Journal of Economic Perspectives Volume 9, Number 1 Winter 1995 Pages 65 75 Using District Magnitude to Regulate Political Party Competition Douglas W. Rae The mere existence of a Council of Economic Advisers has always seemed a nettlesome provocation to political scientists like myself. There is no such thing as a Council of Political Advisers for two very good reasons. First, a democratic polity cannot be expected to maintain a consensually agreed set of political goals comparable to price stability, full employment, and economic growth. Second, whatever goals might be set, political science offers very little in the way of technical leverage. To be effective in this way, political scientists must identify policy instruments that have known and controllable impacts on valued outcomes. With monetary policy, for example, there are tools like the discount rate for manipulating the money supply. There is no analogous tool in a political system. The closest approximation of such a tool would perhaps be the design of an electoral system. We do know quite a lot about the consequences of adopting different electoral systems. In this essay, I will focus on just one electoral system variable, "district magnitude," which is the number of legislative seats assigned to each electoral district. For example, the U.S. House of Representatives has a district magnitude of one, with 435 seats in as many districts. Some parliamentary systems (like Israel's) elect the entire membership in one national district, so that magnitudes reach three figures. Worldwide, most magnitudes fall somewhere between one and 20. I will show how the manipulation of this variable can help regulate some important features of political competition. Douglas W. Rae is Richard Ely Professor of Public Management at the Yale School of Management, and Professor of Political Science at Yale University, New Haven, Connecticut.
66 Journal of Economic Perspectives Two Conflicting Goals Electoral systems clearly affect the shape of political party systems and the rhythms of governmental change. But how can such systems be evaluated? While a virtually unlimited range of goals can be found in the dustbin of political history, two partly conflicting aims are at the center of most electoral engineering projects. The first is minority representation, by which I mean providing representation to groups whose members articulate important shared interests, even if they command a relatively small fraction of a national electorate. The second aim is perhaps best called "defractionalization." Defractionalization simply entails the provision of incentives for the formation of major parties and broad-based electoral campaigns that produce governing majorities without reliance upon fragile interparty coalitions, put together after the election itself fails to pick a majority winner. It is believed, with some reason, that some sort of defractionalizing effect is required to promote stable and effective governance in any but the most homogeneous and docile societies. These two goals are often thought of less as the achievement of something good than as the avoidance of something bad. For example, it is widely agreed that it would be less than optimal to have a single party (like Mexico's PRI) or two parties (like America's Republicans and Democrats) sweep away all other contestants. But on the reverse side, it is also unattractive to have no truly major parties, so that very small parties bargain endlessly in the formation of coalitions (as in Italy, at almost any moment since 1945). Given the evident partisan implications of minority representation and defractionalization, and a strong tension between them, there can be no perfect agreement on such goals. But students of electoral engineering are asked from time to time to plot out an electoral system capable of achieving one of the two without gutting the other. Policy Instrument: District Magnitude It is possible to divide the variables defining an electoral system into three groups: ballot structures, 1 electoral formula, 2 and districting systems. I will concentrate on the third group, setting the others to simple and common values. Let's assume that ballots structure the voting choice by saying, in effect, "Of the p parties contesting this election in your district, which party is to be credited with your entire vote?" This implies a simple total of votes for each party. The district magnitude (m) defined as the number of seats in a district can vary from one up to the total size of a national parliament. 1 Ballot structures can be nominal (for example, here are n alternatives, pick one), or ordinal (rank several), and, if nominal, can require a categorical decision (pick only one) or may allow the voter to divide his vote among two or more parties. See, for example, Rae (1971). 2 Electoral formulae include things like majority rule, plurality voting, and proportional representation (many subtypes).
