Statistical differences between rolling methods

Sean and I have been discussing rolling methods for ability scores, particularly in the context of PCs vs. henchmen. Consider the two following methods for generating a set of abilities (sourced from the 2e PH, p. 13):

• Method II. Roll 3d6 twice and take the higher of the two values.
• Method V. Roll 4d6, discard the lowest roll, and add up the other three.

For player characters, I've long employed Method V (4d6 drop lowest) with a caveat that the player can roll up to three sets; if the player discards one set to roll another, the discarded set cannot be used. This gives the player the ability to discard an underwhelming set or swing for the fences to try to attain a great set, but also adds an element of risk, especially if the third and final set (which must be accepted) turns out to be poor.

For most NPCs and hirelings, I employ Method II, which tends to generate more averages sets. I don't use any re-roll caveat for NPCs. They get what they get.

Because, in our campaign, henchmen have the ability to replace fallen PCs over the long-term, we want to make sure we're in agreement on the rolling method to use. Most AD&D players intuitively know that Method V is more likely to generate high scores (15+) than Method II, but what if we evaluate the methods more deeply? Let's say, for example, that we're looking to create a ranger, which requires two scores of 14 or higher and two additional scores of 13 or higher.

I wrote a quick app to generate random sets of ability scores using both methods. Scores within each set are arranged from highest to lowest, with sets that qualify for the ranger class marked with a letter r. Here are results for ten sets of scores for each method:

Method II results (3d6 twice, take higher):
  16  14  13  13  11  11  (r)
  16  13  12  12  10   8
  15  14  13  11   9   8
  16  15  14  12  11  11
  14  14  13  13  12  11  (r)
  13  11  11  11   9   5
  15  15  15  13  12   8  (r)
  15  14  14  12  12   8
  13  12  12  11  10   8
  16  14  14  13  12  11  (r)

Method V results (4d6 drop lowest):
  15  13  13  12  12  12
  14  14  12  10  10   8
  16  13  12  12   9   4
  17  15  13  11  11   7
  14  14  12   9   9   8
  14  14  12  11   8   7
  17  14  13  12  11   7
  15  12  11  10   9   8
  16  14  13  13  10  10  (r)
  15  15  14  13  10   8  (r)

Four rangers for Method II, only two for Method V. This are very small samples sizes, so let's run them again to observe the variance:

Method II results (3d6 twice, take higher):
  13  12  11  11  11   6
  14  13  13  12  12  12
  14  13  13  12  12  11
  17  16  12  11  10   9
  13  13  13  12  11  10
  15  15  12  10   9   8
  17  16  12  11   9   8
  15  14  14  10   9   9
  16  16  15  15  14   9  (r)
  17  13  13  13  12  12

Method V results (4d6 drop lowest):
  15  12  11  11  10   9
  15  14  13  12  12   7
  15  13  11  11  11   9
  16  14  12  11  10   8
  14  14  13  13  12  12  (r)
  15  15  14  13  11  10  (r)
  17  14  14  14  13  11  (r)
  16  16  15  12  10   8
  15  15  14  13  12   5  (r)
  18  14  13  12   9   9

This time, one ranger for Method II, four rangers for Method V (an opposite result). Obviously, we need more data. I'll have the application roll 100 sets for each method and take the averages (rounded down to whole numbers):

Method II averages:
  15  13  12  11  10   9

Method V averages:
  15  14  12  11  10   8

These are actually really close; in fact, the total number of ability points is the same with both methods. Through another few runs, I was able to verify that these exact averages still hold even with a very high (10,000) number of sets.

But the averages don't tell the full story. Again, we know from intuition that there's going to be tangible variance between the methods. If it's not in the total number of points, then where?

For starters, we know that, in order to get stuck with a score of 3 using Method II, we need to roll six 1s in a row. That means 6^6, or one in 46,656 scores. To get equally unlucky with Method V, you only need to roll four straight 1s, or 6^4, which is one in 1,296 scores.

That's a major difference: you're almost 40 times more likely to end up with a score of 3 using Method V (the 4d6 method) compared with Method II (the 3d6-twice method).

