What is the optimal strategy for the bowling encounter in the AL Epic adventure Last Orders at the Yawning Portal?

Question

The Adventurers League Epic adventure Last Orders at the Yawning Portal (DDEP08-03) has a Tier 4 quest where one or more party members try to beat an NPC at nine-pin bowling. If at least one PC beats this NPC, the party succeeds at the quest. Even if one or more PCs win, all PCs that lose suffer a significant, unavoidable consequence. (A PC that ties is counted as neither winning nor losing.)

Rules

• There are ten frames, during which each participant rolls the ball at most twice.
• Each participant makes an improvised weapon attack using either Strength or Dexterity.
• Like traditional bowling, a participant that knocks down all of the pins on the first roll scores a strike. A participant that knocks down all of the pins in two rolls scores a spare. Otherwise, the participant leaves an open frame.
• All participants roll the ball simultaneously for a given frame. All participants who require a second roll will then roll simultaneously for that one.
• Everyone knows everyone else's score and pin count.
• Any character that scores a strike or a spare is treated to a drink from the bar. Consuming a drink increases the difficulty of future rolls.
• The NPC opponent will only drink if the party does so as well.
• Assume everyone has 1 minute between everyone's first and second roll, as well as 1 minute between frames. Any spellcasting, class features, etc. may be used during these breaks, so long as there is sufficient time. There are no other breaks.
• For the purposes of this question, assume that the NPC does not change their strategy in response to the players. They always roll with the same bonus and buff (listed in the spoiler section below).

Now for the details; full spoilers ahead:

• Base DC 12. For each number above the DC, the participant knocks down a pin.
• If 5+ pins (but not all nine) are knocked down, the DC increases by 2 for the second roll.
• A strike is 15 points. A spare is 10. An open frame is one point per pin knocked down.
• Each drink cumulatively increases the DC of subsequent rolls by 2.
• The NPC opponent has a +10 to their roll, as well as advantage from foresight.
• Any attacks, spells, or other effects directed at the NPC opponent fall away harmlessly.
• The module includes an allowance for the use of spellcasting (like mage hand) to directly assist in bowling. For the purpose of this question, assume that no one does this. Buff spells are okay, though.

Drinking

The decision to drink is the only unique mechanical interaction in this encounter. I believe that it holds the key to an optimal strategy. Unfortunately, the module is a bit vague on when the NPC drinks, so I will formalize their behavior.

1. Start: the party status is Abstaining.
2. All participants complete a frame.
3. If no PCs were offered a drink, their status remains the same.
4. If at least one PC was offered a drink AND all PCs who were offered a drink choose to drink them, then the party status becomes Drinking.
5. If at least one PC was offered a drink AND any PC who was offered a drink choose NOT to drink it, then the party status becomes Abstaining.
6. If the party status is Drinking AND the NPC scored a strike or spare, then the NPC drinks.
7. While there are still frames left, GOTO 2.

This means that a party that drinks on frame 1, but fails to get any strikes or spares on frame 2, will still be considered Drinking.

Constraints

In order to constrain the question, assume that all PCs at the table have the following goals:

1. At least one PC wins. (Thus at least one PC participates.) The party should aim for a 75% chance of accomplishing this goal.
2. As few PCs as possible lose. Ties are okay.

If a general case answer is not possible or feasible, assume the following party (with no magic items or feats unless specified):

• Half Elf Lore Bard 20. +4 bonus from Dexterity. Five Bardic Inspiration dice. Combat may not be initiated to gain additional dice.
• Hill Dwarf Life Cleric 20. +2 bonus from Strength.
• Human Champion Fighter 20. +9 bonus from a belt of storm giant strength.
• Halfling Thief Rogue 20. +6 bonus from Dexterity (and a manual of quickness of action).
• High Elf Evocation Wizard 20. +3 bonus from Dexterity.

Assume that spellcasters know/prepare any spells needed. For the sake of making this question interesting, assume that wish and Divine Intervention are only used for duplicating spells. Additionally, assume every player knows these mechanics and (ab)uses them to the best of their ability.

What is the optimal strategy for this encounter, particularly when it comes to the drinking mechanics? How does requiring all PCs to participate change the optimal strategy?

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Rules over rulings

Most people play Epics at conventions, often with a DM they don't know. Answers should assume that the DM strictly adheres to RAW. If something isn't mentioned in the adventure, don't rely on a favorable ruling.

Can I do X to the NPC?

