STRATEGIC EVALUATION IN MAJORITY JUDGMENT

Jack H. Nagel1

Presented at a Program in Celebration of Michel Balinski Stony Brook University July 18, 2012

[

Because I retired at the beginning of this year, people often ask me if I worry about condemning myself to intellectual stagnation and oblivion. They use more tactful words, but that is what they mean. I always answer that I’ll be able to continue research and writing, if that is what I want to do. Not only that, I say, but it is even possible for a retiree to make contributions of astonishing originality and fundamental importance. I then go on to tell them the story of the existence proof for that claim—Michel Balinski. In his seventies, and just in that fraction of his scholarship that I know about, his work on and , Michel has revolutionized thinking about two absolutely basic topics.

First, he invented Fair Majority Voting (Balinski 2008), which seemingly squares the circle of legislative elections. Like other political scientists, I had always taught my students that proportional representation requires multi-member legislative districts, and is thus incompatible with single-member-district (SMD) elections, such as are conducted for national legislatures in the United States, France, Britain, Canada, and India. But Fair Majority Voting is a system of PR based on single-member districts! There is, to be sure, a little intellectual sleight-of-hand behind Michel’s miraculous result. In FMV, seats are simultaneously allocated at two levels—the district and a larger unit (e.g., a state). The latter functions as an implicit multi-member district. Achieving proportionality at that higher level comes at a price—sometimes it may be necessary to seat a candidate who did not win the most votes in his district. That may be a fatal flaw when it comes to selling FMV as a reform. Nevertheless, I am persuaded that Michel’s plan, if it were adopted, would be far superior to the present gerrymandered, largely non- competitive system of elections for the U.S. House of Representatives—and it would not even require repealing the 1987 statute that mandates SMD elections for the House.

Michel’s second and even more remarkable achievement as a septuagenarian, carried out in collaboration with Rida Laraki, is the re-casting of social-choice theory (and much more along the way) in their splendid book, Majority

1 Professor Emeritus of Political Science, University of Pennsylvania, Philadelphia, USA; [email protected].

1 Judgment. That great work introduces a second novel and practical method for reforming elections (and other competitions), the method of majority judgment.

As Michel knows, I hugely admire the book Majority Judgment, and I am intrigued by majority judgment the reform. Indeed I have already recommended its use on several occasions. Nevertheless, in this paper I want to play the role of sympathetic critic, in the spirit of their own quotation (p. 291) from Charles Darwin: “I had, during many years, followed a golden rule, namely that whenever a published fact, a new observation or thought came across me, which was opposed to my general results, to make a memorandum of it without fail and at once…Owing to this habit very few objections were raised against my views which I had not at least noticed and attempted to answer.” I propose to raise a few objections to the method of majority judgment that may have escaped the notice of Michel and Rida (and that did not occur to me when I read their book in manuscript), with the aim of encouraging them to develop persuasive answers.

I will focus on a key claim that Balinski and Laraki make in favor of majority judgment—that it resists strategic voting better than any other voting system. This is not the only virtue of majority judgment, but it is certainly an important argument in favor of the method. My analysis will be confined to elections and other contests that are designed to choose a single winner and in which the voters care a great deal about who wins. I will have in mind mainly large-scale elections. Because the individual’s probability of casting a decisive vote in such contests is extremely low, one may legitimately wonder whether any voter who also cares about expressing an honest evaluation will ever vote strategically. My answer is that many voters do not act as isolated individuals but as members of groups with similar preference profiles. The individual may follow a voter’s categorical imperative: “I should vote as I would want everyone who shares my opinions to vote.” Usually such a norm is evoked and socially reinforced by group leaders who influence and mobilize many voters. Such leaders are often quite capable of strategic calculation and action.

My analysis will lack the rigor and depth of Balinski and Laraki. That is mainly because I am not a mathematician, but a relatively informal argument may be appropriate for understanding ideas likely to influence the behavior of voters who do not have the time or capacity for deeply sophisticated rational calculations.

The paper will proceed as follows: I begin by classifying voting systems according to balloting methods in order to distinguish the types of strategic voting available under different voting systems. Next I briefly discuss Balinski and Laraki’s case for the superior resistance of majority judgment to manipulation by strategic voting. I then present three overlooked influences that may increase voters’ propensity to evaluate candidates strategically in a majority-judgment election. A final section summarizes factors affecting strategic voting in majority judgment and six rival voting systems.

