This article was started way back when I was submitting articles to Roleplaying Tips, in fact, more than ten years ago, but it was never finished – until now. So “recently” means “relative to 2006″…
I recently read a book describing the calamities that befell Lloyds of London in the early 80s and 90s and it got me thinking back to the days when I was an insurance clerk, and what I had learned then about how insurance premiums were calculated. And I suddenly realized that some of what I had learned back then had not only started to make more sense in the decade or so since, but this new understanding had applications in the world of Gaming, in enabling the GM to adopt a whole new approach to some aspects of his craft. Does it work in practice? That we’ll have to wait and see.
Insurance premiums are set by calculating the risk that the insured party will have a claim of value “X” in the year, and then the risk of a claim of value 2X, and then the risk of one of 3X, and so on. Multiply each risk by the claim value and add up the results and you have the total risk of the insurance, ie (in theory) how much the premium should be for that particular driver or homeowner or whatever.
For example, let’s say that there’s a 1 in 100 chance of a $5,000 claim, a 1 in 1,000 chance of a $20,000 claim, and a 1 in 5,000 chance of a $50,000 claim, per year.
- 1/100th of $5,000 = $50.
- 1/1000th of $20,000 = $20.
- 1/5000th of $50,000 = $10.
So if those are the only risks to worry about, the average claim per year per customer would be $80. Set your premium for that, plus a share of your administrative, operational, & infrastructure costs, plus a profit margin, and all will be well.
In theory, you don’t need insurance – you just need to save $80 a year and you’ll have enough saved when the time comes, even without interest on those savings. But in practice, that’s not the case, as anyone who’s rolled dice knows – that 1 in 100 chance might come up on the first roll, the 15th, the 80th, or the 131st. The likelihood that it will hold off until the one-hundredth, when you will have saved the $5000 to pay the cost of repairs is vanishingly small.
Well, for technical reasons, it’s a lot easier to work with the chance that something won’t happen.
- In the first year, there’s a 99% chance that the first event won’t happen, or 0.99.
- In the second year, there’s a 99%x99% chance that the first event won’t happen, or 0.9801.
- In the third year, there’s a 99%x99%x99% chance that the first event won’t happen, or 0.970299.
…and so on.
Right away, you’ll notice that the chance that something will happen in those three years is smaller than you would get by just adding three lots of the 1% chance together. In the 100th year, the chance that it won’t have happened yet is still 36.6032341273%! Or, to put it another way, the chance that it will have happened at least once by this point is about 63.4%. And, the chance that it will have happened at least two times in that 100-year period is going to be a smidgen less than 63.4% of 63.4%, or 40.2%. There’s a significant chance that this once-in-a-century event will have occurred at least 5 times in the course of a century – just over 10% chance, in fact.
For most people, probability is inherently counter-intuitive (and yes, I’m one of the majority). This shows quite clearly that an insurance company who relied on once-a-century items only happening once a century would go out of business in short order. In fact, there’s a significant risk that the company would have to pay out $5000 five times in the course of that century. To be adequately prepared, the premium would have to be more than five times the $80 a year.
Except that this is impractical and unprofitable. Insurance companies make their product more attractive by dividing that risk by the number of insured that are not likely to have any claims in the year in question, and then insuring themselves against catastrophic events that would cause more people than normal to make claims. So if only 1 in 5 will actually have an accident in a given year, then you can divide the “true premium” by 5.
The Insurance Risk Assessment Shortcut
That’s all well and good in theory, but in practice it’s way too fiddly and it takes too long to be entirely practical. There are far too many combinations; it would take weeks if not months to calculate a single individual’s premium.
Instead, most insurance companies assume that they will have enough customers that they will encounter every possible outcome in the course of a year (within the scope of their coverage, at least) – in other words, that their customer pool represents a statistical universe.
They then use statistical tools to determine the average value of claims expected in the year and simply assess the risk of any given car having a claim of that size; they can then determine the premium to charge (plus a share of expenses and a profit margin). To make things easier for their staff, they use a points-based system to calculate an estimate of that risk – staff simply look up the total points scored on a table and compare it with the value of the insurance to get the premium (or have a computer system do it for them).
This means that staff can determine premiums without being taught really understanding the underlying complexities.
The same approach can be used by GMs for a number of types of Event and Occurrence in RPGs. Wandering monsters, weather, plot complications, etc. The result is a table showing a die roll on one side and an outcome based on the risk.
Why this would be an advantage over the traditional methods of creating such tables are the ability to incorporate a multitude of factors that normally have to be handwaved due to the complexity of calculating probability combinations, as will shortly be seen.
