A mathematical approach to bonus heroes and buying covers

helix72

My current situation:

1. I am in the 4* tier
2. My goal is to max cover every 4* hero as quickly as possible, regardless of order
3. I don’t have enough ISO to champ any 4* at the moment
4. I don’t have a ton of extra time, money, CP, HP, or tokens
   

If this does not describe your situation, what I am working on may not entirely apply for you, but parts might be useful.

Towards that goal, I am focused on answering 2 questions:

1. When should I think about going after specific covers (e.g. Heroes for Hire, Vaults, 3* feeders, spending 120 CP, etc)?
2. Who should I set as my 4* bonus hero?

While saved covers are great later because of the rewards they generate, trainable covers are great now because they increase the abilities of your character and give you an immediate increase in level cap. So the difference between a random 4* cover (example: tokens), a random cover for a specific 4* (example: 4* bonus hero), and a specific cover for a specific 4* (example: heroes for hire) is important.  I have found this really handy for boosted characters and DDQ where even one additional power level and/or partial character leveling can make a huge difference.


I started by taking every possible non-max cover combination (0/0/0 through 2/5/5) and through brute force simulation (maybe there’s a math-y way to do it but my kung fu is not that strong) calculated the average number of random covers it would take to fully cover a character based on what their coverage is now.

Part 1: When should I think about going after specific covers?

Take a character at 1/5/5. A random cover has a 1/3 chance of being the cover you need, so on average you’ll need to draw 6 random covers for that character to get the 2 specific color covers you need. But if you were able to target and acquire that specific color cover, the character would now be 2/5/5 and the average number of random covers you would need for full coverage drops to 3. That’s a reduction of 3 random covers by acquiring 1 specific one. Not a bad investment if you have the HP, CP, tokens, or time.

Now take a character at 1/4/5. The simulations tell us that on average you’ll need to draw 6.4 random covers for that character to get the 3 covers you need to get to 13 (assuming you are impartial to 3/5/5 or 4/4/5). But if you acquired a cover that took it to 1/5/5, you still need to draw 6 more covers on average to get it to 13. That is after the 1 you would have just acquired, so you actually took a step backwards! Bad investment unless you really have a lot of extra resources required to obtain it.

So I applied this to my entire roster. Not surprisingly, getting black covers for my 5/1/5 Deadpool and a red cover for my 5/2/5 Ghost would be my best investments. Below that it gets more helpful, as going after a blue cover for my 4/3/0 Cyclops is a better investment than going after a blue cover for my 5/2/4 Jean Grey.

As you have likely figured out by now, the investment return here is the number of random covers needed prior to acquiring the specific cover minus the number of random covers needed after acquiring the specific cover.

I’ll post part 2 tomorrow with how I’ve answered question 2 and a link to the file with the simulation results.

As always, thanks for reading through and would love to hear feedback.

Jaedenkaal

Given that you can save covers for a character if they're not covers you can use, over time you'll end up with characters at the same level no matter which order you get the colors of their covers in. If you have a 2/5/5 character, and it takes 10 covers before you finally get that last cover in the correct color, you can (ISO willing) immediately champ them and they'll be leveled to 279.

So as to what that means to max-covering all 4*s as quickly as possible, I'd say all you really need to do is get as many covers as you can as quickly as possible. I don't think any particular strategy is needed to use Heroes for Hire to target specific covers, unless it's for a character that you like more than the rest (and have 3600 spare HP, I suppose). I also wouldn't spend CP to get a specific cover for one character, as that will cut into your ability to fully cover everyone else. 

You'll also want to start championing the 4*s you do have (or get) fully covered so that when you inevitably get another cover for them, you get some resources back right away.

Roland113

I'm about 42 champs into the 4 star tier and question these types of things as well.  Here is my follow up question for your simulations.

I interpreted your question as pertaining to individual characters.  How does your model apply to your roster at large?

