You really like not engaging some of the most mathematically salient points.
What's to engage in? There's nothing wrong with your maths. The error is in misapplying it or granting something weight based on whether it is true, rather than if it is true and important. Which is why I made my response about that. Specifically:
Response #3: A +1 in 3d6 is less or similarly meaningful than a +1 in d20 if your target is low or high.
Target numbers are not evenly distributed in usage. Not between the games, not within either game. My statement is about its significance across total dice rolls. I know you understand this mathematically. It's you who is deliberately ignoring what I actually wrote so far as I can see. If something is more significant across 90% of all dice rolls, then you saying 'but in these 10% of cases at the extremes it flips a little' isn't changing my point. It's like when you see a headline saying something has doubled the risk of a disease and the reader gasps "double - that sounds high" even what a study actually shows is your chance of the disease has gone from 0.01% to 0.02%. Yes, it's doubled. It doesn't refute anybody's position who said the chance was low.
And given for some reason you accuse me of not engaging with your maths (I did), how about you use a little more mathematical precision yourself. Let me translate your "if your target is low or high" into actual numbers because it shows the issue in your post. The flip over point is at approximately 4% probability at each end. That is when the odds of success are either lower than 4% or higher than 96%, the +1 becomes more significant in the d20 system.
Is it clear what I am disagreeing with you over? And why I'm "not engaging in the most mathematically salient points" in your opinion? It's because I'm not challenging your maths, I'm saying that a statement that 'it's more meaningful when the target is low" makes it sound like it matters. Casual language hides the truth that except in extremes of probability like <=4% chance of success, what I say is true. And given that what I said wasn't "it's always true" but that it's true in the large majority of cases, I see nothing false in what I said that you seem to take so much issue with.
If you think I'm writing too much for you, fine. But you went to the trouble of producing an actual chart to show that I was wrong and it doesn't do that. You have target numbers one to one across both systems and as already explained (and ignored by you) target numbers aren't arbitrarily set nor could they be and have a meaningful game. Target numbers are derived from desired chance of outcome. If I want a 4% chance of success, I'm going to set a different number in a 3d6 system than I would in a d20 system. So would any GM. The game systems themselves are designed to do so.
A +1 to hit in a 3d6 system is relevant a lot more often than it is in a d20 system. It was a simple enough statement that is provable unless target numbers cluster at the extremes, which they do not in any game system I've ever heard of. Unless there's one out there that relies on flipping coins.
Your specific, objectively wrong claim is that +1 is "meaningless" or "irrelevant."
Objectively? The cut off for what someone thinks is meaningless or irrelevant is a subjective thing and I'm willing to concede that you may convince yourself that giving multiple low-level henchmen +1 to hit and aggregating the effect of that over round after round is relevant, but even you must concede that in comparison to a 3d6 system it's massively
less relevant. Can you accept that much as a starting point?
You accuse me of not engaging with your points but my post responded to every point you made. You just didn't like that my responses weren't "your maths is wrong" but showing play-based reasons why they don't matter. As I said in my last post, and which
you declined to engage with, players are not Monte Carlo simulations. Every dice roll is an active investment by a player. It doesn't matter if you have them roll one action or a hundred because the question
isn't the aggregate effect, despite your insistence, it's the proportion. And a player rolling a die a hundred times and seeing +1 make a difference 5 of those times is not going to be happier than a player who rolled it 20 times and saw it make a difference once. Your argument is akin to comparing both and saying 5 > 1, therefore happiness is greater. Whilst ignoring that 95 > 19 and therefore that +1 has faded into insignificance to them.
Why even bother typing multiple paragraphs if you're going to ignore the central points?
Well it was initially a good faith attempt to explain my point of view. You're welcome to point out anything in your original post that I ignored. But saying "yes but it misses the point" isn't ignoring something you said. It's disagreeing with your premise. The multiple paragraphs in this post? I don't know. Just a hope you'll come back to this and accept it as a genuine attempt to explain. Please do tell me what I actually ignored if there is something. I honestly do not see it.
I guess at the end of the day, sure getting +1 makes you feel really good, since if you absorb lots of internet arguments from people who focus myopically on means and medians, it's a +12% effect close to the mean of 3d6. The problem is, I already agreed with this, but you want to be contrarian, ignore that as well, pretend that the valid point was unacknowledged, and respond to me as though I disagreed with everything you said, rather than one specific part.
Contrarian how? You do not represent some default perspective. I made a comment about d20 vs. 3d6. You say you agreed with it but come in with a whole lot more. If all you're looking for is for me to say that the effect of a +1 in a d20 system is mathematically provable then you can have it. I never argued otherwise. If what you're arguing over is because to you "meaningless" isn't a natural language word to be assessed in the context of a comparison between systems but must be interpreted as "cannot be measured", then you can't. Because I'm happy to say that compared to a 3d6 system, a +1 to hit in d20 largely is. Because as I said in the post you replied to, if something is only coming up 1 time in 20 that doesn't feel significant to players. It's only significant to theory crafters who focus myopically on means and medians.
Since you're going to ignore everything I say and argue with a ghost, there's really no need to continue this discussion.
I genuinely don't see any point you made I ignored. I see points I commented on and then said but they don't change what I said. If I've missed something, point it out and I'll respond.