Unified Efficiency Index - How to compare apples to oranges

[ColorlessGreen]

Win/loss value

Every AI target has a stated value - the number shown as "reward" for AI# targets, and the number of electrum shown by the league for arena. All targets have a win value of between 0.5 the stated value and one less than the stated value. 

Not entirely accurate. While AI1-3 and arena all use the same formula, AI4 and FG have different ones. I calculated their exact values a month ago, and if my memory can be trusted AI4 has a minimum of 15 (instead of 20) and FG has a minimum of 35 (instead of 30).

Can you provide any sort of citation for this? Or for that matter a screenshot of an AI4 win with under 20 electrum provided? In the mean time I'll start paying closer attention and see if I can verify the numbers myself.

Many thanks!


Also, I noticed another error in the formula that doesn't actually require a modification of the formula, so much as a modification of the way XHP is handled. IIRC, finishing with 100+X HP provides X/2 extra electrum, while the guidance on using the XHP variable says to use a factor of 100+X (resulting in it inaccurately providing X extra electrum). If I'm remembering wrong about this, please let me know.

This will be is updated in the OP shortly, though again, it doesn't affect the formula itself, so much as the guidance on XHP will be changed from X to X/2. Additionally, I think this needs more attention in order to separate out XHP as added to regular wins as opposed to XHP as added to EM wins. For now it's just fine as it is IMO due to affecting very few decks and being set up specifically in relation to EM wins currently (as that's where the highest change in electrum is) and being a completely subjective factor anyways, but if we ever get something more scientific for this part of the formula than "pick a factor that you feel reflects the deck", the formula may need to be changed to account for it.

Also, I got like an hour of sleep last night so I apologize if any part of this post was less coherent than normal.

[Tirear]

Win/loss value

Every AI target has a stated value - the number shown as "reward" for AI# targets, and the number of electrum shown by the league for arena. All targets have a win value of between 0.5 the stated value and one less than the stated value. 

Not entirely accurate. While AI1-3 and arena all use the same formula, AI4 and FG have different ones. I calculated their exact values a month ago, and if my memory can be trusted AI4 has a minimum of 15 (instead of 20) and FG has a minimum of 35 (instead of 30).

Can you provide any sort of citation for this? Or for that matter a screenshot of an AI4 win with under 20 electrum provided?

Here's a screenshot of a 2 hp win against AI4, though I don't see a way to tell the difference from an AI3 match.

Testing was done in the trainer using the ability to start with 75 quanta + bolt spells.

[Leodip]

I love it. Would like to work on it, too, if you're okay with it, once we've finished gathering data for all of the formulas.
Subbing to keep track.

[Pella]

Spoiler for Hidden:

A few details from the author of the FGei 1.32 spreadsheet.  These are the constants I used in the spreadsheet and the reasons I used those values.  You may use or not use this information, as you see fit.

- Card value = 1,177 .. This is the average sell price of all upgraded cards available in 1.32.  When I assembled the spreadsheet, all available data predated 1.32, so I calculated it myself.
- Card drop rate = 45.657586% .. Again, this is a value I calculated.  Unfortunately, I no longer remember which pre-1.32 data I used or what calculations I performed on them to arrive at this figure.  I do remember that the entire process was more complex than I would have liked, and it took me about 30 minutes to do the whole thing.
- Skip time = 5 seconds .. Most skips are essentially 0 seconds, while some skips occur up to a minute or more into the duel, depending on your draw, AI draw, cards played, your mood, etc.  Five seconds seemed like a reasonable average, especially since the vast majority of skips are autoskips.

Obviously, in the final formula, skip = loss.  Therefore, skips really have little to no relevance with any target other than FGs.  While true objectivity may require collection of skip data on other targets, I have doubts as to its usefulness.  More to the point, I hypothesize that the ROI* would be negative.

*ROI = Return on Investment .. In this case, the investment is the time required to collect this extra data.  The return is the value of the modification to the formula that the data would justify.  I believe that any skip time-based adjustment to the final formula would be statistically insignificant and, therefore, not worth the time to collect the data and calculate the adjustment.


Amazing work, CG.  +rep

(a) When I refer to the FGEI assumptions, I was referring to the assumptions as listed in the original FGEI thread which you link to in the OP of the 1.32 FGEI thread. I fully support using more accurate assumptions, but I would recommend noting that you are using different calculations than those used by the original FGEI somewhere in the thread/spreadsheet. If you already do note that and I just missed it, I apologize.

(b) Regarding upped card value, did you do any weighting with regards to pillars or anything of the sort? I'm quite glad to have a more accurate number for upped card values, and I'm honestly most concerned with making sure that the UEI assumptions are equivalent to the FGEI assumptions, so I'll likely update this at some point.

(c) Regarding drop rate, that looks about right for FG. It was 47% in the original FGEI thread, but IIRC that was before SoFos were added. However, I don't actually use FG drop rate anywhere in here. AI4 stats are taken from the AI4 study, and all other spin chance stats are taken from Xeno's AI3 simulation (working on the assumption that they will remain sufficiently similar for arena regular spins).

