Evolution of the Phoenix - Event Analysis

[Deleted User - 3932582]

Alright, there were a lot of discussions about how much you can expect from the current event in terms of daily and grand prizes, but I haven't seen a definitive analysis of how probable (or improbable ) different reward outcomes are - given different chest-opening strategies. So I figured I should write one I did a full simulation of the event mechanic, and looked at expected values for different rewards, as well as variability around the expectations.

So take a look here:
https://minmaxgame.com/evolution-of-the-phoenix-event-analysis/

Strategy selection might be too late for quite a few players, but you can take a look and see how likely your actual result is. If you kept records, that is

 

[DeletedUser9601]

I hope this writeup is for a college thesis in statistics or something .

For your analysis, did you assume that each set of 3 chests to open would be random, and have the same probability? Now that I think of it, I can't say definitively whether if I've opened a particular chest (say 18 feather chest), does that chest show up in the next set of 3?

 

[Enevhar Aldarion]

I hope this writeup is for a college thesis in statistics or something .

For your analysis, did you assume that each set of 3 chests to open would be random, and have the same probability? Now that I think of it, I can't say definitively whether if I've opened a particular chest (say 18 feather chest), does that chest show up in the next set of 3?

Yes, I have had the chest I opened appear in the next set of three. Not often, but it happens.

 

[Deleted User - 3932582]

Yes, I have had the chest I opened appear in the next set of three. Not often, but it happens.

Indeed. I usually only pay attention to purple/82 and lightwood/18 chests, and these certainly repeated more than once or even twice in a row.

 

[Deleted User - 3932582]

For your analysis, did you assume that each set of 3 chests to open would be random, and have the same probability?

Yep. The only restriction is that you can't have the same chest in the same draw more than once (i.e. all chests are different).

 

[Enevhar Aldarion]

Indeed. I usually only pay attention to purple/82 and lightwood/18 chests, and these certainly repeated more than once or even twice in a row.

Most of my repeat chests were the 18 and 20-essence chests. I did not pay attention enough to the others to really track it.

 

[DeletedUser17455]

This lines up with my experience. My strategy was 18, 20, 82 and I got ~8 evolutions and ~27 event buildings.

 

[The Unbeliever]

This lines up with my experience. My strategy was 18, 20, 82 and I got ~8 evolutions and ~27 event buildings.

Mine is similar, but I go; 18 -> 82 -> 20 in order of priority.
I personally feel that the .7 difference per feather between the 82 vs. 20 is worth the risk to land those extra essences... I've scored the 200 essence prize 3 times thus far.

I also had a nice run yesterday of landing the 18 chest 5 or 6 times in a row, which has helped save a few extra essences for more rolls!

 

[Pheryll]

The analysis looks decent, however the ordering for optimization begs me to ask what calculation method you used. For example, to optimize grand prizes, you had the gold/orange chest take priority over the bronze. While the two chests do indeed have the same average, the weight of the gold/orange chest (that is purchasing two feathers at an unfavorable rate instead of purchasing one) places it as an inferior choice to the bronze chest, and proper calculations should reflect this. The method I use is to weight the probability of each chest by the following ratio 28:21:15:10:6:3:1, that is the top pick has a rate of 28/84 of appearing, while being stuck with the worst options, having to pick your seventh best is 1/84 chance. To find essences per feather I simply divide essences per chest by feathers per chest.

 

[DeletedUser19458]

That write up looks ... thorough My strategy was get as many 30kps as I could so almost always chose what had the highest daily % chance and if they all sucked (only 5% or 10%) picked which had cheapest feathers per diamond. I landed 9 evolutions with that. Still have 400 purple diamonds but that won't get the 10th.

 

[Ashrem]

Mine is similar, but I go; 18 -> 82 -> 20 in order of priority.
I personally feel that the .7 difference per feather between the 82 vs. 20 is worth the risk to land those extra essences..

Not only that. The 20 chest has a 45% chance of resulting in a single Rune Shard, which to my way of thinking makes it the worst of all possible options. Out of potentially 300+ chest openings, putting the 20 essence chest at number two suggests about a 22% rate of rune shards, or about 70 wasted spins.

 

[shimmerfly]

All this planning and pondering and I had great luck with just my gut feeling at every spin.
I kind of went after the time instants or the 200 essence ones or whichever one I "felt" like.

 

[Deleted User - 3932582]

The analysis looks decent, however the ordering for optimization begs me to ask what calculation method you used. For example, to optimize grand prizes, you had the gold/orange chest take priority over the bronze. While the two chests do indeed have the same average, the weight of the gold/orange chest (that is purchasing two feathers at an unfavorable rate instead of purchasing one) places it as an inferior choice to the bronze chest, and proper calculations should reflect this. The method I use is to weight the probability of each chest by the following ratio 28:21:15:10:6:3:1, that is the top pick has a rate of 28/84 of appearing, while being stuck with the worst options, having to pick your seventh best is 1/84 chance. To find essences per feather I simply divide essences per chest by feathers per chest.

Oh, someone was paying attention Well actually, I haven't used any calculations to define the strategies - I have used definitions for G and D as a given. That late in the event it was more about figuring out the results of using particular strategies, rather than coming up with the best strategies. Most people I know used GoK tables and sorted them either by cost per feather, or cost per daily prize. As both those chests have the same cost per feather there, the tie is resolved by looking at the cost of daily prize, which is significantly better for the Gold chest.

