# The Steem reward system, Part 5: Curation rewards calculus, s = 0 case

in theoretical •  3 years ago  (edited)

This is the fifth post in a series about the Steem reward system. Please read my disclaimer. For previous posts, see my blog.

This post is a mathematically intense post. In order to follow everything in this post, your mathematical toolbox will need to include sharp algebra and calculus skills. Non-mathematically-inclined readers may wish to skip to the conclusion.

I will frequently reference equations and principles discussed earlier in the series. If you have questions about an equation or principle, try reading the previous posts in this series.

### Solving the equation

In the previous post, we left off with the equation:

Readers familiar with calculus should immediately recognize this as a differential equation. Specifically, it is a first-order separable differential equation involving a low-degree rational function, which leads us to suspect that an exact symbolic solution may be possible.

### Solving the equation, s = 0 solution

When s = 0 the solution is:

Let's take a quick look at the graph of this function (with e.g. A = B = 1):

It's clear from this graph that the solution rises quickly early on, but slows as R becomes larger. We can clearly see that the total weight issued, W(R), goes toward B = 1 as W becomes large. Interestingly, B represents the maximum amount of weight the algorithm is ever "allowed" to issue to a post's upvoters, regardless of how many upvoters there are -- it's a "weight ceiling". The fact that the weight does have a ceiling was not one of our starting assumptions; rather it was a consequence of them. I was actually quite astonished when I discovered that the existence of a ceiling actually follows from the principles!

There is an interesting little corner of the graph, which I've outlined with a small box near the origin. Let's blow up the box to a full-sized graph:

So actually W starts off flat, starts to rise at an increasing rate, then (by eyeball, at about R = 0.5) its rate of increase starts to fall. Where the function changes from rising rate of increase to a falling rate of increase is called an inflection point.

To further study W(R), let's use calculus to compute the first and second derivatives:

We can determine the inflection point exactly from this expression. The inflection point occurs when W'' is zero, at R = 1 / (2A). The "eyeball estimate" of R = 0.5 was spot-on!

The monotonic supply schedule principle requires the W curve to have a decreasing slope everywhere. The region before the inflection point, where the slope increases, violates the monotonic supply schedule principle. As a result, the s = 0 solution isn't compatible with the principles listed earlier in this series. In the next post in this series we will examine the s > 0 solution.

### Conclusion

We started out with some principles, which are mostly different specific mathematical ways to quantify the idea of a "fair" reward system which accelerates [1] the reward on a single post as upvotes accumulate. The most basic accelerating scheme simply determines the total curation reward for a post as the square of the number of upvoting R-shares, V(R) = R^2; this is the "curation reward curve".

In today's post, we discovered a problem: The mathematical consequences of the first four principles, applied to the curation reward curve V(R) = R^2, result in a system where sometimes later upvoters get more weight per R-share than earlier voters! In the next post we'll examine a different curation reward curve and show that it doesn't suffer from this problem.

[1] The reward (as measured in V-shares) actually does continue to accelerate forever. But since V-shares are claims on a pool of STEEM whose size is bounded by the allowed inflation, a post accumulating truly huge numbers of upvotes would see the increases in its reward (as measured in STEEM) eventually start to slow due to its V-shares beginning to represent a large majority of the pool.

Sort Order:
·  3 years ago

how and when do I know if my Steem Dollar staked enough to get interest?

·  3 years ago

Good you pointed out that although V-shares accelerate forever, the actual reward growth slows down as it approaches the max inflation.

·  3 years ago

Edited to fix incorrect expression for W'' (the LaTeX was missing a set of brackets, causing the -2 to be typeset in the denominator).

·  3 years ago

nice work keep it up