# D&D.Sci Coliseum: Arena of Data Evaluation and Ruleset

This is a follow-up to [last week's D&D.Sci scenario](https://www.lesswrong.com/posts/WjGcbR3RcRfGtmFfJ/d-and-d-sci-coliseum-arena-of-data): if you intend to play that, and haven't done so yet, you should do so now before spoiling yourself.

There is a web interactive [here](https://aphyer1992.github.io/dndsci_coliseum) you can use to test your answer, and generation code available [here](https://github.com/aphyer1992/dndsci_arena/blob/main/dnd_gladiator.py) if you're interested, or you can read on for the ruleset and scores.

RULESET
=======

A character's class/race/level/items boil down to giving them scores in two separate stats: **Speed** and **Power**.

Stat Calculation
----------------

Each parameter of a character has an effect on their stats:

*   **Class:** Your class provides base stats as follows:
    *   Knight: 2 Speed/12 Power
    *   Warrior: 4 Speed/10 Power
    *   Ranger: 6 Speed/8 Power
    *   Monk: 8 Speed/6 Power
    *   Fencer: 10 Speed/4 Power
    *   Ninja: 12 Speed/2 Power
*   **Race**: Your race modifies stats as follows:
    *   Dwarf: -3 Speed, +3 Power
    *   Human: No Effect
    *   Elf: +3 Speed, -3 Power
*   **Level**: Each level provides +1 Speed and +1 Power.
*   **Items**: Boots of Speed +X grant +X Speed, Gauntlets of Power +Y grant +Y Power.

So, for example, the following characters all have Speed 10 and Power 10:

*   A Level 3 Elf Warrior (Speed 4 + 3 + 3, Power 10 - 3 + 3)
*   A Level 3 Dwarf Fencer (Speed 10 - 3 + 3, Power 4 + 3 + 3)
*   A Level 2 Human Ranger with Boots of Speed +2 (Speed 6 + 2 + 2, Power 8 + 2)
*   A Level 2 Human Monk with Gauntlets of Power +2 (Speed 8 + 2, Power 6 + 2 + 2)
*   A Level 1 Elf Knight with Boots of Speed +4 (Speed 2 + 3 + 1 + 4, Power 12 - 3 + 1)
*   A Level 1 Dwarf Ninja with Gauntlets of Power +4 (Speed 12 - 3 + 1, Power 2 + 3 + 1 + 4)

These characters are, from a mechanical perspective, all completely identical.  They will behave the same in every possible matchup.

Particular congratulations are due here to **simon**, who managed to figure out this whole mapping within 24 hours (actually posting an essentially-complete solution before the scenario even made it to the frontpage).

Combat
------

When two characters fight, each one calculates a Combat Score:

*   In the brutal one-on-ones of Arena combat, which can end within seconds, being *just a bit* *faster* than your opponent is vital.  Whoever's Speed is higher wins initiative and gets the first strike, which counts for +8 to Combat Score.
*   Power is also very useful, but it doesn't have a sharp cutoff the way Speed does.  Each character adds their Power to their Combat Score.
*   Each player rolls 1d6 and adds that to their Combat Score.[^p49aop2cgyp]
*   Whoever ends up higher wins.  In the case of a tie, the winner is decided at random.

So, taking these three characters:

*   P has Speed 15 and Power 5.
*   Q has Speed 10 and Power 10.
*   R has Speed 5 and Power 15.

and considering the matchups between them:

*   If P fights Q, Q has 5 more Power, but P is faster and gets a +8 bonus.  P is therefore 3 points of Combat Score ahead, and will likely win unless Q rolls very well.
*   Similarly, if Q fights R, Q ends up 3 points of Combat Score ahead.
*   But if R fights P, R has 10 more power, and P's Speed still only gives a +8 bonus.  This time R is 2 points of Combat Score ahead, and will probably win.

This means that there are in general two good strategies:

*   Being **a little bit faster** than your opponent (to get the +8 bonus), and then staying close enough in Power that this bonus dominates.
*   Being **much slower** than your opponent (allowing them the +8 bonus), but trying to beat them on Power by enough that you're ahead anyway.