2 Voting Systems Classified by Balloting Methods

Electoral systems have three principal elements: an objective (the output of the election, usually characterized by how many winners are to be chosen), a balloting method (how voters express their judgments about candidates), and a decision rule (how are aggregated to choose the winner or winners). Because my analysis applies only to single-winner contests, the systems to be considered will be defined by balloting methods and decision rules. Those two elements are strongly interactive, so for most purposes their effects must be assessed in combination. However, because the balloting method determines the repertoire of strategic voting options available to the elector, I begin by classifying voting systems according to balloting methods. The analysis is confined to seven methods compared by Balinski and Laraki, although the labels I use differ from theirs in some cases, for reasons to be explained. I classify the balloting methods for these seven systems into three types: binary, preferential, and evaluative.

Binary Voting Systems

In a binary (or categorical) , the voter has just two options with respect to any candidate: The candidate may receive a vote, or not receive a vote. Every vote has an equal value, so these are (1,0) systems. In the familiar single-vote ballot, the elector may cast a vote for one and only one candidate. In contrast, the approval ballot permits the elector discretion to vote for any number of candidates from one to M-1, where M is the number of candidates on the ballot.2 Each of these two variations on the binary ballot may be combined with innumerable decision rules, but we will consider just three.

Single-vote plurality (SVP), better known as first past the post:3 Electors cast single-vote ballots, and the candidate with the most votes wins.

Majority [or] runoff, or two-round system: Electors cast single-vote ballots. If no candidate receives an absolute majority (more than fifty percent of votes), then all candidates except the top two are eliminated. Those two advance to a second round of voting (the runoff), in which electors again cast single-vote ballots. The candidate with the most votes wins, but with just two competitors, the winner will have a majority, if abstentions are ignored.4

2 In strongly collegial elections, an elector might give approval to all M candidates, but we are assuming a competitive situation, in which approving all makes no sense. 3 SVP is also commonly (mis)named “”. Plurality is better understood as a decision rule that can be applied to various balloting methods. 4 Balinski and Laraki call this system “two past the post”, but that label is appropriate only when no candidate wins a majority in the first round.

3 Approval plurality (AP): Electors cast approval ballots, and the candidate with the most votes (a plurality) wins.5

Preferential Voting Systems

In a preferential (rank-order) ballot, the elector assigns numbers from 1 to M to the candidates, in order of preference, with the most-preferred candidate usually marked 1. Two crucial questions in the design of preferential systems are whether tie rankings (weak orderings) are an option, and whether voters are permitted (or even required) to submit a truncated list, as opposed to a complete ordering. In the country that has the most experience with preferential voting, Australia, the prevailing practice is to require a complete strong ordering. Following Balinski and Laraki, I consider two preferential voting systems.

The treats voters’ preference rankings as point scores, and sums them across all candidates. When a score of 1 corresponds to “most preferred”, the candidate with lowest point total wins. Equivalently, if the highest point score (usually M or M-1) goes to the most preferred option, the candidate with the highest point total wins.

The alternative vote (also called preferential majority and instant-runoff voting6) uses the same preferential ballot as the Borda count, but then applies an entirely different decision rule. Rather than provide a score to be summed, as in Borda, the preference rankings determine the order of elimination of candidates and of vote transfers to the remaining candidates. As in the majority runoff, any candidate who receives a majority of first preferences is immediately elected. However, if no candidate wins a majority, the candidate receiving the fewest first- preference votes is eliminated. The votes of that candidate’s supporters are then transferred to the candidates they ranked second. The process of elimination and transfer is iterated until some candidate reaches a majority, adding together original first preferences and lower preferences that are counted because of transfers. The majority runoff and alternative vote have the same goal of electing a candidate with the support of a majority rather than a plurality, and both are elimination systems. However, in the most common runoff system, all candidates except the top two are eliminated simultaneously, whereas eliminations in the alternative vote proceed sequentially.7

5 Approval plurality is usually called “”, but the latter term fosters confusion, because the approval ballot can be combined with many decision rules besides plurality. 6Social-choice theorists often use “” (STV) for the alternative vote, but empirical analysts of electoral systems usually reserve STV for the multiple-winner version of this system, a type of proportional representation. 7 “Instant-runoff voting,” the catchy term used by American reformers to popularize the alternative vote, obscures this difference.