(d) Regarding skip time, we could use 0.5 turns per skip (~5.4 seconds at the current turns per hour value) if it were factored into this calculation, but I also think it pretty much isn't worth worrying about for anything besides FG. I may at some point look into how much factoring in skips would affect AI3 (assuming I can find any skip-related data whatsoever to work with) to see whether or not it actually is worth worrying about for AI3. I am pretty confident it is 100% impossible to do anything at all with skips for AI4/arena that would fit into the principles and standards of any study ever done here.

Many thanks for the input!

edit: Regarding upped card value specifically for FGEI, the StatMasta spreadsheet calculates actual per-god spin rate (which I believe is still known information) and only provides the average for people calculating by hand. I would strongly recommend updating the google doc to process it in the same way.

(a)  I haven't noted that anywhere.  I'll fix that later today.

(b)  I did no weighting; it's just a straight average.  What sort of weighting do you think I should do?

(c)(d)  Very well, I'll keep these figures as they are.

Your edit:  I'm confused.  First you talk about the value of upped cards, then you switch to card spin rate.  I can easily modify the spreadsheet to calculate spin rates per FG and to display the average empirical spin rate.  Ultimately, though, the normalized FGei uses the normalized spin rate I listed in my first post in this thread, which is 45.657586%.  In the interest of keeping the sheet as simple as possible, I opted to calculate very few per-FG numbers.
Somehow, I'm confident that I'm missing your point, so please feel free to restate it.

[ColorlessGreen]

Your edit:  I'm confused.  First you talk about the value of upped cards, then you switch to card spin rate.  I can easily modify the spreadsheet to calculate spin rates per FG and to display the average empirical spin rate.  Ultimately, though, the normalized FGei uses the normalized spin rate I listed in my first post in this thread, which is 45.657586%.  In the interest of keeping the sheet as simple as possible, I opted to calculate very few per-FG numbers.
Somehow, I'm confident that I'm missing your point, so please feel free to restate it.

Rather than using the average spin rate across the board for FGs, the current versions of the StatMasta spreadsheets (to the best of my knowledge) use the actual spin rate for each individual FG when calculating the FGEI. So, instead of saying that one victory against a FG (regardless of which FG) is worth 45.66% (or 47%) of 1,177 (or 1,160), it uses the per-FG spin chance for each FG combined with the normalized win rate against that particular FG to calculate the overall efficiency. In other words, it properly accounts for the fact that a deck that has a 90% win rate against divine glory and a 10% win rate against rainbow is substantially more profitable than a deck that has a 10% win rate against divine glory and a 90% win rate against rainbow (example assumes all other factors are equal).

Basically, instead of taking the overall win rate and multiplying by the average FG card spin chance and the average card value, it takes the individual win rate and multiplies by the specific FG spin chance and the average value and then goes from there.

I got basically no sleep last night so I still might not be explaining properly - please let me know if so and I'll do my best to improve it.

Also, this has no real direct bearing on UEI, as UEI is not equipped to deal with normalization against known opponents in the same way that FGEI is, so this may honestly be a better discussion to move over to the FGEI thread if it continues.


Regarding card value being weighted for pillars: I don't have a good answer for this, but I'm willing to bet it's been discussed elsewhere. If not, we could start a discussion about it either here, in the FGEI thread, or in its own thread. With regards to FG specifically, xenosim can probably get actual tests from millions of simulated runs in the same way it did for AI3 (and actually I kinda think it already did, but that it's outdated) which would be as close to actual data as we're ever likely to have.

[ColorlessGreen]

Spoiler for Hidden:

Win/loss value

Every AI target has a stated value - the number shown as "reward" for AI# targets, and the number of electrum shown by the league for arena. All targets have a win value of between 0.5 the stated value and one less than the stated value. 

Not entirely accurate. While AI1-3 and arena all use the same formula, AI4 and FG have different ones. I calculated their exact values a month ago, and if my memory can be trusted AI4 has a minimum of 15 (instead of 20) and FG has a minimum of 35 (instead of 30).

Can you provide any sort of citation for this? Or for that matter a screenshot of an AI4 win with under 20 electrum provided?

Here's a screenshot of a 2 hp win against AI4, though I don't see a way to tell the difference from an AI3 match.

Testing was done in the trainer using the ability to start with 75 quanta + bolt spells.

Some real quick and dirty unscientific testing shows you're most likely correct about FG as well, which means I should replace the assumption about how much HP you end a game with across the board. I will do some quick and dirty scientific testing at some point here and see if I can come up with exact formulae for all targets and readjust the UEI formula to use FGEI's ending HP assumption.

For anyone curious, this is likely to slightly lower all profit numbers across the board from my projections earlier, but is not likely to have major effects on most decks. Gold/silver arena decks that focus on EMs (and high HP EMs for that matter) will be most affected.

Many thanks again!

[Leodip]

BTW, I'm not sure on how to use the XHP function. For example, let's say that I'm playing PDial, I play only one SoD and I always activate it, in every game, and it's always EM. If I'm going to win without using SoD and without it being an EM with exactly 124 HPs, I'll quit.
This assures that each win is an EM with 124 HPs. What I didn't understand is how to use it in the formula.
Let's say the stats are those:
PDial 1 (always) SoD
WR: 0,55
EMR: 1
TTW: 9
League: Platinum

Can you write me the formula of this (unsimplified)?