With that aside, you're absolutely right that prioritizing bronze over gold would give you a higher expected value for the grand prizes. In purely mathematical sense, that is In reality, this reordering that down the priority list is not material. With 10mm scenarios and running these two variants head to head, the difference comes to 16.39 vs 16.42 of expected GPs. That's a difference of 0.03 GP, and I am not even sure that 10mm scenarios is enough to make it a significant difference. In other terms, with 20 feathers per GP, we're still not breaking even a single feather difference on average.

Probability of hitting 17 GPs or more (i.e. getting enough Artifacts for L10 Phoenix) goes up by ~1.5%, so there is that. Bottom line is, outside of the top 3 chests the order of the rest has a very marginal impact on the GP results - because those chests would be picked fairly infrequently, and variability in feather costs is pretty low.

 

[Deleted User - 3932582]

Not only that. The 20 chest has a 45% chance of resulting in a single Rune Shard, which to my way of thinking makes it the worst of all possible options. Out of potentially 300+ chest openings, putting the 20 essence chest at number two suggests about a 22% rate of rune shards, or about 70 wasted spins.

I agree, I didn't like opening 20 chest as all it's auxiliary rewards are pretty meh. But this is fairly individual as now we're mixing value of different rewards in the same strategy. Those things could be very different (I love time boosts and don't care as much about MM spells, other people can have entirely different views on that etc). That's why looking at pure strategies where you just maximize one reward - grand prizes, daily prizes etc - is at least something most people can agree on. Doesn't mean they would want to follow such strategy

And for the 20 chests, don't think about them as wasted spins - think about them as bringing you closer to that L10 Phoenix than almost any other chest. At the expense of few other things

 

[Pheryll]

With that aside, you're absolutely right that prioritizing bronze over gold would give you a higher expected value for the grand prizes. In purely mathematical sense, that is In reality, this reordering that down the priority list is not material. With 10mm scenarios and running these two variants head to head, the difference comes to 16.39 vs 16.42 of expected GPs. That's a difference of 0.03 GP, and I am not even sure that 10mm scenarios is enough to make it a significant difference. In other terms, with 20 feathers per GP, we're still not breaking even a single feather difference on average.

Any time a move is made from optimum to cater to a second priority, there is a possibility of better moves being made. In this case, placing purple as second priority (above dark wood but below light wood) is actually more cost effective for feathers and daily prizes than the favoring orange/gold over bronze.

Let's look at the three progressions:

Light Wood, Dark Wood, Purple, Bronze, Gold, Red, Green (optimum for feathers)
Essences per feather: 20.593 Essences per daily prize: 284.607

Light Wood, Purple, Dark Wood, Bronze, Gold, Red, Green (move purple to higher priority)
Essences per feather: 20.630 Essences per daily prize: 267.788

Light Wood, Dark Wood, Purple, Gold, Bronze, Red, Green (move gold to higher priority)
Essences per feather: 20.669 Essences per daily prize: 282.258

 

[Deleted User - 3932582]

Any time a move is made from optimum to cater to a second priority, there is a possibility of better moves being made. In this case, placing purple as second priority (above dark wood but below light wood) is actually more cost effective for feathers and daily prizes than the favoring orange/gold over bronze.

Let's look at the three progressions:

Light Wood, Dark Wood, Purple, Bronze, Gold, Red, Green (optimum for feathers)
Essences per feather: 20.593 Essences per daily prize: 284.607

Light Wood, Purple, Dark Wood, Bronze, Gold, Red, Green (move purple to higher priority)
Essences per feather: 20.630 Essences per daily prize: 267.788

Light Wood, Dark Wood, Purple, Gold, Bronze, Red, Green (move gold to higher priority)
Essences per feather: 20.669 Essences per daily prize: 282.258

I am not quite sure what your point is. I already mentioned that indeed for grand prizes strategy 1 is mathematically better than strategy 3, even though in practice the differences would be rather imperceptible. Practically, all 3 strategies are equivalent as far as grand prizes are concerned (difference of less than 1 feather over the whole 7000 essence roll).

Now, if we want to use more than one value for our target optimization (i.e. both grand and daily prizes), then we need to be concerned about their relative value, and that is a personal decision. So that's a hybrid strategy territory. From that perspective, strategy 2 strictly dominates strategy 3, and I would argue also strategy 1 (as I believe cost per feather to be equivalent, but strategy 2 is better on daily prizes).

But from that perspective I wouldn't use strategy 2 either, and will go straight to strategy H or similar (push purple all the way to the top instead of #2). It is not strictly dominant, but you give up something like 2 feathers in H vs strategy 2, but you gain about 1.3 daily prizes on average. I'll take that trade any day if I am even remotely interested in daily prizes. If not, any of the 3 strategies above will do.

 

[Pheryll]

I am not quite sure what your point is. I already mentioned that indeed for grand prizes strategy 1 is mathematically better than strategy 3, even though in practice the differences would be rather imperceptible. Practically, all 3 strategies are equivalent as far as grand prizes are concerned (difference of less than 1 feather over the whole 7000 essence roll).

My point is the Grand prize optimal sequence you chose seems odd as it is technically a hybrid and compared to other trade offs it comes out lacking. Moving the purple up one slot gives 1 extra daily building for less than a feather when 7000 essence budget is considered.

But from that perspective I wouldn't use strategy 2 either, and will go straight to strategy H or similar (push purple all the way to the top instead of #2). It is not strictly dominant, but you give up something like 2 feathers in H vs strategy 2, but you gain about 1.3 daily prizes on average. I'll take that trade any day if I am even remotely interested in daily prizes. If not, any of the 3 strategies above will do.

If you want the comparison numbers, your hybrid gives about 5.5% more daily buildings than my strategy 2, while my strategy 2 produces 0.9% more feathers. For 7000 essences that is a 3 feather trade for 1 daily building.