Douglas W. Rae 67 Let the electoral formula be equivalent to a pricing system in which the parties "bid" their total vote in a district, divided by one more than the number of seats which have already been awarded to that party. Each of the district's m incumbencies goes to the party able to pay the highest price. In a singlemember district (m = 1), this implies the plurality rule since the single "highest bidder" wins outright. If we have two or more seats, then the different parties "bid" for the seats, using their votes as a sort of currency. If two seats were available, for example, then a winning party must have at least twice the votes of the second place party to claim both seats. In the case of many parties and many seats, the number of votes needed to claim a seat will be bid down, since it will immediately be recognized that the price of a seat is set by the smallest party with enough votes to claim a single seat. This system is equivalent to a widely used form of proportional representation, commonly called d'hondt proportional representation after its Belgian inventor. To see how it works, suppose three parties contest a district with five seats. Their vote totals fall thus: Party Red Black Green Votes 120,000 100,000 70,000 Votes/2 60,000 50,000 35,000 Votes/3 40,000 33,333 23,333 Votes/4 30,000 25,000 17,500 The formula asks for each party at each stage: if we give the next seat to this party, will it be paying a higher average price for all its seats than would be the case for any other party if we awarded it the seat? When no party has yet won a seat, we divide all vote totals by one and give the seat to Red, at 120,000. Then for the second seat, we still divide by one for Black and Green, but by two for Red, since it already has one seat and would have two if it won the next seat. The seat goes to Black, which "pays" 100,000 votes. For the third seat, Green's "average" is still 70,000 while the others have fallen to 60,000 and 50,000. For the fourth seat, Red wins with 60,000; for the fifth and last seat Black wins with 50,000. The boldfaced numbers are winners, and the formula has maximized the average price given for the five seats at stake. Limiting the discussion to this sort of voting rule neglects many of the voting schemes discussed by Levin and Nalebuff in this symposium. However, it allows us to concentrate on districting systems as a particularly powerful way of manipulating the effects of an electoral system. It would be possible to substitute many other voting rules for the one studied here without greatly changing the impact of district magnitude on party competition. Districting systems include the drawing of district boundaries and the assignment of the number of seats to each district. They vary in three important respects. First is their apportionment according to population measured by the extent to which the ratio of seats to total population is constant across
68 Journal of Economic Perspectives districts. 3 Second is their degree of fit to target populations such as the Maori of New Zealand, or particular racial and religious groups in the United States. Third is district magnitude, meaning the number of seats in each district. The magnitude can vary by district. At the low end, we find the single-member districts common to English-speaking countries. At the high end, we find national districts with more than 100 seats, such as those in Israel and Holland. Although each of the three factors just named has an importance all its own, I will concentrate mainly on magnitude. Minority Representation in High-Magnitude Districts Consider the first goal giving representation to fairly small minorities. This is achieved in a weak sense if such a minority manages to win at least one seat in the legislature being formed and, in a stronger sense, if it is represented in proportion to its size (in other words, if its adherents amount to one xth of the citizenship, its representatives would amount to one xth of the legislature). As long as we are willing to neglect groups smaller than the size of the electorate divided by the size of the legislature that is, groups whose proportional representation would amount only to some fraction of a seat we know perfectly well how to achieve this goal. The simplest approach to proportional representation is to combine the entire electorate in a single nationwide district and then apply an electoral formula of the sort described above. This election method has been extensively field-tested by Israel, Holland, and Iceland. In addition, it was important in the dramatic 1994 election in South Africa. One of the most robust empirical and theoretical conclusions in analyzing this topic is that the proportionality of electoral outcomes grows finer as the magnitude of districts increases, but that gains diminish at the margin (Rae, 1971; Lijphart, 1994). The electoral formula described here operates as a mechanism of successive approximation, getting closer to proportionality with the awarding of each seat. To understand this point, take the simple case of two parties dividing an electorate. How close will we come to giving each party its proportional due? This will depend on the number of seats at stake and on the division of votes that happen to arise. For district magnitudes of one, two, and three, the options in splitting seats between the stronger and weaker party are shown in Table 1. With just one seat, society may be compelled to make a "proportionality error" approaching 0.5 or half the vote. This would happen if the weaker party had close to 50 percent, but was denied representation. With more seats to deal with, the approximations grow better and better. To be exact, the maximum 3 This precept was an important constitutional issue in U.S. politics, as articulated by the Supreme Court in Baker v. Carr, 369 U.S. 186 (1962), and Reynolds v. Sims, 377 U.S. 533 (1964).
