Unified Efficiency Index - How to compare apples to oranges

[ColorlessGreen]

I've been working on this for quite a while, which I've mentioned in a number of different places. I posted an earlier draft of this concept in the AI3 vs AI4 thread, but since then it's gotten a few different refinements, and I've decided it's time for some peer review.

For quite some time, we've had the False God Efficiency Index (FGEI) in order to accurately determine the average profit for various FG farming decks. However, every other target has been limited to studies of win rate and TTW which are only tracked within that given target. There has been no easy way to determine whether it's better to go for a fast deck with a mediocre win rate, or a high win rate deck with mediocre speed, and where the balance is between speed, win rate, and EM rate. On top of that, there are wildly conflicting opinions on which AIs provide the most actual profit over time.

There is no reason that the principles used in FGEI can't be applied to the other AI targets in order to arrive at an actual number, similar to FGEI, representing net profit over time. Such formulae would make it quite simple to accurately rank decks in the various studies by net profit rather than by either win rate or TTW alone, as has been traditionally done. Additionally, once every deck has an expected profit over time easily attached to it, it becomes trivial to determine the actual advisability of farming the various AI targets. As a side effect, it would answer questions about the breakpoint where it's more profitable to farm gold or platinum (for example) depending on which way the rewards of each have moved.

 

Put as simply as possible, the net profit of a deck is as follows:

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(((Non-EM wins over a time period) * (Average profit per Non-EM win)) + ((EM wins over a time period) * (Average profit per EM win))) * (Average percentage of extra HP)) - ((Losses over a time period) * (Cost per loss)) + ((Wins over a time period) * (Average spins profit per win)) + ((Special spins over a time period) * (Average special spin profit))


Time

First of all, the words "over a time period" are everywhere through the formula, so we'll start there. In the interest of removing as much subjectivity from the formula as possible, the formula will use a number of turns as a unit of time rather than an hour or any actual unit of time. While it is certainly possible to convert this formula to use actual units of time rather than a number of turns, the speed people play is highly variable and is also dependent on computer/ISP/network conditions.

I have not completely finished studying the relationship between turns and actual units of time yet, but looking at the limited information available in various studies which have tracked seconds per game, there appears to be relatively little overall variation within a given target even with different types of decks, but there is great variation between different players with similar decks against identical targets. This leads me to believe we're never going to arrive at a perfect answer here, but as things like the 1.32 FGEI study get further along, I'll have more source data to work from. Ideally we could start getting game time recorded in the various studies (as is already done in FGEI) just so we can have information rather than just opinions.

Personally, I assume that there are 333 turns per hour, which works out to around 10.8 seconds per turn. Looking at (quite outdated) AI3 studies puts AI3 at a bit faster than this on average (though, like I said, there's a good bit of variance based on player). If you assume that decks take the same amount of time per turn (which they don't, but it's actually pretty close), turns per hour is actually pretty much irrelevant unless you want to compare the result of this formula to FGEI. If you're a slower player, 250 per hour may be a better choice for you; in that case, multiply the resulting OUEI of 333 TPH by 0.75.

As far as the difference in time per turn between decks goes, as I said, I hope to have better information later, but for now, this part has to remain one of the two places in this formula which are just plain subjective. If you feel that you play a given deck slower per turn than what you're comparing it to (or than average, or whatever), multiply the number this formula produces by 0.9 or whatever you would like. This formula will (until there's a better way to do this) assume the same number of turns per hour for all decks.

Time TL;DR: Turns per hour ("TPH") = 333. All decks are the same (until we have better information)


Wins/Losses

I hope this is the least controversial part of this entire article. Win rate ("WR") is (Wins / (Wins + Losses)). Loss rate ("LR") is (1 - Win Rate).

Wins/Losses TL;DR: It's like 15 words long.


Turns per Win

The next step is to convert turns to win and win rate into an average turns per win. First of all, this formula assumes that turns to lose is equal to turns to win. It isn't always, but IMO it's the best way to handle the subject, and that also seems to be the general opinion of the community when it's been discussed elsewhere. This is certainly open to debate, but as TTL isn't routinely tracked in various studies, if you decide TTL != TTW, this whole formula is useless until we redo basically all the studies.

