Curves, Did you know?

Back in the day, we had a reward curve that looked like R(r) = (r + c)^2 - c^2 and curation curve C(r) = r / (r + 2c). I wrongly assumed it was just r^2 and r, but the specifics of this got me pondering about what exactly are the properties that the curation curve needs to satisfy.

Right now we have R(r) = r and C(r) = Sqrt(r), which is much simpler form, relatively speaking.

But where is this coming from?

Note that with a post at r rshares, the marginal benefit from voting with h additional rshares to the voter is

(C(r+h) - C(r))/C(r+h) * R(r+h) * K

where K is the ratio between reward pool and recent claims. As more rshares are piled on top, the share becomes even greater.

Let's look at the case where you vote at the tail end, and just focus on your minimum. Note that for h that is significantly less than r, this expression is approximately

h C'(r)/C(r) * R(r) * K

So on a per-rshare basis, the marginal benefit is approximately

(C'(r)/C(r))*R(r)*K

This expression is constant in both sets of curves used previously, and is referred to as the "Ultimate Indifference Principle". But really, another way to see it is that even if you're late to the party, there's still an incentive to vote on what you like. It's just not as great as if you voted earlier before everyone else did. See this post for outline of the curation principles.

Once you accept this principle, it anchors down a specific form for the curation curve.

If (C'/C )*R = a, for example, one can show that

(log C)' = a/R

or

C(r) = exp(integral(a dr/R(r))

which has nice closed forms for various examples of R, which you can just throw into wolfram alpha to crank out, nifty.(And if you carry the exercise of plugging in the formulas for the previous two sets of curves, you will see that they both match this property)

Same applies to 'convergent linear', which appears to have had a preliminary code push. I was curious about it, so was starting to ask around and dig into how the curation curve even works, which is sort of where I am with this post.

There's a series of posts by @theoretical from long ago that have some discussion involving this topic as well as related theory.

May need some edits, as I noticed some images and links are no longer valid. And is very mathematically involved.

After this there's still a fair amount that can be configured and that's something I'm still reading more about to see what the considerations are.

In addition to this indifference property, the other properties motivate the decision to consider a curation curve to begin with. Curation is designed to reward earlier voters for discovery of valuable posts, which is a principle that I believe as essential to a properly functioning vote economy.

What does the change to curation mean if EIP goes through? Well, not much, really. The strategy remains unchanged as far as trying to vote before others. But because of the nonlinearity, it becomes more of a motivating factor to find what will become the valuable posts, and more of a motivation for downvoting those that shouldn't be valuable. EIP gives plenty of reasons to do well on curation.

Some more curved food for thought.

If you take anything from this post it's that there is such a principle at work:

• (Ultimate indifference principle) Even if you are late to a party, if you like the post, you do get something out of voting it! (In today's linear world, 1/8th of your vote value)

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