Using District Magnitude to Regulate Political Party Competition 69 Table 1 Options for Splitting Seats error falls as m rises, with each new seat cutting the maximum error by a little less than the one before. The maximum error equals 1/2m, where m is the district magnitude and where maximum error is defined by the largest lawfully possible difference between a party's vote proportion and its seat proportion. With more than two parties, the pattern is more complex, but not different in kind. And it is clear that gains in proportionality diminish at the margin in all cases. The vote share required for very small parties to gain representation will for the same reason fall lower and lower as magnitudes increase (and will do so at a rate diminishing at the margin). For simplicity, define the threshold of representation to be the share of the national vote that guarantees a party one or more seats, no matter how unfavorably the vote happens to fall among the other parties in a legislature whose (well-apportioned) districts are of size m. The intuition is obvious: this threshold is 1/(m + 1). 4 If a legislature of size 100 were elected in single-member districts, then a total tally that reaches one vote above 50 percent would guarantee the party's first seat, even if the rest of the vote was against a single opposition party with a worst-case distribution of strength in this case, one that is uniform across all districts. 5 If, instead, the same 100 seats were divided among four districts of size 25, then just more than 1/26th of the national vote would assure representation, no matter how unhappily the split fell among other parties and across districts. At the limit, 4 Notice that we are looking here at the threshold defined by the highest total that could still under worst-case circumstances be excluded from representation. For each value of m, we construct the worst-case split among other parties. Another threshold concept asks how small a total could under best-case circumstances win representation. For further discussion, see Loosemore and Hanby (1971), Rae (1971), and Lijphart (1994). The latter provides a very useful discussion of more complex findings and empirical applications of the threshold literature from pp. 25 39. There is also evidence that the birthrate of small new parties responds most dramatically to district magnitude when and where ethnic heterogeneity is greatest. See Ordeshook and Shvetsova (1994). 5 This pattern will often repeat itself across single-seat districts nationwide, so that small parties are systematically underrepresented. A common generalization is the "cube law," stating that the total ratio of seats between two such parties will tend to equal the ratio between the cubes of their respective votes. See, for example, Taagepera (1973).
70 Journal of Economic Perspectives one district of magnitude 100 would push the threshold down to just more than 1/101. The direct consequence of a lower threshold is to help achieve minority representation, but this is not all. Two other considerations must be factored in: bargaining power and legitimacy. Holding a seat is one thing; having real bargaining power in the formation of legislative majorities is quite another. Suppose that a nation has two large parties, each just below majority size, and a smallish third party large enough to push either of the others to majority size (for example, a three-way split of 48 percent, 47 percent, and 5 percent). In the strict logic of coalition power, the small party is greatly overrepresented (as occurs with Israel's religious parties). Even if a small party does not play much of a role in forming majorities, having a seat in the legislature does confer a particular kind of legitimacy. The very first seat won in national elections sets a group apart from others clamoring for attention in any nation's political discourse. Such a party may have no bargaining power in majority formation, yet having a seat may allow it to be an important influence on the actual agenda of government. This again runs counter to the strict numerical ideal of proportionality. Full proportionality flattens all incentives for aggregation of factions into large parties (at least unless party organizing and campaigning are subject to important economies of scale), leading to legislatures that are highly fractionalized and governance that is consequently unstable. This general pattern has been documented across a fairly wide class of nations (Powell, 1982). This brings us to the second goal. Defractionalizing Shifts and the Nurturance of Large Parties It is of course possible to be all too successful in achieving proportional representation. The result is not only representation of minorities, but of minorities within minorities. Consider the Italian election of 1987, summed up in Table 2. The vote-seat transformation is essentially proportional, top to bottom, as might be expected with district magnitude averaging about 20 seats. The incentive for partisan consolidation is consequently nil: no two smaller organizations would gain much from merging, and any larger organization's internal factions could defect without greatly diminishing the pool of seats they share. The fractionalization of votes among a dozen parties of course reflects the operation of such incentives over the decades, and implies a very unstable bargaining process in the formation and maintenance of parliamentary governments. Only two possible coalitions having as few as two partners command a majority. The electoral law effect that matters here is generally described as a defractionalizating shift in the transition from votes to seats. Nearly all electoral systems tend to give more than proportionate rewards to large parties and less
Douglas W. Rae 71 Table 2 Votes and Shares in Italy, 1987 than proportionate rewards to small ones. 6 The strength of that tendency is a key variable, and is usually measured by comparing an index of fractionalization for votes with one for seats. The usual index, F, varies from zero for a one-party system to one for the limiting case of an infinite number of small parties. The index value is defined as the complement of the sum of the squared party shares in the vote or seat outcome: where X i is the vote or seat fraction of the ith party. 7 This can be computed as Fv for votes and Fs for seats. The defractionalizing shift is then seen as the difference between the two results. In the 1987 Italian election example, the defractionalizing shift amounted to a very weak.02 (from.78 in votes to.76 in seats). This was radically altered by the new electoral system introduced in 1993 and put to use in 1994. 6 The obverse case, in which a leading party receives less than a proportional share of seats, is exceedingly rare. One instance is offered by the Australian Labor Party, which was the largest party in many postwar elections, yet got less than its proportionate share of the lower house. In 1949, for instance, the ALP got 46 percent of the vote and just 39 percent of the seats, while the smaller liberal party got 39 percent of the vote and 45.5 percent of the seats. The ALP's heavy concentration in urban areas, combined with single-member districting, explains the anomaly. Essentially, ALP candidates systematically ran up much larger margins of victory. 7 Economists will recognize this measure as a political science complement of the familiar Herfindahl-Hirschman index of concentration. When applied to large numbers, the index can be interpreted as the probability that the two voters selected different parties.
