Wasted Votes: A Conceptual Synthesis and Generalization to the Case of PR Elections*

Wasted Votes: A Conceptual Synthesis and Generalization to the Case of PR Elections* Peter Selb Department of Politics and Public Management University of Konstanz 78457 Konstanz, Germany peter.selb@uni-konstanz.de Bernard Grofman Department of Political Science and Center for the Study of Democracy University of California, Irvine, Irvine, CA 92697 bgrofman@uci.edu * Grofman s work was supported by the Jack W. Peltason (Bren Foundation) Chair, University of California, Irvine, by the UCI Center for the Study of Democracy, and by SSHRCC research grant #410-2007-2153 (co-pis: Stanley Winer and Stephen Ferris). 1

ABSTRACT We look at conceptual underpinnings of the idea of wasted votes and we propose straightforward ways to measure wasted votes in multiseat list PR contests using divisor rules (e.g., d Hondt). We offer six possible definitions of wasted votes, each of which has the nice property of reducing to a more familiar definition for the plurality case when district magnitude equals one. We show the resemblances among these six measures and also provide some analytic results about the relative magnitude of the proportion of wasted votes that we should find under each measure. Then, based on ideal type cases such as lopsided elections and distributions with Duvergerian and non-duvergerian equilibria, we offer illustrative projection results about how, for a fixed distribution of party vote shares, wasted votes vary with district magnitude. Keywords: wasted votes, strategic voting, electoral systems 2

1. Introduction The idea of wasted votes is often used as a way to judge normative features of voting systems, with some electoral rules (e.g. list forms of proportional representation) argued to be less likely to generate wasted votes than simple first-past-the-post (plurality) voting. In a longitudinal perspective, a reduction in wasted votes is frequently considered indicative of processes by which voters learn to cast ballots in efficient ways, i.e., in ways that decrease the likelihood that candidates they dislike will be elected (e.g., Cox 1997, Duch and Palmer 2002, Reed 1990, Tavits and Annus 2006). Thus, the concept of wasted votes is often used to integrate the study of party competition (including party formation and dissolution) with the study of voter choice and electoral system effects (Cox 1997, Downs 1957, Duverger 1959, Riker 1982). In particular, the concept of wasted votes is commonly linked to ways to measure the existence of Duvergerian and non-duvergerian equilibria (e.g., Cox 1994, 1997). 1 The concept of wasted votes is also potentially relevant in understanding incentives for turnout (Cox 1999, Selb 2009). 1 Consider, for example, this quote from Chibber and Murali (2006:5, emphasis added): Duverger s law predicts that two parties will capture all the votes in district level elections in countries with single-member, simple-plurality rules. Research has shown that, at its heart, Duverger s law relies on the assumption that district-level elections are characterized by strategic voting. Following a standard definition of strategic voting that voters prefer not to waste their votes if meaningful and potentially consequential votes can be cast the implication of such an assumption in single-member, simple-plurality elections is that voters prefer to vote for a candidate who has a chance of winning the election, all else being equal. 3

The aims of this paper are to rethink the conceptual underpinnings of the idea of wasted votes, and to see how this concept can be defined when we look beyond just single member district plurality (SMP) elections. We seek to provide a conceptually unified way of thinking about wasted votes applicable to both plurality and PR settings. For electoral rules that reduce to plurality when district magnitude equals one, such as most divisor rules, our proposed measures have as a key feature that they will become identical to standard measures of wasted votes in plurality single member district elections. Thus we offer measures of wasted votes that are conceptually the same for plurality and PR settings. 2. Measuring wasted votes We may usefully distinguish two aspects any approach to defining wasted votes in terms of the answers it gives to the following two questions. (1) Which of the votes not given to winning candidates should be taken be wasted? and (2) Should votes given to winning candidates in excess of what they needed to win be counted as wasted? We begin with definitions that respond only to the first question and which (implicitly) answer the second question in the negative. The argument that we should answer the second question in the negative rests on two claims: (a) large vote margins can have a deterrent effect on subsequent challengers so that an excess of votes for a winning candidate or candidates need not, in pragmatic terms, truly be This quote perfectly exemplifies our proposition that the idea of voters wasting their vote is commonly used in conjunction with the study of strategic voting and Duverger s law. 4