So anyway, from now on, TTL = TTW. In order to get turns per win, all you have to do is take TTW and divide by win rate. For example, a deck with 10 TTW and a 50% win rate averages 20 turns per win (10 / 0.5), and a deck with 10 TTW and 75% win rate averages 13.33 turns per win. Honestly, if you're only comparing decks within a given target, you can just about stop here, as turns per win is generally speaking the biggest determining factor of a deck's overall profit as compared to other decks against the same target.

Turns per win TL;DR: Turns per win ("TPW") is turns to win ("TTW") divided by win rate. Turns to lose are not considered and assumed to be equal to TTW.


EMs

EM rate is EMs divided by wins. It is not EMs divided by total games.

Rather than handling EMs separately, it's easier to combine the profit from EMs into the profit from regular wins. It's completely identical mathematically, but it makes for a simpler formula. This is done by multiplying the total win value over time by (1 + (EM rate) * (Factor representing difference between EM value and non-EM value)). That will allow you to arrive at a total value for all the wins (while considering EMs) as one item in the formula rather than breaking non-EM profit and EM profit into two different steps.

EM TL;DR: EM Rate ("EMR") = (EMs / Wins).


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. Arena and AI3 have a win value of between 0.5 the stated value and one less than the stated value. AI4 has a win value of between 0.375 the stated value and one less than the stated value. AI3 and AI4 have a loss value of 0.5 the stated value. All arena have a loss value of 15 (assuming you rate the deck which beat you). FGEI assumes that an average FG win is worth 47 electrum, which works out to a 50 hp victory. This formula assumes the same, or 75.00% of the stated value for arena and AI3. A 50 hp victory against AI4 works out to 68.75% of the stated value (as it increases linearly).

An EM victory is worth double the stated value. As noted above, EM profit can be factored in with non-EM profit by multiplying base win profit (for all wins) by (1 + (EM Rate) * (Factor representing difference between EM value and non-EM value)). 200% divided by 75.00% gives an EM factor of 1.67 for AI3 and arena, and 200% divided by 68.75% gives an EM factor of 1.91 for AI4. Note that the EM factors are one less than the result of the division in order to avoid double-counting the original win. Therefore, total win value for AI3/arena is ((Stated value * 0.75) * (1 + (EM Rate * 1.67))), and for AI4 is ((Stated value * 0.6875) * (1 + (EM Rate * 1.91))).

Win/loss value TL;DR: Win value ("WV") = ((Stated Value ("SV") * 0.75) * (1 + EMR * 1.67)) for AI3/arena, and ((Stated Value ("SV") * 0.6875) * (1 + EMR * 1.91)) for AI4. Loss Value ("LV") = 15 for arena, and (0.5 * SV) for AI#.


Extra HP

This is the other subjective part of the formula. You need to apply a multiplier to the win value based on whether/how much max HP boost the deck gets. There are no studies about the average number of SoDs that, say, pdials gets off before it wins. I personally have decided, completely arbitrarily, that assuming about 2/3rds of your maxHP boosters are played for a given win, so a 3-SoD pdials would have a factor of 1.24 (since you receive half as much electrum as you have extra maxhp). This is definitely open to discussion, and feel free to pretty much just guess at this factor if you're trying to figure out a deck of this sort. If you're the sort of person who sits around every game and makes sure that you stall until you get all the SoDs/stoneskins out, feel free to count all of them. Like I said, subjective. Also, totally irrelevant to basically everything except higher arena leagues, as the maxhp bonus is pretty irrelevant to everywhere else which gets nearly all its profit from spins. Also, totally irrelevant to about 95% of farming decks which never go above 100 hp.

Extra HP TL;DR: Extra HP Factor ("XHP") is a subjective number (until it gets tested, if ever) based on the average maxhp of a deck (and is calculated by way of 1 + (half extra average maxhp)).


Spin Profit

Put simply, spin profit is the profit you get from selling a won spun card multiplied by the rate of winning a spun card. This is also where we deal with the bonus five electrum from a match-2. To start with, the value from selling a won upped spun card is 1160 electrum, which is the value used by FGEI. The value for selling a won unupped card is 62 electrum per Xenocidus' simulation of the spins from 1.3 million AI3 games. In the absence of any way to get information about the value of unupped cards against arena/ai4, they will use the 62 electrum as well, since it's the best we've got. If anyone has suggestions on this front, please say so. The unupped spin value includes the bonus electrum from matched cards. I don't know if the upped spin value does, but when winning an upped card, the 5 electrum difference in spin value might as well be a rounding error.

