In my D&D 5e games, I use a homebrewed system of handling activities that has proven to work very well for five years of sandbox gaming where the characters want to do lots of things. The core mechanic is that every time a character wants to undertake some significant task or project, they spend one or more hit dice. Hit dice represent an overall pool of energy, vitality, and competence that can be used for many things other than simple recovery. This means that leveling up allows characters to be more competent and productive out of combat, which is especially important if the game ends up focused on politics and skill use.
When a character spends a hit die, they roll a skill or tool check to see the results. Because these checks represent many hours of sustained work, they cannot be boosted by things like Guidance, bardic inspiration, or anything else that would boost a single skill roll. Expertise and Reliable Talent do not apply to skill rolls that use a hit die. (They just apply to things that happen in a single action or while role-playing; Bards and rogues are already really good in a skill-based campaign and this would make them overpowered.) Characters are assumed to be using all class abilities that could be helpful or increase productivity, like spellcasting, Fast Hands, Action Surges, etc., which is part of why high-level PCs can undertake more projects than low-level characters.
With this system, long rests restore half of a character's hit points and all of their hit dice. Characters who are wounded from adventuring must spend hit dice to heal until they are up to half of their hit point maximum before doing anything else, and in times with no combat, characters can undertake a number of projects equal to their character level per long rest.
Investigation
Many quests and plots require the characters to investigate things or learn facts in order to proceed. Few GMs and groups are prepared to run a sandbox-style game where failing these checks means that the quest fails and the party must find something else to do. So, when going through a quest or plot line, every attempt to learn or investigate an important fact requires a character to spend a hit die. This makes failure meaningful, while still allowing them to eventually get through the plot.
Looting
In the field, I give each character one free check after each combat encounter to search and loot the enemies or the environment. If they want to investigate any additional things, they must spend a hit die per investigation. At low levels, or with simple tasks, you can make the cost one hit point instead. (Explain to your players that hp should be thought of as vitality points, not wound points. Losing hp means getting tired and a little scratched or bruised, as you use your battle reflexes and reserves of energy to twist out of the way at the last instant. The only attack that actually hits as intended is the one that dropped you to zero hp.) This forces them to make choices and keeps the game moving.
Using normal guidelines, decide how much loot the encounter would normally give. (I prefer to combine the hoard and individual creature loot and spread them out more equally among encounters, using a method kind of like the 4e treasure parcel approach.) Then double that amount and divide by the number of characters. When the characters do their first free loot check, they get that value of loot, of a type appropriate to what they searched, if they beat a DC 15 check. The DC should increase at higher levels, to reflect the fact that such rare and esoteric loot is harder to properly identify or harvest.
As a general rule, further checks done by spending hit dice will normally give about 25 gp worth of loot on a successful check, similar to using a hit die to earn income as described below (5 gp average for things that just required spending a hit point). You can raise this if they are in a rich environment and/or you want to encourage pushing their luck by spending their potential healing to loot, or lower it if the looting would not result in much or you want to discourage this behavior. However, you might rule that if most or all of the initial free checks failed, they can get their loot by spending hit dice to keep trying.
For a flavorful handling of loot that reduces bookkeeping, have each item of loot, by default, be worth 25 gp, plus another 25 gp per additional word used to describe it (not counting articles or connecting words). So a "belt buckle" is an art item worth 25 gp. (The fact that it is listed seperately means that it is not an ordinary belt buckle). A "turquoise goblin belt buckle" is worth 75 gp. A "fancy diamond studded goblin belt buckle with a really spiffy shiny turquoise inlay" is worth 500 gp. Describe it to the players, and have them copy down your exact words on their treasure listing.
I prefer using Body Part Loot, where most monsters 'drop' interesting magical ingredients rather than gold or art items. These are extracted from the corpse with Medicine or Arcana checks, or an appropriate tool proficiency. They can then be crafted in downtime as described below.
Downtime Projects
I use this system with the 'Gritty Realism' variant (DMG pg 267) where a long rest is several days of recovery and happens once every week or two. All example projects below are based on that system. If you use it with the normal rest variant, make the projects and income smaller to compensate.
In downtime, each hit die spent allows a character to roll a check to accomplish a single task that could be completed in an afternoon of intense work, or several days of intermittent work, like:
*asking around to uncover a piece of information.
*investigating an item.
*negotiating a simple deal.
*writing an important letter or report.
*teaching a simple task to a henchperson.
*finding the right person for a job position.
Larger projects and long-term goals require more checks, ranging from a few successes to over a hundred successes in various tasks.