wasted, and (b) extra votes may not really be extra, in that the degree to which an election is noncompetitive may be exaggerated by the actual outcome, since an anticipation of a losing contest may lead to poorer challengers and less money being spent on campaigning. Equally, supporters of the hopeless competitor(s) may stay at home to a disproportionately high extent to avoid wasting their votes, thereby skewing the election result even further. Thus, we think that a plausible claim can be made to defined wasted votes solely in terms of the first component. But we will remain agnostic on that point, and offer both types of definitions. 2.1. Wasted votes in SMP systems votes not given to winning candidates Let x i be the vote share of the ith largest candidate or party. Let n be the number of candidates or parties. Most empirical studies of wasted votes count all the votes for candidates or parties that did not eventually gain representation as wasted. 2 This notion gives rise to our first (and simplest) definition of wasted votes. Definition 1: x, for all i 11 x i 1 Imagine, for example, a very lopsided SMP election where the winner gains 70 percent of the votes, leaving the first and second loser with 18 and 12 percent, respectively. Now the wasted vote would be 30 percent. 2 See, among others, Anckar (1997), Bakke (2005), Chiva (2007), Siaroff (2000), and Tavits and Annus (2006) for national level, and Johnston (2001), and van der Weyden and Meuleman (2008) for district level examples. 5

As straightforward as this definition may seem, it is rather inconsistent with the original conception of wasted votes, which is closely tied to the notion of strategic voting, i.e., voting in favor of a potentially less preferred, but more viable, alternative. As long ago as 1869, Henry Droop (the originator of the Droop quota that is used for allocating seats in Single Transferable Vote (STV) and some largest remainder proportional representation (PR) electoral systems) recognized the basic logic under single-member district plurality (SMP): As success depends upon obtaining a majority of the aggregate votes of all the electors, an election is usually reduced to a contest between the two most popular candidates. [...] Even if other candidates go to the poll, the electors usually find out that their votes will be thrown away, unless given in favour of one or the other parties between whom the election really lies" (quoted in Riker 1982: 756). Thus, according to Droop, votes for candidates or parties other than the subsequent winner and second-place finisher are wasted. We take this as Definition 2 of the concept of wasted votes. Definition 2: x, for all i 2 1 x x i 1 2 Going back to our lopsided SMP election where the winner gains 70 percent of the votes, leaving the first and second loser with 18 and 12 percent, respectively, now the wasted vote would be only 12 percent. Note that, for two-party competition, this definition gives us no wasted votes. 6

But this definition, too, seems not fully satisfactory in that it neglects strategic incentives. Duverger's (1951) famous prediction that SMP systems favor (local) two-party competition rests on voters not wasting their votes (as the term is used in Definition 2), or on political elites that negotiate on two promising alternatives before the election so there will be no thirdto n-th place finishers to waste one's vote on. As Cox (1994, 1997) observes, situations may occur where the expected vote shares of first and second losers are too close to decide which of the two to desert (a situation that falls in to the category of what Cox calls non- Duvergerian equilibria ). In this case, so argues Cox, no one could seriously accuse supporters of the second loser to have wasted their votes on a hopeless cause. But this approach suggest a further extension of the concept of wasted votes, namely that votes for the third to n-th place finishers should be counted as wasted only to the extent that there is gap between their vote shares and that of the first loser. Cox s argument leads us to what we propose as Definition 3 of the concept of wasted votes: i 2 i i i Definition 3: min x ; x x x x,for all i 2 The first term of this expression is there to assure that we do not get estimates of wasted vote that are larger than the sum of the votes given to third and lower-placed candidates. This can occur, for example, if we have a vote distribution of, say (45, 35, 10, 10), where looking at the gap of 25 percent between the third place and the second place finisher and the identical gap of 25 percent between the fourth place and the second place finisher would lead us to conclude that 25 percent of the votes were wasted when in fact only 20 percent of the vote was given to the third and fourth place candidates combined. The latter term of the second 7

