# D&D.Sci Tax Day: Adventurers and Assessments Evaluation & Ruleset

This is a follow-up to [last week's D&D.Sci scenario](https://www.lesswrong.com/posts/N7ZgcGvG3kfiBryom/d-and-d-sci-tax-day-adventurers-and-assessments): 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_tax) you can use to test your answer, and generation code available [here](https://raw.githubusercontent.com/aphyer1992/dndsci_taxes/refs/heads/main/dndsci_taxes.py) if you're interested, or you can read on for the ruleset and scores.

RULESET
=======

Goods
-----

Each good is assigned a value for tax purposes:

| **Good** | **Value** |
| --- | --- |
| Cockatrice Eye | 6gp |
| Dragon Head | 14gp |
| Lich Skull | 10gp |
| Unicorn Horn | 7gp |
| Zombie Arm | 2gp |

Tax Brackets
------------

Depending on the total value of all goods you have, you determine a tax bracket:

| **Total Value** | **Tax Rate** |
| --- | --- |
| <30gp | 20% |
| 30-59gp | 30% |
| 60-99gp | 40% |
| 100-299gp | 50% |
| 300gp+[^xraricyti2k] | 60% |

Your taxes due are equal to your Tax Rate multiplied by the total value of your goods.

So if you have two Lich Skulls (20gp), your tax rate is 20% and you will owe 4gp of taxes.

If you have three Lich Skulls (30gp), your tax rate goes up to 30% on the **whole value** of your goods, and you owe 9gp of taxes.

It is therefore very valuable to fall just below the various tax-bracket thresholds.

Benefits and Exemptions
-----------------------

There are a variety of special rules:

*   Clerics and Paladins are very popular for their undead-hunting, and the Church is very powerful and well-connected, leading to broad support for a Clerical tax exemption.  However, the Tax Assessment Exactors do not know who is a Cleric and who is not.  Instead, **anyone who only has Lich Skulls and Zombie Arms** is treated as a Cleric, and receives a **special 10% tax rate** regardless of the value of their goods.
*   The Unicorn Ranching Association has very good courtiers.  **Anyone who has at least 5 Unicorn Horns** is classified as a ranch, and their tax rate is **cut to 25%.**[^emy4avaysgb]
*   Cockatrices are a pest, and there is a bounty on killing them, implemented as a tax rebate.  The bounty is **6gp per cockatrice killed**:
    *   If you hand in N eyes, you will receive **N/2 (rounded up) bounties**.  So if two people each hand in 3 eyes, they will each receive 2 bounties, but if one person hands in 6 eyes they will receive only 3 bounties.
    *   This reduces your taxes owed, but **not below zero**.  So if one person hands in 1 Cockatrice Eye and nothing else, they owe only 1.2gp of taxes before the bounty, and the 6gp bounty can only reduce that to 0 and no further.
*   The Royal Society for the Prevention of Cruelty to Dragons is opposed to the killing of dragons, and wants it very heavily taxed.  However, heroes have been able to convince the Tax Assessment Exactors that it is sometimes necessary to kill a dragon menacing a village.  As a compromise, **one Dragon Head** killed for personal reasons is taxed normally, but **any subsequent Dragon Head is taxed at double the usual rate.**
    *   For example, if you hand in 3 Dragon Heads (42gp) and nothing else, you are in the 30% tax bracket.  However, you will pay 30% taxes on the first Dragon Head (4.2gp) but 60% taxes on each of the next two (8.4gp each), totaling 21gp of taxes.

STRATEGY
========

There are three relatively-low tax rates accessible:

*   10% for anyone who can get classified as a Cleric (with only Lich Skulls and Zombie Arms)
*   20% for anyone with <30gp of goods.
*   25% for anyone who can get classified as a Ranch (with 5+ Unicorn Horns).

Optimal play aims to get each adventurer one of those low rates.  There are two ways to accomplish that, which produce nearly-identical end tax bills:

*   With a Cleric (getting a lower tax rate on Lich Skulls and Zombie Arms):
    *   One adventurer brings 5 Lich Skulls and 7 Zombie Arms.[^hr2tgb9j5vr]  They pay a 10% Cleric tax rate on 64gp of goods, for **6gp 4sp** of tax.
    *   One adventurer brings 2 Dragon Heads and nothing else.[^b3agjjm9dmo]  They pay a 20% tax rate on 28gp of goods, but doubled on the second head, for a total of **8gp 4sp** of tax.
    *   One adventurer brings 1 Cockatrice Eye, 1 Dragon Head, 1 Unicorn Horn, and 1 Zombie Arm.[^qnrxs0kcbjd]  They pay a 20% tax rate on 29gp of goods, and receive a 6gp rebate for one confirmed Cockatrice, paying **no tax.**
    *   The last adventurer brings 3 Cockatrice Eyes, 1 Dragon Head, and 6 Unicorn Horns.  They pay a 25% Ranch tax rate on 74gp of goods, and receive a 12gp rebate for two confirmed Cockatrices, paying **6gp 5sp** of tax.
    *   This adds up to **21gp 3sp** of tax.
*   Without a Cleric (getting instead more-efficient handling of Cockatrice Eyes and Dragon Heads):
    *   Each adventurer brings 1 Cockatrice Eye and 1 Dragon Head (getting the maximum Cockatrice bounty and no double-tax on Dragons).
    *   Two adventurers each add 1 Unicorn Horn and 1 Zombie Arm.  They pay a 20% tax rate on 29gp of goods, and receive a 6gp rebate for one confirmed Cockatrice, paying **no tax.**
    *   One adventurer adds 4 Zombie Arms.  They pay a 20% tax rate on 28gp of goods, and receive a 6gp rebate for one confirmed Cockatrice, paying **no tax.**
    *   The last adventurer takes all remaining goods (5 Lich Skulls, 5 Unicorn Horns, and 2 Zombie Arms).  They pay a 25% Ranch tax rate on 109gp of goods, and receive a 6gp rebate for one confirmed Cockatrice, paying **21gp and 2sp** (technically 2-3sp depending on rounding, but it happens that the Tax Assessment Exactors will round down in this case) in tax.[^mb4as75rlo]