STRATEGY
========

The four House Champions had the following stats (including their items):

*   House Adelon: 13 Speed/17 Power
*   House Bauchard: 11 Speed/20 Power
*   House Cadagal: 24 Speed/9 Power
*   House Deepwrack: 14 Speed/17 Power

Your six characters had the following stats (before giving them items):

*   Uzben Grimblade: 14 Speed/10 Power
*   Varina Dourstone, 6 Speed/18 Power.
*   Willow Brown: 11 Speed/13 Power.
*   Xerxes III of Calantha: 13 Speed/11 Power.
*   Yalathinel Leafstrider: 18 Speed/6 Power.
*   Zelaya Sunwalker: 11 Speed/15 Power.

First, you want to decide your Speed matchups:

*   You cannot outspeed House Cadagal's Ninja.  All you can do is send your highest-Power character, Varina.
*   The other three Champions you can outspeed, but you want to do that with the slowest characters you can to save points for Power.  To do that, you send:
    *   Willow, with at least +3 Boots, against House Adelon.
    *   Zelaya, with at least +1 Boots, against House Bauchard.[^c8rh8u95xfq]
    *   Xerxes, with at least +2 Boots, against House Deepwrack.
*   You don't want to use Uzben or Yalathinel (your faster characters) in the optimal solution: since you can outspeed all three of these Champions without them, all swapping them in does is waste Power.
*   You also don't actually need to use the +4 Boots of Speed: they provide some wiggle room in your solution (with them you could e.g. send Willow against House Deepwrack), but don't actually allow you to send any more powerful characters/outspeed anyone you couldn't already.  (This is suspiciously fortunate for you, see the Bonus Objective below).

Then you want to allocate your Power boosts optimally:

*   Before you distribute any of your Gauntlets:
    *   Willow against House Adelon has 13 Power vs 17 (4 lower), but is faster (+8), for a net of +4 Combat Score.
    *   Zelaya against House Bauchard has 15 Power vs 20 (5 lower), but is faster (+8), for a net of +3 Combat Score.
    *   Varina against House Cadagal has 18 Power vs 9 (9 higher), but is slower (-8), for a net of +1 Combat Score.
    *   Xerxes against House Deepwrack has 11 Power vs 17 (6 lower), but is faster (+8), for a net of +2 Combat Score.
*   Since Power boosts have more effect the closer to even the fight is (e.g. going from +1 to +2 is more important than going from +4 to +5), you want to give out your Gauntlets to even this up:
    *   Willow gets no Gauntlets
    *   Zelaya gets +1 Gauntlets
    *   Varina gets +3 Gauntlets
    *   Xerxes gets +2 Gauntlets
*   This brings you to +4 Combat Score against every Champion.

BONUS OBJECTIVE
===============

If you noticed the right things in the dataset, you would find that:

*   Level 1-6 characters are all very common.  There are a variety of characters of every race/class/level combination up to Level 6.
*   Similarly, items up to +3 are very common.  You can find +1 through +3 items used everywhere.
*   Level 7 characters are *not* common.  There are only four different combinations in the dataset: Elf Knight, Human Warrior, Dwarf Monk,[^vvkt5kt1nna] and Elf Ninja.
*   Similarly, +4 items are *not* common.  The only place +4 Gauntlets show up in the dataset is on the Level 7 Dwarf Monk(s).  And the only place +4 Boots show up is on the Level 7 Elf Ninja.
*   Well, +4 Boots **used to** show up on the Level 7 Elf Ninja.  About 97% of the way through the dataset, the Level 7 Elf Ninja downgraded to +2 Boots for some reason.
*   Your set of dubiously-acquired magical items includes +4 Boots.[^x5pbrv237nj]

The Bonus Objective was to realize that your +4 Boots in fact belong to House Cadagal, and that using them is a bad idea:

> If you use the +4 Boots of Speed, the Lady Cadagal will believe that you were responsible for their recent theft from her House. 
> 
> You will earn her lasting enmity, and you'll lose standing with her even if you win. (Counts as -1 win).
> 
> By contrast, if you show your honor by returning them to her, you will win her friendship even if you lose in the Arena. (Counts as +1 win).

Congratulations to everyone who figured this out, particularly to **abstractapplic**, who was the first person to get a reasonably-complete explanation of what had happened.