4 Evaluative Voting Systems

An evaluative ballot asks the elector to appraise candidates using a method intended to convey more information than a mere rank-ordering. The two types of evaluative ballots considered here are superficially similar, but differ deeply in underlying assumptions, which are crucial to the decision rules with which they are combined.

Score voting (aka range voting) presents the elector with a set of numerical scores within a defined range (e.g., integers from zero to ten or zero to 100). The scores are interpreted as meaningful indicators of voters’ cardinal utilities. Thus to determine a winner, the scores are summed or averaged, and the candidate with the highest total or average score wins--the distinction matters only if some voters do not evaluate all candidates. (Balinski and Lariki refer to the system as “point-summing”, a term that could apply equally to the Borda count.)

Majority judgment, the method invented by Balinski and Lariki, requires no introduction to this audience. The evaluative categories or grades are usually verbal, but even when numerical, they are not assumed to be an interval scale. Instead they are qualitative (but ordered) judgments based on a common language of evaluation, distilled into the grades to which the voters assign candidates. Thus decision rules based on summing are rejected in favor of procedures based on candidates’ median grades. For a large electorate, Balinski and Laraki propose the “majority-gauge”, a triplet (p, α, q), in which α is the median grade received by the candidate (the “majority-grade”), 8 p is the number or percentage of the competitor’s grades above α, and q is the percentage or number below α. The candidate with the higher α wins. If two candidates have equal majority grades (αi = αj), a sequence of tie-breakers is applied. If p = q for one candidate i but not for j, then j wins if pj > qj and j loses if pj < qj. If p > q for both candidates, then the candidate with the higher p wins. If p < q for both candidates, then the candidate with the smaller q wins (pp. 236-7).

Strategic Voting Options

Each balloting method determines a set of strategic voting options that are logically possible within the constraints of the values the method permits (or requires) the voter to assign. When a balloting method is combined with a decision rule, the set of strategic voting possibilities may be narrowed farther, because some options may be logically possible but seldom or never desirable for a rational voter. For voting systems considered in this paper, three broad types of strategic voting can be distinguished. Each type in turn has an extreme or limiting version.

8 If there is an even number of evaluators (voters), the majority grade is the lower of the two middlemost grades.

5

Re-ordering and decapitation

Although only preferential ballots explicitly asks the voter for a preference ranking of candidates, all the balloting methods establish a preference order. In the case of binary ballots, the order is weak and degenerate, as the ballot divides the candidates into just two classes, one preferred and the other not-preferred. Evaluative ballots may yield a weak or strong preference ordering, depending on the number of permissible scores or grades in relation to the number of candidates and on the elector’s decisions.

Re-ordering of preferences, the most familiar type of strategic voting, is thus logically possible with all three ballot types. Strong re-ordering occurs when there is at least one pair of candidates, Ci and Cj, such that Ci > (ranks above) Cj when the voter casts a sincere (or honest) ballot, but Cj > Ci when the elector votes strategically. For some systems, it also possible to weakly re-order candidates, as when , Ci > Cj in a sincere ballot but Ci = Cj strategically, or Ci = Cj sincerely but Ci > Cj when voting strategically. The extreme form of strong re-ordering is decapitation,9 which occurs when the displaced candidate is the voter’s sincere first choice, C1. When only less-preferred candidates are involved, I will refer to secondary re-ordering.

Truncation and bullet-voting

If a balloting method permits the voter to assign values (other than a non- vote) to more than one candidate, then another strategic option becomes available. This is truncation, which can be understood most easily in the context of the approval ballot. If the elector would sincerely approve of the first k candidates in his preference ranking, but when voting strategically approves only of the first h candidates, h < k, then truncation has occurred. In the extreme form of truncation, bullet-voting (also known as plumping or plunking), h = 1, and the approval ballot functions as if it were a single-vote ballot. In preferential ballots, truncation is logically possible, but the Australian requirement of a complete ranking prohibits it. In evaluative voting systems, a key design decision is whether or not to permit the voter to refrain from assigning values to some candidates, and if such abstention or truncation is permitted, how to incorporate it into the decision rule.

Spreading and polarization

Evaluative voting systems introduce a new form of strategic voting that is not possible with binary or preferential balloting. This is spreading, in which a pair of

9 “Decapitation” and several other terms used in this paper are drawn from Merrill and Nagel 1987.