part of this expression has been normalized to provide a weighted average of the gap between the first loser (second place finisher) and all other losers. Going back to our lopsided SMP election where the winner gains 70 percent of the votes, leaving the first and second loser with 18 and 12 percent, respectively, under this third definition, the wasted vote would be only 6 percent, since the third place finisher has 100 percent of all the votes of those who finish 3 rd or lower. When there are only two parties this definition becomes equivalent to Definition 2, and thus, for two party plurality competition, this definition gives us no wasted votes. Note, too, that for a distribution such as (x 1, (1 x 1 )/2, (1 x 1 )/2), we again get no wasted votes. In Cox s terminology, this is a non- Duvergerian equilibrium. 2.2. Wasted votes in SMP systems votes not given to winning candidates and unnecessary votes for the winning candidate If we do choose to regard some of the votes given to the winning candidate as wasted, the obvious way to measures those wasted votes for the winner is that proposed by Cohan (1975), namely as the difference between the vote shares of the winner and those of the first loser, i.e., (x 1 x 2 ). Taking into account this notion of wasted votes for the winner, we get three additional definitions of the concept of wasted vote. Definition 1*: x x + x, for all i 1 x x, for all i 2 1 2 i 1 i Since the definition of votes for the winner in excess of the second finisher corresponds to Johnston and Pattie s (2001) definition of surplus votes (also see Ardoin and Palmer 2007, Sickel 1966), another way to state Definition 1* is that wasted votes are all the votes for the 8

losers plus the surplus votes. This is the definition proposed by Cohan (1975). Definitions 2 and 3 may be restated accordingly: Definition 2*: x x x, for all i 2 1 2 i i i i i Definition 3*: ( x1 x2 ) min x ; x2 x x x,for all i 2 Go back to our lopsided SMP election where the winner gains 70 percent of the votes, leaving the first and second loser with 18 and 12 percent, respectively, Definition 1* gives us a wasted vote of 82 percent; Definition 2* gives us a wasted vote of 64 percent; and Definition 3* gives us a wasted vote of 58 percent. Thus, our six different concepts of wasted votes give us a range of answers from 6 percent to 82 percent, and no two answers are the same. Note also that, in Definitions 1* through 3*, we have the striking result that a majority, and sometimes an overwhelming majority, of the votes cast in the election are considered to be wasted. 2.3. Wasted votes in PR systems But how do we extend these rather simple concepts of wasted vote to the PR context? For rules that have the property that they reduce to simple plurality when district magnitude M = 1, such as pure list PR quota rules, what we are looking for are definitions that have the nice property that they reduce to our earlier definitions for voting rules when M = 1. 3 3 Grofman and Selb (2009) consider a related question, how to measure competition in both majoritarian and PR settings, from this same perspective. 9

In what follows we will limit ourselves to quota rules such as d Hondt and Sainte Lagűe. In quota rules, we can identify the M winners in an M seat district by dividing party vote shares by a set of divisors (e.g., 1, 2, 3, etc. for d Hondt; 1, 3, 5, 7 etc. for Sainte Lagűe) and finding the M highest quotients. To make the connection between plurality systems and PR systems we define losers relative to the M-th seat gained. In other words, for M 1 we take the first M seats as winners, and define the first loser as the (M +1)-th largest quotient, except for the first losing quotient of the party that wins the M-th seat (since its highest losing quotient is in competition with itself). We will use the notation, q M, for the lowest winning quotient, i.e., the quotient that wins some party the M-th seat, and we will use the labeling q M +1, q M +2, for the first loser quotients, arranged from highest, i.e., q M +1, to lowest, i.e., q M + n-1, where n is the number of parties competing. The reason we have only n 1 quotients to examine is that we are omitting the first losing quotient for the party that wins the M-th seat. Note, too, that, except for omitting the first losing quotient for the party winning the M-th seat, we are ordering the quotients according to their size, not according to how many total votes the party received, i.e., we are ordering the quotients according to how close the party came to winning a(nother) seat with its highest non-winning quotient. It is easy to see that any of the divisor formulae for multi-seat list PR (e.g, d Hondt or Sainte Lagűe, etc.) reduces to single-member plurality for M = 1. 4 Since we are only dividing by 1 to 4 Grofman (forthcoming) looks at the relationship between multi-seat and single-seat systems in terms of what he calls roots; the root of a multi-seat system is the system that it will reduce to when M = 1. For example, the alternative vote is the root of STV. A given single-seat selection rule can be the root for more than one multi-seat rule, as we have seen in the case of the relationship between simple plurality and PR quota rules. Pure list PR and plurality have 10

