Arguments for Discrete Exponential Score Systems

My guess is that this isn’t a new idea, but there is a voting system concept I’ve been considering that is at the very least interesting to me. The way it operates is that a real number B in the interval (0,1) is chosen, and then voters are allowed to score each candidate independently with any one of the discrete values B^k for integer k from 0 to N, where no score indicated is by default B^0=1. In its most simplistic form, each candidate has all of their scores summed, and the candidate with the smallest sum of scores is declared the winner. Basically it’s just like simple score, but the values of the scores are weighted exponentially, and rather than the highest sum winning, it is instead the lowest sum.

This structure of scores has some interesting properties, in my opinion. Often people will point to min-maxing as a manipulation strategy that reduces the quality of score voting. However, if exponential scoring rather than standard scoring is paired with something like a STAR runoff, I have the present belief that both min-maxing and majoritarianism should in principle be significantly diminished.

Consider the simplistic version without a runoff. As in the normal score system, voters will do best to score their honest favorite candidate most favorably—in this case, giving them a score of B^N. At the same time, they would do best to score their honest least favorite candidate most poorly—in this case, giving them a score of B^0=1.

At this point, the most favorable score acts as a “source,” and the least favorable score acts as a “sink.” There is a strategic incentive for the voter to score all of the rest of the candidates poorly, so as to boost their most favored candidate’s chances of winning the election, or at the very least, there is incentive to categorize the candidates into “winners” and “losers,” and then to score the winners very favorably and the losers very poorly, which unchecked will naturally result in an approval voting process.

However, consider a collection of winners who the voter plans initially to score most favorably. In reality, it is unlikely that the voter is totally indifferent to these candidates, and probably there is a single candidate they most prefer. Hence, since the difference between the most favorable score value and the second-most favorable score value is very small compared to the values of most other scores (since they are exponentially decayed), the voter is much more likely to indulge in expressing this preference, however slight. But again, even the difference between the second place and third place slots are slight in comparison to the lower scores, which are exponentially larger. So there is an extended incentive to spread out their first several top choices in an honest fashion!

On the other end, suppose the voter wants to score all of the “losers” as poorly as possible. Again, it is unlikely that they are totally indifferent between these loser candidates. Probably there is one candidate that they would be least happy with winning the election. Since there is such a large gap between the last place and the second-to-last place slots, and since both slots are still sufficiently large compared to the upper tier score values, the voter can feel secure in indicating this preference in an effort to prevent their least-favorite candidate from winning over any of the others, and they can do this without really compromising the status of their top choices. The same goes for any other sufficiently strong preferences over candidates at the bottom.

So in an exponential score system, I believe there is more strategic contention toward the middle range scores than at either end. But in the middle, there is also less wiggle room for manipulation. I think voters would generally compromise their preferences starting from the top and the bottom, and those should in principle meet somewhere naturally in the middle. Compound these natural incentives and constraints with a runoff that gives even stronger incentives for indicating preferences, and I think such a system could do very well against min-maxing strategies and majoritarianism in the long-run.

Any thoughts are welcome.

I say what I had already said in a similar post:

  • if this is the range of score voting [0,1,2,3,4,5] you say to treat it exponentially as if it were eg. [1,2,4,8,16] or [1,5,9,12,15,16] (the second is better).
    I don’t know how much this can actually reduce the min-maxing, but I think it can give a greater representation of interests to those who vote honestly, without increasing the complexity of the vote (since the range is wider but still has only 5 ratings).
  • more generally, the property of the system you describe is that of favoring winners who may have a lower average but also a lower standard deviation than the candidate who has a higher mean and a higher standard deviation.
    In this context:
    A[5] B[3] C[0] D[0] E[0]
    A[5] B[3] C[0] D[0] E[0]
    A[0] B[2] C[5] D[0] E[0]
    A[0] B[1] C[2] D[5] E[0]
    A[0] B[0] C[2] D[0] E[5]
    Majority: C
    Utilitarian: A
    “Proportional” or “Exponential”: B.
    The problem with methods that win B is that they depend on how much weight you give to the standard deviation with respect to the average.
    At this link you can see which methods win A,B or C in this example (in the site you found also the system below, called “Avg-Deviation”).

A general (and simple) way to describe what I call the proportional winner is this:
X * avg(B) - Y * sd(B) = B total score
avg(B) is the average of the ratings received by candidate B.
sd(B) is the standard deviation of the ratings of B.
X and Y are constant; by default they are (for simplicity): X = Y = 1.
The candidate with the highest total score wins.

I still haven’t figured out what the best value for X and Y is.
Actually, I think there isn’t because it depends on how important you want to give the standard deviation, which is not something objective.
If you aim for maximum simplicity, then in the end I would say:
avg(B) - sd(B) = B total score

Calling it “proportional winner” or “exponential winner” seems wrong to me; a more suitable word would be needed.

That is true, you can transform the system into an equivalent one where the metric is given by something like

SUM{k from 0 to N}(1+B^N-B^k)p(k)

and instead choose the largest metric. You can also approximate the weights to integers and such. I’m not sure how effective it would be either, but I have the feeling that it could do a pretty good job. For the reasoning I gave above I think a fully “strategic” vote would not necessarily vary much from an “honest” or “naive” vote, so it simultaneously tackles majoritarianism and manipulation at least to some degree. Especially with a runoff I don’t think min-maxing would be a very effective strategy.

I want to point out that not only is the range wider, the fact that the score bins are discrete and “self similar” should I believe play a significant role in decision making. Also, there are certain “compromising principles” that exist in this kind of system. For example, it is always true that


and so on for many such combinations that would represent a compromise between two voters who score in opposite directions along a political spectrum.

Regarding the mean and standard deviation metric, I agree with you about that to an extent except I wouldn’t necessarily say it is more general. I also don’t know if I would want to tie the system directly to those metrics. It could be fine. Also you don’t need to vary both X and Y—since the metrics are relative and we can be basically certain X is nonzero, you could divide through by X and find a single parametric weight Z=Y/X for the standard deviation. So the metric you propose is as good as


I think that could be a decent metric but I would guess there are better ways to organize which distributions are superior to which others. That’s probably always true. But this covers at least a very broad class of systems like that. You could basically employ a Taylor expansion in two dimensions to really try to get a handle on a very good metric, but then there are many more parameters.

At the same time, it doesn’t necessarily tackle the min-maxing issue. But on a different note, I think that any function of the form


that is linear in its first argument and everywhere decreasing in its second argument should be a decent metric. For example, your metric can be re-expressed in that form as


Which satisfies those properties.

I agree in many things.
I just add that:
the “proportional” or “exponential” methods that actually favor candidates with ratings closer to the average, make the minimization even more effective (since it reduces the mean and can increases the standard deviation). But, for similar reasons, it reduces strategic maximization.
This is why I think that actually, the best thing is to have higher ratings than low, that is, like this: [1,5,9,12,15,16] rather than like this: [1,2,4,8,16] which I think is similar to what you want.

This formula:
avg (B) (1-Z.std (B) / avg (B))
is equivalent to this:
X * avg (B) - Y * sd (B)
X = 1 , Y = y / x , x different from 0.

I prefer my formula because it makes it clearer that we only consider average and standard deviation, with variable weight (determined by X and Y).

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Indeed that is exactly what I want, it’s totally equivalent to the system I am proposing.

Also as to whether minimization is “effective,” it depends on the ends to those means. The purpose of minimization as I see it is to maintain the status of the most preferred candidates on the ballot as compared to the less preferred candidates. But for the same reason that there is a very small difference between the highest and second-highest slots, there is a very small difference in the amount that the status of those highest slots will be compromised by scoring a less preferred candidate as the lowest versus the second-lowest slot, etc. Both slots are “much worse” than the highest positions, so if the incentives to indicate binary preferences are high enough, voters should tend to indicate their preferences even between the candidates among the lowest slots. Does that make sense? Reducing a candidate by a slot has quickly diminishing functional returns, even though it has increasing marginal returns in terms of the score value itself. But the score value doesn’t actually matter—what matters is the function the score value serves.

Also, the standard deviation is not being considered at all, and the reason for that is that it is unclear whether one is considering the exponential values as scores themselves in some linear space, or as weights placed on the “underlying” scores. Do you take the standard deviation of the function of the scores, or of the scores themselves? This is why I don’t want to consider the standard deviation, or the mean for that matter. They are mixed up philosophically with the “underlying space,” i.e. the “actual utility,” which in my opinion isn’t a real thing. If we could measure the “actual utility,” then those measures might make sense, but the scores are just scores. Do you know what I mean?

To elaborate, utility is just an auxiliary construct that is used to predict human behavior. It is defined in terms of functional human behavior, not the other way around. “People will choose what they prefer” says nothing more than “People will choose what they choose,” since the only empirical measure of preference is what people do in fact choose.

Basically what I’m trying to say is I don’t want to consider ghostly auxiliaries like utility at all. I am most concerned with the functions of these voting systems and the results they would produce for the groups that would use them. I think utility is a useful construct for talking about certain ideas, but when push comes to shove I personally just don’t actually believe in the concept.

That’s true, that’s what I meant—they are equal. I’m just trying to consider alternative abstract forms for a metric. Personally I think a system like that is still going to be very arbitrary anyway. For example, how would you extend your metric to make sense for negative values? Is it true that a distribution with a negative mean is “better” when it has a smaller variance? I’m not so sure. So then are we only looking at positive values? In that case, do you think the metric should change? Should we also consider the skew or higher moments? I don’t know the answers to any of these questions.

A while ago I suggested metrics of the form




etc. to be used for distributions over positive ranges, depending on a single parameter k. But still I don’t know about those, they also seem quite arbitrary to me. And I don’t think they address manipulation anyway.

I’m sceptical of any system that transforms scores but in the “continuous case” is isomorphic to continuous score. (By which I mean that any ballot in one system can be converted to a ballot in the other system in a way that, if the conversion is applied to all ballots in any election, the winner will not change.) I don’t see what is effectively a change in the levels offered to voters being worth the added complexity.

Do consider that min-maxing can occur in contexts not motivated by strategy, such as online reviews, where the most popular ratings are often 0 and 5.

Yes in the continuous case there is no reason to bother with a transformation. But the scores in this system are discrete, and that will have a huge effect on the ways candidates are scored, because rather than interpolating between scores the voter will be forced to select between discrete alternatives. The decision procedure is totally different between continuous and discrete ranges independently of how they are weighted, let alone when the weights are changed between alternative discrete systems.

Skepticism is good, but what do you think of my arguments that attempt to explain the worth in the case of this specific such transformation? At the same time, there is virtually no change in the complexity from ordinary score to this weighted system, so I’m not sure what complexity you are referring to. I mean, if simple addition is out the window, what are we all talking about here?

In the case of this system, strategy should actually compel voters to resist min-maxing to at least some degree, as far as I can tell, for the reasoning I tried to explain. The functional returns of strategies approaching min-maxing diminish quickly, and probably in most typical situations will fall below the functional returns of indicating pairwise preferences before full min-maxing is achieved. This is observably not the case for unweighted scores, which also fail to have the “compromising properties” I pointed to above.