# Why the range 0-99?

Why not 0-9? Or 0-100? Or 0-10? Or 0-5? Or -100 to +100? Well, mathematically it does not really matter very much which one you choose: In fact, if continuum real numbers are permitted (which I prefer!), any range is equivalent to any other – if voters rescale their votes to the new range then the same winner will be elected, and the same ordering of the 1st place, 2nd place, 3rd place... winners will occur. So the only reasons to prefer one range choice over another are related to human psychology or to inaccuracy due to "roundoff errors."

And in fact 0-9 might be superior to 0-99 initially (because it is more convenient for use on more-primitive voting machines) even though ultimately, once good-enough voting machines get common enough, we would prefer 0-99 for more accuracy.

Warning – Revised views: See this newer page for a survey of the psychometric and market-research experimental science literature about rating scales, which instead recommends a 10-level scale as apparently "best."

But the present page, which is older and was largely unaware of that literature, recommended 0-99, for the following reasons:

1. In the range-voting test election we conducted in conjunction with the US presidential election of 2004 using voters exiting from polls (see paper #82 here) we used the range 0-100. That turned out to be a bad idea because some people thought the votes should add up to 100 rather than the truth: each vote is arbitrary within the full range and you can give several candidates full-100 scores simultaneously if you want. (100 is automagically pre-associated in some people's minds with "percentages" and the notion they all have to sum to 100. Even though the instructions refute that, about 5% of the people don't bother reading that part of the instructions because it is "obvious" that things sum to 100 – they heard that in grade-school. In our study we pollsters were there to nip that error in the bud, but in a real voting situation we would not be there. )
2. With 0-99 this psychological sum-bugaboo of "100" is avoided, and we also get the nice feature that everything is a 2-digit number and that every 2-digit number is a valid score (unlike, say, if it were 0-97).
3. Usability issues favoring 0-9 & 0-99 over 0-10 & 0-100: The fewer digit slots for 0-99 instead of 0-100, or for 0-9 versus 0-10, means: fewer voter-errors, fewer illegible scores, fewer invalid ballots, less fraud, no opportunity to screw up by entering "407" or "-89" into the three digit slots for an 0-100 range, (with 0-9, every way to enter digits yields a valid vote), easier & less error prone to enter data into hand calculators, easier to build voting machines, more compatible with blind/disabled voters, etc.
4. Although 0-10 (Olympics Actually since Olympics use 0.1 increments, it is really equivalent to 0-100 for integer scores) and 0-100 (Schools) admittedly seem the most familiar to people, we do not believe the transition to 0-9 or 0-99 will cause anybody any difficulty. It has been suggested 0-10 be used – with 0.1 increments for those needing more precision. (The idea of rating somebody 9.9 out of 10 also seems familiar.) However, the "dot" perhaps could cause problems for visually-impaired or math-challenged people.
5. To see what we mean about 0-100 being more conducive to fraud than 0-99, consider some voter who fills in the three slots with "*97" where * denotes a blank. Some fraudster could then invalidate that vote by filling in the blank with a 1 converting it to "197," an illegal vote. To repair that problem we could demand that voters write an "X" for the * to prevent that, but many would not – leading to more problems (are those votes invalid too?). And some voters might fill in the * with a "–" (I often do that to fill in blank spaces on cheques), which could then be interpreted as a minus sign, again invalidating the ballot. Why invite all those problems? It is better to take the advice of the wise old karate master in the movie The Karate Kid. "Best block," he explained, "is not be there."
6. Which numerical rating scales have pro pollsters used for rating politicians?: I searched the ORS's international historical poll database of over 600,000 poll-questions. The following score-ranges were used in questions seeking ratings of politicians:
 Score range Popularity 1-4 1-7 0-100 1-5 1-100 1-10 0-10 ABCDF ABCDEF 445 880 285 2700 64 1062 829 586 3
but, apparently, no others. [Here "popularity" is a very crude estimate based on "hit count" in a separate too-simplistic automated search among both political and non-political polls.] No polls were found using negative numbers. I did not investigate (≤3)-level scales, but tried every reasonable possibility 0-to-X and 1-to-X with 4 or more levels, also tried "negative X to Y" and "minus X to Y." There also were many nonnumerical verbal scales like "negative view, neutral, positive view" and "favorable, unfavorable" most of which were "both-signed," so in that sense negative numbers happened. Also other numerical scales besides those tabulated above were used for rating things other than "overall quality of politicians."
7. What people want – the data: In the 2004 test election, about 25% of our voters employed only 100 and 0 (and/or intentional blanks); about 40% instead employed other multiples of 10 (such as 60) and blanks in their votes; about 30% also employed multiples of 5 (such as 75); and the remaining 5% used other numbers like 1, 2, or 73 somewhere in their votes. We conclude from that that while 25% of real world voters felt that the range 0-1 (i.e. just two levels) would have been enough expressiveness for them (at least in this election); 40% felt they needed the range 0-10, 30% felt they needed the range 0-20, and 5% seemed to want to express very slight or precise preferences and hence wanted the full 2-digit range 0-99.
8. Martine Aubry in France 2011 shows at least 7 score levels really are desirable: A score-voting-style poll, using a 7-level verbal scale (of 1000 random French adults, telephone) by the professional polling agency OpinionWay, was conducted in April 2011 concerning the upcoming (April 2012) presidential election. The frontrunning presidential prospect at that time, Martine Aubry, was found to have the following score-distribution (at left): Note Aubry's distribution has 3 peaks and 3 valleys. Every one of those peaks and valleys is statistically significant with ≥99.5% confidence for each. In other words, the voters in this case were telling us that they wanted at least 7 scale levels. With fewer, this peak and valley structure, which was genuine, would have been obliterated. In contrast Chevenement and Le Pen had less interesting distributions.