Let's see what the distribution of scores is over ten sets of scores using each rolling method:

Method II totals:
   3s:  0
   4s:  0
   5s:  0
   6s:  0
   7s:  0
   8s:  2
   9s:  6
  10s:  4
  11s:  5
  12s:  12
  13s:  7
  14s:  12
  15s:  6
  16s:  6
  17s:  0
  18s:  0

Method V totals:
   3s:  0
   4s:  0
   5s:  0
   6s:  2
   7s:  2
   8s:  2
   9s:  6
  10s:  3
  11s:  3
  12s:  12
  13s:  9
  14s:  5
  15s:  10
  16s:  3
  17s:  3
  18s:  0

That looks reasonable: fewer high but also fewer low scores when using Method II. How about for 1,000 sets?

Method II totals:
   3s:  0
   4s:  2
   5s:  11
   6s:  36
   7s:  101
   8s:  257
   9s:  424
  10s:  629
  11s:  839
  12s:  979
  13s:  926
  14s:  727
  15s:  513
  16s:  313
  17s:  184
  18s:  59

Method V totals:
   3s:  3
   4s:  25
   5s:  51
   6s:  99
   7s:  194
   8s:  269
   9s:  451
  10s:  585
  11s:  697
  12s:  794
  13s:  740
  14s:  737
  15s:  586
  16s:  430
  17s:  240
  18s:  99

Now we're starting to see the numbers at work. A thousand sets contain 6,000 individual scores; we have three scores of 3 using Method V, and none using Method II. Given that we were expecting only one in 46,656 scores with Method II but about one in 1,296 scores with Method V to result in a lowly 3, these results look pretty solid, though our sample sizes are still small enough that we're hitting a fair degree of variance.

Here are the totals for 100,000 sets:

Method II totals:
   3s:  15
   4s:  184
   5s:  1056
   6s:  3819
   7s:  10642
   8s:  24747
   9s:  43524
  10s:  65439
  11s:  84711
  12s:  94523
  13s:  92145
  14s:  72943
  15s:  51490
  16s:  32525
  17s:  16626
  18s:  5611

Method V totals:
   3s:  473
   4s:  1917
   5s:  4722
   6s:  9581
   7s:  17370
   8s:  29011
   9s:  41925
  10s:  56347
  11s:  68252
  12s:  77520
  13s:  79661
  14s:  74389
  15s:  60690
  16s:  43464
  17s:  25012
  18s:  9666

For Method II, fifteen out of 600,000 scores ended up as 3, or one in 40,000, which is very close to the one in 46,656 ratio that we expect to normalize to over the long term. For Method V, we had 473 scores of 3, which is about one in 1,268... extremely close to the normalized ratio of one in 1,296.

If we add up all the scores of 7 or lower, Method II only generated 15,716 while Method V produced a whopping 34,063. If we add up all the scores of 15 or higher, Method II gave us 106,252 while Method V resulted in 138,832.

The takeaways are that you're more than twice as likely to get bad scores (7 or lower) with the 4d6 method, but only about 30% more likely to get high scores (15 and above). Method V, however, is almost twice as likely to generate very high scores (17 or 18), while Method II is far more likely to hit in the average range of 10 through 14.

Finally, here's a plot graph of the 100,000 set results, which makes everything nice and clear:

Click to enlarge

Though Sean and I still haven't decided exactly how to handle scores for a new henchmen, the above data definitely provides the right ammunition to help us make the best decision for our campaign. As an extra bonus, here are score distributions for two additional rolling methods described in the PH, along with an additional graph that charts all four methods.

• Method I. Roll 3d6 for each score.
• Method IV. Roll 3d6 twelve times, and take the six highest values.

Here are the side-by-side averages of all four methods:

Method II averages:
  15  13  12  11  10   9

Method V averages:
  15  14  12  11  10   8

Method I averages:
  14  12  11   9   8   6

Method IV averages:
  15  13  12  12  11  10

...along with the score distributions of the two new methods:

Method I totals:
   3s:  2817
   4s:  8353
   5s:  16620
   6s:  27915
   7s:  41657
   8s:  58653
   9s:  68882
  10s:  74966
  11s:  74945
  12s:  69277
  13s:  58329
  14s:  41804
  15s:  27916
  16s:  16701
  17s:  8340
  18s:  2825

Method IV totals:
   3s:  0
   4s:  0
   5s:  0
   6s:  1
   7s:  129
   8s:  2063
   9s:  14860
  10s:  50573
  11s:  98940
  12s:  123988
  13s:  114935
  14s:  83276
  15s:  55540
  16s:  33591
  17s:  16596
  18s:  5508

...and the final graph depicting all four methods:

Click to enlarge

(If you made it to the end of this post, congratulations!)

