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RE: Non-lineair rewards: convergent linear vs fish-size bonus
If I understand your formula (and the blue line that represents it) correctly then this will fail horribly with the first sibil attack.
What stops someone from creating many empty accounts and vote with them? They have a scale of 1 which would give them some influence over the reward pool with zero stake. That is not sustainable.
Unless I understand your proposal wrong, in which case please elaborate and compare your proposed curve with the existing curves in a graphical way :)
The curves are scaling curves. That is, it is a curve representing the effective per vest value of account stake. The current linear reward curve as scaling curve would simply be a straight horizontal line slightly below the convergence of the blue and the orange lines in this representation.
If you would take your 500+ MV and split it up into 1000+ 0.5 MV accounts, those three orders of magnitude for the blue line would result in a decrease in effective value to S^-3 times the current effective value. So if S is set to 1.025 for example, the total effective value of your vesting shares would be reduced to 92.86% of your current value.
If you did the same with the current linear reward, the total effective value of your vesting shares would remain at 100.0% of your current value.
With the proposed convergent linear reward (the orange line) that Steemit Inc is currently speaking out its support for, you can pick a value for S (different type of S) to fit a desired incentive level between two set account sizes, but the problem is doing so doesn't create the same incentive at a wider range of account sizes.
If you look at the orange line, tuned for Orca incentive to make my point, Orca accounts are effectively yet modestly incentified not to break up their account into 100 or more smaller accounts. But looking at whales, there is little to zero incentive for a whale not to break up his stake into 100 or more smaller accounts, so no scaling up. Looking at a dolphin account, the incentive for a dolphin not to break up his stake into a hundred or more accounts is over the top and as a result, new accounts are hit really hard by this curve that is tuned for Orca incentive. This I hope shows the orange line is unfair and lacks scaling.
In contrast, the blue line creates equal incentives for dolphins, Orcas and whales not to break up their stake into 100+ puppet accounts.
Hope this clarifies the curves.
Can you plot your scaling curve on the mvest scale? Basically have x be MVEST and y be $ vote value?
I think that is in the end the metric we all can grasp. And any discussion about it needs to show how it differs in this comparison from other curves and why that is good.
Edit: your current plot compares apples with oranges, as the other curves all apply on the resulting rshares and your curve applies per voter. Which makes a difference, as your x axis is personal vests in your blue line but rshares on the post in the orange line.
No oranges, all apples! Comparing reward scaling functions to reward scaling functions, that is what my blog post is about. Maybe read @trafalgar's post for some background on the orange line. @trafalgar talks in terms of n^2/(1+n) that as scaling function translate to n/(1+cn)
In this plot the X axis is in VEST on a log scale (added the fish type graphics to make it understandable to people who don't think in terms of MVEST)and the Y axis represents . The Y axis can ve seen as $ vote value "per MVEST".
You could multiply each of the equation by some c x V, but doing so will just show slightly curved almost linear lines that truly don't communicate what the "per MVEST" graph communicates. That's the whole problem I am trying to address here. If you use a lineair X axis or plot $ against MVEST or both, you end up obfiscating that whatever S you choose for the orange curve, you either allow orcas and whales to not be incentified to good behaviour, or you end up screwing over new accounts so badly that you basically end up stating that we are full and new accounts are no longer welcome.
But in case you want to play with the scaling equations a bit, here is the code I used to make this graph:
Hope this one helps. Removed the floor function as that one seems to truly confuse you while its sole purpose was to take away confusion for the average user, so if it confuses you it loses its purpose.
The blue line is the current linear reward n -> c.
The brown, pink and gray line are the linear convergent rewards Sn/(1+Sn) -> c/(1+Sn) for different values of S.
The orange, green and purple lines are my nS^log(n)-> S^log(n+c) lines for different values of S.
Again, note that the scale for X is logaritmic, while the Y axis is linear and conveys the reward per unit of influence in order to clearly demonstrate the scaling properties of the reward scaling functions.