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Leagues

Sir Derf

Artisan
I'm bored, so can't help myself. If I'm typing ahead of you, and in an incorrect direction, oh well...

Again, trying to understand your position, and in terms of the 20d6 vs. 6d20, are you putting forth the proposition, "It is better to go for the best chance of truly heroic outcomes, regardless of how I might fare otherwise"?

Working out everything about 6d20 and 20d6 is a monumental pain (consider all the different ways to roll a 52 in either system), but I noodled out a little bit. The typically inferior 6d20 does have the advantage over the typically superior 20d6 at scoring the herculean 120...

6d20 - 0.00000156%
20d6 - 0.0000000000000274%

That's just over 57 million times higher, but is only going to happen 1 time in 64 million.

How about the odds of 118-120?

6d20 - 0.00004375%
20d6 - 0.0000000000063181%

That's nearly 7 million times higher, but is only going to happen 1 time in 2.3 million.

Somewhere the two curves are going to cross. There is some value above which you are more likely to get it with the 6d20 than with the 20d6. I don't know where it is, but I'm betting it's not that big a range, and all at a very low chance. Let's say (and I think I'm overestimating just to be generous) that's at 105, and the odds are 1 in 1000. Is it really worth it to say "in the 1 in 1000 chance that either of us rolls 105 or higher, I am likely to roll 1-15 points higher with 6d20 than with 20d6?"
 

CrazyWizard

Shaman
I'm bored, so can't help myself. If I'm typing ahead of you, and in an incorrect direction, oh well...

Again, trying to understand your position, and in terms of the 20d6 vs. 6d20, are you putting forth the proposition, "It is better to go for the best chance of truly heroic outcomes, regardless of how I might fare otherwise"?

Working out everything about 6d20 and 20d6 is a monumental pain (consider all the different ways to roll a 52 in either system), but I noodled out a little bit. The typically inferior 6d20 does have the advantage over the typically superior 20d6 at scoring the herculean 120...

6d20 - 0.00000156%
20d6 - 0.0000000000000274%

That's just over 57 million times higher, but is only going to happen 1 time in 64 million.

How about the odds of 118-120?

6d20 - 0.00004375%
20d6 - 0.0000000000063181%

That's nearly 7 million times higher, but is only going to happen 1 time in 2.3 million.

Somewhere the two curves are going to cross. There is some value above which you are more likely to get it with the 6d20 than with the 20d6. I don't know where it is, but I'm betting it's not that big a range, and all at a very low chance. Let's say (and I think I'm overestimating just to be generous) that's at 105, and the odds are 1 in 1000. Is it really worth it to say "in the 1 in 1000 chance that either of us rolls 105 or higher, I am likely to roll 1-15 points higher with 6d20 than with 20d6?"
That says it all, just because in this example both cases are very slim, the fact that the difference is this big says it all.

20d6 gives out a very stable output of damage. because you throw 20 dies the likelyhoof for extreme results (both upt and down) are very slim.
and you just said yourself with 6d20 the chance for an extreme result is 57 million times higher.
That is not an insignificant difference!!!

If you want a very good result in an event like this, even if the chance is very slim, you will need the 6d20 approach. it's the only probable route to success. of course it has it's risks on the flip side.

But if you throw 20d6 the chance to be one of the higherst rollers you claimed yourself is now 57 million times lower.
So to be able to get a top result, 1 of the goals would be to limit the amount of risky rolls, the less time you roll the more random the result will be and the more likely you are to het into silver / gold.

The more dice you roll the more likely you are to get a similar result than everyone else. because more chances means that extreme results are less likely to occur.
This is why if you want the set, or in former events you always wanted as many rolls as possible to limit score swings and get a stable result.
With silver / gold what you want is the opposith, you want extreme results as only that will get you in the top 1% or close to it. a stable result will never get you in the top 1% or close

Your own example confirms this and shows the difference is huge.

the 27 chests if you are unlucky will cost you 20% currency per staff than the 45 chests.
but 19 vs 19.6 means the costs for the 45 chest on average will be 4% higher. but your risk mitiligation is 20%.
Also in my personal case, less die rolls is a bigger chance on a result swing to a much lower or higher state. but from a risk management point if I want a stable endresult and thats my goal I also take the 4% hit for a 20% less risk factor.
 

Sir Derf

Artisan
The difference in the dice is huge, because the odds we're talking about are tiny. And, I believe are too tiny to satisfy your purpose.

You, my friend, are aiming for a different target than am I, and also I think for a slightly different target then you think you are. And the mathematical particulars behind both the dice analogy and the Elvenar problem we have been wrestling with I think both foil your real goals.

