D&D.Sci Evaluation and Ruleset

by abstractapplic4 min read12th Dec 202010 comments

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D&D.SciWorld ModelingRationality
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This is a followup to the D&D.Sci post I made a week ago; if you haven’t already read it, you should do so now before spoiling yourself.

Here is the web interactive I built to let you test your solution; below is a complete explanation of the rules used to generate the dataset. You’ll probably want to test your answer before reading any further.

(Unless you’d prefer to enter your answer, then read through the rules with a slowly growing sense of vindication and/or concern before clicking “Begin Quest!”; or unless you’d like to completely ignore the spirit of the exercise and use the rules to improve your solution. I’m not the boss of you.)

Ruleset

Generation and Selection

An applicant for Adventurer College gets their stats by rolling two ten-sided dice for each stat and then adding them.

Not everyone who applies gets in. Only those with >60 total points are allowed entry; you, with your 62 points, barely made the cut. Of course, you only get the records for those who were accepted; this produces small but non-negligible selection effects.

Advantage and Disadvantage

On graduation, an Adventurer will inevitably find themselves in a party of 3-6 loyal friends who help each other with their Great Quests. Your chances of success are determined by how well you can contribute to a team like that (your Advantage), and how much you get in their way (your Disadvantage). These quantities are calculated as follows:

  • Everyone starts with 1 Advantage and 1 Disadvantage.
  • You can ignore the Dexterity score; in this world, it’s useless for everything Adventurers do except qualifying for Adventurer College.
  • Having unusually high stats allows you to help take skill checks for your team. For every non-Dexterity skill you have above 12 points, add 2 Advantage for every point it exceeds 12 by. (For example, if you had 15 Charisma, that would give you (15-12)*2=6 extra Advantage.)
  • Having unusually low stats means you cause trouble for your team. For every non-Dexterity skill you have below 8 points, add 3 Disadvantage for every point under 8. (For example, if you had 5 Charisma, that would give you (8-5)*3=9 extra Disadvantage.)
  • If your Strength is greater than your Constitution, you’ll end up committed to physical tasks you don’t have the stamina to see through: that’s +2 Disadvantage.
  • For similar reasons, if your Intelligence is greater than your Wisdom, that’s also +2 Disadvantage.
  • Being able to out-talk the local Vizier (who has Charisma 16) is very useful. Adventurers with >16 Charisma get an additional +5 Advantage.

Success and Failure

Your odds of success are the ratio between your Advantage and your Disadvantage. That is:

P(Success) = Advantage / (Advantage + Disadvantage)

As Advantage and Disadvantage are both always >0, there is always a chance of success or failure.

Strategy

With the above in mind, the optimal strategy given your starting position is as follows:

  • Get STR and CHA to 8.
  • Get WIS above INT.
  • Put your remaining skill points into WIS and/or CON and/or INT, while keeping WIS above INT.

Closing Thoughts

You may have several objections to this scenario. Relevant selection effects were only vaguely alluded to, and the dataset contains phenomena – the discontinuity at CHA>16, and the STR-CON interaction – irrelevant to your situation. To this, I can only plead realism: most datasets in the real world are much messier, have much more dubious relevance to your goals, and contain distortions about which the GM provides no hints at all.

You may also object to the use of random elements in scoring. Even with perfect allocation, you can’t get above a 93.75% chance of success: it is not only possible, but plausible, to do everything right and still lose. Meanwhile, refusing the fairy’s offer leaves you a 25% chance of success, deliberately allocating points badly leaves you about one chance in three, and most random allocations still give better-than-even odds. I plead realism here too, but I can see why it might bother some people; as a compromise, I provide probability-of-success alongside success/failure, in case you’d prefer to keep score that way.

Finally, you may object to the ways that the challenge was unfair in your favour. If you didn’t account for selection effects, you may have correctly avoided boosting DEX because you thought it was actively harmful instead of merely useless. If you didn’t look for interactions, you may have dodged the WIS<INT penalty just because WIS seemed like a better place to put points than INT. And I’m pretty sure even the three people who submitted optimal answers on the last post (good job simon, seed, and Ericf) didn’t find them by using the right link function, just because the linkage I set up between predictors and response was so arbitrary and idiosyncratic.

Here, I not only excuse but congratulate myself. The main benefit of exercises like this over Kaggle analyses – aside from Fate’s unwillingness to show up the following week and explain the algorithm it used to choose who would survive the sinking of the Titanic – is that making real-fake-world decisions based on real-fake-world data trains the ability to make mistakes that don't hurt you.

(If you have any other objections, please let me know. I very much want feedback so I can make the next challenge better.)

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10 comments, sorted by Highlighting new comments since Today at 11:58 PM
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Excellent job on this. I thoroughly approve of basically all of your choices. I would have loved if there was a dex trap, too. Something like you're offered two instances of "pick either 5 stats distributed as you wish, or +6 to one stat and -1 to three other stats". Not sure if this particular construction allows for a good dex trap, though, especially because making it possible draws attention to it.

Seconded. Very happy with the solution and the resulting reflection on my attempt.

Given the complexity of the ruleset my model was entirely too simple. I assumed that higher stats are always better, and that the probability of success was proportional to #success|stat / #entries|stat (with an additional pseudo count each for success and failure to reduce overfitting).

Nevertheless, I got lucky in that I happened to end up in the region of optimal solutions

STR: 8/20 
CON: 15/20
DEX: 13/20
INT: 13/20
WIS: 15/20
CHA: 8/20

The result of my adventure was

You fail to complete your Great Quest.

(you had a 93.75% chance of success.)

