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.

ash (72)· 2 years agohow and when do I know if my Steem Dollar staked enough to get interest?

markopaasila (65)· 2 years agoGood you pointed out that although V-shares accelerate forever, the actual reward growth slows down as it approaches the max inflation.

theoretical (63)· 2 years agoEdited to fix incorrect expression for W'' (the LaTeX was missing a set of brackets, causing the -2 to be typeset in the denominator).

true-profit (53)· 2 years agonice work keep it up