Looking back over this conversation, you seem to be setting a goal of being in the top tier of the Leagues (which makes sense), which is not necessarily the same goal as maximizing your score. And is also not the same as giving the best odds at the highest score. If your goal is Gold League, then your goal is to get the best odds at what will probably be the top 1%.

So you have 57 million times better odds of getting the perfect 120 with 6d20. At 1 in 64 million, that's a slim chance at overkill for trying to maximize being in the 1%. If the question is "Who will roll a perfect 120 first?", 6d20 is your best bet. It might take forever and a day before someone wins, but that should be 6d20.

So you have 7 million times better odds of getting 118-120 with 6d20. At 1 in 2.3 million, that's still a slim chance at overkill for trying to maximize being in the 1%. If the question if "Who will roll 118-120 first?", again go with 6d20, and this time it might only take forever.

Let's take my guess at the crossover. If both methods have equal chance at 1 in a thousand for 105 or higher, 6d20 and 20d6 have broken even at attaining the 0.1%. But 0.1% was not our goal. Having crossed over, that means that 20d6 would have the far superior chance of giving you values that would place you in the top 1%, behavior in the top 0.1% notwithstanding.


And I think it's even worse for the Elvenar situation, because the dice analogy breaks down when trying to think about Elvenar. Dice are like Elvenar beacons that take SK and convert them directly to staffs. Start with 120 SK, you can open 20 d6 chests or 6 d20 chests, and can get up to 120 staffs from the process. But Elvenar beacons don't just convert SK to staffs, they can also give additional SK. Which can compound into still more SK. And not just more, they can possibly give infinite additional SK. A closer Elvenar to dice analogy would be like every net6 let you roll an additional 3d6, and every nat20 let you roll another 2d20. There's also the fact that Elvenar chests aren't smooth even outcomes. 'Unlucky' rolls don't come as 1, 2, 3, 4, and lucky as 5 & 6; You either get the bonus ('80', '200') or you don't ('0', '0'). That also makes a difference.

The 27 chest is superior to the 89 chest under average conditions. The 27 chest is even more superior to the 89 chest with doubled odds and even still more superior to the 89 chest with tripled odds. I thought it would break when, at 733% (6.33-ed odds) the 89 chest had 100% chance of giving bonus (that is, spend 89 SK and every time receive 200 SK bonus), but the 27 chest at 83.33% return rate was still beating it. At all equal 'luck' levels, at all equally unlikely situations, the 27 chest beats the 89 chest, both in maximizing score and in maximizing the possibility of being in the the top 1 %.
 

Paladestar

Enchanter
Pretty sure neither of your two towns will get into the gold league. The lower one will probably not even make silver, unless the server are very different.
I guess what I want to say: Luck is not getting you into the gold league. Unless you have silly big luck.
Luck will be the deciding factor between bronze and silver.

Effort gets you bronze
Luck gets you silver
Money gets you gold
Right now:

Pala has 236 (EN2, Bronze = 208, Silver = 260)
Acha has 301 (EN3, Silver = 260), Gold = 305)

Acha close to gold without spending any diamonds, just total luck.
 

Gargon667

Mentor
Right now:

Pala has 236 (EN2, Bronze = 208, Silver = 260)
Acha has 301 (EN3, Silver = 260), Gold = 305)

Acha close to gold without spending any diamonds, just total luck.
I would say that fits the prediction as good as can be expected:
Effort (Pala) gets you bronze
Luck (Acha) gets you silver
Money (neither) gets you gold.

My own town is pretty much in between the two, but closer to Pala, so I am hoping for silver, but more likely leaning towards bronze. I can´t really guess yet how big the rush will be in the end.

That also means the set will be finished before reaching the silver league (I already got that one, and am guessing Pala doesn´t have it yet?)
 

Gargon667

Mentor
Rather than blindly type further, are you suggesting that it is better to go with the 6d20, because with fewer rolls, it is easier to luckily make a 100+ damage with the 6 larger rolls than the 20 smaller rolls? I think that is the argument you are making, but before I go further, I wanted to verify that.
I would say yes, that was his idea! And also my thought, how it could work out to be. Not my goal though ;)

The difference in the dice is huge, because the odds we're talking about are tiny. And, I believe are too tiny to satisfy your purpose.
That is proof (and I never doubted it) that indeed extreme results are more likely on 80ies.

Also nobody ever doubted that the average is better on the 27.

Which leaves us with the one big question. How extreme does the result need to be? And i think we will be able to answer that question when the event ends. We´ll know how many staffs were necessary to get into gold league and can then calculate backwards.
We will then know if the result needed to be sufficiently extreme to merit the extremist tactics (pure gambling) or if it was sufficiently low to fall below the crossing point of the two curves and one is better of with the higher average (and additional luck) tactics.
 

