The Ranked-Points System Thread: Page 3

The Ranked-Points System Thread

Gocnak

@Gocnak , ask people outside Momentum! Ask people far outside Momentum. I will only strengthen the group-thinking if I continue and I can't waste more time on this

The problem that arises is that the communities I would reach out to that would have an idea of how to go about this would be the speedrunning communities I've taken part in. The issue here is that the only measure of skill there is just purely time. There's no point system with regards to a game's speedrun.com leaderboards, mostly because it'd be like looking at designing a rank system for only one map, which in Momentum's context will be complete and utter tunnel vision. At that point there is no purpose to have an overarching system like this rank points one, because there's only one thing being measured: completion for one category of one game. Compare it to Momentum's thousands of maps across tens of different, pretty unique gamemodes. We have to drive one's improvement across it all, ergo this rank points system on top of a leaderboard that already pretty clearly defines who's "better" than you.

The goal of this rank points system is (or at least, should be) to motivate improvement. Someone's total rank points for a gamemode is just a measure of how consistently good they are at the mode, as one with higher overall points has consistently set higher overall ranks in the leaderboards. It is also the only dynamic system we have, as you can lose points as people beat you and bump you out of groups. If that happens, go back and set a better time, you can do better. Ideally, in the vast competitive world of free-to-play platforming like Momentum will be, you shouldn't set one time on a map and be comfortable if it wasn't the best you could possibly do. Someone will beat your time -> you have less gamemode rank points -> an intrinsic value to go "get those points back" and improve your time.

On the other hand, the goal of the cosmetic XP system is to reward the efforts going into actually playing the entire game, but it doesn't say much about one's skill because you can grind one map for it. Whereas with rank points, you could grind a map all you want, but if you don't improve your rank, then yeah you don't get anything in return for it with regards to rank points. This ties in with what was mentioned before about a 5 hour improvement to a 9 hour PB. This is ranked points we're talking about. If you didn't improve your rank, you don't get any change of rank points, but you will definitely get cosmetic XP for the effort.

zer0k_z

Gocnak The goal of this rank points system is (or at least, should be) to motivate improvement.

Gocnak This ties in with what was mentioned before about a 5 hour improvement to a 9 hour PB. This is ranked points we're talking about. If you didn't improve your rank, you don't get any change of rank points, but you will definitely get cosmetic XP for the effort.

It isn't exactly motivating to give 0 point for a decent improvement, don't you think? Of course this example might be too extreme, but imagine a 5 minute improvement to an one hour run on a very hard map instead. This kind of things happens a lot in harder maps (again, from a kz perspective).

Take a look at this leaderboard of a KZ map for example. This map is tier 6, one tier away from the hardest tier: https://kzprofile.com/maps.html?map=kz_pendulum&mode=kz_timer&run=tp

The last column is the points awarded to the player.

You can see that #56 is faster than #57 by a small margin, so the point difference is small. But #57 is much faster than #56, therefore the point difference is bigger! Also notice no matter how small the margin is, the faster guy always gets more points (see #60 and #61).

Now if you use your current system, the point gap between all of these people would be more or less the same. The point difference between #58 and #59 would basically be the same as #59 and #60, even though their time gap is 120 times bigger than the latter! The demotivating part comes when you are #58, you can improve your run by a 19 minutes and get exactly 0 point as a reward, after 2.5 hours of tirelessly grinding! I don't think that is fair or motivating.

Again, I'm not asking for a total remake of the system, I just want it to not base solely on rank.

879m

zer0k_z The demotivating part comes when you are #58, you can improve your run by a 19 minutes and get exactly 0 point as a reward, after 2.5 hours of tirelessly grinding! I don't think that is fair or motivating.

A 20min improve on a multiple hour time isn't a big improve though. I know there's some fucked up maps (and don't know a lot about kz maps) but if you don't significantly decrease your time (i'm talking hours) then you haven't gotten much better. If you were failing for hours on your last completion, you might just get slightly luckier and complete a hard section quicker on a rerun while still being just as bad.

In rj/sj, the general attitude towards multi-hour completions (with very few exceptions) is that whoever gets them was barely good enough to beat the map and get their completion. These players should go and get better by playing other maps for a while. Then when they eventually return to the hard map they will improve their time by a lot.

The improve in that kz situation is actually worth the pittance it gives imo.

