Breeding

Breeding is the Paragon operation that takes two Tier0 Paragons to create a new Tier0 Paragon.

Two Tier0 Paragons are bred to produce a new Tier0 Paragon. The original Paragons and the new Paragon are unlocked after some time and returned to the wallet. Breeding cost and time depend on maximum of Breed Count attribute value of Paragons being bred according to Table below.

Breed Count	Breeding Cost [HTO]	Breeding Time [hours]
0	10	1
1	20	2
2	30	4
3	40	8
4	50	16

Paragons with Breed Count value equal to 5 cannot be bred any more.

Breeding is the operation that increases the supply of Paragons, and it's applicable for Tier0 Paragons only. The basic schema for breeding is shown in Image 1.

Image 1: Schema of breeding operation, where two Tier0 Paragons are bread to produce a new Paragon of the same Tier. As PFPs, we've used NFTs from DeGods and Spirits of Pappataz collections.

We'll demonstrate breeding on a single core attribute $a$. Let's denote $a_1$, $a_2$ values of $a$ of Paragons being bred, $a_1 \leq a_2$. Breeding is designed to promote breeding of Paragons with the similar values - the closer $a_1$ is to $a_2$, the bigger chance $a_R$, attribute value of the resulting Paragon, will be greater than $a_2$. Because of Tier0 restrictions, all values of $a$ must be from interval $a_i \in \langle 0, 50 \rangle$.

We need to dive deeper into Math to describe this relation exactly.

$$\begin{align} f'_a(x, a_1, a_2) &= \sum_{i=0}^{50} \exp{\left[ - \frac{1}{2} \left( \frac{x-a_1}{\sigma} \right)^2 \right] \cdot \delta(x-i)} \nonumber \\ &+ C \cdot \sum_{i=0}^{a_2-1} \exp{\left[ - \frac{1}{2} \left( \frac{x-a_2}{\sigma} \right)^2 \right] \cdot \delta(x-i)} \\ &+ C \cdot \sum_{i=a_2}^{50} \exp{\left[ - \frac{1}{2} \left( \frac{x-a_2}{\sigma_R} \right)^2 \right] \cdot \delta(x-i)} \nonumber \end{align}$$

where $\delta(x)$ is Dirac delta function, $\sigma = 2$ is the standard deviation of normal distributions being used, $\sigma_R = \sigma \cdot \exp{\left[ 1 + (a_1 - a_2) / 5 \right]}$ is adjusted standard deviation used for $x \geq a_2$, and $C = 1 - (a_2 - a_1) / 50$ is scaling factor.

Probability density function $f_a(x)$ for $a_R$ is defined as

$$f_a(x, a_1, a_2) = \frac{f'_a(x, a_1, a_2)}{\int_{\R} f'_a(x, a_1, a_2)}.$$

Example of this function for $a_2 = 35$ and different values of $a_1$ are shown in Image 2.

Image 2: Probability density function for different values of its parameters. For better illustration, values $\sigma = 5$ and $\sigma_R = \sigma \cdot \exp{\left[ 1 + (a_1 - a_2) / 10 \right]}$ are used.

When generating a new Paragon, every core attribute is processed independently to follow distribution $f_a(x, a_1, a_2)$.

There is one additional restriction for the new Paragon - score attribute $s$, that must be in the interval $s \in \langle 0, 99 \rangle$. In case this condition is not fulfilled, the new Paragon is re-generated (with all of its core attributes) and this repeats until the new Paragon fulfills the condition for $s$.