Longevity

By all reasonable accounts, Zeb probably should be dead. It wasn't a miracle of the dice that stayed him, but weak judgment on my part. At the last minute (after a failed save had been rolled), I called into question whether a blinded dragon could reliably direct its breath weapon at unseen opponents, even with their last known location unchanged. In the moment, I decided this was too contentious to stake a character's life on, so I allowed Zeb to avoid the cloud, even though the full breath weapon damage would have killed him instantly. Erring on the side of the PCs isn't the worst thing in the world, but I probably leaned too far this time. Zeb can rightly acknowledge this as Malar having spared his meager life.

In light of the above, I've given much thought to the prospect of characters dying and what a tragic event like the loss of Zeb would mean for the campaign. If I felt like allowing a PC to die would spell the end of the game, that would severely burden my judgment when it comes to adjudicating deadly situations.

Fortunately, I think we're all on the same page that the campaign can continue, even under the most dire of circumstances. For the benefit of everyone, here are some points to live by going forward. This is effectively a charter as to my long-term vision for the game.

• If a PC dies based on dice rolls, I'm not going to intervene. It's hard, as DM, not to pull strings at times, but doing so deprives the players of a fairly run game and the experience of overcoming the death of a character.


• I don't expect (nor should anyone else expect) the campaign to end if a character gets killed. Obviously a TPK may result in a break in continuity at the very least, but either way, life in the game world goes on, just like in the real world. I see our campaign as being greater than any individual character; there's no critical plot artery that hinges on the survival of Zeb, Audric, or anyone else. We're full-on Game of Thrones, here.


• A player can either create a new character to replace a perished (or retired) one, or elect to take on a henchman as a full-fledged PC. New characters begin with the same starting XP as the original PCs and are expected to advance in level through play. This may mean that a new PC needs to be shielded from harm at times by the rest of the party in order to stay alive.


• Despite entering the game at a lower XP total, a new PC can normalize in level over time thanks to the AD&D experience tables. For example, a 1st-level, 0 XP Fighter and a 5th-level, 16,000 XP Fighter will both be 6th level, 40,000 XP later. (It's also reasonable to expect that a low-level PC will advance more quickly in a high-level party, since the party's XP awards will naturally be larger.)


• In the meantime, it's possible that other, established PCs will die/retire, or new players/henchmen cycle into the game. The original “new PC” may become a veteran before long.


• Under this system, the total power level of the party is likely to fluctuate over time. Many of us are used to games where the party only ascends in level and ability as the game progresses, up until the point that everyone retires and the campaign ends. In my model, we could look back six months from now and recognize this point in time as the peak power level of the campaign. This allows for organic continuity across years of gaming, with the ability to re-experience the early and mid levels again and again. A high-level character that sustains through it all is truly something to be valued.

I could probably write a book on this stuff, but these feel like the most important points, so I'll leave it here for now. Happy to discuss further in the comments.

FR #10: The Burden of Their Spoils

20 Mirtul, sunrise

Toward the end of their first day traveling west from the ruins, the party was set upon from behind by a gnoll warband, which split into flanking groups and rushed in, throwing small axes from amid the trees. The adventurers stood fast, their swords and magic felling several of the dog-faced creatures before the remaining few retreated into the forest.

Injuries to the party were mounting, though not insufferable. As twilight loomed, a suitable campsite was located and watches were arranged without a fire. During the first, Arendeth praised Aranos for his bravery and the proficiency with which the budding warrior fought against the gnolls. In the early morning hours, Rumolt and Berwyn exchanged vagueries regarding their pasts and future aspirations.

The following day was met with no additional threats from their surrounds, and as latefeast approached the party was hailed by a short, round-faced woodsman named Banion, a patroller of the hunt trails from the Slumbering Manticore. Though the PCs initially feared that the encounter may have been more than mere chance, Banion led them safely to a well-trodden path that returned them to the inn before nightfall. Amid the grounds outside, the smell of campfire smoke ushered them in, and Constable Rictus strode out to meet them.