Sir Derf

Artisan
That is proof (and I never doubted it) that indeed extreme results are more likely on 80ies.
You are not paying attention to the math I am describing, and the difference between dice and beacons. Despite the behavior of dice, Elvenar beacons do not have an increased heroic chance for the harder odds.

With dice, 6d20 has the better-but-infinitesimal chance of producing a heroic 120 total because 20s weigh more than 19s weigh more than 18s.
With beacons, 80ies do not have that same better chance, of any size, of producing a heroic total, because there is no weight in their successes.

First, let me change the dice analogy so it is mathematically closer to Elvenar beacons. To better simulate Elvenar beacon numbers, lets consider 50d20 vs 10d100. As straight dice, the larger die still has the microscopic edge on the heroic 500, this time 1x10^-20 vs 8.88*10^-66, as well as at least the next several heroic values. But Elvenar beacons don't scale across all rolls of the dice; Elvenar beacons transform the entire spectrum of rolls into a single binary outcome, bonus or no bonus. The 27 beacon is like a d20, where bonus occurs with 19 or 20, and no bonus with 1-18 (10%); the 89 beacon is like a d100, where bonus occurs with 88-100 and no bonus 1-87 (12%). Look at what we're doing with the d100. A roll of 100 is no more important than an 88; they both collapse into a single value of 'bonus'. A roll of 0 is no less important than an 87; they both collapse into a single value of 'bonus'. The peak output of 50d20 is 50 'got bonus' results, which comes from combined rolls of from 950-1000 (not all 950-998, but bear with me), while the peak output of 10d100 is 10 'got bonus' results, which come from combined rolls of 880-1000 (not all 880-989, but again bear with me). Throwing 5 100s is no longer heroic, it is democratically equivalent to throwing 5 88s, or a 90-91-92-93-94 combo; you get the same 200 bonus in all three of those circumstances.

So, as misguided as I think the strategy of striving for the highly unlikely heroic at the expense of the highly likely typical, which mathematically is possible with straight dice, it is a strategy that is not mathematically applicable with Elvenar beacons.

One really has to be careful with building mathematical models of reality, and with the portability of conclusions from one model or scenario to another.
 

Laurelin

Conjurer
Just to be argumentative - It's 5.2% in American roulette and it's 1.35% in French roulette.
Very interesting to learn the actual odds underlying roulette systems! But yes - it's precisely this knowledge, i.e. that the house will always win long-term in any kind of gambling scenario, that motivates me, non-thrill-seeker as I am, to choose a lesser-but-assured reward (assuming that it's sufficient for my needs, of course) above a greater-but-risky reward (unless only by taking a risk can my needs be met), in video games or anywhere else, and certainly in this particular game, since experience shows that my Elvenar luck is typically below average - not dismal, but definitely poor.

And while I know that there's no reason why my luck in any large-scale equitable (if small-scale patchy) RNG system shouldn't suddenly change to being excellent, or at least average, meaning that I may be depriving myself of an improved overall outcome by taking as few chances as I can (e.g. by choosing Event Chests according to their face value guaranteed return and excluding their chances of winning extra Currency), I'd rather have [what would be in most Events, although not this one, unfortunately] the certainty of winning what I am aiming for (the Event's Grand Prize, in this case) rather than risk losing out on one or more Set pieces (or Artifacts, etc.) by gambling on the chance of winning the full Set plus any additional possible - but not assured - reward(s).

All that said, though, and despite my [unchanged] generally anti-risk stance, I have - to my surprise - finally managed to gain the full Pilgrim's Manor Set, with 14 (!) SK currently to spare, since the last few sets of Beacons available to me while I was spending my final ~1,000 SK yesterday and today happened to represent, four times in a row, the only situation in which I've chosen the 80s or 89s during this Event (i.e. I was given undesirable [by my face-value definition] low- and mid-value Beacons (mostly 30s and 54s) with either an 80 or 89 [which is such a small SK-per-Staff cost difference that taking a chance on winning a good amount of SK from the high-value Chests was, in my opinion, justified, particularly since I would otherwise definitely fail to earn enough SK to complete the Set without paying Diamonds] - and I was, unusually, fortunate enough to win 2 x 200 and 1 x 300 extra SK from only four Beacons). Although one might think that this change in my usual degree of luck may, or even should, convince me to adopt a less cautious strategy in future, I'm still of the view that an assured (but lesser) sufficient return is preferable to a risk-based (but greater) potential return. I suppose it really comes down to the fact that I'm definitely a 'bird in the hand is worth two in the bush' person!