72 Journal of Economic Perspectives Figure 1 Percent of District Vote to Strongest Party Not Winning a Seat, by District Magnitude, Spain, 1989 The defractionalizing shift works in three ways: 1) in the direct mechanism of an electoral system, giving more than proportional rewards to large parties and less than proportionate rewards to small parties; 2) in the alteration of expectations among politicians, leading to differential recruitment in favor of larger blocs; and 3) in the psychology of voting, leading especially to the abandonment of first-choice parties thought to be below the threshold of electability in favor of second-choice parties thought to have a better chance. 8 District magnitude is a predictable determinant of the strength of the defractionalizing shift, with higher m leading to weaker defractionalizing shifts. This is illustrated by comparing Italy's high-magnitude system with an otherwise broadly similar system operating in Spain. The critical difference is that the Spanish law provides for districts averaging only 6.7 seats each (versus about 20 in the Italian system). This is low enough to permit a defractionalizing shift of considerable force. Over the six elections from its introduction in 1977 through the election of 1993, Spain's system of low-magnitude proportional representation has made a mean difference of.11 or roughly 15 percent as it took.74 fractionalization of votes and returned.63 fractionalization of seats. Combined with the less-direct mechanisms mentioned in the previous paragraph, this system has led to the evolution of a near two-party system in which only 15 of 204 directly elected seats escaped the two major parties in the 1993 election of the lower chamber. Seen at the district level, Spanish data show the mechanism of district magnitude at work. In Figure 1, I have arrayed Spanish districts by magnitude, from one at the left to 33 at the right. On the vertical axis, I have shown the district vote percent obtained by the largest party failing to win any seats in that 8 See Gunther (1989) for an impressive empirical case study. For a more theoretical treatment see Cox and Shugart (1994).
Using District Magnitude to Regulate Political Party Competition 73 Figure 2 Party Transactions, Huesca Province, Spain, 1989 district. Parties winning more than a third of the vote are shown failing to win any seats in single-member districts, while only tiny splinter groups fail in the larger districts. 9 The invisible ceiling above the data points is of course the threshold mentioned earlier, requiring a vote share of 1/(m + 1) to guarantee a seat. Figure 2 shows how a major defractionalizing shift occurs in the smaller districts. Here, more than a dozen parties compete in Huesca Province for seats in the 1989 general election, with just three seats to be divided. On the horizontal axis, we see each party's percent of the district vote. On the vertical axis, we see the difference between its percent of the vote and its percent of the seats. Parties above the zero line are getting good deals; parties below it are getting bad deals. The socialists at the upper right exchanged about 40 percent of the vote for two of the three seats (67 percent minus 40 percent yields its + 27 percent transaction). A second party exchanged about 25 percent of the vote for the third seat. All of the smaller parties, including three around the 10 percent mark, were denied representation. These major also-ran parties are presented with a strong incentive to combine in future elections. A strong shift occurs here as the index of fractionalization for vote shares is about.75, and the fractionalization of seats comes in under.45. Managing Trade-Offs In the United States, we have largely confined ourselves to the use of the single-seat districting, which has provided a strong defractionalizing brake on third parties. At the same time, it has led our jurisprudence of minority representation into a nasty cul-de-sac. We find ourselves drawing districts in 9 A result explained in part by the fact that Spain requires a party to have at least 3 percent of the votes before being considered for any seats at all.