Readily accepting of the inn's food, drink, and shelter, the party entered the taproom and began to discuss reconciliation terms with Rumolt. Rekindling the tension felt at the ruins when the topic of the scepter was breached, it became clear that neither the PCs nor Rumolt were willing to part ways with the other having possession of the artifact. Emotions peaked when Rumolt spat on the table and called Arendeth a fool at the dwarf's suggestion that Rumolt exchange his share of the scepter for the wraith's black sword and helm. Wren subsequently dismissed Rumolt from the party under implicit threats of violence before cooler heads prevailed, and all took their sleep for the night.

The PCs set watches outside their rooms, fearful of attempted thievery by Rumolt or someone else. Arendeth imparted to his companions that a fire trap spell was already cast upon his pack, protecting the scepter. Using detect magic, Riwyn discerned that Wren's longsword emitted a minor aura of evocation magic, the wraith's sword and helm a moderate aura of necromancy, and the scepter a powerful aura of an unknown nature.

In the morning, Berwyn met Rumolt in the taproom to further discuss their options. Rumolt committed to no specific course, but his impasse with the party would not be relinquished. The pair convened with the larger group upstairs, with all finally settling on an agreement that Rumolt would continue to accompany them as an equal until the matter of the scepter could be reconciled. Rumolt handed over leather-bound coin rolls owed to the remaining party members, bidding them to rest until all were returned to full strength and their next destination could be determined.

DM's Commentary

Since the moment they crossed paths, the party has been waiting for Rumolt to show his hand. The man's past is unclear, without a doubt, but is not also that of the PCs, in Rumolt's eyes? The distrust between the sides is palpable, and it's been fun to play out in-game. It's interesting to see how an encounter of random chance can be perceived as the first sign of an intricate betrayal. The party is waiting for the other boot to drop. Will it?

In addition to XP earned for the past few sessions, which is detailed below, Aranos is hereby promoted to henchman status under Arendeth. This means that Aranos is now formally a first-level character who will begin to accumulate experience, starting at zero. Hereafter, Aranos will receive a half XP share from the group, and otherwise compensated primarily by Arendeth, to whom the warrior has become devoutly and fanatically loyal. Sean will control Aranos fully, and I will only step in if egregious orders for him are issued.

XP

The following XP awards cover the last three sessions, the entirety of the time that Rumolt has traveled with the PCs.

• Orcs, 20 - 290 XP
• Kobolds, 3 - 22 XP
• Wraith - 730 XP
• Gnolls, 8 - 310 XP
• Longsword +1 - 400 XP
• Wizard scrolls, 6 - 1,100 XP
• Coinage plundered - 170 XP
• Scepter - 10,000 XP
• Story award - 5,000 XP

This amounts to 18,022 points, which is split six ways (as Rumolt, for better or for worse, is entitled an equal share for his significant contributions to the party's efforts). That's 3,003 points per PC, plus one hundred each added for Rumolt's payment, totaling 3,103. Accounting for prime requisite bonuses, I arrive at:

• Berwyn - 11,169
• Arendeth - 11,653
• Riwyn - 5,827/5,827
• Wren - 5,297/5,827
• Lincoln - 9,396

These are definitely substantial gains, and well deserved. Berwyn ascends to level 4, Riwyn to levels 3 and 4 in her classes, and Wren to levels 3 and 4 in hers. Lincoln is close to advancing, and Arendeth is much closer to his next milestone than he was previously. Totals are updated on the right-hand side of the blog.

FR #6: From Brithem to Port Llast

Note: A bit of additional detail regarding Grimmantle is provided below, as I don't think I relayed Mara's findings completely during the game.

4 Mirtul

After its initial victory rush subsided, the fiefdom of Brithem began to pick up the pieces left from the destruction wrought by the dragons. The majority of the party helped where it could, transporting supply carts and fortifying the castle, while Wren sought Mara to learn more about her unattributed, weathered map. Through the tomes in her study, Mara imparted that "Grimmantle" (the name cryptically scrawled upon the parchment) was a kingdom in the eastern Crags founded by humans in 272 DR (curiously, the "Year of the Weeping Kingdom") to guard against orc attacks. The following passage was found pertaining to the year 306 DR (more curiously, the "Year of the Fanged Horde"), three decades later:

The kingdom of Grimmantle in the Mlembryn lands falls to the Thousand Fangs orc horde, which then assails Illusk and Neverwinter. The horde is eventually blunted and scattered by a mercenary army led by Grauth Mharabbath, "the Knight of Many Battles."