If used to rest and recover, spending one hit die in downtime allows you to roll a Constitution saving throw as described in the Recuperating subsection of the Between Adventures section
If used for training, each Hit Dice will allow you to roll an ability check after you find an instructor. Learning to speak a language requires 25 DC 10 Wisdom checks. Learning to read and write a language requires 25 DC 10 Intelligence checks. Learning a tool proficiency requires 50 DC 10 checks of the appropriate ability.
Lifestyle Expenses and Benefits
The higher-end lifestyles of ease and comfort, with access to staff and servants, give characters more hit dice to spend on their tasks and projects and quest investigations. Every 25 gp you spend per long rest gives you one extra hit die to spend on projects. Ordinarily, you can do this up to 4 times (because there is a limit to how much more productive you can become by throwing around money), but building long-term relationships with your staff and the area can increase this cap, as they learn better how to work with you.
Spending 25 or 50 gp in this way gives you access to the Wealthy lifestyle (PHB page 158), and spending 75 or 100 gp (or more) gives you access to the Aristocratic lifestyle.
You can set up an estate by making a one-time payment of 250 gp, and spending 5 hit dice to find trusted managers, organize the operations, and build connections with the local authorities and your tenants. The first estate allows you to maintain a Comfortable lifestyle without spending any money or hit dice, as long as you are in the area. Each additional estate allows you to gain one hit die every long rest thereafter while in the area.
The rental income from your property covers all expenses and maintains the estate and your staff. (You are not actually getting a 10% weekly return on investment; the estate mainly just breaks even, but having a comfortable familiar place with staff who know you replicates the effect of spending money in a new place.)
Estates can occasionally be a source of plot hooks, and may be lost due to war, revolution, intrigue, monster attack, etc., but the GM should not threaten them or take them away long as the characters are doing a decent job checking in every so often and protecting the area from threats. We want to encourage players to role-play and find a place in the world, and if they cannot make reliable investments in such things and get benefits from them, they'll just spend all their money on magic items they can carry with them.
Earning Income and Crafting
For small simple low-risk projects, or practicing a profession, simply spend a hit die, roll a check, and have the gp income or progress be the total check result. This matches the normal rules for crafting or following a profession, assuming an NPC or low-level character with one hit die. When doing something they are skilled in with a good stat, and spending one hit die per long rest to do so, they will earn or make an average of 15 gp per long rest, or about 2 gp per day of income or value added in the crafting.
For example, if a character is attempting to craft a halberd (cost 20 gp), they will need to spend 10 gp to rent a forge and and buy steel, coal, and other supplies. If the check result is 10 or better, they complete it with their hit die of effort. At your discretion, higher check results can be turned into additional item crafting or income, so if the result was a 15, the character completed the halberd early and then earned 5 gp by working as a smith.
To handle lifestyle expenses in a simple way and reduce bookkeeping, you can simply require characters to spend one hit dice per long rest, without any roll, to handle their affairs and maintanin a comfortable lifestyle without spending any money.
A more math-intense option for high-stakes projects like working with volatile ingredients or gambling is to set a breakeven DC of 10 to 15, subtract that number from the check result, and multiply the difference by some number based on the stakes. This multiplier should be at least five and could be much higher.
For example, a character playing games at a medium-stakes table might have a multiplier of 10. Any game involving bluffing is a Wisdom check with the gaming set proficiency, while pure games of skill are based on Intelligence. An average table of adventurers or soldiers would have a breakeven DC of 12 (base stat +1, half of them add a +2 proficiency), so if the character got a 20 on the check, they would gain 80 gold, but a check result of 10 would lose 20 gold.
Magic Items
Characters will often want to use hit dice to craft magic items, especially if they have accumulated an inventory of interesting magical ingredients. I follow the same rules for crafting both magical and nonmagical items, as well as building ships, strongholds, etc. and doing any other activity that earns income. I allow any character with a tool proficiency to craft magical items if they have access to appropriate magical ingredients (In my opinion, any rule system that does not allow a dwarf smith to craft a magic weapon out of exotic materials is simply wrong).
Mark progress for all expensive items and large projects in units of 50 gp. Set the check DC between 15 and 20, depending on conditions and how appropriate the materials are for the desired item. If the character owns the magic item formula, or a good set of blueprints, or has expert assistance, give advantage on the check. Passing the the check makes 50 gp of progress (i.e. turning 25 gp of ingredients into 50 gp worth of completed item). Failure means no progress was made and nothing was wasted (unless a critical failure was rolled). This means that, on average, each hit dice spent will add about 15 gp of value to the ingredients, assuming a decent check modifier.
One successful check will turn 25 gp worth of ingredients into a common consumable item worth 50 gp. Crafting a permanent uncommon items requires 250 gp of ingredients and ten successful crafting rolls, and each level thereafter is ten times as expensive. This matches the item creation cost in the DMG, when you consider the opportunity cost of the character's time.
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