9. The Psychometricians speak: According to chapter 14 (on ratings) of Jum C. Nunnally's book Psychometric theory (2nd ed. McGraw-Hill 1978), many studies show that "increasing the number of points in the scale monotonically increases 'reliability' up to at least 20 points"; albeit after 11 points the gain becomes small. Joy Paul Guilford (Psychometric Methods McGraw Hill, 2nd ed, 1954) in chapter 11 ("rating scales") after examining numerous studies, some using only 2 points on the scale, others 21, others in between, concludes that a 7-point scale is "usually lower than optimal" and "it may pay in some favorable situations to use as many as 25 scale divisions." Guilford also notes (this is the opening sentence of his chapter on ratings)
Of all the psychological measurement methods that depend upon human judgment, rating scale procedures exceed them all for popularity and use.
Here's a list of other claims about ratings made in these two books:
1. Numerical scales are better than non-numeric.
2. In most cases there is a slight advantage in having an even number of points on the scale rather than an odd number.
3. Negative numbers are not recommended.
4. Graphical scales from left to right should be at least 5 inches long and all oriented the same direction (rightward=better).
5. "Good raters are not necessarily self-consistent nor are self-consistent raters necessarily good."
6. Women tend to give lower ratings than men.
10. The distinctness issue: Some voters want to make all of (or many of) the candidates have different scores. Of course, they are allowed, in range voting, to award equal scores, but some voters may fail to realize that, or just want always to indicate a preference by making the scores be unequal. For voters who want to do this, having only a small range like 0-9 or 0-10 as allowed scores would either be inadequate or at least would force a lot of distortion in those scores. With 0-99 or (especially) 0-999: problem solved.
11. Tied elections: The larger the score-range, the less likelihood of a tied election.
12. The Olympics uses 0-100 range voting. (Well, 0-10 with stepsize=0.1, which is equivalent to 0-100; and the Olympics also currently eliminates the two outlier judges from each count.) Also, 0-100 scores are common in academia. We submit to you that there is a reason they chose 0-100 and not 0-10, i.e. that extra precision was important to make the Olympics work well. Indeed the earliest Olympics used smaller ranges like 0-20 and the fact those small ranges later were abandoned evidently indicates they were felt insufficient.
13. Computer simulations of 10-candidate elections show that the range 0-10 is not large enough, in the sense that in 4-to-10% of the test elections, the "roundoff error" inherent in forcing range voters to use single digit range voting (range only 0-9) causes the election winner to differ. [Later note: Other people have done similar computer simulations independently, some of them in published scientific papers, and reached similar conclusions.] While we admit these computer simulations (with random numbers as votes!) were somewhat artificial, this seems a good reason to worry that 1-digit range voting is significantly worse quality than it needs to be – so we recommend going to 2-digit range voting. Still, single-digit range voting (range 0-9) is a good first step, and even "approval voting" (which is range with the smallest possible integer range 0-1; you either "approve" or "disapprove" of each candidate) is a good first step.
14. Starting at 0 is better than starting at +1, or any other nonzero value. Because: Different ranges like 0-9 and 0-99 can be made compatible by rescaling by multiplying (by 11 in this case). But if the ranges started at any nonzero value, then a linear transformation more complicated than a multiplication would be needed. That would be bad. Another reason: A common usage of numbers is that number 1 is best, #2 is second best, etc. But in range voting, the lowest score is worst. Confusion about that would be very bad! We avoid it by starting at zero. Still another reason: starting at zero better enables a possible future switch to the "reweighted range voting" proportional representation system.
15. Negative numbers such as in the range -100 to +100 would be a bad idea. That would lead to more voter confusion. (Imagine if some voter went for -100 instead of +100 by accident! Not good.) With partly-negative ranges, there is more chance for mis-entered and illegible data, and more likely to lead to voters wasting half their votes by only using the positive end of the scale, and harder on blind/disabled, and less familiar; and there seem to be no benefits compensating for these disadvantages. (Here's a strange psychological effect: Range voters like to award the score 0. It is the most popular score. If we make the range -10 to +10, however, then do range voters now like to award "-10"? No! Zero is still the most popular score.)
16. A really stupid usability point for 0-99: Brian Olson had a -10 to +10 scale on his computerized internet voting system betterpolls.com for a while and reported: "someone didn't see the minus sign and asked how they vote on a 10 to 10 scale. Is 10 good or is 10 bad?"
17. Robert J. Richard contends that ranges without an exact midpoint are better:
"Another reason for this is based on experience in public opinion research. Scales with an exact midpoint (0 to 10) encourage people to pick the midpoint to avoid committing themselves. Scales with an even number of choices (0 to 9, or 1 to 10), i.e. without midpoints, force people come down on one side or the other, even if only slightly. Back when I was involved in questionnaire design, researchers had pretty much settled on the format, "On a scale from 1 to 10, how would you rate X?"
18. Michael Poole contends that 0-99 may cause voters to "put more thought into scores" than 0-100.
19. Honeybees employ a large range in their elections. It is not known how great honeybee-voting-precision is, but it is safe to say they use at least a 10-point scale and not more than a 100,000-point scale. (Honeybees have precise built-in clocks that they use to do sun-based navigation; they describe directions to each other relative to the sun and automatically mentally update directions as the sun moves across the sky.)
20. Bonobos rate food on a 5-point scale.

