The cost of acquiring all mythics

freegenz

A few weeks ago, there was a thread asking people how they defined success in MTGPQ. Multiple people answered that it was when they would collect all the cards, or something similar. Right around that time, I was also wondering how long it would take to collect all the cards. Having a mathematically inclined background, I decided to calculate that which would be hardest, but still remain reasonable: what is the cost to acquire all mythics within a given set?

Apart from the various assumptions made, this post is intended to contain only facts. (Only this post, not this thread, feel free to express your opinions, obviously.) I may give my opinion in a subsequent post. In any case, I hope the info presented will guide you in your card and app purchases. Also, all calculations made use probabilities and averages, and are intended to represent the *average* numbers/ amounts/ costs/ etc. Individual experience will inevitably vary.

NOTE: I am not a mathematician, I don’t know everything and this post is lengthy and math heavy, so I probably made some mistakes. Feel free to point them out if that is indeed the case. I also tried to explain as much as I can without giving a probabilities course, but I may have skipped some important stuff, so feel free to ask questions as well.

 

Assumptions

These are not necessarily true or reflective of actual game experience, but are made either to enable or simplify answering the question.

- At the “start”, all other cards below mythic rarity are owned and 0 mythics are owned. This is to assume maximum booster crafting currency (bcc) acquisition. In reality, this will not be the case, and the cost to acquire all mythics will therefore be higher than what is calculated in this post.

- Cards are only acquired through premium pack purchases and booster crafting, and crystals come from buying them for $ in the shop at the max amount (3000). This is mainly to be able to have a monetary value for accomplishing the goal. Also this is the only instantly and infinitely repeatable way of acquiring cards, even if it’s not the most cost efficient.

- All cards within a certain rarity are equiprobable. Not sure if that’s a word, but it means that once a mythic is obtained, the probability of it being any of the possible mythics is the same. For example, if there are 26 mythics in a set, the probability of getting each mythic (once a mythic is obtained in a pack or through crafting) is 1/26. There is no way to know for sure if that is indeed the case unless D3 tells us, and is the best hypothesis to work with imho.

- Each card acquisition is a completely separate event. That means that the probability of getting a card of a certain rarity, or a certain card within a rarity, is not influenced by and of your previously acquired cards. Once again hard to know for sure, but the most simple scenario to analyze.

- Masterpieces are always assumed to be new cards. Depending on the set, the average amount of masterpieces acquired when all mythics are acquired will be around 1-2. I think this is a fair assumption. In any case, it wouldn’t change the conclusions very much and require more math than I think it’s worth.

- Booster crafting is done only when there is enough accumulated currency to buy all remaining mythics. This is pretty obvious, but for those who haven’t figure it out: when you buy a pack you risk getting duplicates. The more cards you have, the higher the chance of getting duplicates. Crafting guarantees new cards. Therefore you should buy packs first and craft second.

 

The math

I’ve calculated the cost for a few sets of the regular format (I started this before M19). Here are the accompanying spreadsheets:

https://docs.google.com/spreadsheets/d/1-MAOzaeMUH4I0FuxGVx8Xe5Kp2y-CpBV3pUVcutQ6iM/edit#gid=465276391

Color code: orange means input dependent on set, and green means final results.

Spreadsheet explanation:

1.       BCC
This section contains the booster crafting currency cost and rewards for each card rarity.

2.       Pack "1 card" probabilities
This section contains the probabilities of getting at least one card of a certain rarity in a pack (shown in game when clicking the % icon on a pack). See next section for more details.

3.       Card probabilities
This sections contains the probability of a single card being of a given rarity in a booster (which I’ll call p). The game gives the probability (which I’ll call P) of getting at least one card of a given rarity (when you click the % icon on a pack) within a pack. This means that the probability of not getting at least one card of that rarity is 1 minus that number, so (1-P). This happens when not getting that rarity happens n times in a row, where n is the number of cards in that pack. The probability of not getting a card of a certain rarity on a single draw is (1-p). The probability of not getting it n times is (1-p)*(1-p)*(1-p)*… n times, so (1-p)^n.

So we have       1-P=(1-p)^n       and therefore   p=1-(1-P)^(1/n).