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

| **Player** | **1** | **2** | **3** | **4** | **Total Tax** |
| --- | --- | --- | --- | --- | --- |
| simon, abstractapplic | 1C 1D 1U 1Z (0.0) | 1C 1D 1U 1Z (0.0) | 1C 1D 4Z (0.0) | 1C 1D 5L 5U 2Z (21.25) | **21gp 2-3sp** |
| **Optimal Play** | **1C 1D 1U 1Z (0.0)** | **1C 1D 1U 1Z (0.0)** | **1C 1D 4Z (0.0)** | **1C 1D 5L 5U 2Z (21.25)** | **21gp 2-3sp** |
| Yonge | 1C 1D 1U (0.0) | 1C 1D 3Z (0.0) | 1C 1D 3Z (0.0) | 1C 1D 5L 6U 2Z | 23 gp 0 sp |
| DrJones | 5L 1Z (5.2) | 1D 3Z (4.0) | 3C 2D 6U 3Z (15.0) | 1C 1D 1U 1Z (0.0) | 24 gp 2sp |
| MaxwellPeterson | 2L 8Z (3.6) | 1C 1D 1U (0.0) | 1D 1U (4.2) | 3C 2D 3L 5U (19.25) | 27 gp 0-1 sp |
| **Even Distribution**[^yzohl1xqmr] | **1C 1D 1L 2U 2Z (8.4)** | **1C 1D 1L 2U 2Z (8.4)** | **1C 1D 1L 2U 2Z (8.4)** | **1C 1D 2L 1U 2Z (9.3)** | **34 gp 5sp** |
| **Random Play** | **??** | **??** | **??** | **??** | **56 gp 9 sp**[^2twoh5lktxq] |

Congratulations to all players, particularly **simon** and **abstractapplic** (who managed to fully solve the problem and turn in a perfect answer, with the first one coming [~~just in time for me to use it on my real-world taxes~~](https://xkcd.com/1566/)).

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

Adventurers come in groups of 1d6 (adventurers range from powerful adventurers who can go out solo to weak ones who need half-a-dozen companions).

An adventuring party goes out and hunts a variety of monsters, and then divides all monster parts as evenly as they can.  (This is why your dataset often contained multiple consecutive rows with very-similar goods).

This standard divide-evenly approach is much better than random (since it gets relatively-low tax brackets for everyone by dividing loot like that), but not close to optimal.

FEEDBACK REQUEST
================

As many players noticed, this scenario was deterministic: the same set of goods would always lead to the same taxes.  I think that this was very much justified given the premise: it's valuable to be able to realize that 'tax calculator' is an area where you shouldn't expect much-if-any randomness.  How did this feel from a player perspective?

As usual, I'm 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?

[^xraricyti2k]: This tax rate existed in theory but was not derivable from the dataset, as the few people with enough goods to be in this tax bracket all had enough Unicorn Horns to get a lower rate below. 

[^emy4avaysgb]: Unlike the other tax rates, which are all divisible by 10%, this one can lead to fractional silver pieces if an odd # of gp is taxed at 25%.  This is resolved using python's round() function...my apologies to simon, who seems to have spent a while trying to figure out when it rounded up or down. 

[^hr2tgb9j5vr]: Not 8, as you will shortly see... 

[^b3agjjm9dmo]: Since the Cleric cannot have a Dragon Head, someone else will need 2 of them...and since the second head will be double-taxed, it is very important to have as low a tax rate as you can on it. 

[^qnrxs0kcbjd]: In order to fully benefit from the cockatrice rebate, they want to get as close to 30gp as possible without going over. 

[^mb4as75rlo]: The no-Cleric approach was meant to be a tiny bit worse than the with-Cleric approach, but a last-minute change that I didn't check properly, plus my confusion about how my own rounding worked, made it end up a tiny bit better instead.  Oops. 

[^2twoh5lktxq]: Dividing the loot as evenly as possible among adventurers. 

[^ublk29s43za]: Monte Carlo distribution.