So is it right that the main reason you judge them worse at wizardry, is the allocation of some first level spell slots to those purposes? Do you mean that judgement to apply over most of their adventuring career? Say those slots weren't important for the majority of their career, would that change anything? Imagine that wizards other than BS also used those slots for identical purposes, would that change anything?
Okay let's look at some numbers, we're assuming a 5 round fight with 2 attacks (Multiattack) every round, +5 attack against AC 22. Due to Shield, by far the most likely way the CR 4 foe kills the BS is the critical. That's why it has become the crux of our debate. Do you agree that over ten attacks, the cumulative probability of that critical = 1-(1-c)^r = 1-(1-.05)^10 = ~40%. (Over two combats, the cumulative probability is about 64%.)
For me, those probabilities strictly indicate that BS is likely to survive two or more combats. Any critical ends the fight so damage isn't at issue: all we need is the cumulative probability of the critical.
Either way, we agree that the BS drops about once per two combats and odds-on survives one combat 6:4. That is what I have been consistently stating all through. BS wins the combat any time they want by Levitating either themselves or the creature and filling it full of heavy crossbow bolts, but I'm charitably giving their allies a job.
You know, this is helpful because it points to another big difference in our understanding. I think you are saying that the BS behaviour is binary, right? They're a fighter, or a wizard, but never both at the same time.
That's not what I think and not what I've seen in play. BS remains a full wizard even when they're tanking with their AC 22 and Shield. They can always win by wizardry and their other benefits are important here - a big buff to Concentration, a big buff to not be grappled, a big buff to speed (when assessing speed buffs, gains are absolute, not a ratio: if X is 10' faster than Y, then Y never catches X unless X lets them).
This is fair. My key objection is that BS just as readily finishes the engagement within 5 rounds. We're being far too charitable to the CR 4 foe to assume it lasts as long as it does.
Also, I believe you are low-balling the PC output, a point I want to come back to in a minute.
I don't believe either of us is entitled to a conclusion yet because I think you've shown that we need to add another layer to our analysis.
You've asserted an asymmetry: Champion takes foe out of combat faster so is exposed to fewer attacks in return. I challenge that asymmetry because it looks to me like you low-balled other PC's contributions and didn't include options that BS has to do likewise, faster. We also need to focus on the majority of BS' career. So far I've gone along with Level 4 for a specific reason - it's probably the last level where Champion has a hope in hell of being better than BS - but we both know that most of an adventurer's career is spent between levels 5 and 12. I think now we do need to play fair on that front too: otherwise your arguments will always be subject to my challenge that you've cherry-picked a fraction of the adventurer's career and called it representative.
Per RAW, the expected career is about 21 days or 30 sessions 1-12, with about 4 days or 6 sessions at levels 1-4, and 17 days or 24 sessions at levels 5-12. What level do you feel is fairest to use? Also, I think I should be making the choices for the BS setup, and you those for the Champion setup, as I don't agree with some of what you chose for BS. I concede that BS can be down-powered through charitable choices; any archetype can be.
Does that all sound constructive?
Firstly, I've multiple times complained that the initial scenario you set up wasn't representative of actual play and presented a few alternatives -- all ignored. You continued referring to your initial scenario (5 rounds, BS using melee attacks, etc), so I obliged and continued using the choices you presented. Now, however, you're laying those choices at my feet and calling them absurd when a full treatment shows they are actually absurd. This is frustrating.
If you did not believe level 4 was a good choice to highlight the bladesinger, why did you pick it?
Secondly, yes, your numbers for the chance of crit over time are correct -- I had an error that propagated through and caused inflation on my spreadsheet. Using the formula would have been cleaner, but I was using cells for multiple things and just wrote a 'next round' formula and copied it over.
However, if you look at the Champion, he barely dies in the second fight. If he wins initiative, he survives. If he gets a healing for around 8 damage, he survives. The champion's fail rate is razor thin, whereas the bladesingers is fixed at the crit chance.
Thirdly, on the binary issue, yes, because it is binary. If the bladesinger runs up into melee to tank and then casts levitate, the tanking is irrelevant if the spell succeeds and deadly if the spell fails (run your crit chance for 6 rounds of combat instead of 5, as it will take another round without bladesinger engagement). Further, a wizard can do that without trying to tank, as successfully, so that's not something that's improved by bladesinging.
Also, if that's the method, then the bladesinger is burning through long rest resources to improve survival and trivialize
medium encounters. At 4th, the 'singer can, at most, do this in two fights. After that, for the remaining two medium encounters to get through the adventuring day, he's tapped and now faces exactly the same math as before. It's not sustainable, even if it wins a single fight.
As to your final point, level 4 is pretty much the last level that the bladesinger maintains superiority over the champion in the tanking/melee role. To exceed the fighter at his role, the bladesinger has to act the wizard, not the fighter. You're going to show that the bladesinger outshines the fighter, but in every case you do so it will be something a non-bladesinger wizard can do as well which shows it's the wizardly bits that excel, not the bladesinger bits. Much like your use of levitate above, which has zero to do with bladesinger tanking.
Bladesingers excel at being the individually defensive wizard. All of their abilities improve their chance of survival as a wizard, but are high risk and resource intensive if used to act like the fighter. The point at which that trails off is also the point where you're obviously more effective acting a straight wizard. But they just don't have the hp pool to be a successful tank past the first tier, and even there it's risky.
Checking the average of the damage spreads in the DMG for each CR, they go up by 6/CR, while the Bladesinger hp increases by 3+CON. You've only broken even at 16 CON and have to hit 18 CON to pull ahead of the curve. This is important because that damage curve means that crits are always going to be deadly to bladesingers. Unless you're dealing with a rolled 3 18 bladesinger, you're always at the mercy of the critical for CRs equal to your level. Figure the chance of crit for the adventuring career and you'll see the downside here.
But, to circle back to your challenge, it's critical that we establish what it means to be the party tank -- that it means actually taking attacks to protect the other members of the party. If your analysis is that, by level 7 the bladesinger can just banish a foe, then we are no longer talking about the bladesinger taking the fighter's tanking shtick, we're talking about what wizards do which has already been conceded as superior to the generic fighter. So, any analysis on the tanking abilities of the bladesinger MUST revolve around the bladesinger taking the same number of attacks as the champion. If we agree there, then we can proceed. We'll find that crit chance dominates for the bladesinger, so survival rates are flat.
