I never actually crunched the numbers on this.... but since its come up in a couple discussions, i decided to finally do it.
To understand why this works the way it does, you have to understand rshares and Vshares.
Rshares are awarded to a post when you vote on it. The number of Rshares awarded varies linearly with the amount of steem power you have. For example, someone with 10,000SP will award twice as many rshares with their upvote as someone with 5000 SP, and ten times as many rsahres as someone with 1000 SP.
A post's vshares is the square of the rshares. Because vshares are an exponential function of rshares, they vary exponentially, not linearly in relation to SP and rsahres. Vshares and reward vary linearly. For example, if my post has half as many vshares as yours does, it will get half the rewards.
So, for example, if your vote awards worth 10 rshares, and you vote on a brand new post, you bring it from 0-100 rshares. 100 squared is 10000, so your upvote awarded 10,000 vshares. Now, Imagine if a post already had 1000 rshares when you voted on it . That would mean it started with 1 million vshares (1000 times 1000). If you used your 100 rshares to vote on that post it would go from 1000 rshares to 1100 rshares. Its vshares would be 1100 *1100 =1,210,000, or 21 times as much. If you have ever voted on a high paying post and noticed your vote awarding more than you're used to seeing, that is why.
With that in mind, i am going to explain to you the most effective way to use your vote to decrease the disparity in blogging rewards, and give everyone a fair share of posting rewards. Am i saying that this is something you should do? No, i don't care what you do, i don't even care enough to do it myself. However, I hear everyone complaining about the feast for a few but famine for most environment here, and speculating about different solutions to the problem. I don't think they believe me when i tell them that there is a far more effective method built into the system already.
Note that when we talk about making it so that most posts pay better, we are, by definition, talking about decreasing disparity. Because the rewards pool is fixed, the money to pay very low or non-paying posts must come from the very high paying posts which get most of the reward. Even Robin Hood had to rob from the rich to give to the poor.
imgagine that post 1 was voted on by big whale like 
post 2 was voted on a small whale like 
FOr the sake of easy math, lets assume a 10K reward fund for blogging. I think its actually a bit more than this. The reward fund says 53k, but that goes for mining, witnesses, blogging and curation... Also for the sake of easy math, I am making up a baseline for R shares. The actual total number of vshares is a 31 digit number, and all the real numbers are in all very big. This is apparently necessary for the system to work properly, but not really relevant to the example.
| post | SP | rshares | mvshares | total mvshares | mvshares/totalmvshares | payout |
|---|---|---|---|---|---|---|
| 1 | 2.5M | 1M | 1m | 1,013,164 | .987 | 9870 steem |
| 2 | 250k | 100k | 10k | 1013164 | .00987 | 98.7 steem |
| 3 | 125k | 50k | 2,500 | 1013164 | .002467 | 24.67 steem |
| 4 | 25k | 10k | 100 | 1013164 | .000098 | .98 steem |
| 5-14 (ea) | 10k | 4000 | 16 | 1013164 | .000016 | .16 steem |
| 5-14 (tot) | 100k | 40k | 160 | 1013164 | .00016 | 1.6 steem |
| 15-114 (ea) | 3k | 1200 | 1.44 | 1013164 | .0987547 | .019 steem |
| 15-114 (tot) | 300k | 120k | 144 | 1013164 | .0987547 | 1.9 steem |
| 115-1114(ea) | 1k | 400 | .16 | 1013164 | .0987547 | .0016 steem |
| 115-1114(tot) | 1m | 400000 | 160 | 1013164 | .0987547 | 1.6 steem |
Its worth while to note that in this model, the bottom 1100 actually get paid nothing (i think), because their payout is less than 2 cents... i have no idea what happens to the 3.5 steem or so that they earned. I assume it gets distributed to the higher paid bloggers.
Now, imagine that another dan sized whale comes along. He decides that he wants to decrease this terrible disparity in rewards. Just to start out, he would like to use only one vote. He can vote multiple times at decreased power, but his total vote strength must add up to one 100% vote.