If you're trying to max cover all 73 four star characters, you have the option of spending 120CP on a specific cover or 120CP for 6 random 4 or 5 star covers (Classic Legend Tokens).  Once you get to a certain point, you are statistically more likely to get a 14th cover for a fully covered toon than iyou are a usable cover for an undercovered toon.  At what point does it make more sense to spend the 120CP on a specific cover rather than 6 Classic Legends?

bbigler

Since your goal is to max cover them all (regardless of order) I would use CP to buy Classics and use extra HP to buy HFH covers for unevenly covered characters (5/5/x), plus open LTs.  Of course, play to get the 4* progression rewards in events too.  That will give you plenty of 4* covers to work with. 

Keep doing this until you've champed 75% of the entire tier and then start buying specific covers for 120 CP.  Why 75%?  Because if 83% of the tier is champed, then opening 6 Classics for 120 CP should give you 1 cover for a non-champed character.  But that 1 "useful" cover may go to a character power that's already maxed, like 5/5/x.  So, open Classics until you have 75% champed, then start buying specific covers.  You could simply track your LT pulls and decide when it's best to make the switch. 

Personally, I wouldn't do this, for 3 reasons: #1 they continue to release new 4*s, making it harder and harder to cover them all, and #2 after you've champed half of the tier, adding more champs isn't going to make a big difference, and #3 I would prefer to continue to add champ levels to everyone else.  I'm guessing you have no desire to transition to 5*champs?

helix72

@Roland113 , love the question! You're much farther along so I haven't had to think about that until now. I wouldn't use the simulations to do it though.

Assumptions:

You don't care about CP "waste" or 5* covers and all you care about is getting trainable 4* covers. I think a lot of folks would advise you against that strategy due to the champion rewards for 4* and the possibility of getting 5* characters, but let me know if I got the question wrong.

Simplifications: In the interest of getting you a fast answer, I'm going to oversimplify and ignore a few things:

First, I'm going to assume each token is worth exactly one random 4* cover. This ignores that you might get one or more 4* bonus heroes and one or more 5* characters from those 6 tokens
Second, that one or more of the 6 covers you get might actually "fill up" one or more of the covers you need, lowering the odds of subsequent trainable covers
Third, that not all characters are available from tokens yet (Dazzler!)

Keeping those in mind:

As you pointed out, 120 CP is worth 6 random covers, and there are 73 total characters who have 3 color covers each or 3 x 73 = 219 unique covers. In that case, when the expected number of useable covers from 6 tokens is less than 1, you would switch from buying tokens to spending CP.

Using a simple binomial distribution, that flips somewhere 182 and 183 "full" covers.

You've got 42 champs, so you have at least 42 x 3 = 126 full covers. Then you need to look at your non-champed roster and add up how many other full covers you have to see if you're at or above 183. Remember to count any full covered character as 3 and then look at which other characters have 5 in one or more colors.

As for the simplifications I made in the process, I'll think about how I can take those into account and where that moves the odds.

BoyWonder1914

Not in your exact situation, but the quandary of knowing when it's worth it to go after specific covers is definitely a relevant one nonetheless. I'm sitting on 65 out of 73 champed. Out of who remains: 

Dino - 5/5/3
Howard - 5/3/5
Jubilee - 5/5/3
Spidey - 3/4/5
Ghost - 3/3/3
Nebula - 0/2/3
Emma - 0/1/3
Dazzler - TBD

You can remove the first 3 for obvious reasons, so it's essentially just 5 characters that I'm going for. In looking at @Dougfresh413 300-pull thread, it was a little unnerving to see all of those pulls distributed so evenly across the tier. Sure there were some oddballs that got 7 covers, but overall he pulled EVERYONE with the single exception of PX, an average of 2-4 times. Combining that with my first experience of not bothering to save CP and just pull every time I got 20 or 25, I'm not very optimistic of covering a specific bunch of the 4s anytime soon. Not unless they redistribute the odds to "latest" characters again in some fashion. Regardless though, burning the hoard and attempting to finish the characters through brute force math is really the only option - as the one cover I may happen to get from a vault, HFH, or progression will still leave most of them quite a ways away from being battle-ready. Even if a character is just one cover away, it seems more fruitful to just make them a BH and keep pulling than to chase after specific deals awarding them. Getting a dupe isn't the end of the world anymore, as its now just a champ level on a hold. 