Spin rate is 41.26% against AI3 per Xeno's simulation mentioned above. I find the general distribution of cards within decks in arena to be broadly similar to AI3, so I use this rate for all regular spins in AI3/arena. Due to the fluid nature of arena, I don't think a perfect number can be found, but I'd love to hear any suggestions if anyone has a better one to use for anything. Total spin profit for AI3/arena is 25.60.

AI4 spin chances are taken from actual tests in the AI4 study. At the time of this post edit on 28-Jan-14, they are 9.08% for upped and 20.27% for unupped. Therefore, an AI4 win has a total spin profit of 117.90.

For the purposes of this formula, it is assumed that all won cards are sold.

Spin Profit TL;DR: Total Spin Profit ("TSP") is 25.60 for AI3/arena and 117.90 for AI4.


Special Spins

Special spins apply only to arena, obviously. For the sake of this study, it is assumed that all won rares are sold. The actual average value of all special-spin-able rares is 156.11 unupped and 1299.30 upped. Special spin success chance is arbitrarily 75% assuming you play optimally. This is absolutely open for discussion, but it's what chat came up with a few months ago when I discussed it, and it sounds about right to me. I would love further opinions on it. is 73.92% per the optimal special spin algorithm study. Because of this, the value of each special spin is 115.40 unupped and 960.44 upped.

Special spin rate is calculated as a function of win rate and TTW. The easiest way is to just use the table from the OP of the arena stats thread, but the actual formula to determine turns per special spin is:

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(TTW * ((WR^-1) + (WR^-1)^2 + (WR^-1)^3 + (WR^-1)^4 + (WR^-1)^5))...for bronze. Truncate the formula after ^4 for silver, ^3 for gold, and ^2 for platinum. Once you have turns per special spin, dividing TPH by turns per special spin will give you special spins per hour. Special spins per hour times special spin success chance times special spin sell value will give you the total special spin value.

Special Spins TL;DR: Special Spins per Hour ("SSPH") = TPH / (TTW * Games/Spin). Games/Spin is taken from the table in the Arena Stats OP or from the formula above. Special Spin Value ("SSV") = 115.40 for unupped and 960.44 for upped.


So, here's the bare bones of the formula from above all in one place:

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Turns Per Hour ("TPH") = 333
Win Rate ("WR") = Wins/(Wins + Losses)
Loss Rate ("LR") = 1 - WR
Turns Per Win ("TPW") = Turns To Win ("TTW") / WR
EM Rate ("EMR") = EMs / Wins
Win Value ("WV") = ((Stated Value ("SV") * 0.75 * (1 + (EMR * 1.67))) <arena/AI3> || ((SV * 0.6875) * (1 + EMR * 1.91)) <AI4>
Loss Value ("LV") = 15 <arena> || (0.5 * SV) <AI3/AI4>
Extra HP Factor ("XHP") = subjective. I like adding 1/3 of (potential max hp - 100).
Total Spin Profit Per Win ("TSP") = 25.60 <AI3/arena> || 117.90 <AI4>
Games Per Spin = ((WR^-1) + (WR^-1)^2 + (WR^-1)^3 + (WR^-1)^4 + (WR^-1)^5), truncated appropriately.
Special Spins Per Hour ("SSPH") = TPH / (TTW * Games/Spin)
Special Spin Value Per Win  ("SSV") = 115.40 <unupped> || 960.44 <upped>
Wins Per Hour ("WPH") = TPH/TTW*WR
Losses Per Hour ("LPH") = TPH/TTW*LR

If we put all that together, we get:

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(WPH * WV * XHP) - (LPH * LV) + (TSP * WPH) + (SSV * SSPH)
This is pretty much just another way of restating what I said at the beginning of the thread. The formula will be most useful if it is expressed purely in constants (per target) and WR/TTW/EMR, as those are the three stats that are tracked in most stats threads. So, if we expand out from that base formula, we get:

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AI3:
((333/TTW*WR) * (SV * 0.75 * (1 + (EMR * 1.67))) * XHP) - ((333/(TTW/(1-WR))) * LV)  + (TSP * (333/TTW*WR))

AI4:
((333/TTW*WR) * (SV * 0.6875 * (1 + (EMR * 1.91))) * XHP) - ((333/(TTW/(1-WR))) * LV)  + (TSP * (333/TTW*WR))