He can pick a post he likes from the bottom 1105 and give it a full power vote. This is what the distribution would look like:
| post | SP | rshares | mvshares | total mvshares | mvshares/totalmvshares | payout |
|---|---|---|---|---|---|---|
| 978 | 2.5M | 1M | 1m | 2,013,164 | .4967 | 4967steem |
| 1 | 2.5M | 1M | 1m | 2,013,164 | .4967 | 4967 steem |
| 2 | 250k | 100k | 10,000 | 2013164 | .00496 | 49.6 steem |
| 3 | 125k | 50k | 2,500 | 2013164 | .00124 | 12.4 steem |
| 4 | 25k | 10000 | 100 | 2013164 | .000098 | .49 steem |
| 5-14ea | 10k | 4000 | 16 | 2013164 | .000016 | .08 steem |
| 5-14tot | 100k | 40k | 160 | 2013164 | .00016 | .8 steem |
| 15-114ea | 3k | 1200 | 1.44 | 1013164 | .0987547 | .005 steem |
| 15-114tot | 300k | 120k | 144 | 1013164 | .0987547 | .5 steem |
| 115-1114ea) | 1k | 400 | .16 | 1013164 | .0987547 | .008 steem |
| 115-1114tot | 1m | 400k | 160 | 1013164 | .0987547 | .8 steem |
Poster 978 is happy, but everyone else is sad. Notice how the whale backed posts now earn individually less, but more as a cohort. In fact, it looks like we pushed the 5-14 cohort below the 2 cent threshold.
The reality of distribution on steem is that there are never 1 or 2 posts that take up 99.9 percent of rewards, but maybe a dozen posts that of posts like 1 and 978 that split up the top tier (preventing insane outliers like the post).
Well,he has a million rshares.. what would happen if he gave them out equally to everyone in the bottom 1000? (lets say he buys an extra 111000 to give them out to 5-114 too, because i messed up and did the math that way), this isnt exactly what curie and other delegated curation guilds to. but its similar. And It doesnt work: CHeck out how little difference it makes:
| post | SP | rshares | mvshares | total mvshares | mvshares/totalmvshares | payout |
|---|---|---|---|---|---|---|
| 1 | 2.50M | 1M | 1m | 1,015294 | .9846 | 9846 steem |
| 2 | 250k | 100,000 | 10k | 1,015294 | .00984 | 98.4 steem |
| 3 | 125k | 50000 | 2,500 | 1,015294 | .002467 | 24.62 steem |
| 4 | 25k | 10000 | 100 | 1,015294 | .000098 | .98 steem |
| 5-14 (each) | 11k | 5000 | 25 | 1,015294 | .000024 | .24 steem |
| 5-14 (total) | 110k | 50000 | 250 | 1,015294 | .00024 | 2.4 steem |
| 15-114 (ea) | 5.5k | 2200 | 4.84 | 1,015294 | .000048 | .048 steem |
| 15-114 (tot) | 550k | 220000 | 484 | 1,015294 | .0048 | 4.8 steem |
| 115-1114(ea) | 3.5k | 1400 | 1.96 | 1,015294 | .0987547 | .0196 steem |
| 115-1114(tot) | 3.5m | 1400000 | 1960 | 1,015294 | .0987547 | 19.6 steem |
He has not increased any of the bottom 1111 significantly in real terms, though they were quite high in relative terms (the bottom 1k got almost 10x as much, but its still less than a penny).
Notice that our whale has far less effect on the payouts when splitting his vote up. His first vote, for post 978, distributed about 5000 steem to the post he voted on. When he broke his vote up and spent it all in small pieces, he only ended up distributing 20 or 30 steem to the bottom 1000.
There are many compromise solutions between splitting his vote 1000 ways and spending it all at once, but they all have the same opposing factors working against them. Dividing his vote makes it weaker in terms of the amount of steem it can distribute, but using his vote all in one place doesn't change the payout for enough people to really make a difference. What he really wants to do is give the 5K he was able to award in the first scenario out to the whole community, not just one person, but there's no way to do that.