obtain the quotients for the M = 1 case (plurality), the only relevant quotients are the values x i, themselves, and so the ways we are proposing below to measure wasted votes for quota rules will be ones that reduce to the six previous definitions when we set M = 1. We propose the following definitions: Definition 1 ( M 1): q for all i : M i M n 1 i Definition 2 ( M 1): q for all i : M 1 i M n 1 i i M 1 i i i Definition 3 ( M 1): min q ; q q q q, for all i : M 1 i M n 1 All the definitions above are to be understood excluding the first losing quotient for the party that won the M-th seat. To see how these extensions of the wasted vote concept would work for D Hondt, where the divisor quotients are the integers 1, 2, 3, M, consider the case of M = 4 illustrated in Table 1, using our previous three party example of (70, 18, 12). The potentially relevant top four quotients for party 1 are 70, 35, 23.33, 17.5. The top four quotients for party 2 are 18, 9, 6, 4.5; the top four quotients for party 3 are 12, 6, 4, 3. The four highest quotients overall are 70, 35, 23.33, and 18, so Party 1 gets three seats and Party 2 gets one, and Party 3 has no seats. The party that wins the fourth seat is Party 2. The lowest winning quotient is 18. The first (highest) losing quotient is 17.5; the second losing quotient is 14; the third is 12, the fourth is in common with one another the fact that they each look only at voter s first preferences (Kurrild-Klitgard 2008), rather than taking into account in some fashion full or partial rankings. 11

11.66. However, it is apparent that some of these quotients come from the same party. We claim that what we want, instead, are the highest losing quotients for each party, respectively, i.e., 17.5, 9, and 12, so that there is no more than one quotient per party Moreover, we also need to exclude the first losing quotient for the party, Party 2, that won the fourth seat. Thus, there are only going to be two n 1 losing quotients that are relevant to our calculations (just as, for the M = 1 case, there were only two (n 1) relevant losing coefficients). In order, the relevant losing quotients are 17.5 and 12, with a gap between them of 5.5. TABLE 1 ABOUT HERE Definition 1 for the case of M > 1 gives us a value of 29.5 percent wasted votes (lower than our answer of 30 percent for the plurality case). Definition 2 for the case of M > 1 gives us a value of 12 percent (the same as our answer of 12 percent for the plurality case. Finally, since there are only three parties, we get a wasted vote value of 5.5 percent (lower than our answer of 6 percent for the plurality case). Note, however, that lower shares of wasted votes in larger districts may still represent larger numbers of voters. Turning to the definitions of wasted vote that allow for the votes of winners to be wasted, it might appear as if there was not a direct parallel to our earlier definitions, since we now have M winners to contend with, not merely one. But, after some reflection, to preserve a parallel structure, the relevant quotient is q M q M+1, i.e., the difference between the quotient needed to win the last seat and the closest loser to that value. Recall that, for M = 1, this is just the familiar surplus vote. 12

