Steem Monsters - Expected Value of an Essence Orb

A few weeks ago, @thomaseamoran created a post arguing that the expected value of an Essence Orb was just $1 and that it was not worth the DEC:

https://steemit.com/splinterlands/@thomaseamoran/hot-take-splinterlands-orbs-not-worth-the-dec

As a response, I wanted to take a more statistical look at the issue.

I am going to summarize the conclusions here:

Without potions, the expected value of an Orb is $1.68 - $2.37.
However, around 30% of this expected value comes from getting a Gold Foil (which is 2%).
And more than 50% of this expected value comes from getting an Epic or Legendary (which is 4.8%)

Depending on how you buy your Orbs they can cost from $1.71 - $2.50 and you are getting $1.68 - $2.37 in expected value.

Using both Brilliant potions, the expected value of an Orb is $2.59 - $3.83.
However, as much as 50% of this expected value comes from getting a Gold Foil (which is around 4%)
And as much as 66% of this expected value comes from getting an Epic or Legendary (which is 5.6%)

Using both potions (which costs 90 DEC without guild discounts) increases the expected value of a card by 54% - 62%, which definitely makes the potions net positive in DEC value.

(The cost of each Orb includes pro rata costs for the potion charges).

Depending on how you buy your Orbs they can cost from $2.04 - $2.95 using potions and you are getting $2.59 - $3.83 in expected value.

If you want to look at the numbers, data, and math, you can continue reading below.

We know that the Card Drop Rates for Reward Cards are:

75.2% - Common
20% - Rare
4% - Epic
0.8% - Legendary

And the chance for a card being Gold Foil is 2% (so a 98% chance of it being Regular).

At the very end of this article I will discuss the Card Drop Rates for Boosters and Orbs which guarantee at least one card of Rare or better.

I created a Google Sheet with the calculations:

https://docs.google.com/spreadsheets/d/1C3tyrN8OKYlURGHR-Cbm9M60VT-fVycvFxXvi6FTToE/edit?usp=sharing

Feel free to Make a Copy and tweak the values of the cards to see how it affects the Expected Value of the Orb. Using Peakmonsters, I took the data based on the lowest single price on the market as well as the highest bid that you could instantly fulfill. This data will constantly change due to market dynamics.

We can see that the Expected Value of one card =

% chance of Regular * (Probability of Regular Common * Average Value of Regular Common + Probability of Regular Rare * Average Value of Regular Rare + Probability of Regular Epic * Average Value of Regular Epic + Probability of Regular Legendary * Average Value of Regular Legendary)

PLUS

% chance of Gold * (Probability of Gold Common * Average Value of Gold Common + Probability of Gold Rare * Average Value of Gold Rare + Probability of Gold Epic * Average Value of Gold Epic + Probability of Gold Legendary * Average Value of Gold Legendary)

Let’s start by analyzing the High Bid value without any Potions:

0.98 * (0.752 * $0.097 + 0.2 * $0.169 + 0.04 * $1.950 + 0.008 * $6.987) +
0.02 * (0.752 * $1.938 + 0.2 * $5.504 + 0.04 * $26.625 + 0.008 * $128)
= $0.3288 for one card, which is roughly $1.64 for one Orb

Of the $0.3288 expected value:
$0.2358 (72%) comes from a Regular card
$0.0929 (28%) comes from the small chance of getting a Gold Foil

If we instead look along rarity lines:
$0.1006 (31%) comes from getting a Common card
$0.0551 (17%) comes from getting a Rare
$0.0977 (30%) comes from getting an Epic, and
$0.0753 (23%) comes from getting a Legendary

So the majority of the expected value comes from getting an Epic or Legendary, even though 95.2% of the time you will just get a Common or Rare.

If we instead look at the Low Buy value without any Potions we have:

0.98 * (0.752 * $0.124 + 0.2 * $0.219 + 0.04 * $3.050 + 0.008 * $8.823) +
0.02 * (0.752 * $2.643 + 0.2 * $6.317 + 0.04 * $41.385 + 0.008 * $280)
= $0.4660 for one card, which is roughly $2.33 for one Orb

Of the $0.4660 expected value:
$0.323 (69%) comes from a Regular card
$0.1429 (31%) comes from the small chance of getting a Gold Foil

If we instead look along rarity lines:
$0.1311 (28%) comes from getting a Common card
$0.068 (15%) comes from getting a Rare
$0.1527 (33%) comes from getting an Epic, and
$0.114 (24%) comes from getting a Legendary

Once again, the majority of the expected value comes from getting an Epic or Legendary.

Now let’s look at the numbers when we add in both Brilliant (+100%) Legendary and Brilliant (+100%) Alchemy Potions.

I’m assuming that when you use a Legendary Potion to increase the chance for a Legendary, that the increased chance results in an equal decreased chance of a Common card.