Or is there?
the following scenario shows our whale downvoting the top paying post at 100%
| post | SP | rshares | mvshares | total mvshares | mvshares/totalmvshares | payout |
|---|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 13064 | .0 | 0 |
| 2 | 250k | 100,000 | 10,000 | 13064 | .00987 | 7654 steem |
| 3 | 125k | 50000 | 2,500 | 13064 | .002467 | 1913 steem |
| 4 | 25k | 10000 | 100 | 13064 | .000098 | 76.54 |
| 5-14 (each) | 10k | 4000 | 16 | 13064 | .000016 | 12.2 |
| 5-14 (total) | 100k | 40000 | 160 | 13064 | .00016 | 122 |
| 15-114 (each) | 3k | 1200 | 1.44 | 13064 | .0987547 | 1.2 |
| 15-114 (total) | 300k | 120000 | 144 | 13064 | .0987547 | 120 |
| 115-1114(each) | 1k | 400 | .16 | 13064 | .0987547 | .122 steem |
| 115-1114(total) | 1m | 400000 | 160 | 13064 | .0987547 | 122 |
Aside from the payouts, also notice that this has made everyone's vote far more powerful. A 25KSP vote that would have awarded around 1 steem as a vote on a new post in the first scenario will now award 76 steem. In total, hes given out 9800 steem.
Of course, as i indicated before, nothing is free. He had to take that steem from somewhere. Its a fixed reward pool. In this case, he took it frompost 1. But any steem he distributes must come from a post, regardless of whether he upvotes or downvotes. For example, when he voted for post 978, he distributed 4967 steem to that post, and nearly all of it came from the same post 1.
If our whale were to flag post 1 at 50%, it would take just as much money away as upvoting post 978 100%, the only difference is that that money would go to everyone, not just that one guy. And that he would spend about 50% of his vote instead of 100%. But we stigmatize doing it that way.
We can actually do better than this for distribution. As we've learned, the effect of your SP is maximized when youre voting on the highest paying posts. This is true of downvotes as well as upvotes. (in most cases, the exception is when youre upvoting on a post that already has an overwhelming share of the reward pool) We start by subtracting post 1 rshares -post 2 rshares.
(p1-p2) , if that is more shares than we have, then we just downvote p1 at 100%
if not, we take the difference between p2 and p3, then multiply by2, we keep going like that until we get to a number thats bigger than our rshares.
(p1-p2)=900000
(p1-p2)+2(p2-p3)
900000+2(50,000)=1M, which is exactly how many rshares we have. Because its exact, we are going to downvote everything down to the level of P3 (50000). we'll downvote post 1 with -950,000 rshares and post 2 with -50,000 rshares. this is what payouts will look like.
| post | SP | rshares | mvshares | total mvshares | mvshares/totalmvshares | payout |
|---|---|---|---|---|---|---|
| 1 | 125k | 50000 | 2,500 | 8064 | .3100 | 3100 |
| 2 | 125k | 50000 | 2,500 | 8064 | .3100 | 3100 |
| 3 | 125k | 50000 | 2,500 | 8064 | .3100 | 3100 |
| 4 | 25k | 10000 | 100 | 8064 | .000098 | 124 |
| 5-14 (each) | 10k | 4000 | 16 | 13064 | .000016 | 19.8 |
| 5-14 (total) | 100k | 40000 | 160 | 13064 | .00016 | 198 |
| 15-114 (each) | 3k | 1200 | 1.44 | 13064 | .0987547 | 1.7 |
| 15-114 (total) | 300k | 120000 | 144 | 13064 | .0987547 | 179 |
| 115-1114(each) | 1k | 400 | .16 | 13064 | .0987547 | .198 steem |
| 115-1114(total) | 1m | 400000 | 160 | 13064 | .0987547 | 198 |
had there been a remainder, we would have split it equally among all of the posts we were flagging. In this scenario, the number of vsahres is as low as we can possibly get it, and as a result, the rewards distributed by each additional vote are also higher. The 25K SP vote that was worth 1 steem on a new post in our original example is now worth about 124.
Also note that this is not only true for whales. If some guy came along with 1000 steem power, he wouldnt be able to have much impact on the system at all, but his impact would be far greater if he used it downvoting overvalued posts than if he used it on upvoting undervalued posts.
Also note that downvotes are not inherently stronger than upvotes. In general, it seems like the the more even the distribution of rewards is, the weaker the downvote is in relation to the upvote. DOwnvotes are stronger than upvotes on steem right now because distribution is so uneven.
That said, this example was deliberately simplified and constructed to demonstrate this fact. One of the reasons we were able to get so much money to give away by downvoting poster 1 is that he already had 99.9% of the reward pool (which doesnt happen in reality). In reality, post 1 will actually be a small cohort of posts that and we would have to devide our downvote power (Even though we could vote more than once, it still wouldnt have the same effect as voting once on a super-mega post like that.)