User:GT/What is your RfA vote worth?

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[edit] What is your RfA vote worth?

Quick reference table:
Support % Value of oppose vote relative to a support vote
10 0.11
20 0.25
30 0.43
40 0.67
50 1.00
55 1.22
60 1.50
65 1.86
66.67 2.00
70 2.33
75 3.00
80 4.00
85 5.67
90 9.00
95 19.0

[edit] Introduction

On Werdna648's 2nd RfA it was asserted that an oppose vote is worth five times as much as a support vote. To a degree that may be generally true, especially as future voters may be more skeptical of a candidate when he has more opposition. Here, however, I examine in a purely mathematical sense the impact of a single vote and the relative impact that vote can have on an RfA depending on whether the voter chose to support or oppose.

[edit] Math

Where,

s is the number of supporters
ɸ is the number of opposers

V(s,\phi) = \frac{s}{s + \phi} (the fraction of support, the measuring stick of an RfA)
\frac{\partial V}{\partial \phi} = \frac{-s}{(s+\phi)^2} = V_\phi (the effect of an additional oppose vote on the fraction)
\frac{\partial V}{\partial s} = \frac{\phi}{(s+\phi)^2} = V_s(the effect of an additional support vote on the fraction)
\frac{V_\phi}{V_s} = \frac{-s}{\phi} (the relative weight of an oppose vote to a support vote)

[edit] Example

If s=30 and ɸ=15,
V(s,\phi) = \frac{30}{30 + 15} = 0.66667
V_\phi = \frac{-30}{(30+15)^2} = -0.014815
V_s = \frac{15}{(30+15)^2} = 0.0074074
\frac{V_\phi}{V_s} = \frac{-0.014815}{0.0074074} = \frac{-30}{15} = -2

This means that when s=30 and ɸ=15, an oppose vote is twice as weighty as a support vote (the negative sign means that the oppose vote affects the fraction in the opposite direction that a support vote does).

Note: If you actually do calculate Vs and Vφ they are not exactly equal to what ends up being the change in support after the vote. The reason for this is that the V function as set up here is continuous for all values of s and ɸ, whereas in reality they can only be integers (since there is no such thing as 0.2 support votes -- at least, I don't think "weak votes are counted that way). For the purposes of this page, this discrepancy is not meaningful.

[edit] Conclusion / Analysis

It turns out that the value of an oppose vote compared to a support vote is very easy to calculate on the fly -- just divide the number of supports by the number of opposes. In a close vote where s and ɸ are approximately equal, voting either way carries the same weight. If the votes are overwhelmingly support, an oppose vote will be much more weighty than yet another support, and vice versa.

Usually, the vote counts are only worth paying attention to if the voting is in favor of the candidate (more than 50%), but below the 80% threshold where bureaucrats will automatically promote. In this range, the "weight" ratio of oppose to support ends up ranging from 1:1 at 50% support to 4:1 at 80% support. A more complete list of values is given in the table at the top of this page.

It remains to be determined what the "psychological" or "social" impact of an oppose vote is in relation to a support vote. I'm sure it would be possible to find it by running an analysis on past RfA's and observing what the voting trend was after both support and oppose votes. If the percentage of support among the votes that follow support votes tends to be the same as it was before those votes, then we could say that support votes don't tend to add much. Conversely, if the support percentage of votes that follow oppose votes tends to be lower than it was to start, then we could say that oppose votes do have an impact greater than their simple statistical one. However, nobody has conducted such an analysis yet to my knowledge.

Of course, the moral of the story is, vote intelligently and vote honestly. This page concentrates on relative weights of support and oppose votes, but really a single vote does not usually have a tremendous statistical impact on the percent of support. While an oppose vote may under certain circumstances be worth four times as much as a support vote, it's usually the case that the oppose vote would have a tiny impact and the support vote would just have an even tinier impact. Therefore, forget about numbers and statistics and focus on the ideas. Don't use the table at the top of this page to attempt to "gauge" the amount of impact you can have on a RfA, or let it dissuade you from adding a relevant point to the discussion. As with any consensus-reaching measure here at Wikipedia, it works best when everybody brings all the relevant arguments they have to the table, so that the community can go through it and reach a common verdict.

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