N. R. Miller
10/18/01
rev. 2/25/03
2nd rev. 10/16/04
STRATEGIC EFFECTS IN VOTING SYSTEMS
AND DUVERGER’S LAW
The Handout on Voting to Elect a Single Candidate makes the point that no voting system for electing a single candidate can be strategyproof if there are more than two candidates in the field. As you would guess, the same is true of voting system to elect several candidates (e.g., in MMDs). Here we present a general way for thinking about, and analyzing the effects of, such strategic calculations in MMDs. We then consider SMDs as a special case. We assume that voters are instrumental — that is, they use their votes to influence the outcome of the election in a way consistent with their preferences and not merely to sincerely express their preferences regarding candidates. Moreover, we assume that their instrumental goal is to influence the outcome of the present election and not, for example, to influence which candidates will enter the field in later elections.
1. Strategic Voting by Individuals
Let us assume that all voters in a district initially intend to cast sincere ballots. If an ordinal ballot is used, a sincere voter ranks the candidates on the ballot exactly in accord with his or her true preference ordering. If a nominal ballot is used, a sincere voter votes for candidate A only if the voter also votes for all candidates he or she truly prefers to A.
Now let us suppose that, at some time before the election, all voters form (on the basis of early pre-election polls, news stories, and other information sources), broadly shared expectations concerning the relative strength of the candidates, i.e., their relative popularity in the electorate and, more particularly, the order in which the candidates will finish in the election (in terms of ordinary votes, approval votes, first preference plus transferred votes, Borda points, etc., depending on the voting system in use).
Suppose this is the commonly perceived ordering, where the n candidates are subscripted in their order of (perceived) electoral strength (remember that the top m candidates are elected):
Candidate Election Prospects
C1
. Leading candidates (very likely to win)
.
.
Cm
Competitive candidates (may win or lose)
Cm+1
.
.
.
Trailing candidates (very likely to lose)
Cn
Neither the prospective margins of victory of the leading candidates (all of whom are expected to win), nor the prospective margins of defeat of the trailing candidates (all of whom are expected to lose), are terribly important. What is important is the relative standing of the competitive candidates, some of whom will win and others of whom will lose. Rather typically (and as shown above) there may be just two competitive candidates Cm and Cm+1 (approximately tied with one another in the ranking) and the question is which one will win a seat and which will lose. (Sometimes, however, there may be three or more competitive candidates clustered in a near tie at the borderline between prospective electoral victory and defeat. On the other hand, there may sometimes be a substantial gap in strength between Cm and Cm+1, so that the outcome of the election is pretty much a foregone conclusion, i.e., all candidates are either leading or trailing.)
To keep things simple, let us focus on the most typical case in which the general expectation is that there are two competitive candidates Cm and Cm+1. This expectation will evidently cause voters to reconsider, in a strategic fashion, how they should vote and, in particular will induce some voters to cast insincere ballots. Voters will be induced to change their voting intentions based on strategic calculation only if both of the following conditions hold: first, the voters must have a clear preference between the two competitive candidates (call them concerned voters) and, second, their sincere ballots must not already (fully) reflect this preference. For example:
(a) Under the Single Non-Transferable Vote (SNTV), a concerned voter who originally intended to cast his or her one vote for a trailing candidate, or for a leading candidate (especially one with a comfortable cushion of support), may be induced to vote instead for his preferred competitive candidate (since a vote for a trailing or leading candidate cannot affect the election outcome and is “wasted,” while a vote for a competitive candidate may affect the outcome of the election).
(b) Under Cumulative Voting (CV), a concerned voter who originally intended to spread his or her votes over a number of candidates or to “plump” them on a trailing or leading (especially one with much support) candidate may be induced to plump them on the preferred competitive candidate.
(c) Under Approval Voting (AV), Generalized Plurality (GP), or Block Vote (BV), a concerned voter who originally intended to vote for both or neither of the competitive candidates may be induced to discriminate between them, voting for the preferred one and not the other.
(d) Under Borda Point Voting (BPV), every voter ranks all candidates and thus already intends
to express a preference between the competitive candidates, but a voter with strong
preference between the two competitive candidates may now be induced to reinforce that
expressed preference by inserting (additional) other (leading and/or trailing) candidates
between the two competitive ones on the ordinal ballot, so as to maximize the point
advantage of the preferred competitive candidate.