74 Journal of Economic Perspectives such a way that members of a particular racial or religious minority will almost certainly win the resulting seat. This form of gerrymandering is unfortunate in its implication that the state should decide which minorities, at what locations, will be so favored. It also yields many inelegant and unlovely districts, having no justification except the rigging of an outcome. The use of proportional representation at the state level, with district magnitudes allowed to vary, might well provide a more effective solution. Under such a system, would-be minorities could sort themselves out and succeed or fail in competition with each other. More generally, it is possible to mix districts of varied magnitudes so as to combine desirable properties. If there are a few high-magnitude seats, it is possible for minorities to compete for modest representation. And if most districts are low-magnitude, it is possible to maintain an effective check on the profusion of smaller groups in the interests of broadly majoritarian governance. Insofar as larger districts are also more heterogeneous, they may also lead to front-loading of political bargaining. Politicians representing small homogeneous districts need not take compromise positions on issues where their constituents are at odds with people residing elsewhere. But if districts are enlarged to encompass many groups, then politicians may in some (but not all) instances have incentives to seek moderating positions. In the case of racial representation, this seems very important as a consideration against the construction of racially homogeneous districts. 10 This is less because of what happens in districts constructed to achieve African-American majorities than because of what then does not need to happen in the remaining districts. If minority members are represented elsewhere, and are numerically unimportant in the remaining four-fifths or more of all districts, the results will tend to encourage legislative majorities whose members have little or no reason to grapple with the concerns of African-Americans. This reasoning holds true for any situation of single-member districts in which the drawing of geographic district lines is the major method of political engineering. In systems allowing multimember districts, it is possible that proportional representation may let groups define themselves. If, say, five seats are contested in a district, and a given minority amounts to something like 20 percent of the district's electorate, the group itself is empowered to define itself as the basis of a candidacy or party, or to combine its number with another bloc of voters. This in turn releases the government from making particularistic judgments about which groups ethnic, religious, political, or issue group require the benefit of political engineering. Proportional representation in multimember districts is a political option that the United States should consider seriously at both the state and federal level. One may, of course, shudder at its impact on an already complex political order, but simplicity has never been a central virtue of American politics. 10 For a broad introduction to the legal and polemical literature, see Guinier (1994).
Douglas W. Rae 75 I wish to thank Alan Krueger, Barry Nalebuff, Carl Shapiro, and Timothy Taylor for their useful and generous comments on this paper. Teri Bimes provided valuable research assistance. References Balinski, M. L., and H. Peyton Young, Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press, 1982. Cox, Gary W., and Matthew Shugart, "SNTV and d'hondt are 'Equivalent,'" Electoral Studies, June 1991, 10, 118 32. Cox, Gary W., and Matthew Shugart, "Strategic Voting Under Proportional Representation," unpublished paper, 1994. Grofman, Bernard, and Arend Lijphart, eds., Electoral Laws and Their Political Consequences. New York: Agathon Press, 1986. Guinier, Lani, The Tyranny of the Majority: Fundamental Fairness in Representative Democracy. New York: The Free Press, 1994. Gunther, Richard, "Electoral Laws, Party Systems, and Elites: The Case of Spain," American Political Science Review, September 1989, 83, 835 58. Hermens, Ferdinand, Democracy or Anarchy: A Study of Proportional Representation. Notre Dame, Indiana: University of Notre Dame Press, 1941. Lijphart, Arend, Electoral Systems and Party Systems: Study of Twenty-Seven Democracies, 1945 90. Oxford: Oxford University Press, 1994. Loosemore, John, and Victor J. Hanby, "The Theoretical Limits of Maximum Distortion: Some Analytic Expressions for Electoral Systems," British Journal of Political Science, October 1971, 1, 467 77. Ordeshook, Peter C., and Olga V. Shvetsova, "Ethnic Heterogeneity, District Magnitude, and the Number of Parties," American Journal of Political Science, February 1994, 38, 100 23. Powell, G. Bingham, Jr., Contemporary Democracies: Participation, Stability, and Violence. Cambridge: Harvard University Press, 1982. Rae, Douglas W., The Political Consequences of Electoral Laws. Revised edition, New Haven: Yale University Press, 1971. Taagepera, Rein, "Seats and Votes: A Generalization of the Cube Law of Elections," Social Science Research, September 1973, 2, 257 75. Taagepera, Rein, and Matthew Soberg Shugart, Seats and Votes: The Effects and Determinants of Electoral Systems. New Haven: Yale University Press, 1989.