Thus, we can construct Definitions 1* through 3* as shown below simply by adding q M q M+1 to our previous three definitions. Note, however, as before, that when the first losing coefficient, i.e., the q M+1 -th quotient, would be for the same party that won the M-th seat, then we omit that quotient from our list of losing quotients, so as to assure that we are always looking at parties that hope to gain a seat. Definition 1* ( M 1): q q q for all i : M i M n 1 M M 1 i Definition 2* ( M 1): q q q for all i : M 1 i M n 1 M M 1 i M M 1 i M 1 i i i Definition 3* ( M 1): q q min q ; q q q q, for all i : M 1 i M n 1 Again, all the definitions are to be understood excluding the first losing quotient for the party that won the M-th seat. In our example, q M q M+1 = 18 17.5 = 0.5 (in general we must be careful to use the first losing quotient for a party that is not the same as the party that is the last winner, but this problem does not arise in this example). Definition 1* for the case of M > 1 gives us a value of 30 percent wasted votes (much lower than our answer of 82 percent for the plurality case). Definition 2* for the case of M > 1 gives us a value of 12.5 percent (much lower than our answer of 64 percent for the plurality case. Finally, for Definition 3* for the case of M > 1, we get a wasted vote value of 6 percent (much lower than our answer of 58 percent for the plurality case). Note that, once again we get six distinct values for our six measures of wasted vote, now ranging between 5.5 percent and 30 percent. Note, too, that for the starred definitions we now get much bigger differences between the M = 1 and M =3 cases, because 13

q 3 q 4 is larger than for the M = 1 case. But this need not hold for all values of M or all vote distributions, e.g., when x 1 x 2 for the M = 1 case is near to or at zero. 3. Analytical and Empirical Results 3.1. Analytical results about the relative magnitudes of the various measures and how they vary with M First, for a fixed distribution of party vote shares, for a plurality single seat contest, it is clear that Definitions 1, 2, and 3 should each always give a lower wasted vote than their corresponding starred versions. Second, for a fixed distribution of party vote shares, for a plurality single seat contest, Definition 1 will always give a higher wasted vote than Definitions 2 or 3, and Definition 2 would always give an equal or higher wasted vote than Definition 3; and, similarly, for their starred equivalents. To see why the claim that the three indexes are ordered 1 > 2 3 is true, we need merely restate the formulas in a directly comparable form: i 2 Definition 1: x, for all i 1 x x, for all i 2 Definition 2: x, for all i 2 i i 2 i i i Definition 3: min x ; x x x x,for all i 2 i Clearly, Definition 2 gives a value that is less than what we obtain under Definition 1, while Definition 3 gives a value that is less than that for Definition 2 unless Σx i for all i > 2 is less 14

than Σ(x 2 x i )(x i /Σx i )), in which case the two definitions give the same value, namely Σx i for all i > 2. Since the corresponding starred definitions only add a constant, the same results go through for the starred propositions. Similarly, analogous propositions go through for the M > 1 case, for a fixed M and a fixed distribution of voter preferences. In the example we have given, for a fixed M (M = 4) and a fixed distribution of party vote shares, we found that, Definitions 1 and 1*, 2 and 2*, and 3 and 3* each gave either the same or a higher wasted vote for the case of M = 1 than for the case of M > 1 under d Hondt. However, in general, things are not quite so simple. Examining these formula we find that, while in general, we might expect wasted votes to go down with district magnitude, M, the pattern need not be monotonic, since changes with M depends upon the relative magnitude not merely of sums but also of differences, e.g., the difference between q M q M+1 and x 1 x 2, a difference that need not be a monotonic function of M. In the next section we examine the magnitudes of these differences for some hypothetical polar cases, e.g., a lopsided election, a (near) Duvergerian equilibrium, and a non- Duvergerian equilibrium. These cases have been chosen to indicate the range of variation in outcomes we might expect in actual data. When we look at these projections, we will observe both a long term negative trend with M and some non-monotonicities 15