6 candidates who would be closer together on the evaluative scale in a sincere ballot are placed farther apart for strategic purposes, without changing the preference ordering of candidates. (The same motive might lead to re-ordering in a preferential ballot, but in an evaluative ballot, spreading does not always require re- ordering.) For example, in score voting the voter’s sincere judgment may be that C1 deserves a 9 and C2 a 7, but to help improve C1’s chances of defeating C2, she may give C1 a score of 10. Similarly, she might downgrade her least favorite candidate, Cn, from a sincere 3 to a strategic zero. Similar spreading is possible with the grades used in majority judgment. The extreme form of spreading between two candidates (typically one’s highest- and lowest-ranking choices) is pairwise polarization, in which the favorite is given the highest possible score or grade and the least-favored is given the lowest possible. In complete polarization, all candidates are assigned either the highest or the lowest score or grade. When that happens, the evaluative ballot degenerates to a binary approval ballot. If only one candidate is assigned the highest value, so that not only polarization but also bullet-voting occurs, then an evaluative ballot collapses into a binary single-vote ballot.

Balinski and Laraki’s Analysis of Strategic Manipulation

When evaluators (e.g., voters) care about who wins as well as the grades that competitors receive, Balinski and Laraki grant that, like all other voting systems, majority judgment is not immune to manipulation by strategic voting. They contend, however, that it “resists” such manipulation better than other systems and is therefore “a more honesty-inducing mechanism” (p. 374). The authors support their claim with intuitive arguments, empirical evidence, and formal analyses. All reach the same basic result: Strategic voting is less likely to succeed under majority judgment than under other voting methods. That implies two conclusions: (a) Voters with good information will less often attempt to manipulate than in other systems, because they will less often anticipate success. In this way, majority judgment induces honesty, even among those who would be “dishonest” if they thought it would pay off. (b) Even if –-because they are ignorant or irrational-- voters attempt to manipulate just as often under majority judgment as under other systems, they will less often succeed in changing the winner, so manipulation will not matter so much in the course of using the system over time.

Unfortunately, the last conclusion does not necessarily follow. Let us posit that the vulnerability (V) of a system to manipulation in practice is given by

V = FS, where F = the frequency with which actual voters attempt to manipulate the system by voting strategically, and S = the conditional probability of success given that manipulation is attempted. Balinski and Laraki show convincingly that S is low for majority judgment. They also implicitly assume that F increases with S. That seems a reasonable assumption to the extent that voters are rational and have good

7 information. But what if other factors influence F, the frequency of attempted strategic voting, so that it is not so closely linked to the probability of success? Could there then be some other voting system for which S is indeed higher than for majority judgment, but F is much lower? Must we reassess Balinski and Laraki’s conclusions about the resistance of majority judgment to manipulation, if there are factors that increase the frequency with which strategic evaluation is attempted, even though its success rate is low?

Three Factors Affecting the Propensity to Manipulate

The likelihood of success surely influences the frequency of strategic voting, but I suggest there are three other factors that affect voters’ willingness to cast a strategic ballot. These are a motive to vote strategically even in the absence of information, violation of the “later-no-harm” property, and the psychic ease or difficulty of the type of strategic evaluation in question.

Is There an Incentive to Vote Strategically in the Absence of Information?

Information that may affect strategic choices in voting situations can arise from two sources: (a) polls (whether or not based on scientific samples) that give the elector some idea about the likely overall distribution of votes, and (b) collusion, in which the elector belongs to a group (or cabal) that he believes (with some degree of confidence) will follow a certain strategy. In the absence of either source of information, the elector must decide in a state of ignorance, or complete uncertainty about the likely behavior of other voters.

Before the advent of evaluative ballots, it seemed to me that strategic voting would never occur in the absence of information—so much so that Sam Merrill and I floridly proclaimed, “Like original sin, strategic voting is a consequence of eating from the tree of knowledge.” (Merrill and Nagel 1987, p. 514). A plausible decision rule for a voter choosing under complete uncertainty is minimax regret. Maximum regret in a voting situation occurs if, by voting differently, one could have prevented the election of one’s least favored candidate, Cn, and caused the election of one’s favorite, C1, or brought about a tie between them when Cn would otherwise have won, or broken a tie between them so that C1 wins. In all the binary and preferential systems considered here, casting a sincere ballot is the minimax-regret strategy. In the binary systems, any sincere ballot will include a vote for C1 and no vote for Cn. A voter can do no more to ensure that her own ballot, if it turns out to be critical, will bring about the victory of C1, or at least create a tie between C1 and Cn. Similarly, a sincere preferential ballot will rank C1 first and Cn last. Nothing more can be done to prevent maximum regret.10