Arena:
((333/TTW*WR) * (SV * 0.75 * (1 + (EMR * 1.67))) * XHP) - ((333/(TTW/(1-WR))) * LV)  + (TSP * (333/TTW*WR)) + (SSV * (333/(TTW*GPS)))

...Which simplifies to the final formula:

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AI3
(333 * WR * SV * 0.75 * (1 + (EMR * 1.67)) * XHP / TTW) - (333 * (1-WR) * LV / TTW) + (333 * TSP * WR / TTW)

AI4
(333 * WR * SV * 0.6875 * (1 + (EMR * 1.91)) * XHP / TTW) - (333 * (1-WR) * LV / TTW) + (333 * TSP * WR / TTW)

Arena
(333 * WR * SV * 0.75 * (1 + (EMR * 1.67)) * XHP / TTW) - (333 * (1-WR) * LV / TTW) + (333 * TSP * WR / TTW) + (333 * SSV / GPS / TTW)

For simplicity, here is all the values of the various constants, a reference table for GPS, and the third or fourth time I define the terms used:

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WR means Win Rate (wins / games played)
TTW means Turns to Win (average)
EMR means EM Rate (EMs / wins)
SV means Stated Value (reward as shown in-game)
XHP means Extra HP (handles maxhp)
GPS means Games/Spin (see below)
LV means Loss Value (see below)
TSP means Total Spin Profit (see below)
SSV means Special Spin Value (see below)

     AI3  AI4    Bronze  Silver  Gold    Platinum   
SV   20   40     ~15     ~60     ~115    ~240
LV   10   20     15      15      15      15
TSP  25.6 104.77 25.6    25.6    25.6    25.6
SSV  0    0      115.4   115.4   960.44  960.44

GPS Table:
Bronze       (WR^-1) + (WR^-1)^2 + (WR^-1)^3 + (WR^-1)^4 + (WR^-1)^5
Silver       (WR^-1) + (WR^-1)^2 + (WR^-1)^3 + (WR^-1)^4
Gold         (WR^-1) + (WR^-1)^2 + (WR^-1)^3
Platinum     (WR^-1) + (WR^-1)^2

Here are the formulas with the constants applied for the various targets:

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Unsimplified:
AI3       (333 * WR * 20 * 0.75 * (1 + (EMR * 1.67)) * XHP / TTW) - (333 * (1-WR) * 10 / TTW) + (333 * 25.6 * WR / TTW)
AI4       (333 * WR * 40 * 0.6875 * (1 + (EMR * 1.91)) * XHP / TTW) - (333 * (1-WR) * 20 / TTW) + (333 * 117.9 * WR / TTW)
Bronze    (333 * WR * 15 * 0.75 * (1 + (EMR * 1.67)) * XHP / TTW) - (333 * (1-WR) * 15 / TTW) + (333 * 25.6 * WR / TTW) + (333 * 115.4 / GPS / TTW)
Silver    (333 * WR * 60 * 0.75 * (1 + (EMR * 1.67)) * XHP / TTW) - (333 * (1-WR) * 15 / TTW) + (333 * 25.6 * WR / TTW) + (333 * 115.4 / GPS / TTW)
Gold      (333 * WR * 115 * 0.75 * (1 + (EMR * 1.67)) * XHP / TTW) - (333 * (1-WR) * 15 / TTW) + (333 * 25.6 * WR / TTW) + (333 * 960.44 / GPS / TTW)
Platinum  (333 * WR * 240 * 0.75 * (1 + (EMR * 1.67)) * XHP / TTW) - (333 * (1-WR) * 15 / TTW) + (333 * 25.6 * WR / TTW) + (333 * 960.44 / GPS / TTW)

Simplified:
AI3       (4995 * WR * (1 + (EMR * 1.67)) * XHP / TTW) - (3330 * (1-WR) / TTW) + (8524.80 * WR / TTW)
AI4       (9157.5 * WR * (1 + (EMR * 1.91)) * XHP / TTW) - (6660 * (1-WR) / TTW) + (39260.7 * WR / TTW)
Bronze    (3746.25 * WR * (1 + (EMR * 1.67)) * XHP / TTW) - (4995 * (1-WR) / TTW) + (8524.80 * WR / TTW) + (38428.2 / GPS / TTW)
Silver    (14985 * WR * (1 + (EMR * 1.67)) * XHP / TTW) - (4995 * (1-WR) / TTW) + (8524.80 * WR / TTW) + (38428.2 / GPS / TTW)
Gold      (28721.25 * WR * (1 + (EMR * 1.67)) * XHP / TTW) - (4995 * (1-WR) / TTW) + (8524.80 * WR / TTW) + (319826.52 / GPS / TTW)
Platinum  (59940 * WR * (1 + (EMR * 1.67)) * XHP / TTW) - (4995 * (1-WR) / TTW) + (8524.80 * WR / TTW) + (319826.52 / GPS / TTW)

