Additional Fairness with Normal Distributions

How the initial supply in the "Riders Mint" follows a normal distribution.

There's one additional element introducing fairness for those using the Rider Mint: When randomly assigning the attributes (which is the case in the Rider Mint) our algorithm uses equal probabilities for all levels, which will yield a normal distribution in the overall distribution of levels. This means it's equally unlikely to mint a lowest level (18) as it is to mint a highest level (90), with a highest chance ("expectation value") of minting a level 54 Rider.

The math behind it: How exactly are levels being distributed?

Recall that each Rider can be equipped with six Gear NFTs, with each Gear NFT having three stats (Speed, Agility, Technique), yielding a total of 6x3=18 stats. Now each of these stats can take on a level from 1 to 5, and a Riders' total value is defined through the sum of all levels. This implies two things:

1. A Riders' lowest level is 18 (all ones).

2. A Riders' highest level is 90 (all fives).

Now the question is this: How many Riders will there be with an overall level of $l$, where $l$ can be any level between 18 and 90. This problem is mathematically analogous to the number of sums of 18 five-sided dice.

Generally, to find the number $m_l$ of combinations summing to $l$, given $N$ dice with $n$ sides each, one has to look at the coefficient of $l$-th power of some random variable $x$ in the polynomial $\left(\Sigma_{i=1}^n x^i\right)^N$.

In our case the corresponding polynomial is therefore $\left(\Sigma_{i=1}^5 x^i\right)^{18}=(x+x^2+x^3+x^4+x^5)^{18}$.

The coefficients $m_l$ belonging to the $l$-th power of $x$ in the above polynomial then yield the normal distribution we were looking for.

Now it is important to note that we can't blindly follow this approach, since we need at least one NFT for each level. As the rarest levels will be 18 and 90 (lowest and highest), we have, using our notation above, that $m_{18}\geq1$ and $m_{90}\geq 1$. But even staying at 1 for both levels, this will result in $m_{54}=248'000'853'348$, meaning around 30 Riders at level 54 for every living human on Earth.

It is therefore rather obvious that we need to flatten the Gaussian in such a way that we can match the number of our initial supply, while up-holding the proportionality and the symmetry of the distribution. We're still trying different algorithms and methods to achieve this, and will update this section once our formulae have been manifested ;)

The reason we do this is two-fold:

• It makes it fairer for those paying 1 AVAX (using the Rider Mint) since then it really is rather unlikely to mint a low level which could have also been achieved by using the Tourist Mint at 0.3 AVAX, meaning risk is decreased and the Rider Mint is made more attractive.

• In the same manner it's also rather unlikely to mint a high level, making it more attractive to actually own a high level Rider and thereby increasing their value.