(e) Under the Single Transferable Vote (STV), somewhat the same incentives exist as under BPV, though the strategic calculations are even more complex.
In general, such strategic adjustments have three different possible effects on the strength of non-competitive candidates:
(1) Under all systems, each trailing candidate is likely to lose some or all of what little support he originally had (which migrates to one or other of the competitive candidates).
(2) Under all systems except AV, leading candidates (especially those far in the lead) lose at least some of their prior support (which migrates to one or other competitive candidate).
(3) Under BPV and STV, some trailing may gain some support and leading candidates may lose support (as they are inserted between competitive candidates on some ordinal ballots).
Effects (a) and (b) are the more general and powerful effects, resulting from the tendency of voters to redirect their votes to “where the action is,” i.e., to competitive candidates. Effect (c) is the peculiar result of the few systems using ordinal ballots.
Now suppose a new round of pre-election polls (and/or other similar information), now reflecting these strategic adjustments in intended votes, becomes available to voters. To continue to simplify matter, let us suppose that no new information has come to light that reflects on the merits of the candidates, no candidate has committed a major gaffe, and so forth. That is, we suppose that sincere preferences are unchanged since the first poll and that any changes in the standing of the candidates reflect only strategic adjustments in voting intentions. In light of the new polls, voters make can further strategic adjustments in their voting intentions, and so forth through several rounds of polls and adjustments.
We can now observe that strategic effects (b) and (c) are self-limiting. For example, suppose that effect (b) is strong enough actually to threaten a leading (with respect to sincere preferences and original voting intentions) candidate A’s status as leading (with respect to revised voting intentions after strategic adjustments). Given a new round of strategic adjustments, some voters will be induced to redirect their votes back to A, restoring A’s leading status (with respect to voting intentions as well as sincere preferences). Effect (c) will be likewise be self-limiting.
However, effect (a) is self-reinforcing rather than self-limiting: as voters desert a trailing candidate B, B’s trailing status becomes even more pronounced and evident, encouraging further desertions, so that B’s trailing status becomes still more pronounced and evident, and so forth.
Thus, after several rounds of polls and strategic adjustments (at least under the nominal ballot systems — the ordinal ballot systems are more complicated), we can expect an outcome that looks more or less like the following: candidates C1 through Cm+1 receive substantial support in the election and m of them are elected (presumably all but Cm+1 or perhaps Cm), while candidates Cm+2 through Cn receive very little support. Thus we have the following proposition that may be called Generalized Duverger’s Law: an MMD with magnitude m generally results in an election outcome with m+1 “serious” candidates who receive substantial vote support (even though additional candidates are on the ballot and may well have some support with respect to sincere preferences). Beyond this, there is some tendency for the vote support for these “serious” candidates to be more evenly distributed than their support in sincere preferences, because some voters who most prefer very strong candidates feel free to desert them in favor of the relatively preferred competitive candidates (but not to the extent that such candidates fail to be elected).
These conclusions rest on the implicit assumption that the candidates represent unique political positions and, in particular, that no candidates share political party affiliations, i.e., either the candidates are non-partisan independents or each is the candidate of a different political party. However, given partisan elections in MMDs, one or several parties may have enough electoral support that they can realistically expect to elect more than one candidate (depending largely on how many “quotas” their electoral support can fill, as suggested on p. 3 of the handout on Voting to Elect Several Candidates). We now examine strategic maneuvers by candidates and the political parties that may sponsor them
2. Strategic Entry and Exit by Candidates and Parties
We now pick up on the discussion of MMD elections in which candidates are affiliated with parties and voters are likely to vote for candidates largely on the basis of their party affiliations, as discussed on pp. 3-4 of the handout on Voting to Elect Several Candidates.
Recall that a political party that commands the loyalty of some number of voters in an MMD can expect to elect zero, one, or more of its candidates, depending on how many “quotas” of support this amounts to. (Remember that the quota is Q ≈ N/(m+1).) On this basis we can distinguish among three types of parties contesting an MMD election:
(a) major parties command the support of a full quota or more of the electorate and can confidently expect to elect at least one of their candidates and perhaps more than one;
(b) marginal parties command the support of a bit under one quota of the electorate and have some chance to elect one candidate but no more than one; and
(c) minor parties command the support of clearly less than one quota and cannot realistically expect to elect any of their candidates (unless there are many minor parties).