I’ve calculated this for every type of pack. Because of the precision of the given P, sometimes the values of p differ from type to type, so I’ve made a sort of rounded guess as to what p probably is give the calculated values. I figure the probabilities are probably rational numbers with max 4 decimals. Obviously I’ve made sure that the sum of all rarity probabilities is 1. Hour of Devastation is the worst one here, I’ve had to fudge the numbers a bit so it makes mathematical sense.

4.       Average bcc reward
The expected bcc reward value of a duplicate for a certain card rarity is the probability of getting that card rarity times the bcc value of that rarity. Assuming all cards are duplicates, the average reward for a single card is the sum of all rarities’ expected rewards. The bcc from the bonus rare is also included here, averaged on 25 cards. Obviously, not *all* cards are duplicates. The mythics will sometimes be new cards. This is accounted for in the “cumulated bcc” column in the following section.

5.       Main table
This is where the cost of each mythic is calculated. Each line represents the acquisition of a mythic through pack purchases. The average number of cards required for acquiring the next new mythic is calculated (mean nb of cards for next mythic). The total amount of cards acquired is accumulated in cumulated cards at next mythic. Then, the cumulated bcc from those packs is converted in mythics (nb of mythics buyable with bcc) using the average bcc reward per card, and that bcc is converted in mythics giving the total mythics acquired + acquirable. The cumulated cards are also converted in premium packs, which are converted in crystals, which are converted in Canadian dollars. Finally, that crystal cost is divided by the total acquired mythics for that line.

Since the goal is to acquire all mythics, I’ve highlighted the lines where the mythics acquired + acquirable go from below to above the total number of mythics in the set. The actual average cost is somewhere between those 2 lines.

6.       Mean nb of mythics before new one
When acquiring a mythic, there is a certain probability that it will be old or new. Since I’ve assumed that all cards within a certain rarity were equiprobable, the probability that it is old is the number of mythics already owned over the total number of mythics. The probability that it will be new is p=(total number-number owned)/total number. The average number of mythics n obtained to get 1 new mythic is n=1/p. (from n*p=1). P changes when getting a new mythic, which is why each line of the table represents getting a new mythic.

7.       mean nb of cards for next mythic
There are two ways of seeing this. It is either 1/(probability of getting mythic * probability of getting new mythic), or average nb of cards per mythic*average nb of mythic per new mythic. Both give the same thing obviously.

8.       Second degree model cost @ total mythics in set
Since the total mythics acquired + acquirable don’t necessarily arrive at the total number of mythics in the set, the total crystal cost actually lies somewhere in between the value of the line before and the one after. So I’ve fitted a 2nd degree polynomial to the data to find the actual average crystal cost. (Not included in the spreadsheets.) The fits weren’t too bad, and I didn’t feel like doing error analysis. I just approximated the error based on the difference between the data and the model. Feel free to improve this if it’s not enough for you.

Results and Conclusion

Average cost of acquiring all mythics within standard (pre-M19) sets.

Set	Average crystal cost	Average $CAN cost
Origins	21600 ± 100	1008.00
HoD	19200 ± 100	896.00
Ixalan	24500 ± 100	1,143.33
Rtl	14780 ± 50	689.73
Dom	21700 ± 100	1012.67

 

Once again, please remember that these are *averages* and that personal experience will vary, for better or for worse. Also, as you all know, there are many more ways of acquiring both cards and crystals, so the real cost, when considering freely acquired resources, should actually be less. Maybe this post will have inspired one of you to build a time cost dependent model using in game rewards, or maybe even a combination of both rewards+purchases. Also interesting would be a proper analysis of the cost variance. The average is a good start, but what are the probabilities of diverging from that average by a certain amount? Finally, whether these costs are reasonable or not is something we probably all have an opinion on, and I would love to hear what you think!

 

So that’s it! I hope you enjoyed this post and that the info contained will help guide you in your decisions.

FindingHeart8

I remember a similar analysis being done before booster crafting.  The cost was like ~$250,000 to acquire all cards.

Or something outrageously high.  I didn't write the specifics down.

This is better, but still pretty dang pricey

James13

At the release of Booster Crafting, I had the option to own every Mythic currently in the game from craftable sets having spent about a hundred or so scattered over the extended lifetime of the game at that time.  Another I know to be fully F2P, but a very active competitive player from the very beginning of the game could do the same.