Note that, due to rounding, the simplified formula will produce slightly different results than the unsimplified formula. There are rounding errors all over the place already, so neither is really more "correct" than the other, and really due to the rounding errors, UEI is probably accurate to around the nearest hundred or so. Not that that will stop me from citing values to the hundredth of an electrum all the time. Note that all studies I manage use the simplified formula.


As I said at the beginning, I am posting this primarily because I would like some peer review. Please, let me know what you think. If you have suggestions on how better to handle any part of this, or if you have suggestions on better values for constants, or if you found a typo, or if you found a fundamental flaw in my understanding of math, please, let me know. I would like as much input about this as I can get. Also, I wrote this more or less in a sitting, so it is very possible I made some error somewhere while transcribing everything. Please tell me if you find any. I will likely be updating this post in the near future.

A big thank you to everyone who has done anything that I drew from here, and a big thank you to anyone who actually managed to make it all the way through this enormous amount of stuff.


Bibliography:

Spoiler for Hidden:

Original FGEI Thread
1.32 FGEI Study
1.32 Arena Stats Study
1.32 AI4 Stats Study
AI3 Spins Simulation
Special Spin Formula
Special Spin Droprate

edit: Corrected game->win in first formula per rob's comment.
edit2: Added bibliography.
edit3: Corrected TTW->TPW error in special spin formula per my comment below.
edit4: Corrected TTW->TPW error in wins/hour formula per my comment below. Ugh.
edit5: Corrected misplaced XHP modifier. Also cleaned up most of the formulae so they are hopefully more readable. Added GPS table.
edit6: Fixed XHP error for now.
edit7: Clarified SSV
edit8: Major update. See here
edit9: Fixed typo in EM factor in unsimplified formula for arena decks.
edit10: Fixed incorrect EM factor and added details regarding UEI(333) and UEI(250).

[ColorlessGreen]

Reserved. For reasons.

[rob77dp]

I'm going to start this now and simply modify as I continue to read through and digest the sheer volume of quality content in your post.

#1:  In your first formula I believe there is a "game" that should be "wins"

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(((Non-EM wins over a time period) * (Average profit per Non-EM win)) + ((EM wins over a time period) * (Average profit per EM win)) * (Average percentage of extra HP)) - ((Losses over a time period) * (Cost per loss)) + ((Wins over a time period) * (Average spins profit per win)) + ((Special spins over a time period) * (Average special spin profit))
where I have changed ((Wins over a time period) * (Average spins profit per game)) to (Wins over a time period) * (Average spins profit per win)).  I'm not sure it would effect your results depending on how it was implemented (game can be taken as wins since spins only happen on won games...).

#2:  Wins/Losses section does not address how skips might be handled.  Against False Gods skips are significant in the discussion for basically every FG grinder.  I would agree that other levels of AI are not so significant but as an example an AI3 that loses to Dim-chains will probably skip the Dim-Shield Elder when drawn as opponent.  How would such skips be handled?

#3.  Extra HP and Win/Loss Value - This may lead (or not) to a revamping of significant sections/portions of your original method.  However, I would like to point out (take it or leave it) that simply assuming 2/3 of max or that 57 HP is the average remaining at end of duels at all levels can lead to errors.  I use PDials SoSac a LOT grinding FG's and recording stats (I record less than I used to for stats ) and have discovered that with just a minute investment of time and record-keeping a player can take a few more stats and cover this aspect extremely accurately (that of XHP and W/L ).