MichaelViera

same. none of us want a total remake ........... i have no opinions about groups or anything, or whether a rank system should motivate or not, whether it should try to be accurate or not, make groups , don't make groups, make a steep system, make a flat system, make time based, make anything, make whatever you want (or i do have some light opinions, but they are irrelevant) but don't try to use # completions as driver for competitiveness without adjusting for tier xdddddddddddddd it's a 2+2=5 error. and a rank system with such an error should never see the light of public.

MichaelViera

879m
"A 20min improve on a multiple hour time isn't a big improve though"
In that argument, you appear to acknowledge that minutes say something about improving, but 20mins is just too little of a marginal on a multiple-hour time. So you assume there is some magic arbitrary baseline where the (relative) marginal gets too little, and first, if you go over this baseline, it matters? Wuuuut. If it is very little, then it rewards with very little ofc. There is something called continuous.

People seem to don't understand the term "everything else being equal".

Everything else being equal, when you improve a run, it is because you have improved; become better at the map. There can be exceptions, but the exceptions (about being lucky, unlucky etc.) are equally likely to happen on both runs. You even acknowledge this yourself "you haven't gotten too much better", again working with this weird arbitrary baseline.

Why should they go to other maps and become better? Everything else held equal, all maps are equally good means to become better ......

Especially multiple hour completions can correlate with haven't learned the map properly (or playing a lengthy map where you tend to spent long time to climb the learning curve), and when you haven't learned a map properly, you are at your lower point on the learning curve where the slope value is very large, thus MORE than anything suggesting that in your next run, you will perm the run better objectively .......

By the way, I have no opinion about this. I just look at arguments that work and that don't work

Gocnak

I believe a while ago there was a discussion about people wanting their rank points in groups to reflect as a % of the time compared to the group leader. That way, if you're bottom of a group and there's a large gap between you and the "top" of the group, there'd be a bigger point difference between you. Any incremental changes to time would calculate off of that group leader's time, off of some fixed value. The more you approach the group leader's time, the more you would have to that static value. The group leader would most likely just have that static value?

This way ranks would still determine groups and sizes, but individual times in groups would also factor in how many actual points you'd get for being in the group. The group # would still impact how many points there can be, but your time compared to the group leader would determine how much you get.

Beetle179

here's some somewhat disjoint thoughts that i somewhat tried to organize:

issue i have with using time as a factor: yes, a 5 hour improve to a 9 hour time is a huge improvement, but how many points is that worth? well, it's a HUGE improve, so it should be worth a lot, right? well... no, because it's still, comparatively, a very slow time, and it didn't improve any ranks. so you award a tiny amount of points because it's still an improve. but... 5 hour improve? tiny amount of points? well that doesn't feel fair, it's such a big improve! but you CAN'T award a lot of points, because then you're going to run into inflation problems.

i'm not grossly opposed to using time as a factor, in fact i like the sentiment. but i don't think it's necessary. the system as-is is already kind of bloated and complicated, but at its heart, it only has one variable -- your maprank. and frankly, in a competitive context, i think that's the only really important thing. yes, it leaves some gaps: maybe SUPER strong records could award bonus points, maybe really close times could award similar points, maybe this or that. but with every new thing you try to address, you're adding more complexity and obfuscation to the system. i learned that lesson while reworking the KSF WR points system: it's hopelessly complex, computationally expensive, and at best, it only KIND OF works.

all that said -- i think there's merit to replacing the "Formula points" element with a time-based element. i think that would be the best way to include time as an element, if we were to do it.

i come from a surf perspective, where times tend to be much more clustered and dense, tending to follow clean distributions and with very few big time gaps between ranks (until you get to the slower end of times). so, improving your PR is pretty much always going to improve several ranks. i understand that's not the case with kz, and maybe other modes as well. STILL, i don't think time-based points are the best solution -- at the end of the day, imo, if i improve 2 seconds on a 2 minute time and don't happen to beat any ranks, well, unlucky for me. and i don't think getting a tiny amount of points for that improve would make me feel very rewarded. (if anything, i'd feel more cheated than i would if i just didn't get anything)

another disjoint thought: i think time as a factor makes more sense in the kz points system, but i also am deeply not a fan of the kz points system, which i feel is incredibly lacking for competitive players. (i.e. rank 1 on bkz_apricity_v3 = 3 minutes, 1000 points; rank 10 = 4 minutes, 950 points. why improve from rank 10, when you'll only get 5% more points? just play more maps!)