Mburn7

James13 said:

At the release of Booster Crafting, I had the option to own every Mythic currently in the game from craftable sets having spent about a hundred or so scattered over the extended lifetime of the game at that time.  Another I know to be fully F2P, but a very active competitive player from the very beginning of the game could do the same.

There were a couple of people in my coalition who could do the same.  I was personally able to complete Origins, but not all of legacy.  Right now I'm missing 95 legacy mythics thanks to rotation, so it'll be a while.

FindingHeart8

James13 said:

At the release of Booster Crafting, I had the option to own every Mythic currently in the game from craftable sets having spent about a hundred or so scattered over the extended lifetime of the game at that time.  Another I know to be fully F2P, but a very active competitive player from the very beginning of the game could do the same.

but not the masterpieces, they were a whole different story

Volrak

Excellent analysis and explanation!  I noticed that the DOM bcc reward and cost were both given a 50% boost; that makes the DOM analysis a fair comparison with other sets for mythics, even though the 50% bcc reward bonus now applies to M19 and not DOM.

While this is unlikely to describe how much any real person has paid or would pay per set, since real people don't pump money in to complete a collection without actually playing the game, it's very useful as a point of reference.

For example, it's instructive to see how much the total cost is slashed thanks to booster crafting; whether players spend cash or earned crystals, it can be argued from these numbers that booster crafting has roughly quadrupled the rate at which players are moving towards completing collections (other than MPs).

babar3355

First off, great post @freegenz!  I haven't dug into your math, but I am also not the best person to do this.

Those numbers don't seem completely ludicrous like the old numbers that were in the 5 figures for mythics and 6 figures for Elites.  If you want to whale out and complete the set, then bring your $720 to the table and get all of the Origins.  It seems like a large sum, but you are also circumventing the progression of the game and would not be an optimal solution for most people.

Of course with all of the resources we win in game, those numbers will be much lower.  And frankly now that they have made Elite packs give guaranteed cards many people will be able to have complete collections of sets for reasonable prices.  We shall see the impact on the game, but I have already dropped about $100 on a set I consider relatively weak because I am excited to get a shot at some of the Elites.

Anyway, I always wanted to know what it would cost a brand new player to install the game and get to a complete collection.  This seems like a solid analysis to help us figure that out.  Thanks again!  

madwren

Thanks for doing this, it's very eye-opening. It's nice to see booster crafting has had such a positive impact overall.

GrizzoMtGPQ

Barring masterpieces, I have all the legacy cards and every standard card except those in M19. I have 24 masterpieces. I am still sitting on 250,000 orbs and 1,600,000 runes.

This was all possible because I started the game pretty early, hoarded duplicate cards, and played Quick Battle every opportunity I could. The game was different back then. 

If D3 would let me, and there was a buyer, I'd sell my collection for $250,000.

(BTW MODS, this is not a solicitation for a sale. I'm joking and realize that selling or transferring an account is against the TOS laid out by D3).

FindingHeart8

GrizzoMtGPQ said:

Barring masterpieces, I have all the legacy cards and every standard card except those in M19. I have 24 masterpieces. I am still sitting on 250,000 orbs and 1,600,000 runes.

This was all possible because I started the game pretty early, hoarded duplicate cards, and played Quick Battle every opportunity I could. The game was different back then. 

If D3 would let me, and there was a buyer, I'd sell my collection for $250,000.

(BTW MODS, this is not a solicitation for a sale. I'm joking and realize that selling or transferring an account is against the TOS laid out by D3).

story of my purchasing power right here:

(BTW MODS: I am also joking and not trying to make an actual purchase.)

freegenz

James13 said:

At the release of Booster Crafting, I had the option to own every Mythic currently in the game from craftable sets having spent about a hundred or so scattered over the extended lifetime of the game at that time.  Another I know to be fully F2P, but a very active competitive player from the very beginning of the game could do the same.

And this is why I'm super scared of ever getting to platinum tier. I'm a relatively new player (currently in gold, started in March) and I've yet to acquire a single origins Mythic... 

freegenz

Thank you for the kind comments and you're welcome @babar3355 and @madwren!