10 Of course, the voter would also experience considerable regret if she could have broken a tie between, say, C2 and Cn; but that is less regret than if the tie were

8

In contrast, evaluative ballots do create a minimax-regret incentive to vote strategically. For example, in a majority-judgment election, suppose that the voter’s sincere evaluation of C1 is “very good” and of Cn is “fair”. If their majority gauges turn out to be identical, with α = “very good”, the voter could have brought about the victory of her favorite by awarding C1 an “excellent”. Similarly, she could break a tie at “fair” by down-grading Cn to “poor”. Extended to other conceivable outcomes, this logic might even result in pairwise polarization of the voter’s most- and least-favored candidates. 11 In score voting, the incentive to polarize is stronger. Maximum regret occurs whenever Cn beats C1 and the score total of Cn exceeds that of C1 by more than the sum of “unused” points in the available range—those above the sincere rating of C1 and below the sincere rating of Cn. For example, if a sincere voter rates C1 a 7 and Cn a 3, and Cn defeats C1 by five points, the voter could have reversed the result by giving C1 a 10 and Cn a zero. Thus electors will be strongly tempted to push scores of their most- and least-favored candidates to opposite poles, even in the absence of information about what others will do.

Does the System Violate Later-No-Harm?

Later-no-harm (Woodall 1994) is a test that applies to all methods that permit the assignment of a value (other than a zero or non-vote) to more than one candidate. Thus it is inapplicable to single-vote plurality and the majority runoff but is relevant to the other systems considered in this paper. A system satisfies later- no-harm if it is not possible to hurt any higher-ranked candidate by assigning a (sincere) value to a lower-ranked candidate. The criterion comes into play most obviously when polling information reveals that the main threat to a voter’s favorite candidate, C1, comes from C2, the voter’s second choice. Approval voting, the Borda count, and score voting all conspicuously violate later-no-harm. As a result, voters using those systems may strategically truncate their ballots where that is permitted (as in approval plurality) or engage in secondary re-ordering. As is well-known, such temptation often arises in the Borda system. To take an academic example (drawn from bitter experience), a professor on a search committee may sincerely judge C2 the second-best candidate, but strategically rank him last in an attempt to prevent him from defeating her favorite, C1. The same logic applies to score voting. Majority judgment can also violate later-no-harm, but—given advance information-- will do so less frequently, and offers less temptation to extreme re-ordering, as opposed to moderate spreading of grades. Of all the multiple-value systems

between C1 and Cn, and in the absence of information, there is no reason to believe that a C2-Cn tie is more likely than a C1-Cn tie. 11 Exaggeration of C1’s grade to “excellent” seems more likely than polarization of Cn’s grade to the lower extreme, because it would seem implausible for two candidates to be tied for the lead with α = “poor”. Still, if the voters are in a foul mood, such an outcome might occur!

9 considered here, only the alternative vote satisfies later-no-harm. A voter’s second preference vote is not counted until his first-preference candidate has been eliminated, so sincerely ranking C2 second poses no threat to C1.

Is the Type of Strategic Voting Psychologically Easy or Difficult?

I conjecture that the various forms of strategic voting outlined above differ in their psychic cost or difficulty, for any voter who cares about who wins but also values casting a sincere ballot. Relying mainly on introspection, but inviting further research, I propose the following hierarchy of difficulty for strategically-motivated departures from sincerity.

Most difficult: Decapitation

Moderately difficult: Secondary re-ordering and truncation

Least difficult: Spreading with weak or no re-ordering

As the only methods where spreading without re-ordering is possible, evaluative systems tempt voters into the kind of strategic evaluation that is probably easiest for them to make and rationalize. In contrast, with single-vote ballots, as in SVP and the runoff, strategic voting can only take the psychically most troubling form, decapitation. In SVP, the voter has incentive to decapitate if polls reveal his favorite is prohibitively behind the two front-runners, in which case it may be prudent to vote strategically for the lesser evil. With the runoff, there is less often incentive to decapitate because if one’s favorite does not advance to the second round, the elector still can choose between the two finalists. However, if C1 has no chance to advance, and the three leaders include one who is repugnant (Cn) and one who is not (say, C2), then the voter may decide to desert C1 for C2 in the first round to help prevent Cn from advancing.12