one more disjoint thought: within-group scaling is interesting, and i think would be a good way to address the issue of formula points scaling weirdly and feeling pointless. my only issue with it is that it's a lot more complex. but, imagine something like this, a combination of formula points + group points: https://i.imgur.com/i1aCLwM.png
you still get healthy points for improving groups, but you get more points for improving within better groups, to reflect the increased difficulty of shaving off the last few seconds. (imagining linear scaling within groups following a simple rule like 5x for group1, 4x for group2, 3x for group3, etc. but could be played with)

zer0k_z

Beetle179 i also am deeply not a fan of the kz points system, which i feel is incredibly lacking for competitive players. (i.e. rank 1 on bkz_apricity_v3 = 3 minutes, 1000 points; rank 10 = 4 minutes, 950 points. why improve from rank 10, when you'll only get 5% more points? just play more maps!)

This makes sense, points in KZ are not meant for competitive players (like at all). Competitive players look at world records, they don't care about points. No one looks at the top 10 players with the most points and thinks that they are the top 10 players in the scene, WR count is a much better indication for this, see the points leaderboard for example.

hex

long post but tl;dr jump != surf
im going to dump some screenshots of tempus (rj/sj) leaderboards of a few maps just to give some context since not everyone is familiar with those gamemodes. this is a small sample size but these types of issues occur very frequently (i.e. on nearly every single map).

rj_christmas, had an offline WR of 3:45ish for years that nobody broke on tempus until boshy ran the map for a few weeks and did break it with a very solid run. why should someone like crackuhjax ever bother rerunning if they don't think they can save almost a minute on their time to get wr and get any points from it?
[upl-image-preview url=//i.imgur.com/9fClO16.png]

rj_beef, was held by kidder for years basically uncontested until boshy grinded a very solid run to beat it and now basically nobody is going to bother getting above 3rd place because its an insane amount of effort given how optimized the map is.
[upl-image-preview url=//i.imgur.com/NT48gZ1.png]

sj_asdf, basically soup (alias go crazy here) and someone else trade a map until the person who isn't soup gives up. then the record dies because they don't rerun it and nobody else (see 3rd+ here) wants to because it's a lot of effort and improvement just to get a handful of points. this same thing can be found on over, scorpion, shiver, drenched, koi, karb, orbital, lanc, aznbob, excavation, etc etc etc lol.
[upl-image-preview url=//i.imgur.com/opONzgZ.png]

unless momentum has 10x the amount of people setting good times and world records this same issue is very likely to come up again. we can wait and see if that happens but the learning curve is so steep for learning and especially trying to speedrun jump it's very unlikely it doesn't.

this is exacerbated on tempus by the horrible point system that we have which works by basically giving x set points for completing any given map, getting a top 10 time then gives a static amount of extra points depending on your placement. getting two #6 times on two maps gives more points than a world record and requires a fraction of the effort for a competent top 5 jumper. obviously the momentum system is already a lot better than that in some ways (like discouraging pure completion spam to gain ranked points, rewarding #11 more points than #2000 lol, nuking awful offclass runs that give tons more points relative to effort required than on class runs) but it really does not encourage grinding a record when you could just play 5 different maps for a few top times to get way more points (see the large time gap between 1st and 10th on basically every map).

Hona

If anyone wants a sizable dataset to model their proposed solutions or modifications, here is a dump of all map times (only map, if it is a multi-course map, then the final total time is counted only).

Tempus Map Times Full Json File (332 MB)
(I made the file indented and formatted so it is easier to explore)

This dataset contains a total of 795113 PRs across the two different classes (which for the sake of analysis can be used as two modes).

It would be good if someone models this data against the current point system and several others - tempus, ksf and your proposed solution for example.

Some actual numbers and graphs will help to move the conversations along.

Zike

Apologies if this was already mentioned in the deluge of information. Adding on to hex's thing about jump though is another problem with the current system for jump is that it is trivial to get 5th-10th place times for a lot of people. WRs on the other hand take the top 5 jumpers a couple hours. Other high level players will take days. The current system eventually also runs into the Tempus problem of the most efficient way to get points and rank up is to spam 5th place top tens instead of getting good times. With the current system 2 5th place times takes people like an hour to get and gets them a bit more than 1 WR.

This problem isn't apparent at first when the map pool is small. As the map pool grows however, this problem becomes far more pronounced when the map pool is large enough to where not every player has ran every map competitively.

Ultimately a lot of players at the mid to high level in jump find themselves with the choice of either spamming bad top tens for maximum point and rank gain (monkey brain like number go up 😄), or smashing their head against a wall for a week to get one wr (that they'll probably lose later that week).

Ark_

Zike Ultimately a lot of players at the mid to high level in jump find themselves with the choice of either spamming bad top tens for maximum point and rank gain (monkey brain like number go up 😄), or smashing their head against a wall for a week to get one wr (that they'll probably lose later that week).

In XJ modes in particular, Being that all times in Momentum are offline, should be easier to be consistent at high level already, so i would expect the gaps to mostly close up as more people are able to get times similar to the best online times. If you include the fact that we may get an influx of players into less popular modes after 1.0.0, it may not be an issue at all.

Even if it continues to be though, grinding for one WR is really no different from grinding for a completion or grinding a new skill, and spamming easy top10's/groups for points only works if nobody else is doing the same thing, and at some point you actually need to get better. Surf has dealt with systems weighted too heavily in both directions there, so I think we will end up with something that works decently.

MichaelViera

yeah. I think also most of us agree that a rank system can be never used to measure the actual rank, but is a tool among other tools to measure actual rank - and then it's for motivation, entertainment, w/e, etc.

competition level should also be noticeable on percent wise differences in time, but I don't know enough about statistics to understand how hundred of ranks intermingle or how to make it simplified so it doesn't require too many calculations. and to my delight, these percent wise differences are independent of tier.

also, I wanna say that humans (and probably all animals) are very concerned with fair rewards, e.g., look up "equity theory" on Google (THIS GOES BOTH WAYS, you don't wanna be treated too good or too bad. people are not cowards). so I don't necessarily support superficial points gaps between groups, it's more important that they can flash their groups (though that can also be unfair because the high ranked players in the same group are equalized with the lower ones, but anything is fairer than farming points)

in the same way, it is more important that the wr takers can flash how many wr's they hold etc.

but I'm NOT arguing for either flat point system, steep point system, or anything now, just making some scattered points

MichaelViera

That with-in group scaling doesn't sound bad, but there is still the problem with top10's; it isn't thorough when only focusing on groups. Make it consistent rather than creating weird exceptions (but I support groups). KSF uses time-based as well for top10s? I can't see the issue, they can be transferred across all game modes perfectly fine - which the current system cannot because KZ has too long map learning curves (which isn't the same as tier by the way) as Hex put it (which is gonna have an impact for all ranks, rank10 to 11 isn't a magic line)! But I guess we lack someone who is educated in statistics?

(in all cases remove the # completions expanding group sizes for god sake. the only way each rank cross all tiers can be pushed upwards equally (in absolute terms instead of relative terms) is if each new member added to the population took a fucking rank1, and with such a tendency map tier wouldn't exist ....................)

edit2: the problem is that it is implying tier is independent of completions. whilst it is true the skill standards can increase a tiny bit, that would also simultaneously have to change the tiers because they are expressions of top%'s, and as a larger and larger top% would beat it, the tier would fall.

Jabra

Don't know why my Discord doesn't work.

Suggestion

Ofc. remove the group size function which is broken in its nature.

Use how many percent away from WR time to determine points with some function/distribution (let's call it the formula).

To determine group sizes, the max rank of the groups should be determined by fixed percents.

For example

Group1 3.333 % (let's call it marginal percent) away from WR (rank11; rank 3.333 percent away from WR)

Group2 6.666 % away from WR (rank 3.333 percent away from WR; 6.666% away from WR)

etc.

(maybe it shouldn't be linear like that)

If rank9 is 3.333 percent away from WR, then it is broken? Well, make group min size 1 (group1 would then be just 1 rank), but despite being in group1, the player will get the same amount of points according to his/her marginal percent.

Give bonus points for entering a new group, but it shouldn't give more than the formula. This won't favor short maps, that's only if you use ABSOLUTE marginals. it will only favor short maps as far as they are more competitive as well (you can try to offset this slightly, I have no opinion).

Like I suggested, the players should be able to flash their highest map rank, as well, they should also be able to flash their lowest marginal percent.

edit: I'm only here because very basic problems have been overlooked. I feel very bad when talking about something slightly related to maths because it is not my main field. Check Cowboyana's beautiful curves. But even those curves don't tackle the main issues. So someone has to say something lol.

edit2: maybe a Cowboyana-ish function can be used as the old formula to give the bonus points for both ranks AND groups. the main formula should be the marginal percent (take that as an independent variable, i don't know how the function should look like, but very steep preferably)

edit3: this will give the players something that is somewhat comparable cross maps and can be easily communicated: o, I got a 3 % today!!!

MichaelViera

Works again.

If we now could estimate the perfect time from the distribution of times, and then use marginal percent away from perfect time!!! That would be ideal.

edit: It can't exactly work because the perfect run does not have to spare his/her raw skills (e.g. it is not equally easy/hard to do a wrcp 2 times in row, so it will discriminate long maps). Look in edit2, it is only a problem for TAS runs.

But we can use the TAS runs! And then maybe learn to compensate for this hmmmm. Maybe the solution can lay in actually mixing the TAS and rank system.

edit2: if we infer the perfect time (which would be the perfect time adjusted for length naturally [so we actually don't have to adjust anything I believe!]), then we shouldn't go into that problem. Only TAS runs are impaired by this issue because they would naturally give the absolute perfect time 🙁

fingerprince

Jabra, please don't use multiple forum accounts, especially when posting in the same thread. You've just sent three posts in a row using two different accounts, essentially bumping your own post twice and confusing anyone who doesn't know you're the same person.

If you've already posted something and want to add something further - especially if it's the latest post in the thread -edit your existing post. If you're having account issues, contact a moderator through Discord, don't post on an alternate account.

MichaelViera

I wrote: "Don't know why my Discord doesn't work."

That's why. I think you have to be a member of Momentum Discord to use the forum. It just didn't work yesterday when I wasn't a member of Momentum Discord. If that's the case, maybe the website should inform one about that - or else it was a bug.

But please, let's think about these things, hmmmm.

edit1

The perfect run is a constant. But the perfect run adjusted for length is not a constant at all. I will try to add more to this soon.

edit2

Competitiveness says something about time put into to maps adjusted for time put into learning (which is NOT tier, but related to length especially, e.g., surf_classics has a long learning curve), rng games, etc. Time put into the map adjusted for learning curve is attempt time, but attempt time needs to be adjusted for rng games etc. - so we have adjusted time put into maps.

Percent wise differences (marginal percent) in run time (for example between a rank1 [10sec] and rank2 [11sec] gives 10 %; how it adds together cross more ranks I don't know, but only the highest ranks are important) are an expression of adjusted time put into maps as well! These things express competitiveness!

It is about ... The perfect run adjusted for length uses some adjusted time put into map ASSUMED (which connects to marginal percent assumed and competitiveness assumed etc.) - and how do we assume the competitiveness - we use the most contested map as a base!

So the distribution of run times in the most contested map should in some way be used to set the base. Perhaps it can only be in the top10 distribution; it is also problematic to use too many low rank run times because non-competitive times can be just as much as an expression of low skill as low commitment ish because it is indirect competition, not tournament competition.

So if the most contested map gets more contested (which has something to do with how close the times are I guess), then all perfect runs adjusted for length cross all maps will become faster, and people will lose points because they will get further away from perfect runs adjusted for length! So yeah, only when shifts in the most competitive top10 happens, large calculations take place I guess.

With these perfect runs adjusted for length, every surfer has like a universal valuta which is easily communicable "I got a 2.5 % today, but you only got a 3 % XD and we played for just as long". "Let's race for a 5 % today, bro?"

I don't know. Something like that. I want to add at last that rewards/points have to be backed up by actual value. Reward systems are not communism, but even under communism you can try to play the game and act like you are something, but it's demoralizing and not very motivating. Points cannot be added together fairly (which is okay, we must accept that), but per individual run times, we can find somewhat fair points (if we are clever enough at least).

Beside: My point about the adjusted time put into map is just that the adjusted time put into the map is like the fuel for the perfect runner adjusted for length. So the perfect run adjusted for length is more variable-ish and less constant-ish than I initially thought. The fuel is determined by the most competitive top10 distribution, maybe the perfect runner is rank1 in that distribution, idk ...

MichaelViera

I have edited it a lot now. There were a lot of mistakes before. The general point should be clear now!