[ColorlessGreen]

BTW, I'm not sure on how to use the XHP function. For example, let's say that I'm playing PDial, I play only one SoD and I always activate it, in every game, and it's always EM. If I'm going to win without using SoD and without it being an EM with exactly 124 HPs, I'll quit.
This assures that each win is an EM with 124 HPs. What I didn't understand is how to use it in the formula.
Let's say the stats are those:
PDial 1 (always) SoD
WR: 0,55
EMR: 1
TTW: 9
League: Platinum

Can you write me the formula of this (unsimplified)?

XHP is pretty much a judgement call right now unless you're actually tracking average max hp for all wins (EM or otherwise).

XHP starts at 1, and for every two average extra maxhp of your deck, you add 0.01 to XHP. If you are assuming that 100% of the time you will play exactly one SoD, your average max HP would be 124, and so your XHP would be 1.12.

Unsimplified formula for platinum using the information you provided would be:

(333 * 0.55 * 200 * 0.7833 * (1 + (1 * 2.55)) * 1.12 / 9) - (333 * (1-0.55) * 15 / 9) + (333 * 25.6 * 0.55 / 9) + (333 * 1290 / GPS / 9)

For convenience, GPS for platinum would be:

(0.55^-1) + (0.55^-1)^2

So the whole formula would be:

(333 * 0.55 * 200 * 0.7833 * (1 + (1 * 2.55)) * 1.12 / 9) - (333 * (1-0.55) * 15 / 9) + (333 * 25.6 * 0.55 / 9) + (333 * 1290 / ((0.55^-1) + (0.55^-1)^2) / 9)

Please note that the 0.7833 and the 2.55 need to be slightly altered to correct for the discrepancy that Tirear mentioned, but I haven't gotten to do that yet, and it won't have a major impact on the final number.

Additionally, if you're going to play around with results from the formula, my recommendation would be to paste the formula into a spreadsheet (excel, openoffice, google docs, etc), then replace the formula's references to "WR" with references to cell A2 (or whatever), and references to "TTW" with references to cell A3, etc. This lets you easily change the numbers for the various variables around easily to see the results.

[Leodip]

BTW, I'm not sure on how to use the XHP function. For example, let's say that I'm playing PDial, I play only one SoD and I always activate it, in every game, and it's always EM. If I'm going to win without using SoD and without it being an EM with exactly 124 HPs, I'll quit.
This assures that each win is an EM with 124 HPs. What I didn't understand is how to use it in the formula.
Let's say the stats are those:
PDial 1 (always) SoD
WR: 0,55
EMR: 1
TTW: 9
League: Platinum

Can you write me the formula of this (unsimplified)?

XHP is pretty much a judgement call right now unless you're actually tracking average max hp for all wins (EM or otherwise).

XHP starts at 1, and for every two average extra maxhp of your deck, you add 0.01 to XHP. If you are assuming that 100% of the time you will play exactly one SoD, your average max HP would be 124, and so your XHP would be 1.12.

Unsimplified formula for platinum using the information you provided would be:

(333 * 0.55 * 200 * 0.7833 * (1 + (1 * 2.55)) * 1.12 / 9) - (333 * (1-0.55) * 15 / 9) + (333 * 25.6 * 0.55 / 9) + (333 * 1290 / GPS / 9)

For convenience, GPS for platinum would be:

(0.55^-1) + (0.55^-1)^2

So the whole formula would be:

(333 * 0.55 * 200 * 0.7833 * (1 + (1 * 2.55)) * 1.12 / 9) - (333 * (1-0.55) * 15 / 9) + (333 * 25.6 * 0.55 / 9) + (333 * 1290 / ((0.55^-1) + (0.55^-1)^2) / 9)

Please note that the 0.7833 and the 2.55 need to be slightly altered to correct for the discrepancy that Tirear mentioned, but I haven't gotten to do that yet, and it won't have a major impact on the final number.

Additionally, if you're going to play around with results from the formula, my recommendation would be to paste the formula into a spreadsheet (excel, openoffice, google docs, etc), then replace the formula's references to "WR" with references to cell A2 (or whatever), and references to "TTW" with references to cell A3, etc. This lets you easily change the numbers for the various variables around easily to see the results.

That's what I did (the spreadsheet), I just needed the XHP one and it'll be complete.
Tomorrow I'll post it, too.

[Leodip]

Here you go

[ColorlessGreen]

Here you go

That's great Leodip. Thanks for doing that.

Do note that the UEI formula isn't necessarily completely solidified yet, so there's a few things that are going to change at some point here as more information comes in, but IMO it's definitely close enough that having a tool like this available is great. +rep

[Leodip]

I indeed posted it as a reference for more fixes and to get some feedback (that's why it is a beta). Thanks, though.
BTW, I was thinking of adding a "Reward" slot, this way you can put the exact value of the League, if you want to.