LEADERBOARD
===========

| **Player** | **vs A (Combat Diff)** | **vs B (Combat Diff)** | **vs C (Combat Diff)** | **vs D (Combat Diff)** | **Expected Wins** |
| --- | --- | --- | --- | --- | --- |
| simon | Willow w/+3/0 (+4) | Zelaya w/+1/1 (+4) | Varina w/+0/3 (+4) | Xerxes w/+2/2 (+4) | 3.778 |
| **Optimal Play** | **Willow w/+3/0 (+4)** | **Zelaya w/+1/1 (+4)** | **Varina w/+0/3 (+4)** | **Xerxes w/+2/2 (+4)** | **3.778** |
| Yonge | Uzben w/+1/2 (+3) | Zelaya w/+3/1 (+4) | Varina w/+2/3 (+4) | Xerxes w/+4/0 (+2) | 3.542 |
| abstractapplic | Uzben w/+3/0 (+1) | Xerxes w/+2/1 (+0) | Varina w/+1/3 (+4) | Yalathinel w/+0/2 (-1) | 2.444 |
| SarahSrinivasan | Zelaya w/+3/1 (+7) | Yalathinel w/+1/2 (-4) | Xerxes w/+2/3 (-3) | Willow w/ +4/0 (+4) | 2.125 |
| **Random Play (with +4 boots)** | ?   | ?   | ?   | ?   | **1.710** |
| **Random Play (without +4 boots)** | ?   | ?   | ?   | ?   | **1.371** |
| Lorxus | Willow w/+2/1 (-3) | Varina w/+4/3 (-7) | Xerxes w/+1/2 (-4) | Yalathinel w/+3/0 (-3) | 0.306 |

Congratulations to all players, particularly to **simon**, whose perfect figuring-out of mechanics led to a well-deserved perfect score!

DATASET GENERATION
==================

The bulk of fights in the dataset were from the Arena's regular tournaments, where 64 competitors take part in a single-elimination tournament.  The initial competitors are biased towards low levels, but by later rounds it's usually higher-level characters who remain.

These tournaments are interspersed with occasional duels (used as part of the charming local system of governance to resolve various issues), which are between relatively high-level characters.  Duels occur a bit more than twice as often as tournaments, but since each duel is only a single fight they're a very small minority of fights in the data.

Congratulations to **SarahSrinivasan** for identifying this!

REFLECTIONS & FEEDBACK REQUEST
==============================

The goal I was shooting for with this scenario was to have the simplest underlying system I could manage that still led to interesting emergent behavior.

I felt fairly happy with how this went!  There were a lot of sneaky interactions in the data - e.g. as **simon**'s analysis said:

> For example, a same-class Elf will tend to beat a same-class Dwarf. And a same-race Fencer will tend to beat a same-race Warrior. But if an Elf Fencer faces a Dwarf Warrior, the Dwarf Warrior will most likely win. Another example with Fencers and Warriors: same-class Elves tend to beat Humans - but not only will a Human Warrior tend to beat an Elf Fencer, but also a Human Fencer will tend to beat an Elf Warrior by a larger ratio than for a same-race Fencer/Warrior matchup???

but once analyzed in detail it was possible to disentangle what led to this.

How did this feel from a player perspective?

I'm also curious how the Bonus Objective felt from a player perspective: did people like it?  Did it seem fair?  Should I clue players to the existence of such an objective in future?

As usual, I'm also interested to hear any other feedback on what people thought of this scenario.  If you played it, what did you like and what did you not like?  If you might have played it but decided not to, what drove you away?  What would you like to see more of/less of in future?  Do you think the scenario was more complicated than you would have liked?  Or too simple to have anything interesting/realistic to uncover?  Or both at once?  Did you like/dislike the story/fluff/theme parts?  What complexity/quality scores should I give this scenario in the index?

I'm currently planning to post another scenario in about a month: I would shoot for Nov 29th, except that's Thanksgiving weekend so I might end up going a bit earlier/later.  Let me know if some times work well/poorly for you!

[^p49aop2cgyp]: Entertainingly, these 1d6 rolls were the only part of the ruleset that simon did not deduce. 

[^c8rh8u95xfq]: You could swap Willow and Zelaya here, but it will make your Power allocation worse in the next step. 

[^vvkt5kt1nna]: There was supposed to be only one of each of these, but due to a bug there were two copies of the Dwarf Monk starting about halfway through the dataset.  simon noticed this was a bug, but I think I'm going to go with abstractapplic's interpretation of there being two Dwarf Monks.  Perhaps he was duplicated by the Mirror Mage. 

[^x5pbrv237nj]: Did you steal them directly from House Cadagal?  Or did you take them from a Thieves' Guild that stole them from House Cadagal?  That's really up to you: the Goddess summons Heroes with a wide variety of moralities.  But House Cadagal is not likely to be amused either way.