For the purpose of this question, the NPC's rolls cannot be altered in any way. The NPC will always roll with the bonus and benefit listed in the spoiler section. This is (vaguely) specified by the module (Last Orders at the Yawning Portal, p. 30):

Attacking Halaster

This is ill advised. All attacks, spells or other effects directed at Halaster here in his nine-pin alley fall away from the wizard harmlessly. Halaster will warn the characters ONCE, asking them to please stop and be civilized. If any of them continue to enact hostilities, cast spells of any kind at his person or his servants, the mad mage disappears in [an] explosion of magical force doing 150 hit points of force damage to all characters in the room.

It seems clear that greataxe attacks, blindness/deafness, and Cutting Words would have no effect on the NPC. Does Portent work? What about darkness targeted near the NPC, but not on them? Rather than pick through each answer to these questions (and make a long question even longer), it seemed simpler to say that the NPC's rolls cannot be affected.

Answer 1

The best I could get was a 99.996% chance of winning

The setup

First we assume the NPC is not going to play optimally, but will always roll with the goal of hitting as many pins as possible.

I could not find any way to gain proficiency in improvised weapons under these restrictions, but if we could my program did not have us lose a single one of its one million matches.

We would want to have the highest modifier we can, our Champion Fighter would seem like the best candidate but we can do ever better. There are two monsters of CR 20 or less with 30 Strength, the Leviathan, and the Walking Statue of Waterdeep. We could use true polymorph to turn any of our party members into either of these monsters and have it compete for us.

We can gain an additional use of Bardic Inspiration through the catnap spell whose description states (parenthetical mine):

If a target remains unconscious for the full duration (10 minutes), that target gains the benefit of a short rest...

Then, looking at Bardic Inspiration's description we see:

Once within the next 10 minutes, the creature can roll the die and add the number rolled to one ability check, attack roll, or saving throw it makes...

We can have the Bard grant our participant a use of Bardic Inspiration, and then immediately have the Bard start napping. We could time the start of the bowling so that after our first roll the Bard will wake up, having benefited from a short rest which will trigger Font of Inspiration whose description states:

Beginning when you reach 5th level, you regain all of your expended uses of Bardic Inspiration when you finish a short or long rest.

And so we now have six uses of Bardic Inspiration available to us.

An easy way to grant ourselves advantage on these attacks would be through the faerie fire spell whose description states:

Each object in a 20-foot cube within range is outlined in blue, green, or violet light (your choice)... Any attack roll against an affected creature or object has advantage if the attacker can see it...

And then we could add the bless spell on top of this whose description states:

Whenever a target makes an attack roll or a saving throw before the spell ends, the target can roll a d4 and add the number rolled to the attack roll...

Note: Instead of using faerie fire as this may not be possible there are many other spell options such as foresight whose description states:

For the duration, the target can't be surprised and has advantage on attack rolls...

The Strategy

We have the same modifier as the NPC, but we also have bless and Bardic Inspiration on our side, thus we are strictly better than the NPC at bowling.
Because of this, if there were any time where we both got a strike/spare, we would want to drink; this makes the game equally more difficult for both parties but we remain better than the NPC.

Should you drink on a frame where you got a strike/spare and the NPC did not? No.

In order for drinking to end up being worth it in these cases there have to be at least two frames where you do not get a spare/strike but the NPC does, and these both have to occur before you get another spare/strike. The chance of this happening is quite small (about 2.2%).
This is primarily because a strike is worth so many points and so you would always want to do as well as you can on your first throw in a frame. This results in us having a very high chance of getting a strike (90%).

Note: There were scenarios like the following for which I guess-and-checked (brute-forced):
If the NPC gets a strike on their first bowl and you don't and your status is DRINKING, should you make your second roll?
I tried for various values of remaining pins and in the end you did the best if you chose not to bowl if you could only stand to make 1-4 points (if there were 3 or less pins standing). This only applies if you are not in the 9th or 10th frame as the NPC's DC going up only by 2 for one frame (or none) is not worth the number of points you would otherwise likely get.
It is interesting though that the chance of winning when you bowl every single time you can and the chance of winning if you never make any second bowls are only .131% apart.
Similarly, I decided that if the NPC did not get a strike, had a 50+% chance of getting a spare, your status was DRINKING, you only had 1-3 pins remaining and it was not the 9th or 10th frame, you should fail your second bowl.

The odds of one of these scenarios happening was 3.5% so the strategies were not employed very often.
Note: I did not delve particularly deep into the math here as it gets incredibly complicated quite quickly, so I stuck with the nice round 50+% statistic.

