Variations on the Usage Die

One of the mechanical darlings of the OSR is the usage die, invented by David Black for his rules-light take on fantasy roleplaying The Black Hack. It's a simple and elegant mechanic - rather than tracking the exact number of rations, arrows, or any other other vaguely consumable resource remaining, these resources are assigned a "usage die" which is then rolled whenever the resource is used (aka when resting for rations, or after combat for arrows). On a roll of 1-2, the usage die moves down one step (d20->d12->d10->d8->d6->d4). On any other roll, the usage die remains as-is. When the d4 rolls a 1-2, the usage die is consumed and the resource is depleted.

Opinions on the original Black Hack version of the mechanic vary - some people enjoy the simplicity and insert the usage die as-written into their games at every opportunity, while others find the randomization of tangible resources such as rations and arrows to be a bit overly abstract compared to just counting them. Regardless of their opinions about the specific Black Hack implementation though, most people I've seen agree that it's an elegant mechanic with some serious potential. Personally, I don't use it for rations and arrows, but I think it works very well for resources that may be inherently a bit fuzzy and hard to precisely define - such as magical power, sanity, or fame.

In this post, I'd like to dig into the usage die mechanic a bit, then present a few variants that change the "feel" of the mechanic a bit, allowing for some different use cases. I'll also compare the expected value (aka average number of uses) resulting from each option, as well as a brief overview of the method I used to calculate it (discrete Markov chain analysis)*.

These are dice

OG (Black Hack) usage die: down 1 step on 1-X 

The original implementation of the usage die (henceforth Ud for short) uses a die chain that moves in only one direction - down, with a 2-in-(die type) chance of moving down a step each time the usage die is rolled. This produces a nice spread of average total uses ranging from 2 (for d4) to 30 (for d20), allowing for the modeling of resources with a wide range of "charges," but not so wide as to make the Ud irrelevant at the table.

One of the simplest tweaks that can be made to the usage die is changing the target number that results in a step down. Dropping it to 1 rather than 1-2 doubles the expected number of uses to 4-60, while increasing it to 1-3 drops the expected number of uses to 1.3-20.

Expected uses for variations of the OG usage die

The OG usage die mechanic has a number of distinctive features that makes it particularly well suited for modeling gradual depletion of an adventuring resource:

• It moves in only one direction: the inexorable march down the chain means that even if you get a lucky streak resources will eventually run out, necessitating a return to safety for rest and resupply - this puts a built-in "clock" on any expedition, which is often preferable for a dungeon-crawling or "expedition" focused game.
• Total depletion is predictable: you will have some warning before you completely run out of a Ud resource, because it only moves down a single step at once - this helps to limits frustration by allowing the players to feel like they have a measure of control and that their decisions matter, despite the randomness inherent in the mechanic. 
• Depletion events are unpredictable: while a player can relatively safely count on being able to have some advance warning before completely running out of a Ud resource, any given roll is still an unknown - this creates some dramatic tension each time the die is rolled, which is generally pleasing to human brains (there's a reason gambling is so addictive to many people and why we RPG players love tossing our shiny math rocks).
• Accelerating depletion rate: as your resources deplete, the chance of moving down on the next roll increases - this helps to build tension and fosters somewhat of a push-your-luck feel, as the players start out feeling relatively well-supplied but will start feeling more and more at risk as they continue to adventure.

Of the above features of the OG usage die, the unpredictability of depletion events and acceleration of the depletion rate are fairly baked-in - the whole beauty of using the mechanic is that it injects drama/tension to the depletion of resources while not requiring the players or GM to do any complicated math at the table or track anything other than the current die type. 

I'd argue any tweak that changes either the unpredictability of depletion events or the steadily accelerating depletion rate would fundamentally alter the mechanic such that it's not really a variation on the usage die any more, but something else entirely. That's not necessarily a bad thing, but it does provide a convenient dividing line for the purposes of this post. For now, I'll be sticking to the basic "die chain w/ consistent target number ranges" structure for the mechanic.

Unidirectionality and predictable depletion, however... those are relatively easy to change while preserving the basic structure of the mechanic - and changing either (or both) of them can greatly alter how the mechanic feels at the table. Let's dive in.

Bidirectional usage die: down one step on 1-X, up one step on highest value(s)  

The first variation I'll explore is the bidirectional usage die - that is, a Ud that steps down on low values, but steps up on high values. This version of the Ud models resources that usually tend towards depletion, but occasionally go the other direction.

We need to be somewhat careful here. Making the chance of a step down equal to the chance of a step up fairly quickly results in a situation where, pending a series of particularly unlucky rolls, the number of expected uses climbs very high, very fast. This is undesirable for a few reasons - not only is it extremely inconsistent (and thus difficult to use to create any sort of predictable gameplay experience at the table), but the high average number of uses means that unless the Ud is being rolled almost constantly, it will almost never be depleted in a typical adventure (and thus would be somewhat pointless as a resource tracking mechanic).

