duration versus consumption of resource management
traditionally, torches have a constant duration of six turns or one hour. what happens when we change this up?
resources with variable duration are in practice not very distinct from those with constant duration, and in fact they might be deemed more complicated. the player randomly determines the duration for which the item will last--2d6 turns for instance--and then counts time until this duration passes.
it is therefore not difficult to make something last for 6 turns on average (if 2d6 with an average of 7 is too great, perhaps 2d6-1) but there's no reduced bookkeeping here. the main benefit is the injection of chance should the players actually want it.
random consumption is a totally different beast than constant or variable duration, and i am trying to be very precise with my word choice. variable duration, or random duration if you prefer, randomly determines the duration for which something will last. that is to say that once this duration is determined, and barring any extra circumstances, that duration is set in stone. however, random consumption does not determine the duration of the thing: it only determines the chance that the thing will become consumed given some opportunity (e.g. each turn).
often i read rules for torches that prescribe a 1-in-6 chance per turn that the torch will extinguish. the reasoning is that a 1-in-6 chance means that given six opportunities (i.e. six turns), one of those will result in extinguishing. this is definitely true, but is this the outcome actually desired if players wanted to retain 1-torch-per-hour while reducing bookkeeping? keep in mind that once one torch is extinguished, another must be lit, and so there can actually be anywhere between 0 and 6 torches used every 6 turns. what i am getting at is that the resultant "duration" of the torch using random consumption is indeed smaller than when using constant duration , if we assume a random consumption chance of 1-in-6 and on the other hand a constant duration of 6 turns.
when we say that a torch has a 1-in-6 chance of extinguishing per turn, the more useful information is that a torch has a 5-in-6 of not extinguishing per turn. this is because what we are looking for is the duration for which the torch will last. there is a 5-in-6 chance that a torch will last past one turn, and a 5-in-6 chance squared that it will last pass two turns. the probability that a resource will last for n turns is a function of n and of the probability that the resource will be randomly and instantly consumed, p.
f (n, p) = (1 - p) ^ n
let us look at the probability f that a torch (p = 1/6) will last given different possible numbers of turns n.
- n = 1 --> f = 83.33% (duh)
- n = 2 --> f = 69.44%
- n = 3 --> f = 57.87%
- n = 4 --> f = 48.23%
- n = 5 --> f = 40.19%
- n = 6 --> f = 33.49%
- n = 7 --> f = 27.91%
- n = 8 --> f = 23.26%
- n = 9 --> f = 19.38%
- n = 10 --> f = 16.15%
- n = 11 --> f = 13.46%
- n = 12 --> f = 11.22%
look how the probability switches from greater than 50% to less than 50% right between n = 3 and n = 4. in other words, after 3 turns, the chance that your torch will extinguish becomes sooner than later. in this respect, this method is not very different than saying that a torch lasts for 1-6 (d6) turns, except that since consumption is determined on-the-fly, there is no set point at which the torch definitely lasts or is consumed.
and this is where i return to the point i made: just because each torch is likely to be extinguished once per hour, does not mean that you are likely to spend one torch per hour. consider again that when one torch is extinguished, you light another one. since there is about a 50% chance that a torch will last 3 or 4 turns, you're more likely to go through two torches every six turns rather than just one torch.
this is not to say that random consumption is a bad thing. really, i think it just means you oughta make torches twice as easy to buy and carry than when they last for simply six turns. :)
in another post (link), i compare statistics of different strategies of usage dice, which are basically a way to prolong random consumption mechanics via extra bookkeeping (basically allowing there to be multiple consumptions). there is also no reason why any of these mechanics should be mutually exclusive! my partner had the cool idea of guaranteeing a certain amount of uses/turns for a resource, and then after that consumption becomes random. i think something like this helps players (referee included) make decisions based on what they are guaranteed to have versus what becomes a gamble. these opportunities to introduce decision-making are what make games interesting, or what make games games at all.
finally, you might see some similarities between this post and my previous one (link) about different ways of handling combat. that post was ultimately about handling hit points, which is itself ultimately about managing the ultimate resource: your body! so consider this an appendix to that post, or vice versa.
 i don't see a need to compare variable duration with random consumption, since that's ultimately a matter of play preference and (apropos mathematics) fine-tuning random factors.