Noting that at most 30% of French voters cast strategic ballots in 2007, despite the trauma of 2002 when Jean-Marie LePen (Cn for many voters) advanced to the runoff, Balinski and Laraki (pp. 370-1) infer that winning matters less than rendering honest judgments to most voters, even sophisticates: “We know firsthand of many sophisticated voters—economists and social choice theorists— who, in the 2007 French presidential election, preferred Bayrou to Sarkozy, were sure Sarkozy would be in the second round and that in the second round he would defeat Royal and be defeated by Bayrou, and yet voted for Royal in the first round.” (p. 352) It may be, however, that Balinski and Laraki over-generalize from this observation. The French system requires would-be strategic voters to be seriously

12 A second motive for decapitation might occur if voters attempt to exploit the potential non-monotonicity of the two-round system (cf. Balinski and Laraki, p. 96). I discuss this possibility below in connection with the alternative vote.

10 “insincere” by casting no vote at all for their true favorite (e.g., Royal). If the election had been held using majority judgment, would so many voters be inhibited about exercising the easier strategic choice of upgrading Royal (and perhaps also Bayrou) and downgrading Sarkozy?

Where do our other voting systems stand with respect to the psychic difficulty of the strategic-voting opportunities they offer? For approval plurality, if the voter believes there are two front runners, an attractive strategy is given by the Polling Assumption (Brams and Fishburn, 1983, p. 115): Vote for the more preferred of the two leaders, Cf, and any trailing candidates whom one prefers to Cf. This strategy ensures that the voter will not decapitate her favorite, C1. Instead it results in truncation of the approved list, and bullet-voting if C1 is polling first or second.13

The Borda count can induce decapitation if C1 appears out of the running, and one’s preferred front-runner, Cf, might benefit from the one extra point that would result by ranking him first, displacing C1. More frequently, the Borda count encourages the easier strategy of secondary re-ordering, in which the least-liked front runner is pushed to the bottom of the preference ranking, so that C1 has a better chance of winning.

If the alternative vote is conducted under Australian rules, truncation is not permitted; and because the system satisfies later-no-harm, rational voters will not engage in secondary re-ordering. There can be incentives to practice the only remaining strategy, decapitation, under two circumstances.

The first is closely parallel to the situation already discussed with respect to the runoff system—hardly surprising, since the alternative vote is touted as an “instant runoff”. This occurs if one’s first choice, C1 (e.g., Royal), is expected to advance to the final round of counting but will probably lose there to a disliked candidate, Cn (Sarkozy), who could be beaten by a second choice, C2 (Bayrou), if only C2 could gain enough first preferences to survive to the last round. In that case, supporters of Royal might do better to decapitate, giving their first preferences to Bayrou instead of Royal.

13 This strategy works out quite well if C1 is a definite also-ran, so the voter casts an approval vote for Cf as well as C1. However, if there is a non-Duvergerian equilibrium, with three or more competitive leaders, including both C1 and Cf, who seek approval from the same bloc of voters, members of that group confront what I have called the “Burr Dilemma,” which creates great tension between the camps of C1 and Cf and may well result in the degeneration of approval plurality into single- voter plurality. (Nagel 2007)

11 The second, thoroughly Machiavellian, strategy exploits the much-discussed possibility of non-monotonicity under the alternative vote.14 A candidate who has first-preference votes to spare might benefit if some of his supporters switch their first preferences to a candidate he is sure to defeat (a “pushover”) in the one-on-one final round of counting. For example, if the 2007 French election had been conducted under the alternative vote, and the Sarkozy camp feared Bayrou more than Royal, they might have diverted enough of their votes to Royal to ensure that she rather than Bayrou survived to the final stage. An even better pushover would be LePen if he (or now she) were sufficiently competitive, as LePen père was in 2002. The exploitation of non-monotonicity in a large electorate using the alternative vote is a hugely demanding tactic. It requires all of the following: accurate advance information about voters’ full preference orderings, not just their first choices; a high degree of coordination to ensure that just enough (but not too many!) first-preferences are diverted; and a willingness on the part of the strategic voters not just to decapitate their favorite but also to vote for someone who is not even their second choice (as in the usual instances of decapitation) but instead is probably strongly disliked. These requisites make it unlikely that such a manipulation will be attempted and still less likely that it would succeed.

