Sunday, September 23, 2012

Some Numbers, for the Playtest Crowd

I never know how much to post in math threads, so if this is over-simplified, please forgive me.

To grasp the problem of "difficulty" in the playtest, I ran 20,000 first rounds with a total of 10 different monster conditions. To make this a "fair" test, the five player characters (none with a score above 18) used the same attacks in each one (Warhammer, Great Axe, Magic Missile, Radiant Lance, Sling) and I made the assumption for that there were an equal number of monsters and player characters (in 16,000 rounds) and there would be no possibility of a second hit on the same monster, and that a monster with multiple attacks only attacked once. I picked the monster's strongest non-magical attack where there was more than one. I tried to select monsters with the same combat style, so that I wasn't comparing something that drops poison with something that punches you. I also only ran each condition only 2000 times with each monster, which is good for getting the general 'shape' of the data. Someone w/better software can probably run it tens of thousands of times...which would get way better data. I would assume most percentages in this are +/- about 5%. Note that simulation doesn't generate the best shaped data, but is good for giving general concepts. I used excel to track the simulations. (I know, most people know my view of excel.)

To reiterate, this was a grossly simplified combat scenario. It only covered the first round, and had conditions in which the characters and the bad guys could only hit once, each. I was interested in four questions:

1. Is there ever a time where the risk of one-hit kill in the first round is equal for the players and the monsters?

2. What percentage of the time does an attack miss in different scenarios?

3. What kind of difference is the advantage making (when that is used.)

4. Does the xp-based difficulty mean a damn thing?

The 10 different monster scenarios our 1st level characters faced were:
5 fire beetles, 5 bug bears using melee weapon, 5 bug bears using ranged, 5 dark acolytes, 5 dark adepts, 5 dark priests, 5 drow (attacking once only), 5 ogres, 2 bug bears wielding ranged weapons and 2 bug bears wielding melee weapons. These last two conditions were suggested by the playtest module, in which two bug bears being sent after the party at a low level is repeatedly used as a threat.

For a graphical representation of the relative difficulty of these conditions, the chart below shows the difficulty (based on xp budget) of the 10 conditions, along with bars for the easy, average, and tough xp budgets. I also added a fourth difficulty level, supra, which is just Tough+Easy. In theory, anything above a supra difficulty should be an instant wipe out for the party, unless they run like hell. Taking the difficulty values at face value, something far above supra should be impossible to hit and should kill the party fairly easily.

Based on these numbers, you would think that the last three of these attacks would be the ones in which the party was unable to kill the bad guys and in which the party could, conceivably be killed instantly, squashed into pieces. It turns out it is almost impossible for even the ogre to kill any member of the party in one hit, coming in at only killing a party member 5.5% of the time (about the chance of rolling a natural 20 on a d20).

In fact, other than the ogre, the only chance for a single PC kill comes from the bugbears, when wielding melee weapons. These creatures, so feared in the module presented with the play test, only kill a PC 1.6% of the time per round, and can never do enough damage to kill the two players with the highest hitpoints in one attack. The party can never one-hit kill the Drow, the Dark Priests, or the ogres, but these creatures can be killed with two hits (Drow, Dark Priests) or three hits (Ogre.)

For a comparison of the party's chance of being wiped out in the first attack when facing an enemy that should smush them into oblivion instantly versus the chance of wiping out said monsters, here's another chart...showing both the chances when the party uses Advantage and when the party does not.
Here's the data for the chart.

Monsters % monsters killed (without Advantage) % Monsters killed (with Advantage) % party killed
Fire Beetle (5) 57.42 81.6 0
Bug Bear (5)-Melee 0.9 1.24 1.26
Bug Bear (2)-Melee 8.388888889 8.722222222 0.166666667
Bug Bear  (2)-Ranged 8.222222222 8.555555556 0
Bug Bear (5)-Ranged 1.02 1.3 0
Dark Acolyte (5)-Mace 24.3 34.64 0
Dark Adept (5)-Mace 7.26 9.7 0
Dark Priest (5)-Mace 0 0 0
Drow-(5)-1 attk 0 0 0
Ogre (5)-greatclub 0 0 5.5

And here's your chart:

Note that you can extrapolate these numbers to figure out how many rounds it would take to kill each of these creatures, based on how often the party misses (or hits) in the 20,000 first rounds (10,000 with the party having advantage, and 10,000 with the advantage dice being thrown out altogether.) I suspect the differences between bugbear numbers are within the margin of error, by the way)

So, how frequently does the party miss?
The problem with the frequency of the party's misses is that it varies based on the character. Magic Missile, for example, always hits. This is why I chose to run the combat in a "perfect" first round where there are five monsters and 5 PCs, and they magically all attack and damage at the same time, something that could never happen in a game situation (also makes the math REALLY EASY to follow.) The problem is that this magical combat scenario should hamper the party in most scenarios. In a 'real' battle, the party is smarter than the enemy (except, maybe, the drow) and can do things like fight in a doorway (where the ogres could only hit one or two of them, and only one ogre at a time.) So bear in mind before I put the percent of missed attacks up that most parties are going to have even a better chance of hitting than given here...and that should disturb you.

