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A letter regarding Steven J. Brams’s review of Majority Judgment

Steven J. Brams’s review in the September–October 2011 issue of American Scientist, “Grading Candidates,” is more a defense of a scheme he defends for political elections than a judicious assessment of the new paradigm, theory and method for voting developed in our book Majority Judgment (he accepts majority judgment for ranking contestants with small juries). But, as he confesses, he is “not an unbiased observer, having long been an advocate of another system, approval voting.” This leads him to ignore what we believe are facts:

• voters have more on their minds than merely comparing candidates, or approving of some and disapproving of others, and wish to express it,

• voters care about all the results of an election (the distributions of the votes, who is in second, third, down to last place, the spreads between candidates, and so on) and not merely who is the winner, and

• voters are dissatisfied with election results that do not reflect their true opinions.

The standard system, first-past-the-post (FPTP), permits voters to give one vote to at most one candidate. Approval voting (AV) permits them to give one vote to as many candidates as they wish. In both cases, the order-of-finish is determined by the candidates’ total votes, the winner the one with most votes.

Majority judgment (MJ) permits voters a much fuller expression of opinion. They are asked to answer a question—for example, “To be President of the United States of America, having taken into account all relevant considerations, I judge, in conscience, that this candidate would be”—by assigning to every candidate a grade of either Excellent, Very Good, Good, Acceptable, Poor, or to Reject. With 12 candidates, FPTP offers a voter 13 different possible votes, AV offers 4,096, and MJ offers more than 2 billion (and yet, experience shows, takes no more than a minute or so of the time of voters). A candidate’s majority-grade (M-G) is the grade a majority of voters supports against any other grade. Suppose a candidate’s M-G is Good: then a majority of voters gave the candidate a Good or lower grade (meaning a minority believes she merits a higher grade) and also a majority of voters gave the candidate a Good or higher grade (meaning a minority believes she merits a lower grade). The order-of-finish, called the majority-ranking, is determined by the candidates’ M-Gs (the theory imposes a straightforward tie-breaking rule). The new paradigm—voters evaluate candidates—is that voters judge the merits of candidates in a common language of grades that constitutes an ordinal scale of measurement. This change in point of view—this new model of the problem of voting—leads to fundamental changes in all of the analysis, in particular the role of voters’ utilities, and all of the results.

Although Brams expresses the opinion that MJ is not suited for elections with many voters, his critique revolves about three examples with two candidates, A and B, and but a scanty five voters who assign one of the six grades given above. The commentary reveals allegiance to the centuries-old paradigm of voting theory—voters compare candidates—and ignores the new paradigm, for Brams assumes that “voters would choose Approve only for the candidate to whom they gave the higher grade and whom they therefore presumably prefer.” By that criterion one candidate (call her W) wins in comparisons (4 to 1) and by AV (4 to 1) in each example. Elsewhere (in the preface to the 2010 book Handbook on Approval Voting, edited by J.-F. Laslier and M. R. Sanver) Brams claims the new paradigm—“the idea of judging each and every candidate as acceptable or not”—is “fundamentally different” from either FPTP or “allowing voters to rank candidates.” When Approve has absolute meaning, in our book we call the method approval judgment, because AV becomes MJ with two grades, and it has never before been suggested that Approve has an absolute meaning. Quite the contrary, AV has always been presented and analyzed as an extension of FPTP: voters are asked to “tick” candidates, and the one with the most ticks wins. The new paradigm gives the opposite results: If in the first two examples the judgment Approve means a grade of Good or above, AV makes W the loser. The same is true of the third example when Approve means Very Good or above. AV elects B with two Approvals to A’s one Approval. MJ elects A with a M-G of Good to B’s M-G of Acceptable.

The review’s crowning criticism is this third example, used to show that MJ admits the “no-show paradox” (half of a chapter of our book discusses such examples and explains why the paradox is of no importance in real elections, and close to an entire chapter develops the regrettable logical consequences of insisting on methods that avoid it). The no-show paradox is avoided when, in Brams’s words, “more support helps.” Here the MJ-winner is A with a M-G of Good to B’s Acceptable. Two new voters who both give A an Excellent and B a Very Good arrive, making B the winner with Very Good to A’s Good (because a majority gives B at least Very Good and a majority gives A at most Good).

In Brams’s conception, only the winner matters: Without that assumption, there cannot be a no-show paradox (the two new voters help to increase B’s M-G from Acceptable to Very Good, exactly what they believe it should be). When only the winner matters, a vote helps in AV only if it is “decisive”—that is, it either breaks a tie or creates one. So in particular, if a voter arrives and he judges A above B and A trails by one tick, AV should put A into at least a tie with B if the no-show phenomenon is to be avoided. But it does nothing of the kind in the third example (recall Approve means Very Good or above): A thousand voters who judge A to be Excellent and B to be Very Good could be adjoined to the five, yet B would remain the AV-winner with a score of 1002 to A’s 1001 (whereas A is the easy MJ-winner). The no-show phenomenon cannot occur with AV or MJ when the voter sees a real difference between the candidates (for example, when the two give Very Good to A and Poor to B); it can occur only when the voter sees little difference between the candidates (Excellent and Very Good in the example). In a real political election, however, a single vote is never decisive, so the “paradox” is empty. To this Brams would no doubt retort (correctly) that a bloc of voters with identical votes could be decisive together. True, but then—as with the thousand voters above—AV changes nothing, whereas MJ makes A the winner. With MJ a higher grade necessarily always helps. But then Brams might switch the grounds of the argument and invoke strategy, saying: in fact voters have information concerning likely outcomes (for instance, polls, media commentary) so the two voters or the bloc of voters who believe A is Excellent and B is Very Good would Approve A but not B. With MJ the same is true—such voters could give Excellent to A and Acceptable to B, thus avoiding the no-show paradox when there are two voters—except that giving Acceptable to B is closer to their true evaluation than giving B a 0. Conclusion? As a practical matter, the no-show phenomenon is of no consequence. As a technical matter, AV admits it in spades. As a practical and technical matter, MJ does better than AV.

