I think we may be approaching the concept of balance from different directions, and talking past each other as a result. Based on your proposed analogy and your description of what I would need to show to continue the conversation, it looks like you're focusing on (and perhaps defining?) in-combat balance as PCs having similar "output" over time.
Close, but still approaching it from the other side.
I am not trying to show combat balance. I am trying to show combat imbalance.
In other words, this can't show balance. Having four good tires doesn't mean your car is good to drive. It can show imbalance. Having a flat tire can show that a car is not good to drive.
I actually completely agree that lower or higher numbers of encounters are going to differently affect the "output" of different PCs. As I previously discussed, I think the relationship between number of encounters and PC average effectiveness is more complicated than you presented (because you're not accounting for how often a PC can actually make good use of their highest level spell slots, a factor which I think campaign style influences), but I agree on the basic premise.
I'm sorry, but I'm going to handwave this away. This is a general evaluation where I don't care about any particular situation. Over the course of play enough times will come up to use high level spells, or the spells would be replaced in what the character is prep/known if it doesn't fit the style. I played a bard who picked up Heat Metal at 3rd, one of the best low-level damage spells on the bard list, and trained it away at 6th, never having come across a combat where my concentration was best used against a single opponent, and that opponent was wearing metal. It self corrected to match the style. When
My original point, however, was instead looking at how unusually low or unusually high numbers of encounters affect whether or not any particular table experiences balance issues. In other words, I'm talking about whether the players at any particular table view any imbalance they percieve during their campaign as actually rising to the level of an issue that detracts from their play experience.
Eh, you can go this route, but if I can compare something at least someone rigoriously (though I admit it makes huge assumptions about effectiveness when it's not just damage that's easier to add up and compare), vs. a fully subjective rating, I'll go for the former.
For ladder is just anecdotal. I was in a Theros campaign where we all ended up getting weapons of the gods. I was playing a rogue, and by sneak attack damage, which was situational and often wouldn't occur in the first round of combat, was completely overcast by characters with extra attack who were getting a bonus 3d8 damage every hit. My subjective view was that in combat I contributed the least - I did less damage in a round, I couldn't tank, I didn't have spells. Talking to some of the other players at one point, I had early on (before the Legendary weapons) been "the damage dealer" and when they saw high single hits later on that's what stuck in their minds. So they all thought I was the big damage dealer of the party, while the battlemaster fighter regularly out damaged my character as well as providing other aspects like tanking. Subjective evaluation is just that, subjective evaluation.
That's a much broader (and inherently less quantifiable) view of balance than I think you're focused on.
Again, this is the crux of talking past each other. I am not talking about game balance. I am talking about game imbalance. You can have clear signs that things are not balanced, but the lack of those particular signs is not an indication that they are balanced.
You are trying to talk about a car ready for a journey. I am trying to talk about a car not ready for a journey. Missing a tire and the check engine light off is not ready. All tires and the check engine light blinking is not ready. A flat tire can be a sign that the car isn't ready, but the lack of the flat tire does not say anything about the check engine light.
This isn't a measure that can show balance, just imbalance. All of the discussion points you make about balance may be valid, but they don't invalidate this as a measure of imbalance. Especially once you are looking outside mechanical aspects like spotlight time
Would you agree that campaign style can impact whether any percieved imbalance in "output" at any particular table rises to the level that's it's perceived as a problem by the players?
Regardless of if this is true, it doesn't impact what I was talking about which was a mechanical evaluation. Subjective evaluations like this are important, because they impact the fun a player is having, but that's a separate type of evaluation.
From there, because I agree with you that number of encounters can impact relative PC "output", I think it follows that campaign style can influence whether that impact rises to the level that it's perceived as a problem.
Which is all I was saying. Though perhaps more strongly than you - this is why I'm not picking up the 2024 books. I find when the characters get to T2 and T3 that both running and playing we have so few encounters per adventuring day that it is a large enough flaw that I really don't want to play anymore. I was thinking about running with the Gritty Realism variant from the DMG, but now I've kickstarted 13th Age 2nd Edition and it have a full heal-up (the equivilent of a long rest) once per arc, and that fits my running style even better.
If you can agree with that, then I think we're in broad agreement regarding "balance" as we're each using the term. If you can't agree with that, where do you see our evident disconnect as arising?
I think the disconnect in our discussions was I was talking about something
inhernet in the mechanics/math of the system to measure an
imbalance or not
across all tables, while you were looking at
subjective measurement trying to show
balance based on style, which is
not inherent and the same across all tables.
BTW, don't take this as a criticism of what you are talking about. Frankly, the subjective feeling from all of the players that they are contributing is very important. Critical even. I agree with all of it. This is a game, we should all be having fun. It just is measuring different things and therefore can be used in conjuction with what I was talking about, much like a spedometer and engine RPM gauge are related conceptually but show different things. Having both of these tool makes things even better.