Wandering Monsters & Other Chance Encounters
Every game system, whether it’s level based or points based, has a method by which the power level of an encounter can be measured. In points based systems, it might be set multiples of character points in the encounter, for example 100, 125, 150, and so on. In level based systems, it’s levels. But levels are not a linear measurement in most systems; it takes more experience to go from level 17 to level 18 than it did to go from level 7 to level 8. This distorts things, because it means that there will be far fewer encounters of higher levels than the straightforward “by level” system would allow for. The answer (again) is to use the experience-point equivalent of the level, instead of the level directly: 1000, 2000, 3000, and so on. D&D 3rd Ed (and 3.5) uses an Encounter Level system to allow for the non-linearity.
Because most people will be familiar with it, I’ll be using the D&D / Pathfinder model for this discussion.
Encounter Table Structure
Most encounter tables assign specific encounters to entries on their encounter tables. I think that’s actually counterproductive, because it means that the table contents have to be continually revised as characters increase in levels.
A better approach would be to employ a more abstract system that doesn’t need such revision.
For example, you might list on the table:
You will have noted the empty column – we’ll populate that shortly. Right now, it’s the entries on the right that we’re interested in – the first of them is EL-3/- which should be reas as “Encounter Level minus three, or less”. That’s followed by EL-2, EL-1, EL, EL+1, and so on all the way up to “Encounter Level plus three or more”. The term “Encounter Level” in this table refers to the EL of the PCs, collectively.
The table functions as an index of encounter levels relative to the level of the party, so “EL+2” means the “encounter level value of the party, plus two”. As a general rule of thumb, encounters should rarely be less than 2 below the party’s level or more than 2 above, according to the DMG.
Using this table requires and assumes that you have a separate record, possibly even a completely separate document that lists all the actual encounters that you have prepared, and that meet the specified encounter level; the first page might be labeled EL0, the next EL1, then EL2, and so on. On each page you list encounters, an entry for each monster type that exists in your campaign world that can meet the EL target. The first entry for EL1 might be “1 Orc”, then on page EL3 would be “2 Orcs”, EL4 would have “3 Orcs”, EL 5 “4 Orcs”, and so on.
In the 3.x system, EL rises with numbers as a multiple of the square root of 2: 1, 1.4 (no such thing), 2, 2.8 (call it 3), 4, 5.6 (call it 6), 8, and so on. One representative of a given creature will have its base EL given in its writeup as its CR, or Challenge Rating. You keep going until the numbers no longer make sense to you, as in “there is no way that many of them would be found in one place”.
Depending on the creatures, that might be 1, 2, 3, 4, or 32. That’s up to you. The beauty of this system is that you can keep adding to it as you go, and you can populate the lists with as many customized encounters as you like. If you decide that a creature that is normally CR2, such as a bugbear, becomes CR4 if it has this ability or that equipment package, you know that EL4 should list one creature with that package, EL6 should list 2 of them, and so on.
As a very rough rule of thumb, for example, you might decide that if you total the magical plusses of armor and weapons that a creature wields, you get the increase in the creature’s CR. You might also decide that increases in Hit Dice follow the same 1.4-factor progression, so that a CR4 creature with 4 levels in a character class is a CR 9, as shown below:
The top row is the number of creatures of CR1 that are needed to reach the EL designated in the third row, after appropiate rounding. The middle row shows the real values with a decimal place, the “raw data”. To permit the chart to be a reasonable size, I’ve split it into two triple-rows – the first deals with ELs 1 to 10, the next, ELs 11-20. (If I were doing this for real, I would extend it to at least EL30 and possibly 50).
Some of the “rounded” values have a + symbol. If you look closely, you will see that if the raw value is something-point-two, or in fact anything less than something-and-a-third, it has no plus symbol and is simply rounded down; if there is a decimal higher than 2/3, it is rounded up. The plus is there for those in-between values like “1.4” and “5.6”.
The “+” symbol also confers some kind of advantage to the creatures – so a group of creatures whose EL has a + next to it needs a minor advantage beyond the number of creatures encountered and any other form of EL adjustment.
As you can see, character levels are really easy when you make the assumption described earlier – simply find the cell with EL that has everything else taken into account and move to the right 1 space for each character level. (Note that you don’t have to use this shortcut method if you don’t want to, it is NOT canon. My experience is that it comes pretty close to “reality”, though, in terms of relative effectiveness in combat).
Or perhaps the table is telling you that you need eleven creatures of a given CR to reach your target EL but you want fewer creatures and to give them a couple of character levels instead – just count one space on the table left for each character level on the top line to find how many of the modified creatures should be encountered. Two character levels, two spaces left, so 11 to 8 to your choice of 5 creatures with an extra little “+” advantage or 6 creatures without.