Armed with this information, the PCs set their eyes south to Neverwinter, in hopes that the resources of a larger city might reveal more detail still.

As they prepared to leave, they were approached by a blond-haired soldier, notably one of the ballista operators at the northeast farm. The man bowed, giving the name Aranos Dalebriar, and he humbly petitioned the party for employment. Aranos explained that he had first come to Brithem more than a decade earlier with a lone surviving family member, an older brother, who most recently was slain during the dragons' initial assault upon the castle, before the PCs arrived. Asking for nothing more than food, shelter, and the opportunity to learn (the party declined to offer a wage), an agreement was made.

The morning of their departure, Lady Deidre met the heroes inside the castle gates, gifting them a small, leather pouch containing a polished gray stone with inclusions of deep crimson: "bloodstone," she declared it, and an amount of which valued upwards of one thousand gold pieces, this far west.

Exchanging well wishes, the party took to the rolling trails leading back to the crossroads, which they reached before sunset that same day. No campsites were occupied, at present, though evidence showed that some were recent; fearing Whisper's bandits, they continued south along the High Road for an hour before sleeping, fireless and with watches kept. The next three days passed uneventfully, though Arendeth took the opportunity to converse frequently with Aranos and impress upon the warrior his values. Aranos spoke of having been raised in a small village far south of Neverwinter, many years ago; his father was a stonecutter.

On the fourth night of their journey, rain fell long into the evening, and the party was forced to scavenge for usable wood and set a fire for warmth with the aid of lamp oil and a casting of produce flame. The first watch heard skittering sounds emanating from the underbrush, though several minutes later the noises passed. The following day was dry again; late in the morning, a giant hawk attacked the party from behind, assailing Riwyn (possibly due to the presence of Merlin the owl on her person) multiple times before the bird could be driven off by volleys of arrows and a light spell directed at its eyes.

A few hours later, they crested a hill looking down upon a walled town, and arrived at the gates of Port Llast in the waning afternoon of the Ninth day of Mirtul, 1354 DR (the Year of the Bow).

On Hirelings and Henchmen

Aranos is formally a hireling (albeit an unpaid one, currently). The party's newfound prestige and the ex-soldier's situation brought the sides together, come what may. His actions will be mainly decided by the characters (within reason), and his morale and loyalty influenced heavily by the treatment he receives. This is far from a new concept in AD&D, though it's not one we've employed much in the past. I want to state clearly that Aranos is not intended as a "DMPC" and neither does he consume a share of the party's awarded experience.

Looking forward, depending on how their mutual relationship blossoms (or wilts), there may be an opportunity for Aranos to become something more. Henchmen are a resource we've dealt with even less over the years we've played, but an ally taking on such a status could prove a valuable asset, indeed.

On the Experience System

As posted earlier, I did record damage dealt and received according to the Tao-of-Dnd experience system (my spreadsheet worked quite well). This is only to serve as a proof of concept for the time being, and while I see a lot of merit in the approach, I think certain questions are likely to arise. In the case of the light spell cast by Arendeth against the hawk, that seems to fall under the paragraph quoted below:

"Finally, there are various spells that do not provide an x.p. bonus, though they affect the combatant's attack or defense. Partly this is because of the difficulties in tracking the damage done, but largely it must be understood that casting a spell is very often not a risky action for a spellcaster. This is especially true of spells that can be cast well ahead of a battle, such as bless, armor, phantom armor, barkskin or stoneskin, etc."

(I ended up having a quick email exchange with Alexis to confirm his intentions for spells like light. Casters can take solace in the fact that blinding an opponent still makes it easier to hit, which should lead to greater and easier XP for everyone in the party.)

Again, I'm only dabbling with this system for now; I'll make a formal announcement if I decide to move to it permanently.

A Final Word

I always try to be attentive to the players' interest levels and the varying degrees of distress or frustration around the table when we play. While joking and kidding around during D&D games are inevitable (I'll be the first to admit that I regularly partake), I want to make sure everyone understands that I put a good deal of work into preparing for (and recapping!) our sessions, and that players put a lot of thought and effort into creating and developing their characters, both in and out of game. I think we all need to remember that lines are easy to cross when having alcohol-supported fun, and make sure that our enjoyment never comes at the expense of others' or detracts from the game itself. I've seen some of that over the past couple sessions and kindly ask everyone to please keep these points in mind. I'll work to help set the right example as well.