The review is also misleading in its report on experimental results that assess various methods’ bias in favor of centrists and their manipulability. FPTP is grossly biased against centrists and highly manipulable. When Approve means a grade of at least Very Good, AV is hugely biased against centrists and highly manipulable. When it means a grade of at least Good, AV is hugely biased in favor of centrists but resists manipulation. Range voting and Borda are grossly biased for centrists and are highly manipulable. MJ is the most even-handed with regard to a left-right spectrum and consistently best resists manipulation. As William Poundstone said in his 2008 book, Gaming the Vote, “We want a system that doesn’t automatically exclude [centrist] candidates from winning. We also want a system that doesn’t make it easy for any goof who calls himself a moderate to win.”

Brams contends that in political elections “winning is everything.” That is quite obviously false. In U.S. presidential elections, most states have candidacies beyond Democratic and Republican nominees who have absolutely no chance to win, yet they receive votes, and the same is true in all presidential primaries. In the 2000 Florida presidential election, Ralph Nader received almost 100 thousand votes, yet those who voted for him knew he could never win. Presidential elections occur in other countries too: In France, for example, upward of 10 candidates are voted on, yet often more than half of the votes go to candidates who cannot possibly win. Why? Voters care intensely about not only the winner, but also the order-of-finish, the spread of votes among candidates, as well as every aspect of the statistics of whatever system is used. Indeed, it is entirely rational for a (say) Democratic voter to wish a modest Democratic win rather than a sweeping one. Given the opportunity of voting with majority judgment voters will also care about the candidates’ majority-grades, signals that reveal the electorate’s esteem for each of them.

AV is Brams’s choice. Yet his primary criticism against MJ—the purely academic no-show phenomenon—may be made against AV as well. Moreover, contrary to oft-made claims, practice shows the AV-winner is often of doubtful legitimacy for lack of a majority of Approvals. With MJ, in contrast, a majority always decides on every candidate’s M-G; should the winner’s be Poor or worse there is no acceptable candidate: the electorate demands another slate of candidates. FPTP, AV, and all the systems based on voters ranking candidates give results that are meaningless because voters’ opinions are inadequately expressed: Voters’ ticks carry widely divergent meanings (strong vs. weak support, strategic vs. honest choice, comparison vs. evaluation, as polls and experiments have shown) as do voters’ rankings (for example, one voter’s 2nd-ranked candidate may be highly regarded, another’s 2nd-ranked lowly regarded). These systems all make affirmations tantamount to: 1 inch + 1 foot + 1 yard + 1 mile = 4. Moreover, they are wide open to strategic manipulability on the part of candidates and voters, and they do not elicit honest expressions of opinion, so the results they give are far from accurate expressions of the electorate’s will. Majority judgment was developed precisely to overcome these defects in addition to overcoming Arrow’s and other impossibilities.

Michel Balinski and Rida Laraki
École Polytechnique

Reviewer Steven J. Brams responds:

In their letter, Balinski and Laraki do not mention several positive comments I made about their book, including that it changes the terms of debate about voting systems from ranking to grading candidates. Also, contrary to the impression that they leave, approval voting is fundamentally a grading system—in effect, leaving to the voter the question of where to draw the line between acceptable and unacceptable candidates—whereas majority judgment and range voting give voters more latitude. I believe there are circumstances in which more fine-grained judgments are indeed desirable (for example, small groups evaluating candidates according to shared standards)—and said so in my review—but in many public elections, wherein appeals are highly partisan and there is not a common language, this will not be the case.

To be sure, if there were a standard whereby only a “good,” “very good,” or “excellent” candidate were acceptable, and voters agreed on this standard, Balinski and Laraki would have a strong case. Unfortunately, most public elections come down to two-candidate contests, in which many voters end up choosing between the lesser of two evils (97 percent of voters chose between George W. Bush and Al Gore in the infamous 2000 U.S. presidential election, in which surveys indicate there was a good deal of dissatisfaction with both candidates).

As for my “crowning criticism” that majority judgment is vulnerable to the no-show paradox, the authors fail to distinguish how this property is qualitatively different from nonmonotonicity (voters can hurt a candidate by grading him or her higher). In their book, Balinski and Laraki say, “Monotonicity is, we believe, a very important property; any method that is not monotonic should be disqualified.” I believe the onus remains on the authors to show why the no-show paradox, which they call a “purely academic” phenomenon in their letter, is acceptable, whereas nonmonotonicity is not, because both properties tap the same underlying phenomenon—more support hurts rather than helps.

In my view, both nonmonotonicity and the no-show paradox are manifestly undesirable—approval and range voting are vulnerable to neither—except possibly in the kinds of small groups alluded to above, wherein the robustness of the median against manipulation by single voters may be more important. Incidentally, nonmonotonicity and the no-show paradox actually occurred in recent public elections in Burlington, Vermont, and Aspen, Colorado, which each tried out single transferable vote (a system that is vulnerable to both problems) and then promptly repealed it on this basis, so these problems cannot be dismissed as purely theoretical phenomena, of interest only to academics.

Steven J. Brams
New York University
New York City, New York
United States

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