# Summoning (Breeding of Bezogi)

3 Darkzogi chant their horrific dark magic to summon a Bezogi from the Darkpool.
Mix and match Bezogi to summon powerful and unique Bezogi. Hoard your Bezogi like the famous FUD King of Bezogia, or start your own summoning business and sell your summoned Bezogi at Bull Market.
Each Bezogi has its own unique attributes based on the coloring of its body parts, each representing a different percentage of skill boost. The breed and DNA of Bezogi is directly derived from the two Bezogi used to summon, the purer the blood of the summoners, the higher the chance of getting a rare, epic or even purebred Bezogi.

### Summoning Cost

In order to add additional value to early generations of Bezogi, early generations of Bezogi can summon at a cheaper cost than later ones. There is no limit to the number times a Bezogi can summon but the price will grow exponentially each time. The cost of summoning also grows with the supply of Bezogi in circulation so it is cheaper to summon early on.
The summoning cost for a Bezogi
$A$
is:
$\Large C_{A}=P \cdot 1.05^{G} \cdot 1.1^{S} \cdot\left(\frac{N}{4096}\right)^{0.15}$
here
$P$
is the base price which is around \$100, the actual cost in MBLK and BEZOGE will be updated periodically.
$G$
is the generation of the Bezogi starting at 0 for the first 4096 Bezogi.
$S$
is the number of times the Bezogi has been used for summoning and
$N \geq 4096$
is the total supply of Bezogi. The
$0.15$
exponent reduces the impact of that variable.
The actual cost to summon is the average of the cost of the two Bezogi
$A$
and
$B$
used to summon:
$\Large C=\frac{C_{A}+C_{B}}{2}$
So for a first generation Bezogi and the first time it is used for summoning, the summoning cost as a function of the total supply is as follows:
Following the formula above, the cost of summoning depending on the generation and the summon count of a Bezogi when the total supply is the original 4096 is the following:

### Summoning Mechanics

The algorithm to generate a Bezogi starts from determining the color of the nose, then the eyes, the fur highlights and finally the main fur color. The probability distribution
$p(0)$
for the nose will be based on the breed of the Bezogi used for the summoning by taking the average of their attributes. So for each color
$i$
:
$\large p(0)_{i}=\frac{1}{2}\left(\sum_{j=A, B} \delta_{i, \text { nose }_{j}} \cdot 0.1+\delta_{i, \text { eyes }_{j}} \cdot 0.2+\delta_{i, \text { highlight }_{j}} \cdot 0.3+\delta_{i, \text { fur }_{j}} \cdot 0.4\right)$
$j$
represents each Bezogi
$A$
and
$B$
used to summon,
$\delta$
is the Kronecker delta (
$\delta_{a,b} = 1$
if
$a =b$
and
$\delta_{a,b} = 0$
if
$a \neq b$
).
$nose_j$
,
$eyes_j$
,
$highlight_j$
,
$fur_j$
are the colors of the Bezogi
$j$
.
We then sample that probability distribution to obtain the color of the nose
$c_0$
. Once the color of the nose is selected, the probability distribution is updated by dividing by two the probability to pick again the color of the nose
$c_0$
. Change in probability is added equally to all other colors.
$\large p\left(t \mid c_{0} \ldots c_{t-1}\right)_{i}=p(t-1)_{i}-\delta_{i, k} \cdot \frac{p(t-1)_{k}}{2}+\left(1-\delta_{i, k}\right) \cdot \frac{p(t-1)_{k}}{14}$
​The eyes color
$c_1$
is sampled from that probability distribution
$p(1)$
and the same procedure continues for the fur highlight and the main fur.
The age of the summoned Bezogi will be randomly selected with equal probability between baby, child, adult and old.
With the algorithm above, the Bezogi used to summon will have an important impact on the summoned Bezogi but chances of having purebred, mixed breed or epic Bezogi are low.

### Summoning simulations

We ran Monte-Carlo simulations for several combinations of Bezogi used to summon. Each simulation was run 1M times to obtain a precise estimation of the probabilities.
The optimal way to summon a purebred is with two purebreds of the same breed, in that case the probability is 1.5625%. Mixing two purebreds of different color is one of the ways to obtain the most mixed breed possible with a probability of approximately 4.5246%.
By summoning with two 90% pure epic mintzogis, the probability of having a purebred is reduced to 1.025 %.
When using two Bezogi with random colors (different at every simulations) we obtain the following probabilities:

### Example

If we use the following two Bezogi to summon:
Left: 100% Gold Right: 60% Green, 30% White, 10% Black
The probability distribution for the color of the nose will be:
The color of the nose will be randomly sampled from the distribution above, in this case it will be gold.
The probability of gold is then divided by two and redistributed equally among all other colors for the next step. We randomly sample the resulting distribution and obtain red eyes. This procedure is done again for the fur highlight and the main fur.
The resulting Bezogi will be this rare (four different colors) Darkzogi:
The probability of summoning each level of rarity with the Bezogi used based on a simulation run 1M times is as follows: