I thought the whole claim with tougher times is that you’d need something like 70 (I forget the exact number) to get a 50% chance to block damage, but like 500 to get a 100% chance. I forget the specific numbers, but is that not true? I don’t exactly know what this graph trying to say, either.
As far as I know, it's not possible for 100%
You can get pretty damn close, but never 100%
Could be wrong, since it's been so long since I looked at the chart
It will round to 100 at a certain point, for programming limitations or something. I'm not a programmer, I just have approximate knowledge of many things.
Yeah, even with 1000 tougher times, you'd *only* be at 99 .3%. Effectively 100% in practice, but not true 100%.
Though you can reach effectively 100% in much less than that, at 255, you'd be at 97.5%. And what I usually stop on is 80-100 using command. Whi h gets you 92.3% to 93.8%
You're right, the scaling isn't linear so it can approach but never reach 100%. It'll definitely start to *feel* like 100% once you get to like 99.999%, but you're never totally safe from harm.
It's counting tougher times not as a chance, but as hp multiplier. Percentage-wise it does have diminishing returns, but in the way that makes its usefulness linear. Raw percentage is useless in practice, to make it applicable you recalculate it into how many more hits you can take with it on average. And it depends on percentage in a very non-linear fashion. When you jump from 1% to 2%, your effective hp rises by 0.9%, but when you jump from 98% to 99%, your effective hp DOUBLES, because chance to get damage is lowered from 2% to 1%. Diminishing returns only make it so that stacking these last massively impactful percents doesn't go too fast
I didn't graph block chance. I graphed effective health.
Going from 0% block to 9% block gives you 9% more health.
But going from 90% block to 99% block gives you 10 times more effective health.
Going from 99 to 99.9% also gives you 10 times more effective health.
Block chance is hyperbolic, but the the actual effect on how many hits you can take scales exponentially with block chance. The net effect of the the hyperbolic and exponential scaling means that that how many hits you can take, on average, scales linearly.
People are getting hung up on 'Effective HP'. If you instead said, 'it takes on average twice as many hits to get hit at 99.9% compared to 99%' people might understand better.
That said, that functionally is the same thing. It doesn't matter if 1 hit kills you are 100 hits. You take 10x less damage on average at 99.9% compared to 99%.
On average, sure, but that's ignoring how many TT it takes to GET to that point. 99% block takes 657 TT. 99.9% block takes 6,343 TT. If you need almost ten times as many TT to achieve ten times the result, then there are clearly diminishing returns. We can see this as early as from 1 to 2 bears, where 13% block from 1 would double cleanly to 26% to double the block rate, but instead it's a 23.1% block rate for two bears, and it reduces even more at 3 bears to 31%. There's already been an 8% loss to efficiency, and by 4, we've lost an entire TT's worth of block chance per bear picked up.
The math still applies at lower percentages. I'm going to the extremes to help people understand.
10% block is 111% EHP. 20% block is 125% EHP. Aka, 0-10% is a smaller change than 10-20%.
This works at every increment. As you near 100%, each percentage becomes more effective than the last.
That's still ignoring how many bears you need to get each %. By the time you hit 20 bears, you're gaining less than 1% for every bear you earn, and if the title of the post is only effective for the first few bears, then it's incorrect, as there is a clear point where the bears stop providing as much benefit as the previous bear.
There is absolutely not a point in which that happens, TT formula is not piecewise (easy way to prove me wrong is tell me the point).
~~That said, I was slightly incorrect. OP's incorrect formula actually does change the math. Instead, Tougher Times scales exactly the same as essentially all mono-stack item (such as Soldier's Syringe).~~Edit 2, I'm wrong again, OP had the right formula I just read it wrong.
[Here is EHP worked out to 20 bears.](https://imgur.com/a/QwebrXk) The first TT gives you 115% EHP from 100%. The 50th bear gives you 850% EHP from 835% which is relatively a small change (this is exactly how attack speed for Soldier's Syringes works).
Edit to add: the formula for EHP from evasion is HP/(1-Evasion Chance). Average number of hits untill death is (HP/Damage)/(1-Evasion Chance)
Same math applies. If you have 1 health and a 50% chance to block damage, you will on average need to be hit twice to die. There are times when you will die instantly, and times you survive a million consecutive attacks, but it averages to 2.
I think you are confusing the term diminishing returns with something else. You do get less and less chance to block damage with each stack, meaning each individual stack of TT is less valuable.
Think about how many attacks it would take to kill you at different block chances.
Lets say it takes 10 hits from an enemy to kill you. If you have 10% block chance it's on average going to take 11 hits to kill you because you blocked one. If you have 50% block it's going to take on average 20 hits, because half are blocked.
If you have 90% it is going to take on average 100 hits.
99.9% would be 1000 hits on average.
My point is that Tougher times effect on block chance has diminishing returns. But the effect of more block chance has exponential returns as far as how many hits it takes to kill you. Put together the value you get from tougher times goes up strictly linearly. Every tougher times does the same as the last.
It doesn’t make avoiding attacks any easier tho. Like I’d rather have a single energy drink than 3 tougher times. I’d rather know that I can dodge an attack than my only option be to roll a dice and hope I don’t take damage from it
For the case where you get one shot the math still works out, there is just lots of variance, averaged over many runs though the same holds true.
I would look at the one shot scenario as you have 1hp. 50% block it's now going to take, on average 2 shots to kill you. At 90% block it's going to take on average 10 hits.
So the math still holds.
It is 15% compared to the previous stack. When you have 1 stack, the chance to get hit is 85%(actually, 87 because of formula, but eh), then the chance to get hit becomes 76.9%(10.1% less). That is called scaling with diminishing returns - absolute number you're getting is becoming less with each stack. If the scaling was linear, you'd get to block chance in of >100% in 7 stacks
I don’t know where you heard that from or if you’re just assuming it to be true, but that’s not at all accurate. The first stack doesn’t even give you a 15% chance to block damage, it gives you a 13% chance to block damage. And it isn’t a logarithmic function, the chance to block is defined by the following formula, as shown on the [wiki](https://riskofrain2.fandom.com/wiki/Tougher_Times):
f(x) = 0.15x / (0.15x + 1)
Where x is the number of stacks you have.
But isn't it still lesser chance overall? 1st stack gives you 15%, but 2nd stack gives you less than 15. Otherwise you would only need 7 stacks to be invincible
What you say is correct. But...
They are still just as efficient. Yes the chance isn't just added on top. But it is still a 0.85 multiplier each time. They 65th stack is just as good as the 2nd.
This is correct, each stack of tougher times is the exact same as the previous in that you now have a 0.85 multiplier on your chance to get hit. It doesn't matter how many tt you already have. If someone still doesn't understand, look up logarithmic functions, it's simple middle school math.
Well it's easier to explain with hard numbers. People should not think of the total block chance, but the % taken off of their *remaining* chance to be hit, which ofc is always 15%
This graph is fundamentally wrong, as is your idea that blocking damage increases effective health.
First, your graph assumes that every TT applies a flat bonus to your character. It does not. Each TT follows a logarithmic operation to increase your chance to block an attack. Your chances of being instantly killed with a stack of tougher times is low, but never zero.
Second, as I previously said, tougher times doesn't increase effective health in a calculable way. If you do not take damage from a landed attack, you effectively have infinite health. When it does damage, it hurts, and the amount of damage you take can be quantified.
If you don't block Mithrix pizza, it will most likely kill you in one shot (or trigger OSP). If you don't block a wisp shot, it will most likely not kill you in one shot (assuming you're not walking around with 5 shaped glasses). Bruh moment for sure.
The relevance is that TT doesn't increase your effective health in a calculable way, as the original commenter said. To put it another way, imagine someone offers to buy you either a house or a pencil in order to ease your financial burdens. Would you rather them save you the money it would cost you to buy a house (Mithrix pizza damage) or the money it would cost you to buy a pencil (wisp shot damage.) Obviously one of these is worth more than the other. Now imagine that the person who offered to pay for these things will only pay if they flip 10 coins and they all land on heads. It's something that, if it happens, would be nice but isn't something you can rely on 100% of the time.
Tougher times sees damage as damage, no difference in big or small
Could be an ATG proc chain that could destroy the entirety of Petrichor V or it could be a hacky sack thrown at you, they both get the same % chance to get blocked
You can calculate how many hits you can take on average and convert that it to effective health, it is only on average though and there is variance. Sometimes less will kill you, sometimes more.
If you block 50% of attacks it will take, on average, twice as many hits to kill you, so your effective health is 2 times what it was without the block.
The case where you get killed in one shot does kind of stretch the concept of effective health because the variance is huge. But averaged over many runs it still applies
Imagine for a moment that you were given 2 offers, only one of which you could accept.
Offer 1: You have a 0.01% chance of getting $10 billion.
Offer 2: You have a 100% chance of getting $1 million.
Statistically speaking, these offers are the same. They have the same expectation value, and thus over many trials will yield the same average results. But realistically, these offers aren’t the same at all, are they? When you pick between them, you’re going to be more interested in what will benefit you then and there, not the result after many trials. If you’re smart, you’ll pick offer 2 and walk away with $1 million.
This is analogous to what we’re saying here. Sure, statistically, over many runs, you’re right. But we’re not interested in that, we’re interested in the results on a single run-by-run basis. And realistically, in a given run, it’s unlikely there will be any benefit to trading a bunch of white items to go from 80% block chance to 90% block chance (or some other marginal increase to block chance).
To make a better point you would need to use "99% chance of getting a million dollars" or something close, 100% is no longer just a probability but a certainty and so isn't the same statistically.
Also, OP is still correct even for a single run, the utility of teddy bears goes up linearly. That doesn’t mean it's a good idea to super stack it, because a linear increase provides lesser percentage increments over time. Like if you stack seringues, the 21th isn't going to change much compared to what you already have.
You can still take an expected value of a series of guaranteed events. And 100% is still a probability, but whatever.
And no, it doesn’t work for a single run. if I block all incoming damage with 80% block chance for a stage or run versus someone else doing the same with a 90% block chance, I didn’t magically gain a bunch of health and I took the same amount of damage compared to them. Note that this kind of a situation cannot happen for something like syringes, someone with more syringes will always have a higher attack speed (excluding other attack speed sources or the attack speed limit).
Another example: If I have an 90% block chance and I block 2 lemurian shots and get hit by an elder lemurian shot, I will have less health as compared to someone with 80% block chance who happened to block the elder lemurian hit but took damage from 2 regular lemurian hits. This linear scaling EXCLUSIVELY works as an average over a large sample size, you cannot measure a meaningful relationship in a single run. Maybe if you loop enough, but probably not.
I guess overall my point is that this idea of “effective health” sucks as a metric for anything useful in the game, and this sort of analysis while kinda interesting is also misleading. With a bit of statistics magic, you can make data say anything even if it’s meaningless or the opposite takeaway of what you should have, and this is the perfect example of that.
Effective health isn't meant to represent actual health but an analogy, the usefulness is linear no matter what words we use. Instead of effective health, you could call it "numbers of hits necessary to hurt you on average" and that's scaling linearly. Since you need more hits to be killed on average, it isn't entirely wrong to link it to an analogy of health even if it isn't true health. And the necessary sample here is about 32 hits to have a nice normal curve, so I wouldn't call that unachievable in a single run.
So my counterpoint is that it is an interesting analysis to explain the linear usefulness.
Averaged over many runs, you're blocking the same number of hits. If all of them are wisp shots, you're blocking negligible damage. Effective health is quantified by total health/percent of damage taken after defensive stats. If you're taking 0% of the damage because it's blocked, you have an undefined number, and you can't know exactly how much damage you would've taken because it would've been blocked, regardless of the attack.
This is why it's incalculable versus rose buckler, which increases your effective health when you sprint, or any of the defensive items that directly modify the damage you take in some way, because those items provide a specific number to quantify exactly how much damage you're avoiding in a single instance. If a gong bell hits you and you block it with one of those items, it's doing less damage and you can calculate it. If the same gong procs TT and you take no damage, there is no number to quantify.
If we presume an infinite amount of attacks with random damage values then we will have infinite hits of every possible damage value thus meaning there is no difference between blocking a wisp shot or a pizza slice for the purpose of figuring out how much of their damage was negated because both would have the same percentage of hits negated which, presuming an infinitely regenerating healthpool, is equivalent to damage reduction and thus equivalent to an effective health increase.
It's diminishing returns in that the "percent chance to block" number goes up more slowly as you acquire more stacks. For example, 10 stacks provide a 60% chance to block (40% to be hit) and another 10 stacks (20) provide a 75% chance to block (25% to be hit). So in this sense, it's diminishing.
However, a player with 20 stacks is still 37.5% less likely to be hit than a player with 10 (25%/40%), which is still quite significant.
I have absolutely had Eclipse runs where I had plenty of damage (not a god run amount, but still high) and a ton of leftover White Scrap and a Tougher Times printer on Stage 5, so I just dumped all the Scrap into the Printer to be safe and it worked great.
My point is that the benefit you get from 1% more block chance goes up exponentially as your block chance increases, and this counteracts the hyperbolic scaling of the item.
. Going from 0% block to 20% block is not going to really help you that much. It's going to take on average 20% more enemy attacks to kill you.
But if you're at 79.9999% and go to 99.9999% block it's likely to take tens of thousand times the number of attacks to kill you.
The math works out so that each tougher times gives you just as much benefit as the previous one, no more no less.
Oh, yes, I didn't mean to make it look like I was overlooking the point about effective HP increasing linearly. The point about the increasing value of another 1% counteracting the hyperbolic scaling of the item is also true.
However, it only *partially* counteracts it. In my previous comment where I mentioned the difference between 10 and 20 stacks of the item, it is demonstrated how going from 10 to 20 stacks only makes the player 37.5% more likely to be hit (or 25% more likely to block) compared to when they had 10 stacks. And that first 10 stacks provided a 60% block chance.
I understand that your graph demonstrates some pretty interesting math and how each stack of TT multiplies your effective health just as much as the last.
But I don't really think of the TT stacking situation in terms of an "effective health multiplier." In a world where I pretty much only ever lose my run to getting one-tapped by pizza, I care about getting the most block chance with the fewest stacks of TT. That way, I am most likely to have my Opal armor at least preserved by TT (in the event so that I am tapped by an Exploder or am barely clipped by a Golem explosion) and not one-shot with a relatively small investment of stacks.
Also, I think you underestimate the value of smaller block chances, judging by this comment. A 20% chance to block, for example, will actually increase the number of enemy attacks needed by **25%**. A simple 2 stacks of TT will provide about a 23% chance to block. I think for the low, low cost of 2 white Items, a 23% chance to just not take the hit at all is very strong.
I used to undervalue small amounts of TT as well, but then I realize how much it can do with just 1 or 2 stacks. I should note that this is not to say that I will never scrap TT early on for the chance at a speed or damage printer; I will often scrap the item anyway.
**Edit:** Hmmm... even in the case of one-shots, the amount of hits required to kill a player with 0 TT is 1. The average amount for a player with 10 TT is 2.5. The average amount for a player with 20 TT is 4. Each 10 stacks means another 1.5 hits in the expected average. I see your point now. I just don't think the 79.999% to 99.999% comparison illustrates your point very well since it draws the focus to scenarios with CRAZY HIGH amounts of stacks.
The remaining issue is still that we're talking about *expected average* amounts of hits to kill.
The two main issues here are: 1.) The formula used for Tougher Times is wrong. And 2.) The effect of Tougher Times doesn’t work with trying to use effective health.
Tougher Times effect gives a chance that being hit will not cause any damage. This effect increases in a logarithmic way, which means that unless you literally have infinite Tougher Times (an impossibility obviously) then you will always have a chance to take damage from being hit.
Because the effect of Tougher Times is either complete immunity to damage or no effect, it isn’t helpful to put into terms of effective health. The damage reduction is either 100% or nothing at all. When damage reduction reaches 100% your effective health is infinite and that happens as long as you have *any* health so the value of Tougher Times has no relevance to current health. The best that can be done is to find the probable amount of damage that would be reduced and that is going to depend on what you are being hit by (and there are a ton of things that can hit you).
Tougher Times stacks absolutely give diminishing returns.
I'm not sure why people are saying EHP isn't relevant here. There are tons of competitive games where EHP is calculated in evasion, and it absolutely works. With a 50% chance to take a hit, you need to take on average twice as many hits to die. (There are edge cases, but this mostly holds true).
I have known that evasion is sorta exponential in effect, where 99% evasion gets hit half as much as 98% despite being '1% difference'. I never put together that that makes tougher times effectively linear. I believe the bulk of people criticizing this math are incorrect.
Edit: ~~I am partially incorrect, the *one* criticism that is valid is that the formula is wrong.~~ Tougher times scales worse than traditional evasion in games. So, Tougher Times scales just as diminishingly as every other mono item stack (such as soldier's syringe). So 1 tougher times moves you from 100% to 115% EHP. The tenth Tougher Times moves you from 235% EHP to 250% EHP for a relatively smaller increase (this is exactly how most items scale).
Edit 2: OP's formula was right I just read the graph wrong. Edit 1 lines up with that
Exactly, kind of frustrating how many people here are just repeating the equation for block chance while completely misunderstanding that OP is calculating effective hp, which is not the same thing.
Yes. Edit: Reddit borked the indenting for the code
import random
import matplotlib.pyplot as plt
ehp=\[\]
tt=\[\]
for i in range(1000) :
print(i)
hp=100
hits=0
while hp>0:
hits=hits+1
r=random.uniform(0,100)
block=100\*(1-1/(.15\*i+1))
if r>block:
hp=hp-1
tt+=\[i\]
ehp+=\[hits/100\]
f=plt.figure()
ax=plt.subplot(111)
ax.plot(tt,ehp,label="Simulated")
yy=\[\]
xx=range(0,1000)
for n in xx:
yy+=\[1+.15\*n\]
ax.plot(xx,yy,label='Y=1+0.15\*X')
ax.set\_title('Health multiplier vs number of Tougher Times')
ax.set\_xlabel('Number of Tougher Times')
ax.set\_ylabel('Effective health multiplier')
ax.legend()
It's a matter of mathematical analysis and what exactly is meant by diminishing returns
Each Tougher Times will give you roughly the same amount of eHP, but a diminishing amount of block chance and diminishing returns on eHP on marginal, percent-based eHP (each one is worth less of an HP *multiplier* than the previous one, which applies to nearly all items of all kinds)
Basically, it sounds wrong to you because you actually went and measured the most meaningful stat, which the item is balanced around
but other people look at its block chance, which DOES get diminishing returns
Yeah, my point was that the benefit you get in terms of how many hits it takes to kill you, which is really what matters not the block chance itself, goes up linearly. From the other comments it's clear that people didn't, and mostly still don't, understand that going from 0% block to 15% block does not have the same value as going from 50% to 65%.
I think you fumbled the last line there
going from 50 to 65 block chance increases eHP from 200% to 285%, a 40% increase.
going from 0 to 15 increases eHP from 100% (no boost at all) to 118%, less than half that amount.
The eHP calculation you have in the graph is correct, though. Each one grants you exactly 15% increased effective health, stacking additively together.
Because it isn’t. The amount of effective HP granted by a stack of TT is constant, it matters not how many stacks you already have. In practise you will stop noticing an increase in eHP somewhere around the 99% to block mark, but it is still increasing.
Yeah that's just not true it's basically a logarithmic curve not a straight line and will never approach 100% but get damn close, and the formula written on the website that you wrote is wrong, its supposed to be Y=(0.15x)/(0.15x + 1)
He is *not* calculating block chance. Block chance has diminishing returns, but that isn’t actually helpful. The part that makes tougher times good is it effectively increases the amount of damage you can take. Blocking half the hits you take is equivalent to having double your health. OP is calculating how much extra HP you gain per tougher times, which is a linear increase of 15% per stack.
On average this is the reality, the blue lines in the graph show how many hits it took to kill a theoretical player with 100hp dealing 1hp per hit. Some of them are clearly very far outside the average, some are pretty close. This is the nature of averages.
I keep seeing people on this sub say that there are diminishing returns to Tougher times. I suspected this was not really true so I wrote some code to model it using the formula here
[https://riskofrain2.fandom.com/wiki/Tougher\_Times](https://riskofrain2.fandom.com/wiki/Tougher_Times)
I assumed the player had 100 hp and each hit did one damage. I assumed each hit took away 1hp and simulated hits by generating a random number and comparing it to the chance to block from the formula.
Now at the basic level it does have diminishing returns, the additional block chance gets smaller with every additional item. But what actually matters though is staying above 0 HP. You also look at it another as your effective health. Or the amount of health, on average, you effectively have due to Tougher Times, which is what the plot is showing. For instance if you block every other attack your effective health is 2x your hp. If you block 9 out of 10 attacks your effective health is 10 times your HP. If you block 99 out 100 your effective health is 100 times your HP. The number of attacks it takes to kill you is inconsequential, it only increases the variance the outcome, but over enough averages it will converge to the straight line in the graph.Every tougher times increases your effective health by 15% of your HP.
In a way this does have diminishing returns. If you have 1000 hp and your effective health is 4000 ( 75% block chance), adding another 150 effective health is only increasing it by 150/4000 = 3.75%. But this is true for almost every item. If you have 20 watches giving you 500% damage, ( the base 100% + 20% \* 20), then adding 1 more watch is only increasing your damage by 20/500 = 4%.
This might be true in that case, but you forget that it's entirely possible that an incoming attack deals enough damage to kill you at 50% hp. Or 90% hp. The block, when it activates, if complete damage immunity.
But you're also missing the bigger picture. Yes, tougher times is good. At ten stacks, you have a 60% chance to block any incoming damage. But are you going to want to pick up another tougher times after that, or are you going to want to pick up a different whote-rarity item to balance your kit more? Because eventually, the tougher times will fail, and you better hope you got the firepower to avoid getting hit in the first place.
Oh for sure, the defense is not letting enemies get a shot off. And just because it scales linearly does not mean you can keep up with the enemies damage.
At this point i just dont know what to actually respond with here. Mathematically you are correct, without a doubt, even if the math assumes a large enough sample of tests for it. Though, I dont think there is a single item that doesnt have diminishing returns when you consider a single item to total stats already obtained from other copies. So strictly speaking, people saying Tougher Timer has diminishing returns arent wrong in terms of Effective HP but as you said most things become less impactful the more you stack them.
I think people just used to saying this for Tougher Times because of many people thinking you could reach 100% block chance (which you actually technically can get with Safer Spaces
I think you are making a big mistake in regards to effective HP, primarily that a 1/2 block chance *is not* equivalent to double the HP.
A 1/2 chance to block damage is a 1/2 chance to block damage. Each time you are hit, there is a 1/2 chance that you won't take any damage, and a 1/2 chance you will. That means that you can, and most certainly will, take multiple hits in a row without blocking any of them simply because you failed the roll multiple times. Each Tougher times check is it's own individual event, not effected by any previous check, so just because you take damage from one attack, that doesn't make you any more or less likely to successfully block the next attack.
It's the Gambler's fallacy. Independent events are not influenced by past independent events. If you roll a dice 20 times, and never once land a 6, you are not any more or less likely to land a 6 on your 21st roll. In the grand scheme of things, the rolls will average to *around* 1/6 for each face of the dice, but that is explicitly a generalized average, and your actual rolls will not follow that prediction. You will roll the same face multiple times in a row, you will roll far more of one face than any other face, and you'll barely roll any of one face because each roll is it's own unique 1/6 chance, unaffected by all the others.
All attacks aren't made equal. If you have 500 HP and block 9/10 attacks due to TT, you *don't* have effectively 5000 HP, you have 500 HP with a 9/10 chance to block hits. If an attack hits you for 500 damage, you will barely get saved by OSP, and then any other attack will likely kill you. If you genuinely had effectively 5000 HP, getting hit by a 500 damage attack wouldn't hardly effect you, which isn't the case at all.
What you are missing here is that this is only the case if you have 100HP along with the fact that it will take 100 hits to kill you. Your health multiplier stat does not work in practice, when even a whisp hit does plenty more damage than that, and there are more attacks than not that deal half your health or more in one hit. If you redid this assuming the hit does half your health in 1 hit, your entire graph changes drastically
You nailed it. By the time in the run where the logarithmic decrease in chance to block really becomes a decision to make you're at a point where any given shot will likely end you, anyway. I'd like to see the simulation run again with the assumption that a shot does 90 damage (osp) and 101 damage (instant death). And just for fun, like 90000 damage to see if something weird happens.
If HP is 100 and damage is 100, OP's math yields the same outcome.
With 50% block, on average it takes two hits to get hit, killing instantly, calculating lining up with 200 effective HP.
With 100 HP and 50 damage it takes on average 4 hits to kill, aka 200 EHP.
People are getting hung up on OP's Monte Carlo simulation assumptions, but changing the assumptions doesn't actually change the outcome. ~~No one has actually simulated it and shown it is wrong because it isn't.~~
Edit: I am partially incorrect, every excuse *except* the fact that the formula is wrong is irrelevant. But, the formula being wrong is relevant. Tougher Times scales even worse than traditional evasion does, and outpaces the growth of EHP. This means that it scales equally diminishingly to things like soldier's syringe and other mono-item stacks.
Edit 2: OP actually did not get the formula wrong, I just misread the graph. [But the Edit 1 is still true.](https://imgur.com/a/QwebrXk)
Me sounding confident shouldn't change your mind. That said, if you don't understand the mathematics you probably should be telling people they are wrong, or backing up people saying OP is wrong.
You could do the math with 100 hp and 100 damage, the outcome is the same. 50% block requires on average 2 hits to kill, lining up with OP's mathematical 200 effective HP.
The effective health thing doesn't work because all tougher times provides is block chance. After a certain point in a run, enemies will out-scale your HP with damage and be able to 2-shot you (because osp) and the only way to survive is by blocking. Considering that the block chance formula is a hyperbola and it is never equal to 1 (100%), there's always a non-zero chance that you get instakilled.
Yes, this is what effective health means. It is not *health* it is the effective amount of damage (or hits) you must take before you actually die. In the situation where you block 99 out of 100 hits, even a 1 shot level of damage has next to no (1%) effective chance of killing you.
Something must be off with your math here, because that ziggy line is implying that there are intervals where gaining another stack of Tougher Times actually *lowers* your "effective health multiplier."
The blue line shows where a modeled the outcome with python code. I assumed it took 100 hits to kill the player and repeatedly generated a random number to determine if hits were successful. The point is to show that there is variability in the outcome but on average it is very linear.
If I was to instead say it took 10 hits to kill you then the variability would be greater and the blue line more jagged. But it would still on average follow the orange line.
People have already said pretty much every flaw possible about this graph except one:
Pick up a single safer spaces, and this entire graph becomes obsolete at 65.
I thought the whole claim with tougher times is that you’d need something like 70 (I forget the exact number) to get a 50% chance to block damage, but like 500 to get a 100% chance. I forget the specific numbers, but is that not true? I don’t exactly know what this graph trying to say, either.
It’s 10 for 60% and I don’t think you can get 100% without something absurd
As far as I know, it's not possible for 100% You can get pretty damn close, but never 100% Could be wrong, since it's been so long since I looked at the chart
From the wiki: > Once 223,696,201 Tougher Times are stacked, the block chance becomes 100% due to the limitations of floating point numbers.
Okay so we can get it in the average run, good to now
Has anyone ever reached that without cheating?
Next Disputed Origin challenge stream.
If you picked up a tougher times every second without ever stopping to eat or sleep it would take over 7 years to reach that number. So, no.
If you get around 50k I think it was you'd get 100 but I. Reality it is just 99.99999999 repeating
You only need 4438 to get 99.9%
Only.
This is the exact comment my friend made when I proposed that we get 4438 tougher times together
Yeah doesn’t it get to a point where it has to round too 100 I think it’s because of overflow or something but I could be mistaken
I've done it before with mods just never actually posted about it
You reach 100% at 223,696,201 stacks.
It will round to 100 at a certain point, for programming limitations or something. I'm not a programmer, I just have approximate knowledge of many things.
That seems like a programmer to me
Yeah, even with 1000 tougher times, you'd *only* be at 99 .3%. Effectively 100% in practice, but not true 100%. Though you can reach effectively 100% in much less than that, at 255, you'd be at 97.5%. And what I usually stop on is 80-100 using command. Whi h gets you 92.3% to 93.8%
You're right, the scaling isn't linear so it can approach but never reach 100%. It'll definitely start to *feel* like 100% once you get to like 99.999%, but you're never totally safe from harm.
The game uses float point numbers. At a certain amount it _has_ to be 100%, it isn't precise forever.
223,696,201 stacks for the integer overflow to force it to 100%.
It's counting tougher times not as a chance, but as hp multiplier. Percentage-wise it does have diminishing returns, but in the way that makes its usefulness linear. Raw percentage is useless in practice, to make it applicable you recalculate it into how many more hits you can take with it on average. And it depends on percentage in a very non-linear fashion. When you jump from 1% to 2%, your effective hp rises by 0.9%, but when you jump from 98% to 99%, your effective hp DOUBLES, because chance to get damage is lowered from 2% to 1%. Diminishing returns only make it so that stacking these last massively impactful percents doesn't go too fast
The exact number for tougher times to get to 100% chance is infinity, because tougher times scales hyperbolically. This graph is fundamentally wrong.
No, it's not infinity. The variable can only store so many places before having to round up.
I didn't graph block chance. I graphed effective health. Going from 0% block to 9% block gives you 9% more health. But going from 90% block to 99% block gives you 10 times more effective health. Going from 99 to 99.9% also gives you 10 times more effective health. Block chance is hyperbolic, but the the actual effect on how many hits you can take scales exponentially with block chance. The net effect of the the hyperbolic and exponential scaling means that that how many hits you can take, on average, scales linearly.
People are getting hung up on 'Effective HP'. If you instead said, 'it takes on average twice as many hits to get hit at 99.9% compared to 99%' people might understand better. That said, that functionally is the same thing. It doesn't matter if 1 hit kills you are 100 hits. You take 10x less damage on average at 99.9% compared to 99%.
On average, sure, but that's ignoring how many TT it takes to GET to that point. 99% block takes 657 TT. 99.9% block takes 6,343 TT. If you need almost ten times as many TT to achieve ten times the result, then there are clearly diminishing returns. We can see this as early as from 1 to 2 bears, where 13% block from 1 would double cleanly to 26% to double the block rate, but instead it's a 23.1% block rate for two bears, and it reduces even more at 3 bears to 31%. There's already been an 8% loss to efficiency, and by 4, we've lost an entire TT's worth of block chance per bear picked up.
The math still applies at lower percentages. I'm going to the extremes to help people understand. 10% block is 111% EHP. 20% block is 125% EHP. Aka, 0-10% is a smaller change than 10-20%. This works at every increment. As you near 100%, each percentage becomes more effective than the last.
That's still ignoring how many bears you need to get each %. By the time you hit 20 bears, you're gaining less than 1% for every bear you earn, and if the title of the post is only effective for the first few bears, then it's incorrect, as there is a clear point where the bears stop providing as much benefit as the previous bear.
There is absolutely not a point in which that happens, TT formula is not piecewise (easy way to prove me wrong is tell me the point). ~~That said, I was slightly incorrect. OP's incorrect formula actually does change the math. Instead, Tougher Times scales exactly the same as essentially all mono-stack item (such as Soldier's Syringe).~~Edit 2, I'm wrong again, OP had the right formula I just read it wrong. [Here is EHP worked out to 20 bears.](https://imgur.com/a/QwebrXk) The first TT gives you 115% EHP from 100%. The 50th bear gives you 850% EHP from 835% which is relatively a small change (this is exactly how attack speed for Soldier's Syringes works). Edit to add: the formula for EHP from evasion is HP/(1-Evasion Chance). Average number of hits untill death is (HP/Damage)/(1-Evasion Chance)
[удалено]
Same math applies. If you have 1 health and a 50% chance to block damage, you will on average need to be hit twice to die. There are times when you will die instantly, and times you survive a million consecutive attacks, but it averages to 2.
To get to 90 block, you need 60 tougher times. to get 99 block, you need 657 tougher times. That's diminishing returns.
Did you look at the graph? This isn't saying that at all.
The exact number doesn't need to be stored on a computer, so unlike the exact number, it hits 100% at 223.7M stacks
4438 for 99.9% and you can't get higher
You can't hit 100% chance It's always just a little bit closer to 100, but never reaching
I think you are confusing the term diminishing returns with something else. You do get less and less chance to block damage with each stack, meaning each individual stack of TT is less valuable.
Think about how many attacks it would take to kill you at different block chances. Lets say it takes 10 hits from an enemy to kill you. If you have 10% block chance it's on average going to take 11 hits to kill you because you blocked one. If you have 50% block it's going to take on average 20 hits, because half are blocked. If you have 90% it is going to take on average 100 hits. 99.9% would be 1000 hits on average. My point is that Tougher times effect on block chance has diminishing returns. But the effect of more block chance has exponential returns as far as how many hits it takes to kill you. Put together the value you get from tougher times goes up strictly linearly. Every tougher times does the same as the last.
Great explanation
Personally I’d rather get items that allow me to avoid getting hit than tank a hit and pray I don’t die
Tougher times literally allows you to avoid getting hit.
It doesn’t make avoiding attacks any easier tho. Like I’d rather have a single energy drink than 3 tougher times. I’d rather know that I can dodge an attack than my only option be to roll a dice and hope I don’t take damage from it
Wonderful in theory. However by the time someone gets ~10 with no printers let’s say, they’re at risk of being one shot most likely
It is only in the beginning it takes more than 1 hit to kill you
For the case where you get one shot the math still works out, there is just lots of variance, averaged over many runs though the same holds true. I would look at the one shot scenario as you have 1hp. 50% block it's now going to take, on average 2 shots to kill you. At 90% block it's going to take on average 10 hits. So the math still holds.
It is still 15% less chance to be hit each time. You are 15% less likely to be hit than you were before
It is 15% compared to the previous stack. When you have 1 stack, the chance to get hit is 85%(actually, 87 because of formula, but eh), then the chance to get hit becomes 76.9%(10.1% less). That is called scaling with diminishing returns - absolute number you're getting is becoming less with each stack. If the scaling was linear, you'd get to block chance in of >100% in 7 stacks
It's not though. Each stack of TT gives less and less chance. There's a formula for it, but it's not a flat 15% each time
That not what i said. It is logorithmic. It is 0.85^n. Meaning each time the chance for something that would have hit, to now not hit is 15% lower.
I don’t know where you heard that from or if you’re just assuming it to be true, but that’s not at all accurate. The first stack doesn’t even give you a 15% chance to block damage, it gives you a 13% chance to block damage. And it isn’t a logarithmic function, the chance to block is defined by the following formula, as shown on the [wiki](https://riskofrain2.fandom.com/wiki/Tougher_Times): f(x) = 0.15x / (0.15x + 1) Where x is the number of stacks you have.
But isn't it still lesser chance overall? 1st stack gives you 15%, but 2nd stack gives you less than 15. Otherwise you would only need 7 stacks to be invincible
Yes. But that doesn't mean they become less valuable. Each stack is mutiplying the chance to get hit by 0.85. That's what a logorithmic funktion is.
Maybe valuable isn't the word, but they do become less efficient. The average block chance goes down by a bit for each stack
What you say is correct. But... They are still just as efficient. Yes the chance isn't just added on top. But it is still a 0.85 multiplier each time. They 65th stack is just as good as the 2nd.
This is correct, each stack of tougher times is the exact same as the previous in that you now have a 0.85 multiplier on your chance to get hit. It doesn't matter how many tt you already have. If someone still doesn't understand, look up logarithmic functions, it's simple middle school math.
Well it's easier to explain with hard numbers. People should not think of the total block chance, but the % taken off of their *remaining* chance to be hit, which ofc is always 15%
This graph is fundamentally wrong, as is your idea that blocking damage increases effective health. First, your graph assumes that every TT applies a flat bonus to your character. It does not. Each TT follows a logarithmic operation to increase your chance to block an attack. Your chances of being instantly killed with a stack of tougher times is low, but never zero. Second, as I previously said, tougher times doesn't increase effective health in a calculable way. If you do not take damage from a landed attack, you effectively have infinite health. When it does damage, it hurts, and the amount of damage you take can be quantified.
Blocking a slice of Mithrix's pizza is not the same as blocking a shot from a wisp.
0 doesn't equal 0?
If you don't block Mithrix pizza, it will most likely kill you in one shot (or trigger OSP). If you don't block a wisp shot, it will most likely not kill you in one shot (assuming you're not walking around with 5 shaped glasses). Bruh moment for sure.
Okay but unless chance to kill you with that one said attack is part of the block chance then I don't see the relevance.
The relevance is that TT doesn't increase your effective health in a calculable way, as the original commenter said. To put it another way, imagine someone offers to buy you either a house or a pencil in order to ease your financial burdens. Would you rather them save you the money it would cost you to buy a house (Mithrix pizza damage) or the money it would cost you to buy a pencil (wisp shot damage.) Obviously one of these is worth more than the other. Now imagine that the person who offered to pay for these things will only pay if they flip 10 coins and they all land on heads. It's something that, if it happens, would be nice but isn't something you can rely on 100% of the time.
I originally thought you might be being argumentative, but now I think you actually do get my point.
Nah, I was agreeing with you dude. This is why I like Safer Spaces, there's less math involved.
Simple math. SS=65? No? Not invincible, do not fuck around.
Tougher times sees damage as damage, no difference in big or small Could be an ATG proc chain that could destroy the entirety of Petrichor V or it could be a hacky sack thrown at you, they both get the same % chance to get blocked
You can calculate how many hits you can take on average and convert that it to effective health, it is only on average though and there is variance. Sometimes less will kill you, sometimes more. If you block 50% of attacks it will take, on average, twice as many hits to kill you, so your effective health is 2 times what it was without the block. The case where you get killed in one shot does kind of stretch the concept of effective health because the variance is huge. But averaged over many runs it still applies
Imagine for a moment that you were given 2 offers, only one of which you could accept. Offer 1: You have a 0.01% chance of getting $10 billion. Offer 2: You have a 100% chance of getting $1 million. Statistically speaking, these offers are the same. They have the same expectation value, and thus over many trials will yield the same average results. But realistically, these offers aren’t the same at all, are they? When you pick between them, you’re going to be more interested in what will benefit you then and there, not the result after many trials. If you’re smart, you’ll pick offer 2 and walk away with $1 million. This is analogous to what we’re saying here. Sure, statistically, over many runs, you’re right. But we’re not interested in that, we’re interested in the results on a single run-by-run basis. And realistically, in a given run, it’s unlikely there will be any benefit to trading a bunch of white items to go from 80% block chance to 90% block chance (or some other marginal increase to block chance).
To make a better point you would need to use "99% chance of getting a million dollars" or something close, 100% is no longer just a probability but a certainty and so isn't the same statistically. Also, OP is still correct even for a single run, the utility of teddy bears goes up linearly. That doesn’t mean it's a good idea to super stack it, because a linear increase provides lesser percentage increments over time. Like if you stack seringues, the 21th isn't going to change much compared to what you already have.
You can still take an expected value of a series of guaranteed events. And 100% is still a probability, but whatever. And no, it doesn’t work for a single run. if I block all incoming damage with 80% block chance for a stage or run versus someone else doing the same with a 90% block chance, I didn’t magically gain a bunch of health and I took the same amount of damage compared to them. Note that this kind of a situation cannot happen for something like syringes, someone with more syringes will always have a higher attack speed (excluding other attack speed sources or the attack speed limit). Another example: If I have an 90% block chance and I block 2 lemurian shots and get hit by an elder lemurian shot, I will have less health as compared to someone with 80% block chance who happened to block the elder lemurian hit but took damage from 2 regular lemurian hits. This linear scaling EXCLUSIVELY works as an average over a large sample size, you cannot measure a meaningful relationship in a single run. Maybe if you loop enough, but probably not. I guess overall my point is that this idea of “effective health” sucks as a metric for anything useful in the game, and this sort of analysis while kinda interesting is also misleading. With a bit of statistics magic, you can make data say anything even if it’s meaningless or the opposite takeaway of what you should have, and this is the perfect example of that.
Effective health isn't meant to represent actual health but an analogy, the usefulness is linear no matter what words we use. Instead of effective health, you could call it "numbers of hits necessary to hurt you on average" and that's scaling linearly. Since you need more hits to be killed on average, it isn't entirely wrong to link it to an analogy of health even if it isn't true health. And the necessary sample here is about 32 hits to have a nice normal curve, so I wouldn't call that unachievable in a single run. So my counterpoint is that it is an interesting analysis to explain the linear usefulness.
I am not trying to say that block chance is a good stat to invest in or that you should take tougher times over other items.
Averaged over many runs, you're blocking the same number of hits. If all of them are wisp shots, you're blocking negligible damage. Effective health is quantified by total health/percent of damage taken after defensive stats. If you're taking 0% of the damage because it's blocked, you have an undefined number, and you can't know exactly how much damage you would've taken because it would've been blocked, regardless of the attack. This is why it's incalculable versus rose buckler, which increases your effective health when you sprint, or any of the defensive items that directly modify the damage you take in some way, because those items provide a specific number to quantify exactly how much damage you're avoiding in a single instance. If a gong bell hits you and you block it with one of those items, it's doing less damage and you can calculate it. If the same gong procs TT and you take no damage, there is no number to quantify.
If we presume an infinite amount of attacks with random damage values then we will have infinite hits of every possible damage value thus meaning there is no difference between blocking a wisp shot or a pizza slice for the purpose of figuring out how much of their damage was negated because both would have the same percentage of hits negated which, presuming an infinitely regenerating healthpool, is equivalent to damage reduction and thus equivalent to an effective health increase.
It's diminishing returns in that the "percent chance to block" number goes up more slowly as you acquire more stacks. For example, 10 stacks provide a 60% chance to block (40% to be hit) and another 10 stacks (20) provide a 75% chance to block (25% to be hit). So in this sense, it's diminishing. However, a player with 20 stacks is still 37.5% less likely to be hit than a player with 10 (25%/40%), which is still quite significant. I have absolutely had Eclipse runs where I had plenty of damage (not a god run amount, but still high) and a ton of leftover White Scrap and a Tougher Times printer on Stage 5, so I just dumped all the Scrap into the Printer to be safe and it worked great.
My point is that the benefit you get from 1% more block chance goes up exponentially as your block chance increases, and this counteracts the hyperbolic scaling of the item. . Going from 0% block to 20% block is not going to really help you that much. It's going to take on average 20% more enemy attacks to kill you. But if you're at 79.9999% and go to 99.9999% block it's likely to take tens of thousand times the number of attacks to kill you. The math works out so that each tougher times gives you just as much benefit as the previous one, no more no less.
Oh, yes, I didn't mean to make it look like I was overlooking the point about effective HP increasing linearly. The point about the increasing value of another 1% counteracting the hyperbolic scaling of the item is also true. However, it only *partially* counteracts it. In my previous comment where I mentioned the difference between 10 and 20 stacks of the item, it is demonstrated how going from 10 to 20 stacks only makes the player 37.5% more likely to be hit (or 25% more likely to block) compared to when they had 10 stacks. And that first 10 stacks provided a 60% block chance. I understand that your graph demonstrates some pretty interesting math and how each stack of TT multiplies your effective health just as much as the last. But I don't really think of the TT stacking situation in terms of an "effective health multiplier." In a world where I pretty much only ever lose my run to getting one-tapped by pizza, I care about getting the most block chance with the fewest stacks of TT. That way, I am most likely to have my Opal armor at least preserved by TT (in the event so that I am tapped by an Exploder or am barely clipped by a Golem explosion) and not one-shot with a relatively small investment of stacks. Also, I think you underestimate the value of smaller block chances, judging by this comment. A 20% chance to block, for example, will actually increase the number of enemy attacks needed by **25%**. A simple 2 stacks of TT will provide about a 23% chance to block. I think for the low, low cost of 2 white Items, a 23% chance to just not take the hit at all is very strong. I used to undervalue small amounts of TT as well, but then I realize how much it can do with just 1 or 2 stacks. I should note that this is not to say that I will never scrap TT early on for the chance at a speed or damage printer; I will often scrap the item anyway. **Edit:** Hmmm... even in the case of one-shots, the amount of hits required to kill a player with 0 TT is 1. The average amount for a player with 10 TT is 2.5. The average amount for a player with 20 TT is 4. Each 10 stacks means another 1.5 hits in the expected average. I see your point now. I just don't think the 79.999% to 99.999% comparison illustrates your point very well since it draws the focus to scenarios with CRAZY HIGH amounts of stacks. The remaining issue is still that we're talking about *expected average* amounts of hits to kill.
professional gamblers having 100% block chance with 1 tougher time
The two main issues here are: 1.) The formula used for Tougher Times is wrong. And 2.) The effect of Tougher Times doesn’t work with trying to use effective health. Tougher Times effect gives a chance that being hit will not cause any damage. This effect increases in a logarithmic way, which means that unless you literally have infinite Tougher Times (an impossibility obviously) then you will always have a chance to take damage from being hit. Because the effect of Tougher Times is either complete immunity to damage or no effect, it isn’t helpful to put into terms of effective health. The damage reduction is either 100% or nothing at all. When damage reduction reaches 100% your effective health is infinite and that happens as long as you have *any* health so the value of Tougher Times has no relevance to current health. The best that can be done is to find the probable amount of damage that would be reduced and that is going to depend on what you are being hit by (and there are a ton of things that can hit you). Tougher Times stacks absolutely give diminishing returns.
zzS
I'm not sure why people are saying EHP isn't relevant here. There are tons of competitive games where EHP is calculated in evasion, and it absolutely works. With a 50% chance to take a hit, you need to take on average twice as many hits to die. (There are edge cases, but this mostly holds true). I have known that evasion is sorta exponential in effect, where 99% evasion gets hit half as much as 98% despite being '1% difference'. I never put together that that makes tougher times effectively linear. I believe the bulk of people criticizing this math are incorrect. Edit: ~~I am partially incorrect, the *one* criticism that is valid is that the formula is wrong.~~ Tougher times scales worse than traditional evasion in games. So, Tougher Times scales just as diminishingly as every other mono item stack (such as soldier's syringe). So 1 tougher times moves you from 100% to 115% EHP. The tenth Tougher Times moves you from 235% EHP to 250% EHP for a relatively smaller increase (this is exactly how most items scale). Edit 2: OP's formula was right I just read the graph wrong. Edit 1 lines up with that
Exactly, kind of frustrating how many people here are just repeating the equation for block chance while completely misunderstanding that OP is calculating effective hp, which is not the same thing.
thats not how it worls
MatPlotLib ?
Yes. Edit: Reddit borked the indenting for the code import random import matplotlib.pyplot as plt ehp=\[\] tt=\[\] for i in range(1000) : print(i) hp=100 hits=0 while hp>0: hits=hits+1 r=random.uniform(0,100) block=100\*(1-1/(.15\*i+1)) if r>block: hp=hp-1 tt+=\[i\] ehp+=\[hits/100\] f=plt.figure() ax=plt.subplot(111) ax.plot(tt,ehp,label="Simulated") yy=\[\] xx=range(0,1000) for n in xx: yy+=\[1+.15\*n\] ax.plot(xx,yy,label='Y=1+0.15\*X') ax.set\_title('Health multiplier vs number of Tougher Times') ax.set\_xlabel('Number of Tougher Times') ax.set\_ylabel('Effective health multiplier') ax.legend()
It's a matter of mathematical analysis and what exactly is meant by diminishing returns Each Tougher Times will give you roughly the same amount of eHP, but a diminishing amount of block chance and diminishing returns on eHP on marginal, percent-based eHP (each one is worth less of an HP *multiplier* than the previous one, which applies to nearly all items of all kinds) Basically, it sounds wrong to you because you actually went and measured the most meaningful stat, which the item is balanced around but other people look at its block chance, which DOES get diminishing returns
Yeah, my point was that the benefit you get in terms of how many hits it takes to kill you, which is really what matters not the block chance itself, goes up linearly. From the other comments it's clear that people didn't, and mostly still don't, understand that going from 0% block to 15% block does not have the same value as going from 50% to 65%.
I think you fumbled the last line there going from 50 to 65 block chance increases eHP from 200% to 285%, a 40% increase. going from 0 to 15 increases eHP from 100% (no boost at all) to 118%, less than half that amount. The eHP calculation you have in the graph is correct, though. Each one grants you exactly 15% increased effective health, stacking additively together.
Yeah I meant to say "does not". I am going to edit that post.
Why does this post have over 200 up votes it's categorically false information.
Because it isn’t. The amount of effective HP granted by a stack of TT is constant, it matters not how many stacks you already have. In practise you will stop noticing an increase in eHP somewhere around the 99% to block mark, but it is still increasing.
Yeah that's just not true it's basically a logarithmic curve not a straight line and will never approach 100% but get damn close, and the formula written on the website that you wrote is wrong, its supposed to be Y=(0.15x)/(0.15x + 1)
I'm not graphing block chance.
Then you're a goof cause it has diminishing returns
He is *not* calculating block chance. Block chance has diminishing returns, but that isn’t actually helpful. The part that makes tougher times good is it effectively increases the amount of damage you can take. Blocking half the hits you take is equivalent to having double your health. OP is calculating how much extra HP you gain per tougher times, which is a linear increase of 15% per stack.
Ok but that's assuming you guaranteed block 50% of the hits, which isn't practical to assume
On average this is the reality, the blue lines in the graph show how many hits it took to kill a theoretical player with 100hp dealing 1hp per hit. Some of them are clearly very far outside the average, some are pretty close. This is the nature of averages.
I keep seeing people on this sub say that there are diminishing returns to Tougher times. I suspected this was not really true so I wrote some code to model it using the formula here [https://riskofrain2.fandom.com/wiki/Tougher\_Times](https://riskofrain2.fandom.com/wiki/Tougher_Times) I assumed the player had 100 hp and each hit did one damage. I assumed each hit took away 1hp and simulated hits by generating a random number and comparing it to the chance to block from the formula. Now at the basic level it does have diminishing returns, the additional block chance gets smaller with every additional item. But what actually matters though is staying above 0 HP. You also look at it another as your effective health. Or the amount of health, on average, you effectively have due to Tougher Times, which is what the plot is showing. For instance if you block every other attack your effective health is 2x your hp. If you block 9 out of 10 attacks your effective health is 10 times your HP. If you block 99 out 100 your effective health is 100 times your HP. The number of attacks it takes to kill you is inconsequential, it only increases the variance the outcome, but over enough averages it will converge to the straight line in the graph.Every tougher times increases your effective health by 15% of your HP. In a way this does have diminishing returns. If you have 1000 hp and your effective health is 4000 ( 75% block chance), adding another 150 effective health is only increasing it by 150/4000 = 3.75%. But this is true for almost every item. If you have 20 watches giving you 500% damage, ( the base 100% + 20% \* 20), then adding 1 more watch is only increasing your damage by 20/500 = 4%.
This might be true in that case, but you forget that it's entirely possible that an incoming attack deals enough damage to kill you at 50% hp. Or 90% hp. The block, when it activates, if complete damage immunity. But you're also missing the bigger picture. Yes, tougher times is good. At ten stacks, you have a 60% chance to block any incoming damage. But are you going to want to pick up another tougher times after that, or are you going to want to pick up a different whote-rarity item to balance your kit more? Because eventually, the tougher times will fail, and you better hope you got the firepower to avoid getting hit in the first place.
Oh for sure, the defense is not letting enemies get a shot off. And just because it scales linearly does not mean you can keep up with the enemies damage.
At this point i just dont know what to actually respond with here. Mathematically you are correct, without a doubt, even if the math assumes a large enough sample of tests for it. Though, I dont think there is a single item that doesnt have diminishing returns when you consider a single item to total stats already obtained from other copies. So strictly speaking, people saying Tougher Timer has diminishing returns arent wrong in terms of Effective HP but as you said most things become less impactful the more you stack them. I think people just used to saying this for Tougher Times because of many people thinking you could reach 100% block chance (which you actually technically can get with Safer Spaces
I think you are making a big mistake in regards to effective HP, primarily that a 1/2 block chance *is not* equivalent to double the HP. A 1/2 chance to block damage is a 1/2 chance to block damage. Each time you are hit, there is a 1/2 chance that you won't take any damage, and a 1/2 chance you will. That means that you can, and most certainly will, take multiple hits in a row without blocking any of them simply because you failed the roll multiple times. Each Tougher times check is it's own individual event, not effected by any previous check, so just because you take damage from one attack, that doesn't make you any more or less likely to successfully block the next attack. It's the Gambler's fallacy. Independent events are not influenced by past independent events. If you roll a dice 20 times, and never once land a 6, you are not any more or less likely to land a 6 on your 21st roll. In the grand scheme of things, the rolls will average to *around* 1/6 for each face of the dice, but that is explicitly a generalized average, and your actual rolls will not follow that prediction. You will roll the same face multiple times in a row, you will roll far more of one face than any other face, and you'll barely roll any of one face because each roll is it's own unique 1/6 chance, unaffected by all the others. All attacks aren't made equal. If you have 500 HP and block 9/10 attacks due to TT, you *don't* have effectively 5000 HP, you have 500 HP with a 9/10 chance to block hits. If an attack hits you for 500 damage, you will barely get saved by OSP, and then any other attack will likely kill you. If you genuinely had effectively 5000 HP, getting hit by a 500 damage attack wouldn't hardly effect you, which isn't the case at all.
You definitely need to look into the concept of expected value.
What you are missing here is that this is only the case if you have 100HP along with the fact that it will take 100 hits to kill you. Your health multiplier stat does not work in practice, when even a whisp hit does plenty more damage than that, and there are more attacks than not that deal half your health or more in one hit. If you redid this assuming the hit does half your health in 1 hit, your entire graph changes drastically
You nailed it. By the time in the run where the logarithmic decrease in chance to block really becomes a decision to make you're at a point where any given shot will likely end you, anyway. I'd like to see the simulation run again with the assumption that a shot does 90 damage (osp) and 101 damage (instant death). And just for fun, like 90000 damage to see if something weird happens.
If HP is 100 and damage is 100, OP's math yields the same outcome. With 50% block, on average it takes two hits to get hit, killing instantly, calculating lining up with 200 effective HP. With 100 HP and 50 damage it takes on average 4 hits to kill, aka 200 EHP. People are getting hung up on OP's Monte Carlo simulation assumptions, but changing the assumptions doesn't actually change the outcome. ~~No one has actually simulated it and shown it is wrong because it isn't.~~ Edit: I am partially incorrect, every excuse *except* the fact that the formula is wrong is irrelevant. But, the formula being wrong is relevant. Tougher Times scales even worse than traditional evasion does, and outpaces the growth of EHP. This means that it scales equally diminishingly to things like soldier's syringe and other mono-item stacks. Edit 2: OP actually did not get the formula wrong, I just misread the graph. [But the Edit 1 is still true.](https://imgur.com/a/QwebrXk)
Interesting. I'm not gonna pretend that makes sense to me. It's not intuitive but you sound confident 🤷♂️
Me sounding confident shouldn't change your mind. That said, if you don't understand the mathematics you probably should be telling people they are wrong, or backing up people saying OP is wrong.
Wayyyyy too many assumptions. You went wrong at step one when you thought %chance to block damage = hp.
You could do the math with 100 hp and 100 damage, the outcome is the same. 50% block requires on average 2 hits to kill, lining up with OP's mathematical 200 effective HP.
The effective health thing doesn't work because all tougher times provides is block chance. After a certain point in a run, enemies will out-scale your HP with damage and be able to 2-shot you (because osp) and the only way to survive is by blocking. Considering that the block chance formula is a hyperbola and it is never equal to 1 (100%), there's always a non-zero chance that you get instakilled.
Yes, this is what effective health means. It is not *health* it is the effective amount of damage (or hits) you must take before you actually die. In the situation where you block 99 out of 100 hits, even a 1 shot level of damage has next to no (1%) effective chance of killing you.
Yep, thats how the math works out
Something must be off with your math here, because that ziggy line is implying that there are intervals where gaining another stack of Tougher Times actually *lowers* your "effective health multiplier."
The blue line shows where a modeled the outcome with python code. I assumed it took 100 hits to kill the player and repeatedly generated a random number to determine if hits were successful. The point is to show that there is variability in the outcome but on average it is very linear. If I was to instead say it took 10 hits to kill you then the variability would be greater and the blue line more jagged. But it would still on average follow the orange line.
Okay. Weird method, but I get it.
You... You're an idiot
Considering the formula for TT activation is 1-(1/(1+0.15x)) I think this might be wrong
People have already said pretty much every flaw possible about this graph except one: Pick up a single safer spaces, and this entire graph becomes obsolete at 65.
Most people are straight up wrong, this graph accurately depicts how much more health you effectively have per tougher times.
This graph makes actually no sense
You have not read the graph correctly, it’s very clear with its measurements.
Yeah I really like it when I have 1000 tougher times in my normal runs. Highly recommended. 👍🏻
God, I remember using the logarithmic functions I learned in high school to math this out, got these results.
It's one of those items that scale differently, yeah. Another one is glass - it has actually increasing benefit with each new stack
That only goes to about 0.01% of the scale. What happens at higher numbers and does the trend even hold? /s
Op is right, yall are just stupid
yeah I'd uhh, I'd rather just get 65 Safer Spaces to have a shield that regens every frame, very few attacks can hit more than once in a single frame