What I did was use Excel to make 9 graphs: "% successful" vs. each stat individually, "highest stat," "lowest stat," and "total stats" under the assumption that there would be some sort of mapping of stats -> success, but that there wouldn't be any dependencies between stats -whoops-!

The discontinuities at 8 STR & CHA were visually obvious, as was the negative slope of Dex... I actually inverted everyone's DEX (trueDEX = 22 - givenDEX] for the highest/lowest/total graphs. The data also showed big gains in the 12-16 range for CON and WIS, so that's where I put the extra points.

This was cool! Definitely looking forward to the next challenge and using what I learned from this.

I got 70% chance of success, and succeeded with these stats and strategy.

thanks for making this, it was a lot of fun! Would love to have a go at your next challenges :)

Well, I think that the Neural Net and Decision Forest I used in the last post both saw pretty much what you were going for; with the exception that they both put one too many points into CHA, bumping it up to 9, instead of into WIS.

All in all, a success for throwing lots of data into an ML model you don't fully understand and walking away... except that I had two other models which performed abysmally. 

If you didn’t account for selection effects, you may have correctly avoided boosting DEX because you thought it was actively harmful instead of merely useless. 

I immediately considered a selection effect, but then I tricked myself into believing it did matter by a method that corrected for the selection effect but was vulnerable to randomness/falsely seeing patterns. Oops. Specifically I found the average dex for successful and failed adventurers for each total non-dex stat value, but had them listed in an inconvenient big column with lots of gaps. I looked at some differences and it seemed that for middle values of non-dex stats, successful adventurers consistently had lower average dex than failed ones, while that reversed for extreme values. When I (now - I didn't at the time) make a bar chart out of the data it's a lot more clear that there's no good evidence for any effect of dex on success: 

 

If you didn’t look for interactions, you may have dodged the WIS<INT penalty just because WIS seemed like a better place to put points than INT. 

Yep. Thing is, I *did* look for interactions - with DEX. I had the idea that DEX might be bad due to such interactions, and when I didn't find anything more or less stopped looking for such interactions.

And I’m pretty sure even the three people who submitted optimal answers on the last post (good job simon, seed, and Ericf) didn’t find them by using the right link function

For sure in my case. I calculated the success/fail ratios for each value of each stat individually (no smoothing), and found the reachable stat combo that maximized the product of those ratios. This method found the importance of reaching 8. I was never confident that this wasn't random, though.

When I did later start simming guesses what I simmed would have given smoothed results: a bunch of stat checks with a D20, success if total number of passed stat checks  greater than a threshold. The actual test would have been pretty far down in the list of things I would have checked given infinite time.

Cheers to simon, ericf and myself, for offering an optimal solution! And cheers to abstractapplic for organizing the challenge.

The leaderboard (if you're not here, I couldn't figure out what your final decision was, or you added more than 10 points):

simon, ericf 0.9375 

[('CHA', 8), ('CON', 15), ('DEX', 13), ('INT', 13), ('STR', 8), ('WIS', 15)]

seed 0.9375 

[('CHA', 8), ('CON', 14), ('DEX', 13), ('INT', 13), ('STR', 8), ('WIS', 16)]

Samuel Clamons 0.8095 

[('CHA', 8), ('CON', 17), ('DEX', 13), ('INT', 13), ('STR', 7), ('WIS', 14)]

Asgard 0.7857 

[('CHA', 9), ('CON', 16), ('DEX', 14), ('INT', 13), ('STR', 8), ('WIS', 12)]

Measure 0.7308 

[('CHA', 8), ('CON', 14), ('DEX', 13), ('INT', 13), ('STR', 6), ('WIS', 18)]

kiwiakos 0.6774 

[('CHA', 7), ('CON', 15), ('DEX', 13), ('INT', 13), ('STR', 6), ('WIS', 18)]

Alexey 0.6500

  [('CHA', 11), ('CON', 14), ('DEX', 13), ('INT', 13), ('STR', 6), ('WIS', 15)]

newcom 0.6471

  [('CHA', 11), ('CON', 16), ('DEX', 13), ('INT', 13), ('STR', 7), ('WIS', 12)]

AABoyles, Pongo, GuySrinivasan 0.6389 

[('CHA', 6), ('CON', 14), ('DEX', 13), ('INT', 13), ('STR', 6), ('WIS', 20)]

Yongee 0.6364 

[('CHA', 5), ('CON', 14), ('DEX', 13), ('INT', 20), ('STR', 8), ('WIS', 12)]

Deccludor 0.6098 [('CHA', 5), ('CON', 20), ('DEX', 13), ('INT', 13), ('STR', 6), ('WIS', 15)]

Randomini 0.4688 [('CHA', 4), ('CON', 14), ('DEX', 13), ('INT', 13), ('STR', 16), ('WIS', 12)] 
 

From plotting the data, I saw that:

  • looked like the stats were independtly randomly generated and cut off at sum >= 60.
  • dexterity was useless
  • looked like there was a big advantage to having stats >= 8. For strengh, strength=8 was almost as good as strength=20

I fit a regularized logistic regression and a neural net, but couldn't get validation accuracy greater than 70%, which was only a little better than the 65% baseline of random guessing. I realized that the data is not very informative and I don't know how results are calculated, so I better stick with a conservative model like Nearest Neighbors classifier, and try a few different models. I fit a KNN classifier, gradient boosting on decision trees, and regularized logistic regression (all with validation accuracy 70%), and chose a point which gave near the top scores for all three classifiers. (It had all stats >=8, too.)

I failed, but at (WIS +6, CHA +4) I had a 73.07% chance for success, which I think is at least a local maximum.

Actually adding even 1 point to STR is an improvement, though not as much as STR +2.