One question left is "When should we use Bardic Inspiration?"
I though this question was going to be an absolute nightmare but we are so incredibly good at bowling it's nearly a non-issue.
Strikes are worth the most points, so we would want to use Bardic Inspiration if it stood a significant chance of getting us a strike. However, as it turns out, if we spend a Die every time our roll would get us anywhere between eight and negative three pins knocked over we would want to use it. (You cannot actually knock over a negative number of pins, but if we were three off from hitting even a single pin Bardic Inspiration rolling a 12 will get us a strike).
This is the go-to strategy because a strike is worth more per additional point to our attack roll (what Bardic Inspiration does) than anything else. (A strike grants 6 additional points, while a spare grants only 1).
However, we are so good at bowling that if this was the only time where we spent Bardic Inspiration dice we would, on average, still have 2.5 unspent dice.
We are currently spending Bardic Inspiration dice even if it has only a .0833% chance of getting us a strike (if we need a 12). This would give us an average of .4998 additional points on those attempts. Looking at the statistics for spares and d12s we can see that there is a 41.67% chance of rolling an 8 or higher on a d12. Thus if we needed to roll an eight (or above) to get a spare we would be getting an average of .4167 additional points.
This is close to our expected number from the strikes so I used it as a baseline. We would expend Bardic Inspiration dice if a roll of 8 or higher would give us a spare.
After doing so we would still on average have 1.2 Bardic Inspiration dice left over.

Extended this range further uses calculations which take into account the fact that if there are few enough remaining frames, you would spend your Bardic Inspiration dice differently in an attempt to not have any unused.

Having an additional circumstance where we roll the excess dice even if only a 12 would get us points increases our chances of winning and our average number of remaining Bardic Inspiration dice would become only 0.3. (Expending dice more often resulted in us losing matches more often. This is because in the final frames we would be using up dice on spares too quickly and wouldn't have then for the more valuable strikes)

Random stats: There is only a 14.38% chance we use Bardic Inspiration on the first bowl.
On average we run out of Bardic Inspiration dice on frame 9.
On average we score 143.9 while our opponent scores an average of 80.9

The maximum DC that will ever occur is 26.
If the NPC has a DC of 24 then they would need a 35 to get a strike; however, their highest possible roll is a 30, which would leave 5 pins left. Fortunately, this keeps the same DC as before and so they would need a 29 or 30 to knock over all 5 remaining pins and get a spare. Thus their DC would increase to 26. If this happened then they could never increase it again.
Even if they roll a 30 they will only knock over 4 pins, rolling another 30, again, knocks over 4 pins. So they can never get a spare or strike and thus will never drink.
This means that letting our own DC go above 26 is pointless (or letting it go up at all once the NPC's DC is 26). Thus the maximum DC for a strike for us is 35 and even then we have a 41.32% chance of getting a strike.
To showcase how incredibly good Bardic Inspiration is (as well as how well it works together with bless and advantage on our attack roll): If we did not have the d12, our chance of getting a strike when the DC is 35 would be 0% (though only barely so).

A recap of how to do this:

True polymorph into a Leviathan, or a Walking Statue of Waterdeep.
Use foresight to gain advantage on our bowls.
Use catnap to gain an additional Bardic Inspiration die.
Use bless to gain an additional 1d4 on each roll we make.

Only drink if both you and the NPC got strikes/spares and your current DC is 24 or less.
If the NPC gets a strike and you have 1-3 pins left (and it is not the 9th or 10th frame), do not make your second bowl.
The 50+% chance of the NPC getting a spare is a more complicated section that you could use but it doesn't increase the odds of winning by a significant amount so I'll leave it out of this part.
Anytime your first bowl wouldn't get you a strike, spend Bardic Inspiration Dice.
If rolling an 8 or above on the Bardic Inspiration die would get you a spare, roll the die.
If you are on your first bowl for frame X, and you have 10-X or more Bardic Inspiration dice left, and the die has a chance of getting you points, roll it.
If you are on your second bowl for frame X and you have 11-X or more Bardic Inspiration dice left, and the die has a chance of getting you points, roll it.

Thus I arrived at what is currently my optimal solution for bowling, giving us a 99.996% chance of winning.

A lot of modifications here only increased our chances of winning by .05% or less.
If instead you played much more simply and bowled every single time you could and used Bardic Inspiration even if only a 12 could get you even a single point, you would still have a 99.85% chance of winning.

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On the truly ideal "solution" if the party could be anything

In the truly ideal solution we would always always win.
We would need to have proficiency in improvised weapons. (such as through feats)
We would need many Divination Wizard's... (One 3+ roll to guarantee a strike on our first frame, one 5+ for the second, one 7+ for the third... one 15+ for the seventh, and three 17+ for the eighth/ninth/tenth frames). By the eighth frame, worst case scenario, the DC has already capped out so we only need 17+ for all three remaining frames.

This increases our odds of winning to 100% (It guarantees that we always bowl a perfect game, which the NPC is physically incapable of doing as the DC will become high enough that they cannot bowl a strike).