Please, I'm trying my best inb4 "pseudointellectual" :d

Edit2:

I did a huge blunter. If 2 surfers with two different skill levels put the same adjusted time put into a map (ATPIM) into a track, then that will yield a marginal percent, if they increase the ATPIM, it will still yield the same marginal percent, but the absolute marginal will become smaller (so will STD as well ofc.). So the ATPIM will say something about how small the absolute marginal becomes, but not affecting marginal percent.

I don't even know what use it has.

edit3:

So as ATPIM approach infinity, they will approach the perfect run, and their absolute marginal will approach zero, but relative marginal will stay the same

There is something about euler's number I think ...

I don't know.

EVERYTHING SUMMED UP

Figure out to measure the competitiveness of top10 distributions.

Let the most competitive top10 set the base for the competitiveness of a perfect run adjusted for length across all maps.

You can call these perfect runs adjusted for length for the king (singular because they are united by a base determined by the most competitive top10) of the kings! A meta WR! Maybe the king of kings should be directly connected to the rank1 in that ultra competitive top10.

The player's goal is to get as close as possible percent-wise to that meta WR (close his/her marginal percent). That will determine the competitive value of his/her run time (which is independent of the top10 distributions that only determines the meta WR). So marginal percent will determine how many points a run gives, not ranks or groups. But whenever you increase in rank, group, etc., you will get extra points, but these points should be small; however, the bonus points for top10 and WR especially should not be too small.

Can you pass the perfect run adjusted for length? No idea. But that should obviously make a new most competitive top10.

Whenever changes in the most competitive top10 happens, and when the most competitive top10 falls to another top10, then large calculations take place because that will affect the points on all map ranks.

The groups' sizes are determined by upper limits that are determined by fixed marginal percents (and work as the lower limit for the next group, but group1 has rank11 as its lower limit ofc.)

If, e.g., the upper limit of group1 is placed in the top10, then group1 has a minimum size on 1 rank, so group1 is just rank11. If the upper limit of group2 lays in either top10 or group1, then group2 as well has a min size on 1 rank, so group2 is just rank12, etc.

4 groups are too few. Make 10 groups.

Besides, maybe we can use the TAS runs to gather information about perfect runs adjusted for length and issues in general related to determine competitiveness. By using TAS runs and continuous compounding, we can maybe determine the most competitive time

edit4

A big problem is that on some maps, even adjusted for length, it will naturally be easier to approach the perfect time. Some maps have what I call "flats". Max vel surfing is the best example of a flat. Surfers can do it just as good as the TAS on flats. So there will be issues in regards to max vel surf maps (e.g. as surf_utopia) being favored. The best surfing for competition is between peaked and flat, rampstrafes are the perfect peaks for competition because they are completely blind angle free, and you have a perfect overview - otherwise many peaks can increase rng games. Maps with rampstrafes will be heavily discriminated.

Daaaaaaaaaaaaaaaaaaaaaaaaamn, it's just hard. then adjust for peaks/flats etc., etc. Maybe in 100 years we have the brain for all that.

zer0k_z

MichaelViera

How does one define "most contested map" or "most competitive map"? How do you adjust for map length?

There are some key issues with this:

There is no perfect run, even if you are assuming TAS = perfect run, in reality TAS runs are not absolute. Some TAS runs are faster than others. Ultimately, there's a person behind TAS runs, and that person is susceptible to flaws. TAS can also go to the opposite side of the spectrum and do skips that are not humanly possible where you can possibly skip massive section of the map. This is a mess and should not be in any kind of ranking calculation for human.
Map length is not always known. Disregarding the "perfect" part, every "perfect run adjusted for map length" is one skip away from being completely false. This becomes a bigger trouble when people eventually argue about whether or not to remove the skip. In hard maps where people fail thousands of times, who can say what is the actual map length?
You believe that relative difference of time (in the top 10) indicates competitiveness which is not always true. I present to you "the most competitive course ever", according to the % difference in time.
Anyone playing this course knows this is complete bullshit because it literally is just a fall and the way to get WR is to play on an unstable server.
This is nitpicking, but

MichaelViera So as ATPIM approach infinity, they will approach the perfect run, and their absolute marginal will approach zero, but relative marginal will stay the same

Relative marginal only stay the same if time is continuous, which obviously is not the case here.

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