Also, you're all right that F2P gameplay can speed up this process by a certain amount, but like I said in my previous post, I started playing back last March and I only have 6 ixalan mythics, which is the most I have of any set, as I've done almost every across ixalan event. It's the next set to rotate out, and it would still cost me upwards of 600 dollars to complete...

So yeah, I think the cost is ridiculous, especially considering it's for acquiring unresellable game content. At the current rates, using money to get mythics through packs offers an extremely poor return, content/$ wise, and I would (and have) only spend money on guaranteed content, such as those 3 card packs appearing in the shop or PWs, but even those are ridiculously expensive. Compare that to buying characters in mobas...

Mburn7

freegenz said:

Thank you for the kind comments and you're welcome @babar3355 and @madwren!

Also, you're all right that F2P gameplay can speed up this process by a certain amount, but like I said in my previous post, I started playing back last March and I only have 6 ixalan mythics, which is the most I have of any set, as I've done almost every across ixalan event. It's the next set to rotate out, and it would still cost me upwards of 600 dollars to complete...

So yeah, I think the cost is ridiculous, especially considering it's for acquiring unresellable game content. At the current rates, using money to get mythics through packs offers an extremely poor return, content/$ wise, and I would (and have) only spend money on guaranteed content, such as those 3 card packs appearing in the shop or PWs, but even those are ridiculously expensive. Compare that to buying characters in mobas...

You should not feel compelled to spend money in order to compete.  One thing Oktagon has been doing very well in their sets is increasing the power level of uncommons and rares in sets, so that competitive decks can be made without a large mythic collection.

Being completely or almost completely free to play and still competing at a high level is definitely possible with some time and a good coalition (if you are looking for one, send me a PM).   I think the purchases are purposely made on the expensive side to reduce the pay-to-win aspect of the game, which is something I definitely appreciate.

Rhasget

freegenz said:

And this is why I'm super scared of ever getting to platinum tier. I'm a relatively new player (currently in gold, started in March) and I've yet to acquire a single origins Mythic... 

A good investment is to buy Origins Premium Packs and not just the current Standard sets. And craft rares and mythics from it. Since it's the core set it won't rotate out.

freegenz

@Mburn7
Yes, I've had very good success until being virtually F2P (purchases I've made have had no impact yet). I did have some good luck on which cards I got though. We'll see how it goes in gold.

@Rhasget
I totally agree, this is what I've been focusing on with my spare crystals, I just haven't had any luck there yet. Also, not only will they not go out of rotation, but there are also no events awarding them, so packs are their main (and almost only) source. 

Larz70

Great job @freegenz !  I did the same analysis for ORI, XLN, and RIX back before HOD came out and I used Monte Carlo simulation to calculate the median cost to get all the mythics if you buy only PPs and craft the mythics using the orbs obtained from duplicates.  I posted in the tips and guides section of the forum, but I realized later that the General Discussion is where the action is.  You've inspired me to dust up my old simulator and run it for DOM and M19.  At the very least, it can help provide the USD cost not just CAD, ;-P. 

Here's the link to what I did btw, https://forums.d3go.com/discussion/72612/how-long-will-it-take-to-get-all-the-mythics-in-the-ori-rix-xln-premium-packs-at-the-vault#latest

freegenz

Larz70 said:

Great job @freegenz !  I did the same analysis for ORI, XLN, and RIX back before HOD came out and I used Monte Carlo simulation to calculate the median cost to get all the mythics if you buy only PPs and craft the mythics using the orbs obtained from duplicates.  I posted in the tips and guides section of the forum, but I realized later that the General Discussion is where the action is.  You've inspired me to dust up my old simulator and run it for DOM and M19.  At the very least, it can help provide the USD cost not just CAD, ;-P. 

Here's the link to what I did btw, https://forums.d3go.com/discussion/72612/how-long-will-it-take-to-get-all-the-mythics-in-the-ori-rix-xln-premium-packs-at-the-vault#latest

Nice alternative way of doing it! Hadn't thought of it. I'm glad to see we arrived at very close numbers. The differences are actually because the % of each rarity were different when you did it. I look forward to your results for the new sets!