I'll simply state the values/stats I record that may help here:
Start of or before duel:
* AI-level (0, 1, 2, 3, 4, 5, B, S, G, or P)
* opponent name (if normalizing anything it is a must - like FG's, Arena etc is optional since normalizing is difficult or impossible)

After duel before spinning (or when used to it during spins):
* HP Remaining (0 indicating loss)
* EM ('x' for yes and blank for no, or similar method)
* Turns to complete duel (0 indicating skips)
* Time to complete duel (in seconds; 1 to 5 for skips - I haven't settled/standardized this yet...)
* Spin results
   ~ Spin 1 ('x' for won a card; 'X' for won an upgraded card; '-' for first two reels matching for +5 bonus; blank for no-gain)
   ~ Spin 2 ('x' for won a card; 'X' for won an upgraded card; '-' for first two reels matching for +5 bonus; blank for no-gain)
   ~ Spin 3 ('x' for won a card; 'X' for won an upgraded card; '-' for first two reels matching for +5 bonus; blank for no-gain)
   ~ Spin Special (Arena-only: 'x' for won a card; '-' for first two reels matching for +5 bonus; blank for no-gain)

[ColorlessGreen]

Many thanks for the comments, rob! I'm also going to respond to things by modifying my post as you modify your post, because that sounds like fun.

(1) It should be "per win". I just wrote down the wrong thing. The formula is entirely built around calculating the value of a win, then calculating the average time to get that win and putting them together. This will be is corrected shortly.

(2) Skips have not been included in the formula because they have been ignored in every non-FG study ever, and they are pretty outside case for non-FG farming. I agree that they can come up in AI3 and such, but they're quite hard to quantify anywhere else, and they make a relatively low difference in AI3 due to the absurdly quick games and low number of required skips (and personally I don't even bother with autoskips in AI3, I just quit an AI3 game when I know I've lost, and I more or less consider this more a part of the TTW=TTL discussion honestly). With regards to FG, this formula isn't really aimed at FG (though it works in FG), since FGEI is a far better measure due to being focused specifically on the FGs and being able to normalize and such in ways that aren't possible in the other AIs (except AI3, but it isn't really worth the effort there IMO).

With that being said, I have played around with applying this formula to FGs (or whatever with skips) and it's not hard to modify the formula to account for it. Skips can be handled by tweaking the Turns Per Win formula and adding to the total loss value formula. Basically, calculate a skip rate, which is skip FGs divided by total FGs. Figure games per hour (i.e. turns per hour divided by TTW), multiply by skip rate, and add those extra losses into the total loss value section of the formula. Then change win rate to be non-skip win rate by changing win rate to be (wins / (wins + non-skip losses)). It adds a couple of extra steps if you're doing it by hand. This assumes a skip time of effectively zero, so it would probably be a good idea to modify turns per win by adding (say) one turn modified by skip rate (in some way which I can confirm after I go look at my notes after work).

(3) The constant used for electrum profit rate (0.7833) can be easily modified to whatever someone chooses to use, including an actual average from their tests. It is presented as a constant due to actual electrum being recorded basically nowhere that isn't FGEI. However, as EM profit is recorded as fully and accurately as possible, and non-EM post-game electrum profits are a pretty tiny part of the total profit from a deck (and the variation from an average isn't really that large), I don't personally consider this worth the extra bookkeeping.

As I said, though, the formula (at least the non-simplified final formula where you can still play with constants easily) can accommodate complete stats for post-game profit quite easily by substituting your actual stat for the assumed constant (and correcting the EM rate constant while you're at it). The assumed constant is available for situations where that information is unavailable, and uses the same method of determining this average as FGEI.

As FGEI1.32 develops, I'll take a look at the difference between the actual electrum reported in that study and the average used, and (if necessary) come up with a new assumed constant.

[andretimpa]

I see a flaw with trying to make something like this for gold and plat. The players that put decks in these arena levels are already experienced players, hence they have a very good idea of what they will face and build decks to counter the most popular farmers. This creates a metagame where the most efficient decks are not necessarily the most efficient farmers. Also, once a new "best farmer" arises the arena makes a move to counter it, reducing its efficiency. How can we take this into account?

[ColorlessGreen]

I see a flaw with trying to make something like this for gold and plat. The players that put decks in these arena levels are already experienced players, hence they have a very good idea of what they will face and build decks to counter the most popular farmers. This creates a metagame where the most efficient decks are not necessarily the most efficient farmers. Also, once a new "best farmer" arises the arena makes a move to counter it, reducing its efficiency. How can we take this into account?