So, in 10,000 first rounds, against monsters ranging in difficulty from a cakewalk to a monster that should destroy the party, how frequently did the party miss? Except for that magic missile, they should miss a lot, right? (Note these numbers are not the same as the "didn't kill instantly" numbers, because an instant kill and a hit w/out a kill is still a hit)

Monster #times players missed w/out advantage # Times players Missed with advantage
Fire Beetle (5) 2050 556
Bug Bear (5)-Melee 2230 539
Bug Bear (2)-Melee 2155 530
Bug Bear  (2)-Ranged 2280 575
Bug Bear (5)-Ranged 2270 506
Dark Acolyte (5)-Mace 2425 694
Dark Adept (5)-Mace 2480 655
Dark Priest (5)-Mace 2915 1033
Drow-(5)-1 attk 2460 681
Ogre (5)-greatclub 2475 703

Now, I should add a caveat.  These aren't how many times the party missed on an attack out of 10,000. Each of those conditions represent 5000 attacks. To make the data clearer, let's ignore the ones where the party is not facing an equal number of monsters to the party. Since everyone is attacking simultaneously in our impossible melee, you end up with things like the same bugbear being fatally killed twice at the same time. So, let's ignore those altogether, and convert them into percentages for ease. .

Monster % missed w/out advantage          % missed with advantage
Fire Beetle (5) 41 11.12
Bug Bear (5)-Melee 44.6 10.78
Bug Bear (5)-Ranged 45.4 10.12
Dark Acolyte (5)-Mace 48.5 13.88
Dark Adept (5)-Mace 49.6 13.1
Dark Priest (5)-Mace 58.3 20.66
Drow-(5)-1 attk 49.2 13.62
Ogre (5)-greatclub 49.5 14.06

Or, put graphically:
People better than I have discussed the problems with the advantage dice. The answers to my 4 questions, based on this LIMITED example of first round of combat only, are as follows. Again, bear in mind that I just did this with simulations, so (for example) for a party member to die, the monster needed to both roll a hit AND roll a random damage die that was higher than the PC's :

1. Is there ever a time where the risk of one-hit kill in the first round is equal for the players and the monsters?
Not in the materials currently provided. In the worst case scenario, in which a party somehow encounters an equal number of monsters that should be at least three times the difficulty of the party, the party only has around a chance of a 1 on a 1d20 to lose a member in each round, and are three times more likely to hit the ogres than they are to die from the ogres. Such a party could win against an equal number of ogres by concentrating their attacks on one or two ogres at a time, which would lower the chance of an instant kill each round that they eliminated an ogre. This, the hardest possible scenario using the hardest possible monster, and even avoiding spells like Sleep and Command, is a fairly easy to win scenario, and based on the XP "value" of the difficulty of the encounter should be all but impossible.Your characters are basically never in danger. Running my party though the playtest module, they never got above a 2.5% chance of dying.-with an encounter that they should have higher levels for.

2. What percentage of the time does an attack miss in different scenarios?
Some attacks never miss. When those attacks can be constantly repeated, a party with fast feet can easily win any encounter. But based on nothing but simulation, a character in an "impossible" scenario can hit slightly less than half the time, and in an easy scenario can hit slightly more than half the time. Unless he has advantage dice in play, in which case he hits around 85% of the time in the worst case scenario the game allows.

3. What kind of difference is the advantage making (when that is used.)
It reduces the risk to the party down to negligible. When I got the chance of a beastie killing a member of the party at 5.5%, that was with the ogre not having any advantage or disadvantage. If he's at disadvantage, it's less than half a percentage point. Any time a chance is so low you can't get there by percentiles it bugs me as a DM.

4. Does the xp-based difficulty mean a damn thing?
In simulation? Nope, not a damn thing. The party is at very little risk, regardless of the monsters chosen. I think that's what bores me so much.

I know some people have said that the new generation of gamers isn't comfortable with characters dying, but I find the no-risk to the characters thing to be really boring. As a DM, I am a cupcake. It's almost impossible to die permanently in one of my campaigns....and I find these numbers troubling.

Different things can matter differently to different people. I tend to focus on the roleplaying, not the rolling, but if the chances of risk to the characters is so low, let's all play a diceless game instead.

It's completely possible there are data errors in my simulation. Running stuff by hand my numbers are even worse. It seems to me the simulation method actually is too parsimonious and gives too much benefit of the doubt. All I know is that the second I ran numbers I felt like it backed up what I saw play testing-too little risk makes too great a bored. :)

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