For High Bid with Potions:

0.96 * (0.744 * $0.097 + 0.2 * $0.169 + 0.04 * $1.950 + 0.016 * $6.987) +
0.04 * (0.744 * $1.938 + 0.2 * $5.504 + 0.04 * $26.625 + 0.016 * $128)
= $0.5102 for one card, which is roughly $2.04 for one Orb

Of the $0.5102 expected value:
$0.2839 (56%) comes from a Regular card
$0.2262 (44%) comes from the small chance of getting a Gold Foil

If we instead look along rarity lines:
$0.1270 (25%) comes from getting a Common card
$0.0765 (15%) comes from getting a Rare
$0.1175 (23%) comes from getting an Epic, and
$0.1892 (37%) comes from getting a Legendary

Using the Potions resulted in a 0.5102 / 0.3288 = 1.55 --> 55% increase in value

Not factoring in the Guild discount, a Brilliant Legendary Potion costs 20,000 DEC for 500 charges while a Brilliant Alchemy Potion costs 25,000 DEC for 500 charges (which is enough for 100 boosters or orbs). That means you are spending 90 DEC for each card flipped to increase the value by 55%. Several guilds right now have a 3%-5% discount so that would actually be 87.3 and 85.5 DEC respectively. Once you reach the maximum 10% discount it will just be 81 DEC per card.

Finally, we will look at Low Buy with Potions:

0.96 * (0.744 * $0.124 + 0.2 * $0.219 + 0.04 * $3.050 + 0.016 * $8.823) +
0.04 * (0.744 * $2.643 + 0.2 * $6.317 + 0.04 * $41.385 + 0.016 * $280)
= $0.7579 for one card, which is roughly $3.80 for one Orb

Of the $0.7579 expected value:
$0.3833 (51%) comes from a Regular card
$0.3746 (49%) comes from the small chance of getting a Gold Foil

If we instead look along rarity lines:
$0.1672 (22%) comes from getting a Common card
$0.0926 (12%) comes from getting a Rare
$0.1833 (24%) comes from getting an Epic, and
$0.3147 (42%) comes from getting a Legendary

Now, half the expected value comes from getting a Gold Foil. And 42% of the expected value comes from getting a Legendary. This makes sense since Gold Foil Legendaries command a premium when you look at Low Buy prices.

Using the Potions resulted in a 0.7579 / 0.4660 = 1.63 --> 63% increase in value for using the 90 DEC (per card) charges.

Since each Orb costs 2500 DEC (without discounts), a card should be worth 500 DEC. Increasing its expected value by 55%-63% for the cost of 90 DEC is definitely worth it.

If you did a bulk Orb purchase (buying 100 at a time to get 10 bonus Orbs) and you used a 10% guild discount, then the average cost of each Orb would be: 250,000 DEC * 0.9 / 110 = 2046 DEC. If you had the 10% discount using the potions would cost 81 DEC per card, which is still worth it for the 55%-63% increase in expected value.

One final note. In our calculations we looked at each card individually and then multiplied that by five to get the expected value of an Orb. The average value of an Orb (which contains 5 cards) will actually be slightly higher because of the guarantee that at least one card will be Rare or better so you wouldn’t have the scenario where you end up with 5 commons.

@tcpolymath gave me some of @cryptoeater 's numbers. The adjusted drop rate for boosters and Orbs taking into account the "one card rare or better" guarantee is:

Orb Drop Rate without any Potions

Regular Common: 0.689825
Regular Rare: 0.243135
Regular Epic: 0.0392
Regular Legendary: 0.00784
Gold Common: 0.014078
Gold Rare: 0.004961
Gold Epic: 0.0008
Gold Legendary: 0.00016

Substituting these values in the above equations we get:

High Bid value without any Potions:

(0.689825 * $0.097 + 0.243135 * $0.169 + 0.0392 * $1.950 + 0.00784 * $6.987) +
(0.014078 * $1.938 + 0.004961 * $5.504 + 0.0008 * $26.625 + 0.00016 * $128)
= $0.3356 for one card, which is roughly $1.68 for one Orb

Low Buy value without any Potions:

(0.689825 * $0.124 + 0.243135 * $0.219 + 0.0392 * $3.050 + 0.00784 * $8.823) +
(0.014078 * $2.643 + 0.004961 * $6.317 + 0.0008 * $41.385 + 0.00016 * $280)
= $0.4740 for one card, which is roughly $2.37 for one Orb

Orb Drop Rate with Potions

Regular Common: 0.6680669388
Regular Rare: 0.2381730612
Regular Epic: 0.0384
Regular Legendary: 0.01536
Gold Common: 0.026878
Gold Rare: 0.009922
Gold Epic: 0.0016
Gold Legendary: 0.00064

High Bid value with Potions:

(0.6680669388 * $0.097 + 0.2381730612 * $0.169 + 0.0384 * $1.950 + 0.01536 * $6.987) +
(0.026878 * $1.938 + 0.009922 * $5.504 + 0.0016 * $26.625 + 0.00064 * $128)
= $0.5185 for one card, which is roughly $2.59 for one Orb

This is a 54% increase in value using potions.

Low Buy value with Potions:

(0.6680669388 * $0.124 + 0.2381730612 * $0.219 + 0.0384 * $3.050 + 0.01536 * $8.823) +
(0.026878 * $2.643 + 0.009922 * $6.317 + 0.0016 * $41.385 + 0.00064 * $280)
= $0.7668 for one card, which is roughly $3.83 for one Orb

This is a 62% increase in value using potions.

Feel free to provide feedback and definitely let me know if I made any mistakes in my calculations.