Had I not had such good luck, though (and I very much sympathise with any other players who have not had such a last-minute reprieve), I was [unhappily] thinking about making this the one and only occasion on which I would spend enough Diamonds to effectively buy the final Forge piece of the Set. This would be against my better judgement under most circumstances, since it's hardly a good idea to encourage a F2P game to start gating Event (or any other) rewards behind paywalls (and in the specific case of InnoGames, they're not short of a bob or two - to say the least - even without this new mission to generate even more revenue...), but I need the Pilgrim Manor's full production more than any other Grand Prize in any Event so far, in order to compensate for (a) the lack of Boost+2 Relics which is about to ensue in my City due to the latest (and unexpected) Tournament changes, and (b) my lack of enough T1 Goods [in particular] to increase my Catering levels in order to earn said Relics under the new 'cycling Relics' system - and I doubt I'm alone, either. While the Set appears, primarily, to be intended to provide new players with something to replace and/or support the single Moonstone Set they can now achieve, I don't find it coincidental that the Tournament change, with its potential to reduce many players' ability to build and/or maintain a stockpile of Boost+2 Relics sufficient for Crafting as well as for their Ascended Goods Boost, has been released at the same time as a Set which produces exactly those Relics... just as I also don't believe it's coincidental that the competitive Leagues system, designed as it is to encourage spending, has been introduced alongside such a desirable Grand Prize.

Of course, there's always the chance that the next Grand Prize, and the next, and so on, in an increasing spiral, will somehow be at least equally or maybe even more desirable - perhaps because of yet more currently unforeseen changes to the game, but although I don't underestimate the capacity of many F2P games to blithely scupper their own gameplay quality and/or even knowingly alienate the less financially rewarding portions of their playerbase if greater overall profits will (sometimes merely theoretically) result, I think InnoGames are far too experienced to do any such thing.

P.S. : Many thanks to @Sir Derf - again! - this time, for explaining in words as well as in figures alone how you've reached your conclusions; it's this which genuinely makes the difference, in my case, between broadly understanding [I think?] what your calculations show, and having quite literally no comprehension of them... so it really is much appreciated. I wish my Maths teachers at school had been as assiduous as you are in explaining their calculations - I may then actually have been able to understand algebra more complex than 2x = 4 ...! :)
 

Gargon667

Mentor
And while I know that there's no reason why my luck in any large-scale equitable (if small-scale patchy) RNG system shouldn't suddenly change to being excellent, or at least average, meaning that I may be depriving myself of an improved overall outcome by taking as few chances as I can (e.g. by choosing Event Chests according to their face value guaranteed return and excluding their chances of winning extra Currency), I'd rather have [what would be in most Events, although not this one, unfortunately] the certainty of winning what I am aiming for (the Event's Grand Prize, in this case) rather than risk losing out on one or more Set pieces (or Artifacts, etc.) by gambling on the chance of winning the full Set plus any additional possible - but not assured - reward(s).

All that said, though, and despite my [unchanged] generally anti-risk stance, I have - to my surprise - finally managed to gain the full Pilgrim's Manor Set, with 14 (!) SK currently to spare, since the last few sets of Beacons available to me while I was spending my final ~1,000 SK yesterday and today happened to represent, four times in a row, the only situation in which I've chosen the 80s or 89s during this Event (i.e. I was given undesirable [by my face-value definition] low- and mid-value Beacons (mostly 30s and 54s) with either an 80 or 89 [which is such a small SK-per-Staff cost difference that taking a chance on winning a good amount of SK from the high-value Chests was, in my opinion, justified, particularly since I would otherwise definitely fail to earn enough SK to complete the Set without paying Diamonds] - and I was, unusually, fortunate enough to win 2 x 200 and 1 x 300 extra SK from only four Beacons). Although one might think that this change in my usual degree of luck may, or even should, convince me to adopt a less cautious strategy in future, I'm still of the view that an assured (but lesser) sufficient return is preferable to a risk-based (but greater) potential return. I suppose it really comes down to the fact that I'm definitely a 'bird in the hand is worth two in the bush' person!
Now I would interpret these facts you present quite a bit differently

Choosing the lower risk (=lower average gain) beacons has nearly cost you finishing the set. Lucky for you, the luck was forced on you and you managed to finish the set. If you instead had had the chance to not gamble and chose the boxes you prefer, you would certainly NOT have gained the full set.
Picking the higher average (risk) boxes is not thrill-seeking at all, it is rather the opposite. The sensible choice if you wish to complete your grand prize (there is no point in aiming any higher than that anyway now for non-paying players) is the chests with the highest average. And you basically have the proof for that in your own data. Choosing face value as much as possible did NOT get you the set. Only choosing the higher risk boxes did. of course it could have gone wrong with those boxes, in which case you would have ended up the same way as choosing face value would have: An incomplete set.
Your tactics of face value have worked in the past, due to a bigger margin on gaining the full grand prize. The past was more forgiving to people who choose less than the best average. And I have to admit in that old scenario your tactics even had merit, as you could have a guaranteed grand prize finish with little else instead of a better average result without guarantees of finishing. That however does not seem to be true anymore: From now on choosing face value will be a guarantee to NOT complete the grand prize, while choosing the best average will give you a chance of completing the grand prize. Nothing (but spending money) will give you a guarantee to finish the grand prize.