Summary and Discussion

The conclusions of the preceding arguments are summarized in Table 1, which shows how the seven voting systems rate on the four factors that affect the likelihood of strategic voting. In the table, the absence of shading indicates a factor encourages strategic voting, light shading means intermediate resistance, and darker shading highlights an influence more likely to inhibit or discourage strategic voting.

The first column, the likelihood that strategic voting will succeed if attempted, is my distillation of simulations reported by Balinski and Laraki (pp. 346-9). Using data from majority judgment ballots submitted by voters in Orsay during the first round of the 2007 French presidential elections, the authors inferred the ballots that respondents would have cast if they voted sincerely using single-vote plurality, approval plurality (under three different assumptions), Borda, and score voting. They then tested, under various scenarios, the rate of success of manipulators who would have wanted the second-place candidate to defeat the experimental winner. The runoff and alternative vote were not included in the simulations, because both would have advanced the first- and second-place candidates into a final round, where there would be no need for strategic voting. I have somewhat arbitrarily shaded those two systems at an intermediate level in this column—assuming that in a wider range of situations, they would reward strategic voting less often than single-vote plurality, but more often than majority judgment.

14 All elimination systems can be non-monotonic (Smith 1973). A similar stratagem could also occur in the two-round system.

12 I have also put approval plurality in this category, because the Orsay results for approval ballots vary widely depending on the assumptions made about which grades constitute approval.

Table 1. Factors Influencing Resistance to Strategic Voting

Likelihood that Is there an Does the system What type of strategic voting incentive to vote satisfy Later-No- strategic voting is will succeed strategically in Harm? most likely? (Orsay the absence of experiment) information? Single-vote High No N/A Decapitation plurality Majority N/A No N/A Decapitation runoff Approval Ambiguous No No Truncation plurality Borda count Secondary Very high No No Re-ordering Alternative vote N/A No Yes Decapitation

Score voting Pairwise Highest Yes No polarization Majority Lowest Yes No Spreading judgment

Comparing across voting systems in Table 1, the most clear-cut result is that majority judgment dominates score voting with respect to resistance to strategic voting. This conclusion reinforces the many analytic results to the same effect reported by Balinski and Laraki. If one desires the richer information that an evaluative ballot can convey, majority judgment is, I am persuaded, unequivocally superior to score voting.

However, when one takes into account the three considerations introduced in this paper, conclusions about the resistance of majority judgment to strategic voting in comparison to other systems become less unequivocal. None of the binary or preferential systems gives an incentive to vote strategically in the absence of information, but majority judgment does. The form of strategic voting that majority judgment is most likely to encourage—spreading of grades—is, I suspect, easier for voters to indulge in than the truncation or re-ordering (especially decapitation) required by preferential and binary systems. Like approval plurality and the Borda count, majority judgment violates later-no-harm; but there is one option that satisfies that criterion. This is the alternative vote, which happens to be the option that has attracted the strongest support among electoral reformers in the U.S. (despite its rejection in the 2011 British referendum).

13 How important would these new factors turn out to be if majority judgment were put into widespread use? I don’t know. The best test is experience. I would very much like to see majority judgment adopted for continuing use in some settings, as well as additional experimental tests, such as the Orsay ballot. However, when it is proposed as a reform, decision-makers’ anticipation that majority judgment may encourage strategic evaluation in the ways noted here may affect their willingness to advocate or adopt this novel system. I would therefore encourage Michel and Rida, as well as other proponents of majority judgment, to give more attention to the factors I have highlighted; and I wish them success in refuting the concerns I have raised.

References

Balinski, Michel. 2008. “Fair Majority Voting (or How to Eliminate Gerrymandering),” American Mathematical Monthly, 115: 97-113

Balinski, Michel, and Rida Laraki. 2010. Majority Judgment: Measuring, Ranking, and Electing. Cambridge, Mass.: MIT Press.

Brams, Steven J., and Peter Fishburn. 1983. Approval Voting. Boston: Birkhauser.

Merrill, Samuel, and Jack Nagel. 1987. “The Effect of Approval Balloting on Strategic Voting under Alternative Decision Rules.” American Political Science Review. 81:2, (June), 509-24.

Nagel, Jack H. 2007. “The Burr Dilemma in Approval Voting.” Journal of , 69:1 (February), 43-58.

Smith, John H. 1973. “Aggregation of Preferences with Variable Electorate.” Econometrica, 41: 1027-41.

Woodall, D. R. 1994. “Properties of Preferential Election Rules.” Voting Matters, Issue 3 (December). 8-15.

14