I understand I have a bit of luxury, as with an expansive list of 4-star champs, champing the next one gets less and less exciting unless you know they're going to make a big impact on your roster. The more champs you have, the better you get at gauging that impact, so you know almost right away how big a difference someone will make once they're champed.  The only one of this crop I really care for like THAT is Dazzler - she seems she'll be quite the beast, and won't need to rely on tanking to dish out her best stuff. To someone still getting into the 4-star tier though, I'd say only bother going out of your way for a specific cover if the character is a beast, and will immediately change things for you. 

1. If you've got 5 champs, and a vault/H4H will finish your 5/3/4 Peggy or CM4, by all means go for it. That's going to be huge for you. 
2. If you've got 30 champs, and are in the same situation, then that might be more debatable for you. They're still top-tier characters, but you've likely already got some, and they won't have the same impact for you as a brand new transitioner. 
3. If you've got 50 champs, then clearly you've been just fine without either of them, and it's really all just a matter of personal preference at this point. 

The people in situations 2 & 3 will likely be just fine until another chance to obtain this character in free way comes around. Someone who's newer to the tier and are eager to become more competitive however might love the opportunity. If all you truly care about is eventually getting the character covered, regardless of exactly when, then I'd DEFINITELY say its not worth it to drop resources in the heat of the moment. Hoarding resources and and just playing the RnG long game is what most of us settle into, and while it may take longer for some than others, it does still eventually work. I'm hoarding currently until I get 1000 CP again (currently at 700), and then I will pull from classics , BH these guys and hoping to finish who I can. If not, I'll just hoard and try again, I know I'll eventually get them all. Patience is honestly the hardest thing about this, because it can feel like if you don't get them NOW, you don't know if  you'll ever see them again. 

crackninja

There's never a point where spending 120cp for a single 4* cover is a valid strategy.  It can be a smart move if it allows for an appreciable change to your gaming experience, but unless you don't plan to actually play the game after championing everybody, it's a massive resource loss. 

Warbringa

crackninja said:

There's never a point where spending 120cp for a single 4* cover is a valid strategy.  It can be a smart move if it allows for an appreciable change to your gaming experience, but unless you don't plan to actually play the game after championing everybody, it's a massive resource loss. 

From a pure game play/economics side of the argument you are correct crackninja (and I agree with you).  For a completionist though, whose only goal is to cover and champ all 4*, it isn't.  The OP seems to be a completionist first who is then looking at the economics of how to fill out the 4* tier in the most economical way with that primary goal in mind.  I am guessing at this because 5* characters were not even brought into the OP's analysis and he mentioned his goal in the post.

bbigler

Everyone, please remember that Dino and Howard are not in packs!  So, calculate based on 71 characters, not 73. 

Just wanted to clear that up.

Theoretically, you can never champ all 4*s because they keep releasing new ones.  But for the sake of argument, if you could get 25 x 4* covers per week with somewhat even distribution, it would take about 43 weeks to cover them all from scratch, but during that time they would release another 14 new 4*s.  So, it seems like a pointless endeavor.

shardwick

bbigler said:

Everyone, please remember that Dino and Howard are not in packs!  So, calculate based on 71 characters, not 73. 

Just wanted to clear that up.

Theoretically, you can never champ all 4*s because they keep releasing new ones.  But for the sake of argument, if you could get 25 x 4* covers per week with somewhat even distribution, it would take about 43 weeks to cover them all from scratch, but during that time they would release another 14 new 4*s.  So, it seems like a pointless endeavor.

