Expected Boosters.md

Expected Boosters for Magic: the Gathering Arena

When figuring out the cost of a deck in Magic: the Gathering, people playing tabletop Magic and Magic Online have measures for the value of the deck that correspond to their immediate experience (currency and tickets, respectively). However, Magic: the Gathering Arena obscures its economy; the easiest way to categorize the cost of acquiring the cards necessary to a deck is using the count of wildcards needed to craft it, but acquiring wildcards is a random process.

This document describes a more fungible and immediately understandable measure: the number of expected boosters to crack to get the cards and wildcards needed for the deck, starting from scratch. I propose a simulation to estimate this value at the end of this document.

For reference, applying an implementation of the simulation to the decks mentioned in @yoman_5's "5 of 50, Brewing for Strixhaven Standard and Historic" article yields the following:

 == Golgari Aristocrats by Yoman5.txt
 - 118 boosters
 - (for a cost of 23,400 gems + 1000 gold, about $119.98)
 
 == Jadzi Transmogrify by Yoman5.txt
 - 322 boosters
 - (for a cost of 64,200 gems + 1000 gold, about $329.94)
 
 == Lutri Pact by Yoman5.txt
 - 161 boosters
 - (for a cost of 31,800 gems + 2000 gold, about $169.95)
 
 == Mardu Winota by Yoman5.txt
 - 260 boosters
 - (for a cost of 51,600 gems + 2000 gold, about $269.94)
 
 == Mono-Red Ramp by Yoman5.txt
 - 267 boosters
 - (for a cost of 53,400 gems + 0 gold, about $279.94)
 
 == Mono-Red Storm by Yoman5.txt
 - 157 boosters
 - (for a cost of 31,200 gems + 1000 gold, about $164.96)
 
 == Simic Turns by Yoman5.txt
 - 204 boosters
 - (for a cost of 40,800 gems + 0 gold, about $209.96)
 
 == Temur Genesis Lithoforming by Yoman5.txt
 - 210 boosters
 - (for a cost of 42,000 gems + 0 gold, about $214.96)
 
 == Temur Twin by Yoman5.txt
 - 218 boosters
 - (for a cost of 43,200 gems + 2000 gold, about $219.96)
 
 == Tendrils Mizzix Mastery Combo by Yoman5.txt
 - 243 boosters
 - (for a cost of 48,600 gems + 0 gold, about $254.94)

How to Calculate

The count above is calculated by running 2,000 simulations of a player who:

• starts with an empty collection, no wildcards, and no filled segments in the wildcards counter;

• plans to acquire the least rare print of each card (see below);

• never opens the Vault, if available; and

• can acquire up to 1,000 boosters. (If they are ever forced to acquire more than 1,000 boosters, the simulation ends unsuccessfully.)

The simulation has the player acquire and open one booster pack of the expansion most likely to yield any of the cards in the deck that haven't been acquired yet; then, they check if they found anything useful, setting aside any wildcards obtained by cracking the booster. The player repeats acquiring the now most-likely pack and opening it until they acquire enough relevant cards and wildcards to craft the deck. The final result is the average booster count cracked during all of the simulations that succeeded.

The player may end up in a situation where there is no booster pack that they can acquire that contains cards for their deck. (For example, the remaining cards may have been introduced with Historic Anthology, which now can only be crafted using wildcards.) If they do, they will instead acquire packs of Core Set 2019 until they have enough wildcards to craft what's left.

To simplify the simulation and make it more consistent, the player will only ever be interested in a specific print of a card. The following rules apply:

• The player will refuse to look at prints of a card that are rarer than the minimum rarity that card is available in. For example, if the player needs Cultivate, which is a common in Core Set 2021 and an uncommon in Strixhaven Mystical Archives, they ignore all Mystical Archive Cultivates — they will only be satisfied when they find, or are able to craft, the Core Set 2021 copy (and not acquire any Strixhaven boosters at all unless they need other cards found there).

• If there are multiple prints at the same rarity, the player will go in alphabetical order of English name for each card that has this problem, then choose a print of that card that is in the set whose boosters are most likely to contain cards of interest. (Note that this will make that set more likely when choosing the next print; this ensures that the player concentrates their effort into buying boosters that are likely to have the cards they need.) For example, if the player needs Fabled Passage, they will look for the Throne of Eldraine Fabled Passage if they are more likely to find other relevant cards by opening Throne of Eldraine packs, or the Core Set 2021 Fabled Passage if they are more likely to find other relevant cards by opening Core Set 2021 packs. They will make this choice before they make the choice of whether to go for Throne of Eldraine, Core Set 2021 or Theros: Beyond Death for Thrill of Possibility, since that card comes alphabetically later.

• If they end up in a situation where they need a card whose minimum rarity has multiple prints, and all the prints are in sets that are not going to be bought otherwise, then the player will always choose the latest version of a card. For example, if a player needs Fabled Passage and the deck doesn't need other Throne of Eldraine or Core Set 2021 cards, they will go for the Core Set 2021 Fabled Passage.

I've found empirically that it is rare for successive sets of simulations to have a difference of more than 1-3 boosters, which I find acceptable.