3.2. Empirical results about the relative magnitudes of the various measures and how they vary with M In Figure 1 we show results for each of our three (non-starred) definitions. Vote distributions are fixed in four scenarios: a near Duvergerian equilibrium (49, 48, 2, 1), a lopsided election (70, 20, 6, 4), a non-duvergerian equilibrium (34, 33, 32, 1), and the running example from Table 1 (70, 18, 12). In Figure 2 we show results for each of our three starred measures. FIGURE 1 ABOUT HERE FIGURE 2 ABOUT HERE In Figure 1, the wasted vote measures according to Definitions 1 and 2 exhibit monotonic patterns across all the scenarios as expected. Moreover, even the third definition, which does not have to be monotonic in M is roughly monotonic, with the exception of the non- Duvergerian equilibrium case. However, the same doesn t generally hold for the starred definitions illustrated in Figure 2. While wasted vote values according to Definition 1* are still monotonic in M for all the scenarios, both Definitions 2* and 3* show nonmonotonicities in all cases. 4. Discussion What we have done in this essay is to show how to provide a unified perspective on the concept of wasted vote, with measures that apply to both the plurality and the list PR case and which vary with M. Our measures draw on insights from a range of scholars from Henry Droop to Gary Cox. For a fixed distribution of party vote shares we have developed analytic 16

results about the relative magnitudes of these six measures and about how they can be expected to change with M. We have also provided illustrative results for M = 1, 10 for three polar types of elections: a lopsided election, a Duvergerian equilibrium and a non- Duvergerian equilibrium. There is, however, one important caveat we wish to alert the reader to in thinking about the concept of wasted vote, namely that we must be careful when we are comparing (or aggregating) wasted votes across districts. As Grofman (2001) points out, if we are interested in the actual number of voters who must change their mind to affect election outcomes, we need to take population differences into account. Within any given country, an M seat district can be expected to have roughly M times the population of a single seat district. Thus, if we are looking at a measure of wasted votes aggregated across districts of different sizes then we must normalize by weighting each district s contribution to the overall amount of wasted votes the legislative elections by M / S, where S is the size of the legislature ( = M). In this fashion, we can create a measure which is comparable across legislatures. 17

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TABLE 1. Distribution of seats according to the d Hondt formula of seat allocation in a fictitious district with M = 4. Ranks in parentheses indicate the order of seats won. Numbers in bold represent the parties highest losing quotients. Divisor Party 1 Party 2 Party 3 1 70.0 (1) 18.0 (4) 12.0 2 35.0 (2) 9.0 6.0 3 23.3 (3) 6.0 4.0 4 17.5 4.5 3.0 21

Wasted votes (%) Wasted votes (%) Wasted votes (%) Wasted votes (%) FIGURE 1. Wasted votes versus district magnitudes in 4 scenarios (vote distributions in parentheses): a near Duvergerian equilibrium, a lopsided election, a near Non-Duvergerian equilibrium, and the running example used throughout the text (see Table 1). Lines represent alternative definitions of wasted votes: Definition 1 (solid line), Definition 2 (dashed line), and Definition 3 (dotted line). Duvergerian equilibrium (49-48-2-1) Lopsided election (70-20-6-4) 70 60 50 40 30 20 10 0 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 District magnitude 1 2 3 4 5 6 7 8 9 10 District magnitude Non-Duvergerian eq. (34-33-32-1) Running example (70-18-12) 70 60 50 40 30 20 10 0 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 District magnitude 1 2 3 4 5 6 7 8 9 10 District magnitude 22

Wasted votes (%) Wasted votes (%) Wasted votes (%) Wasted votes (%) FIGURE 2. Wasted votes versus district magnitudes in 4 scenarios (vote distributions in parentheses): a near Duvergerian equilibrium, a lopsided election, a near Non-Duvergerian equilibrium, and the running example used throughout the text (see Table 1). Lines represent alternative (starred) definitions of wasted votes: Definition 1* (solid line), Definition 2* (dashed line), and Definition 3* (dotted line). Duvergerian equilibrium (49-48-2-1) Lopsided election (70-20-6-4) 70 60 50 40 30 20 10 0 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 District magnitude 1 2 3 4 5 6 7 8 9 10 District magnitude Non-Duvergerian eq. (34-33-32-1) Running example (70-18-12) 70 60 50 40 30 20 10 0 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 District magnitude 1 2 3 4 5 6 7 8 9 10 District magnitude 23