows them to be more honest, which would presumably make voting more meaningful and encourage greater participation in elections.

3. It would help elect the strongest candidate. Today the candidate supported by the largest minority often wins, or at least makes the runoff. Under approval voting, by contrast, it would be the candidate with the greatest overall support--such as the moderate candidate alluded to above --who would usually win. An additional benefit is that approval voting would induce candidates to try to mirror the views of a majority of voters, not just cater to minorities whose votes could give them a slight edge in a crowded plurality contest.

4. It would give minority candidates their proper due. Minority candidates would not suffer under approval voting: their supporters would not be torn away simply because there was another candidate who, though less appealing to them, was generally considered a stronger contender. Because approval voting would allow these supporters to vote for both candidates, they would not be tempted to desert the one who is weak in the polls, as under plurality voting. Hence, minority candiates would receive their true level of support under approval voting, even if they could not win.

5. It is eminently practicable. Approval voting can readily be implemented on existing voting machines (unlike the preferential systems discussed earlier), and it is simple for voters to understand. Moreover, because it does not violate any state constitutions in the United States (or the constitutions of most countries in the world), it needs only a statute passed by a state legislature to become law.

Although approval voting encourages sincere voting, it does not eliminate strategic calculations altogether. Because approval of a less-preferred candidate could hurt a more-preferred approved candidate, the voter is still faced with the decision of where to draw the line between acceptable and nonacceptable candidates. A rational voter will vote for a second choice if his or her first choice appears to be a long shot--as indicated, for example, by the polls--but the voter's calculus and its effects on outcomes is not yet well understood for either approval voting or the other procedures discussed herein (Nurmi, l987; Merrill, l988).

Approval voting is now used in many universities and in several professional societies with collectively over 325,000 members (Brams, l988; Brams and Fishburn, l988). Among other officials, the secretary general of the United Nations is elected by approval voting (Brams and Fishburn, l983).

Bills to implement approval voting have been introduced in some state legislatures in the United States; in l987, a bill to mandate approval voting in certain statewide elections passed the Senate but not the House in North Dakota. Approval voting has been used in internal elections by the political parties in some states, including Pennsylvania, where a presidential straw poll using approval voting was conducted by the Democratic Party state committee in l983 (Nagel, l984). Beginning in l987, approval voting has been used in some competitive elections in the Soviet Union (Shabad, l987).

7. Conclusions

There is no perfect voting procedure. But some procedures are clearly superior to others with respect to satisfying certain criteria. Among nonpreferential voting systems, approval voting distinguishes itself as more sincere and more likely to select Condorcet candidates than other systems, including plurality voting and plurality voting with a runoff.

Although preferential systems, notably STV, have been used in public elections to ensure proportional representation of different parties in legislatures, the vulnerability of STV to preference truncation illustrates its manipulability, and its nonmonotonicity casts doubt upon its democratic character. In particular, it seems bizarre that voters can hurt a candidate's chances by raising him or her in their rankings.

Although the Borda count is monotonic, it is more readily manipulable than STV. Whereas it is difficult to calculate the impact of insincere voting on sequential eliminations and transfers under STV, the strategy of ranking the most serious opponent of one's favorite candidate last is a transparent way of diminishing a rival's chances under the Borda count. Also, the introduction of a new and seemingly irrelevant candidate, as we illustrated, can have a topsy-turvy effect, moving a last-place candidate into first place.

Additional-member systems, and specifically ADV that results in a variable-size legislature, provide a mechanism for approximating PR without the nonmonotoncity of STV or the manipulability of the Borda count. But the no-benefit constraint on the allocation of additional seats to underrepresented parties under ADV--in order to rob them of the incentive to throw district races--vitiates fully satisfying PR. Although cumulative voting offers a means for parties to guarantee their approximate proportional representation, it requires good predictive abilities and considerable organizational efforts on the part of parties to ensure that their supporters concentrate their voters in the proper manner.

Because of the impossibility of satisfying a number of desiderata simultaneousy, trade-offs are inevitable in the search for voting procedures that best meet different needs (Niemi and Riker, l976; Straffin, l980; Dummett, l984). We have tried to show how an understanding of certain characteristics of alternative voting procedures--especially those relating to their strategic properties--can facilitate the selection of practical procedures that satisfy the criteria one deems most important.


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