### A contrary argument for 0-5 score-range (Ted at Dodecatheon media)

1. You can make the following simple interpretation (as in the example of the exit polling following the French elections):
```5 = Excellent     4 = Very Good     3 = Good
2 = Acceptable    1 = Poor          0 = Reject
```
2. The score is easily convertible to "percentage": just multiply by 20. Yes, any scale can be converted, but multiplying by 1.0101... to convert 0-99 to percentage would probably be more confusing.
3. You can fit 7 bubbles for 0 through 5 plus No Opinion easily on one line of an optical scan ballot, even in 2-column format.
4. 0-5 is already in moderately wide use (number of "stars" for movie and recipe ratings)... although usually that is 1-5, with "zero stars" forbidden.
This scale also sends an implicit message that a candidate winning with 40% or more average score is viewed as "acceptable" by the population. (I don't know whether that's a good or bad thing.)    Unfortunately, 0-to-5 does not offer very fine discrimination.

### A contrary argument for {-1, 0, +1} score-set (Professor Brian W. Goldman)

1. Minimum possible number of options before it reverts back to approval voting, making it the easiest range voting system to implement using existing voting machines.

2. Values can be expressly mapped to words like "Reject", "Neutral", and "Approve" to get rid of the negative value problem, as voters would be hard pressed to mistakenly mark "Reject" where they mean "Approve."

3. Third parties get all the benefits of normal approval voting, and because of the "Neutral" value, third party candidates will likely be less "Reject"ed than the less liked major party. This is especially true in gerrymandered districts where a large number of voters are going to default vote for one major party, against the other, and likely be in the middle about third party candidates. As such this should still allow for the nursery effect.

I admit {-1, 0, +1} doesn't have the same discriminatory power of larger ranges, but it feels easier to implement, with less change from our existing voting system. It achieves many of the goals desired in Range Voting, even if it doesn't have quite the same power.

### Which scale do voters like? Poll-based answers

French voters in an exit poll study (see this) preferred both 3-point scales {-1, 0, +1} and {0, 1, 2} (about equally) over {0, 1} (approval voting). However, they disliked the 21-point scale {0, 1, 2, ..., 19, 20}. The authors of that poll study conjectured (without explicit evidence) that {-2, -1, 0, +1, +2} would be even more popular. Untested were {0, 1, 2, ..., 8, 9} and {0, 1, 2, 3, ..., 98, 99}.

### Other reduced ranges (coarser than 0-to-99)

We do recommend the reduced range 0-9 ("single digit range voting") or even 0-1 ("approval voting") as stopgap measures more suitable for some of today's voting machines. (The cost of building and buying new voting machines is extremely tiny compared to the benefits Range Voting will bring us, but at first of course we want to use today's machines.) In that case, districts using 0-9 would have to rescale their vote totals by multiplication by 11 when used in combination with other more modern districts using the full 0-99 range. Similarly approval-voting districts using the 0-1 range would need to rescale their totals by multiplying by 99. But it is important that these be only stopgap measures, i.e. that the granularity be required to be above some floor which keeps rising with time until at some known point it reaches 99. That is because the net effect would be a somewhat unfair system if different granularities were used in different places – one of the places would probably effectively have less political clout. The clout-unfairness would be pretty minor with a mixture of 0-99 and 0-9 districts, but might be pretty major with a mixture of 0-1 with 0-99 or 0-9 districts, so we recommend avoiding the latter mixture.

About this unfairness:

1. It should be mandated to go away after the end of a specified transition period – all machines would be required to upgrade to 0-99 (or 0-9999, or whatever the maximum was) uniformly by the end of that period.
2. It would be a small unfairness. Certainly less than 5% and probably well below that. This is far below the unfairness levels that happened in 2000 in Duval County Florida due to differing policies in different locations about rejected ballots – the most pro-Gore districts had 20% rejection rate, the most pro-Bush had below 3%. We would prefer that it be mandated that voting machines be assigned randomly to districts. And random means random.
3. The voters in 0-9 districts would be motivated to upgrade to 0-99 because that way they would have more voting options (all the ones they had before, plus more) which presumably gives them more power (or anyhow cannot hurt).
4. Although this unfairness is bad, it would not be as bad as keeping everything at 0-9 forever. (The numbers show that would lead to more damage.)

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