The way it's (optimally) handled in the arena stats thread where testing of arena farming decks goes on is (ideally) to have multiple ongoing tests and re-tests, and paying attention to the age of the stats you're looking at. I know of no other way to handle this. Stats expire over time due to shifts in the meta and/or patches - arena (particularly the high leagues) just moves a bit faster than other targets like FG. I personally try to re-test the top arena farming decks in the arena stats thread every few months (at least when I'm spending my time testing arena farming) to account for meta shifts.

It would be best to consider any result of this formula to mean "this is the net profit of this deck at the time it was tested by the person/people who tested it."

[Pella]

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

[ColorlessGreen]

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.

[treebeard xiii]

So much quality content I'm glad someone has found a satisfactory formula for either proving or disproving which is the most productive farming target. You got a +rep from me

[ColorlessGreen]

Just caught a transcription error. Special spin rate is actually turns TO win times games/spin, rather than turns PER win. Games/spin already considers win rate, so there's no need to compensate for it twice by using TPW instead of TTW. My excel spreadsheet has this taken care of (since I actually have already made that mistake before) but apparently I screwed up when writing it out. Again.

That will be fixed (and will result in tweaking of basically all the final formulae) sometime tonight. Fixed.

[ColorlessGreen]

Holy crap, you guys. I'm starting to think the most important lesson of this thread is that if you name two of your variables "TTW" and "TPW" and then do all the final formula work literally immediately before going to bed after a long day, things are going to go poorly. Just caught another transcription error in the wins/hour section, where I incorrectly used TPW instead of TTW (again). So, I went and corrected all the formulae (again). I swear I'll find all the typos eventually. I'd like to think they're basically done with now because the results of the formula I'm getting from the spreadsheet I just whipped up at work here are pretty much similar to the results I'm remembering from the actual working model I have in my spreadsheet at home.


Anyways, speaking of results, here's some very preliminary results for some various things purely as a matter of interest. I don't consider these numbers final until everyone has had a chance to discuss the formula and things get adjusted based on that (and I figure out where else I managed to make tiny slip-ups with major consequences). These are pretty much just some random results from arenastats and ai4 stats. The ai3 stat is simulated (taken from kev's ai3 vs ai4 thread).

Unupped

Code: [Select]

Deck                  Net Profit/Hr                   

AI4/Ghostmare         2,230.91
AI4/HBGB              2,011.83
AI4/Psiontal          1,971.00
AI4/Monoaether        1,880.82

Silver/Antabow        1,586.14
Silver/Monodark       1,586.01

AI3/Monodeath         1,442.32

Bronze/Monodark       1,409.66   *bronze stats do not include TTW/EM. rate from silver used.
Bronze/Antabow        1,343.54
Bronze/Monodeath      1,275.68
Bronze/Psiontal       1,172.93

Upped

Code: [Select]

Deck                  Net Profit/Hr                   

Plat/PDials          11,617.92   *plat stats do not include TTW/EM. rate from gold used. see below.
Gold/PDials           9,284.97   *assumes 148 maxhp on average.
Gold/Chaos Wyrms      6,809.24
Plat/Swallow          6,643.00   *NOTE pdials results reported here are wrong due to
Gold/Pestal           5,771.36    using an incorrect maxhp factor
AI4/OffenseIsDiamonds 4,061.71

Note regarding pdials in plat: the pdials plat test was done a long time ago and does not represent pdials' ability currently (or pdials ability for quite a while) according to any time I've played lately, and according to basically everyone I've discussed it with. It also doesn't include TTW or EMR stats, so for the formula I used the gold stats (as I noted) and I'm sure that both TTW and EMR are worse in plat than they are in gold. It badly needs re-tested. It's included just because if it wasn't included, everyone would ask for it. Don't assume that number is accurate.

My favorite part of that is that there's a tenth of an electrum difference between the two unupped silver decks.

Like I said, those results are from whipping up a quick spreadsheet at work just to play around with the formula. When it gets more finalized, I'll post some more complete tables. If you're wondering about any of the decks in question, please see the stats threads that I took them from linked in the bibliography in the OP.


edit: Awesome, just found another error. Fortunately, it only affects pdials, and only by a very tiny bit. When I put together the final formula, I somehow managed to move XHP over so it also modifies losses. It shouldn't - XHP should just modify regular win value. Will be fixed in a few hours, since I'm leaving work right about now and have to do some stuff before getting home. Fixed. Note that the pdials stats above are about 50/electrum per hour less than they should be.

[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).