People will be swimming in Dino covers pretty soon and even though Howard isn't in covers people can get him here and there in Webbed Wonder. As for getting 4s covered it really depends on the player. If you're a competitive player in a top 100 alliance then you could essentially get 6+ covers for a new 4* during their release. I got 7 just for Nebula and if they introduce a new feature soon that helps to greatly lower dilution, like custom packs that would let you pick a certain amount of 4s to pull from like 12-20 versus all but limited/newest, then it would be even easier to get newer 4s, or even just specific older ones that you're more interested in, fully covered much faster than the current system.

bbigler

shardwick said:

 if they introduce a new feature soon that helps to greatly lower dilution, like custom packs that would let you pick a certain amount of 4s to pull from like 12-20 versus all but limited/newest, then it would be even easier to get newer 4s, or even just specific older ones that you're more interested in, fully covered much faster than the current system.

That's just a hopeful dream of getting smaller LT pools, which is an idea I posted about a week ago.  The current system makes it very difficult to cover new 4*s.  Let's say you get 6 covers off the bat for a new 4*, then you continue to open LTs.  To get those remaining 7 covers, it would take about 500 more pulls, and that's assuming even distribution, but they continue to release more 4*s making it take longer than that.  You could set them as your BH, but what about all those other new 4*s that aren't covered yet?  It's futile.

shardwick

bbigler said:

shardwick said:

 if they introduce a new feature soon that helps to greatly lower dilution, like custom packs that would let you pick a certain amount of 4s to pull from like 12-20 versus all but limited/newest, then it would be even easier to get newer 4s, or even just specific older ones that you're more interested in, fully covered much faster than the current system.

That's just a hopeful dream of getting smaller LT pools, which is an idea I posted about a week ago.  The current system makes it very difficult to cover new 4*s.  Let's say you get 6 covers off the bat for a new 4*, then you continue to open LTs.  To get those remaining 7 covers, it would take about 500 more pulls, and that's assuming even distribution, but they continue to release more 4*s making it take longer than that.  You could set them as your BH, but what about all those other new 4*s that aren't covered yet?  It's futile.

I've been advocating for custom packs since classic 4s were vaulted. I got tired of getting lots of Spider Woman covers and wanted to focus more on getting more covers for older character that at the time I needed to roster and build up. It's not overly difficult to get more covers for newer characters. I quit the game earlier this year and since coming back about two months ago I had to roster Iron Spidey, Wiccan, Widow, Valkyrie and Jubilee from scratch. Spidey is max covered, Wiccan's at 12, Valkyrie at 10, Widow at 9 and Jubilee is at 4. Once I get Widow to 10 then I'll pour covers into Jubilee. With how often newer 4s get featured in events it doesn't really take me long to get them to at least 10 covers especially with the help of bonus heroes.

helix72

@bbigler, right you are. If we exclude Dino and Ducko, still using my original simplifications, the math is still the same just on a smaller population: once you have 5/6 of 4* universe covered, or 71 x 3 = 213 * 5/6 = 177.5, round up to 178, you're more likely to get the covers to finish covering faster by spending the 120 CP versus buying 6 tokens.


And thanks @Warbringa, you got the point. It's not an approach for everyone, just for completionists low on patience.

Pongie

crackninja said:

There's never a point where spending 120cp for a single 4* cover is a valid strategy.  It can be a smart move if it allows for an appreciable change to your gaming experience, but unless you don't plan to actually play the game after championing everybody, it's a massive resource loss. 

I did this once when I first championed hawkeye and tip my toes into 5*. The first couple of events were brutal. I was struggling in a field of better team compositions. Coulson was the pve reward and I failed to earn 3 covers (only got 2). He was 1 cover shy of 13 and those 2 were already maxed at 5. It was a no brainer to cough up the 120cp and champion Coulson, apply the 2 champion levels and have a much easier time. Could even beat teams of 550s with this pairing.