Had I not had such good luck, though (and I very much sympathise with any other players who have not had such a last-minute reprieve), I was [unhappily] thinking about making this the one and only occasion on which I would spend enough Diamonds to effectively buy the final Forge piece of the Set. This would be against my better judgement under most circumstances, since it's hardly a good idea to encourage a F2P game to start gating Event (or any other) rewards behind paywalls (and in the specific case of InnoGames, they're not short of a bob or two - to say the least - even without this new mission to generate even more revenue...)
And this is exactly the reason why this is happening. this event would have made even you spend money on it! Just imagine how difficult it is to "convince" a person to spend money on something, they have never before spent money on! They have tried for years now and in this event they finally succeeded and you caved! I know you didn´t actually pay this time, but you were willing for the first time. It will be the same for a LOT of people out there and they may not be as lucky as you were to get away with a bit of gambling. This will probably be the most money generating event in Inno´s history! And that means it is highly unlikely they will give us more event currency next time. They are most likely still looking for that sweet spot where the event is bad enough to make people spend money to make it better, but it can´t be too bad so people completely ignore the whole thing.
 

Gargon667

Mentor
You are not paying attention to the math I am describing, and the difference between dice and beacons. Despite the behavior of dice, Elvenar beacons do not have an increased heroic chance for the harder odds.
I think I do, although I am not 100% sure.

First, let me change the dice analogy so it is mathematically closer to Elvenar beacons. To better simulate Elvenar beacon numbers, lets consider 50d20 vs 10d100. As straight dice, the larger die still has the microscopic edge on the heroic 500, this time 1x10^-20 vs 8.88*10^-66, as well as at least the next several heroic values. But Elvenar beacons don't scale across all rolls of the dice; Elvenar beacons transform the entire spectrum of rolls into a single binary outcome, bonus or no bonus. The 27 beacon is like a d20, where bonus occurs with 19 or 20, and no bonus with 1-18 (10%); the 89 beacon is like a d100, where bonus occurs with 88-100 and no bonus 1-87 (12%). Look at what we're doing with the d100. A roll of 100 is no more important than an 88; they both collapse into a single value of 'bonus'. A roll of 0 is no less important than an 87; they both collapse into a single value of 'bonus'. The peak output of 50d20 is 50 'got bonus' results, which comes from combined rolls of from 950-1000 (not all 950-998, but bear with me), while the peak output of 10d100 is 10 'got bonus' results, which come from combined rolls of 880-1000 (not all 880-989, but again bear with me). Throwing 5 100s is no longer heroic, it is democratically equivalent to throwing 5 88s, or a 90-91-92-93-94 combo; you get the same 200 bonus in all three of those circumstances.

So, as misguided as I think the strategy of striving for the highly unlikely heroic at the expense of the highly likely typical, which mathematically is possible with straight dice, it is a strategy that is not mathematically applicable with Elvenar beacons.

One really has to be careful with building mathematical models of reality, and with the portability of conclusions from one model or scenario to another.
Say we have 6000 SK (so I can do it in my head) Wouldn´t you agree that it is much more likely to get 60 bonus out of 60d100 than it is to get 300 bonus out of 300d20?

The point I am trying to make is that repeating the roll more often will generate a more average result. Yes the average result will be better, but that is completely besides the point. I need an extreme result and my chances for an extreme result are higher if I roll less often. Picking the big boxes lets us roll fewer times.

I agree that mathematical models are simplifications of reality and not always correct :) But I have actually found a good example of where this principle works perfectly in real life, a whole industry is based on it: The lottery!
Everybody knows how it goes: you put in your 10 € and you get back nothing. Yet there are millions of people putting in their 10€ every week! They do it because they have this tiny chance to win 10 000 000 €, which of course realistically everybody knows will never happen.
Same here: we throw away our SK, just for the 1-in-a-million chance to win big.

As I said before: I believe the only real question here is: do we need the jackpot from the lottery to get into the gold league or is less enough? That crossing point you mentioned earlier where the higher chance for the least likely event crosses the line of the better average outcome. I do believe this crossing point exists somewhere, no matter what kinds of dice or beacons we talk about. If the result we need is above the crossing point we need to gamble at maximum risk possible and have a near guarantee to fail of course.
 