All the same, the idea of a usage die that steps both up and down has promise, especially if the step up frequency is limited. Infrequent but very good events feel really compelling at the table from a psychological perspective - there's a reason critical hit mechanics are so often hacked into old school D&D despite being (mostly) absent from the original rulesets. People also tend to disproportionately remember these unlikely good events; a gaming group will often talk for years about the time their character dealt 50 damage to one-shot the boss due to exploding damage dice in Savage Worlds, or saved vs the necromancer's death ray 3 rounds in a row before putting him down.

My preferred tuning of this variant is probably "3 down, 1 up" - that is, on a 1-3 the Ud steps down and on its highest value the Ud steps up. This results in a spread of average total uses of 2.5 (for d4) to 28.8 (for d20), which is surprisingly close to the 2-30 spread of the OG usage die. Note that despite the averages being similar this is still quite swingier than the OG usage die - particularly if you're starting at a d4, where you'll either deplete entirely or vault up to a considerably higher number of average uses on the first roll. I probably wouldn't use this variant of the mechanic for a resource that could start at d4 - it works better starting around d6 or d8. The average number of uses for a few versions of the bidirectional Ud are presented below.

Expected uses for various bidirectional usage die variations
*(3 down 2 up is 2 down 1 up on the d4)

In my opinion the bidirectional Ud works really well for modeling things like reserves of magical power (as a replacement for spell slots or mana) - particularly for something like a wild mage. It could also work well with a sanity mechanic, or (in a reversal of the resource depletion paradigm) for the severity of a wound or disease that gradually heals over time (but with the chance of suddenly worsening).

Jumpy usage die: down multiple steps on 1, down one step on 2-X

The other major feature of the usage die we can mess with is its (relative) predictability. By default, you're not in danger of fully depleting the Ud until it has been reduced to a d4. This is easy enough to change; simply change it such that rolling the lowest value results in moving down multiple steps - either 2 steps (for a little bit more unpredictability) or full depletion (for a lot more unpredictability). 

This change has a few effects. The most obvious effect is that the number of expected "charges" is  significantly reduced. This can be helpful if you're trying to model something with few charges, but still want to make use of a wide range of the dice chain. The other effect is that from a psychological perspective, the Ud is less "safe" - it is more prone to run out unexpectedly. This heightens the feeling of tension (or for a more high stakes resource, dread) promoted by the usage die. Obviously, this will be more appropriate for some applications than others - but it's a nifty tool to have in the toolbox.  The expected number of uses for both of the options mentioned above are shown below.

Jumpy usage die expected number of uses

This variant of the Ud is especially helpful for modeling situations where there's a steady decay, but always with a chance of everything immediately going off the rails. I'd use it sparingly, but it serves well in cases where you don't want the players to feel completely secure at any point. Example applications include magic spells that are unraveling in an unpredictably chaotic manner, or perhaps the attitude of a king whose appetite for extended conversation with the party is rapidly running out. 

Wrapping up

These variations can be combined, of course - one can imagine (for example) a usage die that depletes 2 steps on a 1, 1 step on a 2, and increases 1 step on the highest value. Below is a summary table of the various options discussed in this post, a few new combos, and (bonus) 2 versions of a reverse Ud  - aka a usage die that steps up instead of down, and "finishes" after d20, creating a decelerating (rather than accelerating) depletion rate.

Summary of all discussed usage die variations + a few
*(3 down 2 up is 2 down 1 up on the d4)

I said at the start of the post that I'd present a brief overview of the analysis method I used to derive the expected number of uses for each of these. This post has already run fairly long, so I'll keep it brief. Each of the usage die rulesets I examine in this post can be represented mathematically as discrete Markov chains with one absorbing state. That is to say - they are state machines in which the probability of the next state depends only on the current state. As it turns out, with a little matrix math (using your preferred programming language - mine is Excel :P), it's really easy to calculate all sorts of information about these state machines - including the expected number of cycles before "absorption" (aka depletion). 

For a more in-depth explanation of this analysis method, see my past post walking step-by-step through a Markov chain analysis of the "clock puzzle" from Xanadu. If anyone really wants a walkthrough of the method as applied to a usage die, though, feel free to leave a comment. It's hard to overstate how useful Markov chains are for analyzing of RPG mechanics - I expect to be returning to them not infrequently. 

Well - that's it! I hope this sparks some ideas regarding new ways to implement the usage die in your games! David Black did the OSR a major favor in popularizing the mechanic, and I think with some tweaking it becomes an extremely versatile mechanic appropriate for all kinds of applications. Please comment if you've got other ideas for innovative modifications to the Ud mechanic! 

*You don't really need Markov chain analysis to calculate expected uses w/ the "one-way" versions of the usage die, but it becomes much more difficult to do so ad-hoc as the mechanics get more complicated - the Markov chain analysis on the other hand makes this very, very easy with any arbitrary set of usage die rules as long as they actually do form a Markov chain.