However, to realize this expectation parties must make different kinds of strategic calculations, dependent on the type of electoral system, with respect to (i) how many candidates they should nominate and (ii) how they should urge or instruct their supporters to vote.
Under SNTV “small” major parties (supported by little more than one quota of the electorate), as well as marginal and minor parties, clearly should nominate a single candidate and can count on their supporters to vote for this one candidate even without special instructions. (The single candidate of a “small” major party will be elected, the single candidate of a marginal party may or may not be elected, and the single candidate of a minor will almost certainly not be elected.)
However, a “large” major party (supported by clearly more than one quota of the electorate) faces more difficult calculations. For example, consider Party A that commands the support of about 32,000 voters in an MMD with 100,000 voters from which five candidates are to be elected. The quota is about 17,000, so if Party A nominates just one candidate A1 (and all of A’s loyal voters vote for A1), party A will certainly elect this candidate. But perhaps Party A should be more aggressive and nominate and try to elect two candidates A1 and A2. But suppose A1 is better known or more appealing than A2, with the result that most of A’s loyal voters vote for A1 rather than A2. Then A2 probably will fall below fifth place in the ranking of candidates by votes and so not be elected (while A1 is elected by a wastefully large margin). So Party A must try somehow to induce its supporters to split their votes as equally as possible between the two candidates (so as to maximize the votes for the weaker candidate), but then (given the numbers above) there is some risk that both candidates will fall below fifth place and fail to be elected. Certainly Party A, facing the arithmetic described above, would be shooting itself in the foot if it nominated three or more candidates.
Under CV, a large party faces a similar strategic dilemma with respect to deciding upon the most expedient number of candidates to nominate, though it faces a less critical coordination problem in getting its supporter to vote in the most expedient way (since individual voters can be instructed to split their votes among the party’s candidates). Note that, in the event that all votes “plump” their votes, CV is effectively equivalent to LV.
Under AV, Party A wants its supporters to vote for all of its candidates and no candidates of other parties and, under STV (or BPV), Party A can allow its supporter to rank its candidates in any order but needs them to rank all Party A candidates above all other candidates. If Party A can command this kind of loyalty from its supporters, it will not be particularly penalized by nominating “too many” candidates (i.e., more than it can expect to elect) as it would be under LV or CV since, under AV, voters can vote for any number of candidates and, under STV, the votes for losing Party A candidates will generally transfer to other Party A candidates. (Thus under AV and STV, a party can in effect hold a “primary” within the “general election.”) Finally, under BPT (as noted in the Appendix to Voting to Elect a Single Candidate), Party A can actually gain from nominating “too many” candidates, because the latter may be used to build greater Borda point margins over candidates of other parties.
Next note that minor parties have a clear incentive to collaborate (or engage in fusion) by nominating a common candidate so as to (try to) pool their electoral support into a bloc that approaches a quota and may allow them to elect the candidate. Or a marginal party can collaborate with a minor one to gain a full quota of support and be assured of electing least one candidate. Even a major party may be anxious to collaborate with another party in order to move firmly up to the next quota. (For example, if Party A as described three paragraphs above could expand its electoral base from 32,000 to about 34,000 if could nominate two candidates and, even under SNTV and CV, be assured of electing them.) Such collaboration could extend from a temporary expedient in which one candidate and/or party makes a strategic exit from the present election and endorses another candidate and/or party to a full and permanent merger of the parties.
Of course, such collaboration can profitably occur only among partners that are ideologically proximate or, in any case, not entirely opposed in their policy goals. Suppose there are three parties in an MMD with 100,000 voters in which two candidates are to be elected (so the quota is 33,334) by SNTV. Suppose the Left Party is supported by 24,000 voters, the Center Party by 56,000, and the Right Party by 20,000. Only C has a full quota of support and even C is guaranteed to elect only one candidate. In the absence of inter-party collaboration, however, C will elect two candidates (provided C can coordinate the votes of its supports so that they are sufficiently equally divided between the two candidates) and L and R will elect none. L and R could take the second seat away from C if the pooled their support but — given that they represent opposite ideological extremes — it is unlikely that they could agree on a common candidate they would both prefer to a C candidate.