Sir Derf

Artisan
I may be willing to concede when it comes to simply rolling and summing dice, but I think we need to be more precise on defining what the goal of the dice game is, in order to decide which set of dice (6d20 or 20d6)is the best choice.
  • Are we competing to get a perfect 120 first, rerolling until someone succeeds? 20d6 is best, but I'm not playing.
  • Are we competing to get the largest number in a single mass roll of dice? 6d20 is best.
  • Are we competing to try and get above x in a single mass roll of dice, and if so, what is x? The answer depends on x?
  • Are we competing to try and get in the top x% of all n players in a single mass roll of dice, and if so, what is x and n? The answer depends on x and n?

But, Elvenar beacons do not behave like summing up points on multiple rolls of the dice. There are similarities, but there are also huge differences.

When the internal RNG (or d100) operates to determine the prize when you select an 89 beacon, there is no increased benefit if the RNG returned 89 or 100; they both fall in the 12% range of 89-100, and both give the exact same bonus of 200 SK. Similarly, there is no difference if it returned a 1 or an 88; they both fall in the 88% range of 1-88 and both give the exact same lack of bonus. You could use your initial 6000 SK and roll 67 89s or 67 100s or any of the 43,416 permutations of 67 rolls between 89 and 100, and in all cases get the same 67 bonuses. There is no benefit for the heroic 67x100.

When the internal RNG (or d20) operates to determine the prize when you select a 27 beacon, there is no increased benefit if the RNG returned 19 or 20; they both fall in the 10% range of 19-20, and both give the exact same bonus of 80 SK. Similarly, there is no difference if it returned a 1 or an 18; they both fall in the 90% range of 1-18 and both give the exact same lack of bonus. You could use your initial 6000 SK and roll 222 19s or 222 20s or any of the 444 permutations of 222 rolls between 19 and 20, and in all cases get the same 222 bonuses. There is no benefit for the heroic 222x20.


And what happens if you were to get all your prizes? Well 67 successes with the 89 beacon would get you 13,400 SK in bonus. That's impressive. That's a return of more than double your original 6000 SLK. On the other hand, 222 successes with the 27 beacon would have gotten you 17,760 SK in bonus. A 27 Sweep gets you more than an 89 sweep.

But what, what about the odds, you say? For the 89 beacon, being successful with 67 runs at 12% odds is 2.02*10^-62, while For the 27 beacon, being successful with 222 runs at 10% odds is 1*10-222. Wow, look at that, sweep 89 is 160 orders of magnitude better than sweep 27, isn't it? That means it is the better choice, doesn't it?

Except, remember that the more likely 89 sweep doesn't give the same rewards as the less likely 27 sweep. The 27 sweep achieves way more in return. It isn't necessary to do a 27 sweep, and still do better than an 89 sweep. For a set of 222 27 beacons to obtain at least 13,400 SK bonus, it only needs to have at least 168 successes. It can fail to get a bonus on 54 of its 222 attempts and still beat an 89 sweep. What are the odds on getting at least 168 bonuses out of 222 27 beacons? (scribble, scribble, mutter, mutter, carry the 14...) 7.45*10^-43. 27 beacons can equal or better a sweep of 89 beacons 19 orders of magnitude more often.


There is no condition of picking 89 beacons where it is more likely to give out a higher return of bonus SK than an equally probable outcome from picking 27 beacons. That's the math the way I calculate it. Feel free to point out where I have made a math or logic error. Or to put forth an example that you think contradicts this; but if you do, please attempt to show the math to back it up.
 

Sir Derf

Artisan
The flow of this conversation demonstrates exactly why probability and statistics are so damn frustrating. You bring up the lottery, and that is yet a third system that has similarities and differences, and attempting to draw conclusions from the one system and then try to map it into the other just isn't appropriate. There is information to be learned by looking at different games, or scenarios, or whatever, but most of these games don't map exactly onto each other, and any insight, strategies or conclusions gained might not have cross-game applicability.

I would like to point out a huge difference between the lottery and Elvenar.

With the lottery, you are faced with three options...
  • Don't buy a lottery ticket, keep the $10 (sorry, I is American and don't have the fancy 'e' symbol handy) - Total $10
  • Buy a lottery ticket and loose - Total $0
  • Buy a lottery ticket and win - Total $10,000,000
From there, you can make all the arguments about 'risk' and 'personal assessments' and 'aspirational chances' and 'small loss/big reward' you want.

But, that doesn't transfer to the world of Elvenar quests. SK has no use if you don't play. Choosing not play the event does not leave you with the Elvenar equivalent of $10 dollars in your pocket.

I tried to build a simple lottery as Elvenar analogy, but every detail that I left out was a detail that materially changed the nature of the game.
 

