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 CountBreeding 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.

We'll demonstrate breeding on a single core attribute aa. Let's denote a1a_1, a2a_2 values of aa of Paragons being bred, a1≤a2a_1 \leq a_2. Breeding is designed to promote breeding of Paragons with the similar values - the closer a1a_1 is to a2a_2, the bigger chance aRa_R, attribute value of the resulting Paragon, will be greater than a2a_2. Because of Tier0 restrictions, all values of aa must be from interval ai∈⟨0,50⟩a_i \in \langle 0, 50 \rangle.

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

fa′(x,a1,a2)=∑i=050exp⁡[−12(x−a1σ)2]⋅δ(x−i)+C⋅∑i=0a2−1exp⁡[−12(x−a2σ)2]⋅δ(x−i)+C⋅∑i=a250exp⁡[−12(x−a2σR)2]⋅δ(x−i)\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 δ(x)\delta(x) is Dirac delta function, σ=2\sigma = 2 is the standard deviation of normal distributions being used, σR=σ⋅exp⁡[1+(a1−a2)/5]\sigma_R = \sigma \cdot \exp{\left[ 1 + (a_1 - a_2) / 5 \right]} is adjusted standard deviation used for x≥a2x \geq a_2, and C=1−(a2−a1)/50C = 1 - (a_2 - a_1) / 50 is scaling factor.

Probability density function fa(x)f_a(x) for aRa_R is defined as

fa(x,a1,a2)=fa′(x,a1,a2)∫Rfa′(x,a1,a2).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 a2=35a_2 = 35 and different values of a1a_1 are shown in Image 2.

When generating a new Paragon, every core attribute is processed independently to follow distribution fa(x,a1,a2)f_a(x, a_1, a_2).

There is one additional restriction for the new Paragon - score attribute ss, that must be in the interval s∈⟨0,99⟩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 ss.

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