In general, since by Generalized Duverger’s Law we expect there to be just m+1 serious candidates, we expect there to be no more than m+1 “serious” parties under candidate-oriented systems. Indeed, we probably expect fewer than m+1 parties if m is at all large, since several parties are likely to elect more than one candidate. At the same time, we would expect no fewer than two serious parties, because there should always be a contest for the mth seat in the district.
Under party-list PR, strategic incentives for parties are greatly reduced — though not eliminated — because list PR allows very large magnitude districts (even national districts), which in turn make the quota very small, with the result that many parties — even very small ones — can elect at least a few candidates on their own, with the result that almost all parties with any degree of support in the electorate are “major” parties. Moreover, the seat-winning capacities of these major parties depends on their electoral support in an “almost continuous” fashion, instead of in the conspicuously “stepwise” fashion that results when a few large quota thresholds are surpassed. The principal exception to this generalization occurs when a list PR system imposes a substantial threshold requirement for a party to win any seats. For example, Germany has a 5% threshold, so a party with the support of 4.99% percent of the electorate wins zero seats while a party with 5.01% support wins about 40 seats. (Such a requirement also implies that a large party [e.g., the CDU] may want to try to shift some of its electoral support to a small party [e.g., the FDP] that is a prospective coalition partner if that small party is at risk of falling below the threshold.)
3. Duverger’s Law: Strategic Effects in SMDs
Finally, we reconsider the special case of SMDs. SNTV, LV, GP, BV, CV, and list PR all become equivalent to Simply Plurality Voting, which we here assume. With m = 1, the quota is a simple majority and the perceived ordering of candidate strengths takes this form:
Candidate Election Prospects
C1
Competitive candidates (may win or lose)
C2
.
.
.
Trailing candidates (very likely to lose)
Cn
Of course, if there is something like a three-way (or more extensive) tie for first place, more than two candidates are competitive. Conversely, if C1 has a substantial lead over C2, C1 has leading status and C2 through Cn are trailing. With respect to strategic adjustments, effect (2) cannot occur because either there are no leading candidates or, if C1 is leading, there are no competitive candidates (and of course effect (3) is irrelevant because an ordinal ballot is not in use), so only the self-reinforcing strategic effect (a) occurs. Thus we get the original Duverger’s Law: an SMD election generally results in an election with just two “serious” candidates fielded by two “serious” parties (even though additional candidates and parties may well be on the ballot and have some support with respect to sincere preferences).
In the case of SMDs, there is no tendency for electoral support for these “serious” candidates to be more evenly distributed than their support in sincere preferences, because — given that only one candidate can be elected — there is no reason for supporters of the strongest candidate migrate elsewhere. (However, as we shall see in the next handout, electoral competition between two office-oriented or policy-oriented candidates or parties may blunt difference between them and thereby tend to equalize their support in sincere preferences.)
Under special circumstance, there may be “non-Duvergerian equilibria” involving three or more candidates with significant support in the final vote.
(a) C1 may be so far in the lead with respect to sincere preferences (i.e., C1 may be a majority winner) that there are no competitive candidates to whom supporters of trailing candidate may be induced to migrate. For example: C1 supported by 55%, C2 by 35%, C3 10%.
(b) C1, C2, and C3 (and perhaps additional candidates) may be essentially tied with one another, so it isn’t clear who should migrate where or who should make a strategic exit. For example, C1 supported by 34%, C2 by 33%, and C3 by 33%.
(c) C1 may have a modest lead over C2 and C3 (and perhaps additional candidates) who are essentially tied with one another, so even though supporters of trailing candidates may have an incentive to migrate to one of C2, C3, etc., it isn’t clear which one they should to migrate to or which one of C2, C3, etc., should make a strategic exit. For example, C1 supported by 35%, C2 by 30%, C3 by 30%, trailing candidates by 5%.
(d) Many supporters of trailing candidates may be essentially indifferent between C1 and C2 (perhaps because of strategic convergence) and so have no incentive to migrate to either of them.
(e) The ideological configuration among candidates (e.g., a strong centrist candidate bracketed by somewhat weaker leftist and rightist candidates) may induce equilibrium in the manner described earlier. For example, C supported by 45%, L by 30%, and R by 25%