Gargon667

Mentor
  • Are we competing to try and get above x in a single mass roll of dice, and if so, what is x? The answer depends on x?
  • Are we competing to try and get in the top x% of all n players in a single mass roll of dice, and if so, what is x and n? The answer depends on x and n?
The goal is to get into gold league based on luck and no money spent, so I will be able to tell you what x is in 1d11h.
Which btw I still agree with you is most likely the 27 over the 80ies, because i think it is not all that difficult to achieve. BUT I am trying to also keep the other theory going that more rolls means more likely average outcome and that the opposite effect (fewer rolls on more expensive chests) could be used for gambling on extreme outcomes.
In an elvenar equivalent, say we want to not only be gold league, but beat a guy that spent (completely arbitrary number here) 1000€ on extra event currency following the best average return strategy (and for simplicuity´s sake let´s say he has exactly average luck), purely by luck without spending a diamond. I am mostly curious how way out there the result would have to be to make the crazy gamble the better option than the sensible gamble.

I would like to point out a huge difference between the lottery and Elvenar.

With the lottery, you are faced with three options...
  • Don't buy a lottery ticket, keep the $10 (sorry, I is American and don't have the fancy 'e' symbol handy) - Total $10
  • Buy a lottery ticket and loose - Total $0
  • Buy a lottery ticket and win - Total $10,000,000
From there, you can make all the arguments about 'risk' and 'personal assessments' and 'aspirational chances' and 'small loss/big reward' you want.

But, that doesn't transfer to the world of Elvenar quests. SK has no use if you don't play. Choosing not play the event does not leave you with the Elvenar equivalent of $10 dollars in your pocket.
Not really a difference: in both cases you have only 2 options:
lottery
1. Success: you win 10 mio
2. Failure: you do not win 10 mio (we don´t care if we keep or lose the 10, we don´t even care if we win 1000, because we NEED the 10 mio, remember the crazywiz´s bodyparts?)
elvenar
1. Success: we win a random completely unrealistic amount of staffs
2. Failure: we do not make the goal, we do not care how horribly we fail either. 0 staffs is exactly as good as 1000 staffs if the goal is 1001 staffs.

So the question is a completely different one: We do not care which strategy is giving us the better results in terms of expected outcome. So we don´t care which one probably returns 500 and which one probably returns 600. The guy we owe money doesn´t care if we give him nothing or 500 or 600, he is going to chop off bits just as happily either way. The only way he is NOT going to start chopping is if he gets 10 mio. We want to know which one has the higher chance for keeping our bodyparts, which is really not the same question at all as the question of 500 or 600.

Back to the beacons: I am completely with you on the beacons, we really don´t care if we hit the bonus on the first or last % of the chance, so no we do not need 100%, what we do need is to hit the bonus every single time and never not hit the bonus. Actually that may be the better way of putting it.
Actually I just saw something I hadn´t considered before: Nobody will ever need to hit the 80 or the 27 continuously, even if you hit the prize only every other time will genrate infinite rewards. As hitting it once gives you more SK than you need to buy 2 more chests. So we only need to get close to a 1/2 ratio. Since that makes it much easier I would guess the 27 chest will have an easier time winning the race. But who knows, we can input actual numbers on wednesday :)
 

Sir Derf

Artisan
Let's try this again, @Gargon667 . Your 'theory' that " more rolls means more likely average outcome and that the opposite effect (fewer rolls on more expensive chests) could be used for gambling on extreme outcomes" is not applicable to Elvenar beaons.

Look at it this way. When rolling dice for an aggregate score, more rolls means more likely to cluster in the middle is because the outcomes of individual dice rolls produce a continuum of results, and that collectively highs and lows combine into the same effective result as an equal collection of middles. Elvenar chests, as I've pointed out multiple times, and done the math to back it up, don't behave like that.

First, let's talk about dice rolls. Roll a d20 and get a 20. Awesome, that's an unlikely high, good outcome. Roll it again and get a 4. We've now got a total of 24, an above-average average of 12. Roll it again and get a 6, and while you started out incredibly, you've now got a total of 30 and a down-the-middle average of 10. You might as well had rolled 3 successive 10s.

Now translate that same sequence into Elvenar 27 beacons. Pick the 27 beacon, Internally the RND gives you 20/20. Awesome, that's in the top 10%, you get a bonus 80. Pick the 27 again, RND gives you 6, you get nothing. Pick the 27 a third time, the RND gives you 4, and again you get nothing. Now internally, you appear to have accumulated a total of 30, which is an average of 10, but you also have made a reward of 80 once, or you could think of it as having averaged a reward of 26ish per pick. This is not the same outcome as if you had internally rolled 3 10s. In that case, you would have got nothing three times, and ended with a total of nothing.

