The expected value of Covers and Tokens

MadScientist

A few weeks ago I had 2 Moonstone not rostered and about 10 covers of her about to expire and I was thinking about rostering her to collect champion rewards. A quick back-of-the-envelope calculation shows that 63 of her covers will yield a total of 375 hero points so every cover is worth about 6 hero points. After that I wanted to know if I could calculate the value of a cover or a token more exactly and I wanted to share my findings here.

If I am using some wrong numbers anywhere, please report it and I will fix the errors.

Cycling 2 champions

We roster a 2 character, level it to 94, champion it, level it to 144 and sell it to repeat the process.

We invest
63 2 covers
69529 to level the character from level 15 to level 94
5000 to champion the character

We receive
24 Rank XP for applying covers 2-13
20 Rank XP for championing the character
17500 , 250 , 5 , 5 , 3 3 covers, and 200 Rank XP as champion rewards
65000 , and 125 from selling the character

Written as equation:

63 2 = 7971 + 375 + 5 + 5 + 3 3 + 244 XP (1)

Championing 3 characters

We assume that every 3 character is championed.

We invest
100 3 covers

We receive
57500 , 1000 , 50 , 3 4 covers, 2 , 5 , and 1000 Rank XP as champion rewards

Written as equation:

100 3 = 57500 + 1000 + 50 + 3 4 covers + 2 + 5 + 1000 XP (2)

Championing 4 characters

We assume that every 4 character is championed.

We invest
100 4 covers

We receive
112500 , 4000 , 250 , 10 , and 2000 Rank XP as champion rewards

Written as equation:

100 4 = 112500 + 4000 + 250 + 10 + 2000 XP (3)

Championing 5 characters

We assume that every 5 character is championed.

We invest
100 5 covers

We receive
205000 , 20000 , 625 , 25 , and 4000 Rank XP as champion rewards

Written as equation:

100 5 = 205000 + 20000 + 625 + 25 + 4000 XP (4)

Opening a Heroic Token

A heroic token gives a 2 cover with a probability of 70%, a 3 cover with a probability of 24% and a 4 cover with a probability of 6% as well as 3 Rank XP.

Written as equation:

1 = 0.7 2 + 0.24 3 + 0.06 4 + 3 XP (5)

Opening a Legendary Token

A legendary token gives a 4 cover with a probability of 85% and a 5 cover with a probability of 15% as well as 10 Rank XP.

Written as equation:

1 = 0.85 4 + 0.15 5 + 10 XP (6)

Buying a Legendary Token

You can buy a legendary token for 20 command points.

Written as equation:

20 = 1 (7)

Scenarios
Given these equations we can solve them under various assumptions. The details can be seen here: https://docs.google.com/spreadsheets/d/ ... sp=sharing

Scenario 1: 3 phase and collection

You have all 2 and 3 characters rostered and championed. You open every heroic token. You use every 2 and 3 cover. You collect 4 covers, 5 covers, ISO, hero points, legendary tokens and command points.
Solving equation (1), (2) and (5) using 2 , 3 and as unknowns yields:

2 = 174.907 + 7.019 + 0.120 + 0.004 + 5.072 XP
3 = 580.234 + 10.147 + 0.504 + 0.050 + 10.180 XP
= 261.691 + 7.348 + 0.205 + 0.015 + 8.993 XP

Using these values, you can, for inctance, calculate the value of the new elite token, which gives a 2 cover with a probab8ility of 76% and a 3 cover with a probability of 24% (and most likely 3 XP):

Elite Token = 272.186 + 7.770 + 0.212 + 0.015 + 0.013 4 + 9.298 XP

Scenario 2: 3 -> 4 transition
You have all 2 and 3 characters rostered and championed. You open every heroic and legendary token. You use every 2 and 3 cover. You buy legendary tokens with command points. You collect 4 covers, 5 covers, ISO, and hero points.
Solving equation (1), (2), (5), (6) and (7) using 2 , 3 , , and as unknowns yields:

2 = 174.907 + 7.019 + 0.015 4 + 0.001 5 + 5.167 XP
3 = 580.234 + 10.147 + 0.096 4 + 0.011 5 + 10.935 XP
= 261.691 + 7.348 + 0.094 4 + 0.004 5 + 9.241 XP
= 0.85 4 + 0.15 5 + 10 XP
= 0.043 4 + 0.008 5 + 0.5 XP

Scenario 3: 4 -> 5 transition
You have all 2 , 3 and 4 characters championed. You open every heroic and legendary token. You use every 2 , 3 :star, and 4 cover. You buy legendary tokens with command points. You collect 5 covers, ISO and hero points.
Solving equation (1), (2), (3), (5), (6) and (7) using 2 , 3 , 4 , , and as unknowns yields:

2 = 196.278 + 7.779 + 0.002 5 + 5.589 XP
3 = 713.250 + 14.876 + 0.015 5 + 13.566 XP
4 = 1391.036 + 49.459 + 0.042 5 + 27.512 XP
= 392.037 + 11.983 + 0.008 5 + 11.819 XP
= 1182.380 + 42.040 + 0.185 5 + 33.385 XP
= 59.119 + 2.102 + 0.009 5 + 1.669 XP

Scenario 4: 5 champions
You have all 2 , 3 :star, 4 and 5 characters championed. You open every heroic and legendary token. You use every 2 , 3 :star, 4 and 5 cover. You buy legendary tokens with command points. You collect ISO and hero points.
Solving equation (1), (2), (3), (4), (5), (6) and (7) using 2 , 3 , 4 , 5 , , and as unknowns yields:

2 = 202.555 + 8.296 + 5.725 XP
3 = 759.675 + 18.701 + 14.571 XP
4 = 1517.537 + 59.879 + 30.350 XP
5 = 3031.342 + 249.698 + 65.626 XP
= 415.163 + 13.888 + 12.320 XP
= 1744.607 + 88.352 + 45.557 XP
= 87.230 + 4.418 + 2.278 XP

Additional calculations

If you have sufficiently high Shield Rank, every point of XP is worth around 13.5 ISO. You can use this to increase the ISO values accordingly.

You can calculate the expected value of various tokens, e.g.

1 Taco Token = 1/300 + 2/300 4 + 34/300 3 + 129/300 2 + 40/3 + 925/3 + 3 XP
1 Event Vault Token = 1/80 + 3/80 4 + 22/80 3 + 54/80 2