Here's a bigger example.

Let's throw 10d20s.

Scenario 1. You throw 6 11's, a 10 and three 8's. All near-average individual rolls, totaling 100 and averaging to exactly the middle 10.0.
Scenario 2. You throw 4 20's, 2 6s and 4 2s's. Now, you threw multiple heroic values, but those were subsequently canceled by other low values, totaling 100 and despite the heroic initial rolls, ending with the same average 10.0.

There is your theory borne out with aggregate dice throws.


But what happens when you translate those exact same numbers, those exact same probable sequences, into Elvenar beacons?

Scenario 1. None of the throws give a bonus. Internal average 10.0, No Bonus.
Scenario 2. The first four beacons give bonuses, you're up 320 SK, average bonus 80 per becaon with 100% success rate from a 10% likely beacon. The other 6 beacons give no bonus, and you're at the same internal average of 10.0 as Scenario 1. However, you're still up 320 SK, average bonus 32 SK per beacon with a success rate of 40% from a 10% likely beacon.

There is your theory stillborn with aggregate Elvenar beacons.


Despite having an internal average of 10.0, Elvenar beacon rewards don't average out like the internal average of the RNG. Having more attempts at lower value, lower odd, lower payback beacons might produce more likely average outcomes when looking at the the average of internal RNG values, but the rewards aren't generated based on that average of the internal RNG, but on the individual successes or failures of the individual beacons.
 

Sir Derf

Artisan
My four bulleted descriptions of games goals were for the dice game, because the answer will be different depending on what the goals are in the dice game.


As I keep explaining to you, it doesn't matter what your goals were in the the Elvenar beacon game. The answer is always the same.
  • If you want the best chance to get 200 staffs, 27 beats 89.
  • If you want the best chance to get 400 staffs, 27 beats 89.
  • If you want the best chance to get 2,000,000 staffs, 27 beats 89.
  • If you want the best chance to get in the top 10%, 27 beats 89.
  • If you want the best chance to get in the top 1%, 27 beats 89.
  • If you want the best chance to get in the top 0.001%, 27 beats 89.
  • If you want the best chance to beat a player who spent $10 on extra SK, 27 beats 89.
  • If you want the best chance to beat a player who spent $1000 on extra SK, 27 beat 89.
  • If you want the best chance to beat a player who spent $1,000,000 on extra SK, 27 beats 89.
  • If you want the best chance to get the highest score, 27 beats 89.
What I'm trying to say is, 27 beat 89 every time, all the time, for equal input and equal luck.

The odds of you achieving any of the above goals will vary. It will be easy to get 200 staffs, very difficult to get 400 staffs, and camel-through-the-eye-of-a-needle difficult to get 2,000,000 staffs (It probably involves an industrial-strength blender, a very thins straw and a heavy-duty suction pump), hut in all of those cases, the 27 will have a better chance of accomplishing it than the 89. That's what the math appears to say.
 

Sir Derf

Artisan
As to your discussion of the lottery analogy, flawed, flawed, flawed.

Playing the lottery recreationally versus playing the lottery to save @CrazyWizard from the gangsters are two very different goals from the same game, and neither is really applicable for drawing conclusions about a strategy for Elvenar beacons.

As I described the recreational play, there are three outcomes. Don't play, keep $10 dollar. Play and either likely have $0, or unlikely have $1,000,000. There's risk, and emotion, and 'how much does $10 mean to you on a weekly basis', and all kinds of soft, fuzzy, subjective weight applied to the decision. In the rigged fantasy world of Gangster-motivated lottery, you have basically placed infinite weight on the need for the $1,000,000, as there is no weight on the don't-play-and-keep-the-$10 option. This is no longer a probability question, or even an ethics question; this is a forced choice.

And again, it all falls apart because of the nature of the Elvenar beacon game. It is not offering you the choice between different increasingly averaging payouts; it is offering you the choice between different increasingly accumulating payouts. The best chance of logging into the Gangsters account, spending $10 dollars for an initial stake of SK (I guess the Gangsters haven't played all month) and playing in an attempt to earn 1,000,000 staffs to save @CrazyWizard (some day he'll have to tell us how he got in Elvenar-debt to the Gangsters, but that should probably go in a different thread) is to play 27 over 89. You probably have comparable, if not worse, odds then if you bought a lottery ticket, but that's your play.
 

Stucon

Enchanter
Roll a d20 and get a 20. Awesome, that's an unlikely high, good outcome.
No, NOT unlikely. It is a 1 in 20 chance. Same as throwing any other face number on the dice. You can have high and good though (unless you needed a different number). I hate statistics, bleagh!
Just saying!