The chart can be used in a host of other ways. Suppose you were to have a group of 8 CR2 creatures who you wanted to take up to EL10. Look across the top row until you find 8, then down to the third row to get the CR of 8 CR2 creatures – a seven. That means that you need to give each of them +3 CR’s worth of advantages. Those advantages can be the same for all, or you could decide to give this one +3 STR Bonuses (i.e. +6 STR), that one +3 CON Bonuses (+6 CON), a third gets +3 in magical weapons and armor, a fourth gets three extra hit dice, and the other 4 receive 2 class levels each – in different character classes. Your eight CR2 creatures start to look like a population of individuals and not a homogeneous, generic, monster “with benefits”.
Okay, let’s turn our attention to the empty column. What we want here starts off as a dumbbell probability curve, such as you might get from 3d6. But we want one that gives us some multiple of 7 result categories, because our table has 7 slots to fill.
A few minutes playing around at Anydice.com (my go-to site for this sort of thing) gives me the results as percentages. But there’s some messy rounding involved – the numbers are 1.16%, 8.56%, 23.84%, 32.87%, 23.84%, 8.56%, and 1.16%.
That curve is a little sharp for my tastes, though. I’d like to roughly halve those percentages and apply a flat +7% to each. 7 plus-7-percents is 49%, so the curve component is providing 51% of the roll, so that’s what those numbers have to be multiplied by, as shown below.
When I did the calculations, as you can see, I ended up with a 3% error unaccounted for.
I flattened the top result 1% more to make it an evenly divisible 4% error, and then gave 1% each to the two values on either side of the peak to arrive at the final numbers in the table.
Encounter Probability – base chance
So far, it’s all been fairly conventional going, but here’s where the risk assessment element comes into play.
Most adventures simply assign a percentage chance of an encounter. Surely we can do better than that?
Let’s start by saying that there’s a base chance of an encounter that we want to assign based on the levels of monster inhabitation in the area.
Right away, that incorporates the climate and the levels of non-monster population (who would drive hostile creatures away) and a host of other factors. From there, it’s all about conditional modifiers.
Those modifiers all go in another table, one with a whole bunch of headings. There are three possible types of adjustment: An increased chance with an increased risk, an adjusted chance (up or down, evenly balanced) with an bias on the encounter table, and a straightforward encounter chance adjustment with no increase in risk.
Every time you think of a factor that you want to track individually to tweak the results, all you have to do is add another set of lines to the modifier table.
For example, you might want to track time of day. Monsters are more likely to be active late at night if nocturnal or in the early morning and late evening if not. At first, it might seem that only unintelligent monsters would follow this pattern, but for that very reason, intelligent monsters would be out hunting at that time of day. Noonish is typically the quietest time of day.
Right away, there’s a complication: nothing we have so far distinguishes nocturnal from diurnal creatures. Our list of potential encounters certainly doesn’t. So, we either complicate our nice simple process, or we use risk assessment techniques to balance the books. No normally nocturnal creature would operate during the day unless they were desperate, or had some advantage that made them diurnal instead, one that most members of the creature’s race don’t have. Similarly, diurnal creatures don’t operate at night unless desperate or they have an advantage at night that most such creatures don’t have.
Instead of a straight adjustment to the encounter chance, we are better served by converting some of the adjustment into an increase in risk posed by the encounter. +1 EL worth is the standard, and it’s worth -5% in my book.
Using this information as a basis, I created the following:
Let me walk you through the table and the process that it demonstrates.
- I started by breaking the day up into eight of broadly-defined time periods. That makes it easy to make symmetrical “probability curves”.
- I populated the Nocturnal Modifiers column with a peak of 12%, diminishing to a minimum of 0%. Then, because there were two Diurnal columns, I doubled those values, and finally, because those two diurnal columns were going to be distributed without a lot of overlap, I reduced the nocturnal values to 3/4 of their interim values.
- That scaled the nocturnal readings to match diurnal peaks of 12%, so I added them – one with a peak in the early hours and one with a peak in midafternoon. I reasoned that more creatures would stir in the very early morning than in the afternoon because those who had already found what they needed would not be active; this bias shifted the morning peak to earlier in the day.
- Adding those up gave me the first subtotal column.
- Next, I allowed for normally nocturnal creatures becoming diurnal because of an advantage and vice-versa using the “-5%” column – I’ll talk a little more about that in a moment.
- Another subtotal.
- ..Which was needed so that I could calculate a relative adjustment. Because our base value incorporates the overall impact of every variable, the average modifier needs to be +0%. If you total the values in the second subtotal column, you get 83%. 80 divided by 8 gave a base adjustment of -10%; I then tweaked the after-dark adjustment to -9%, giving me a total of 4% to deduct. Two of it was allocated to the Evening timeslot to make it more closely resemble the “after dark” total, while the other 2% were split up and applied to mid-morning and “noonish”.
- Adding the Relative Adjustment and second subtotal together gives the final adjustment values. But note that five of these adjustments refer to creatures with an added advantage and three don’t; so this table is a mixture of straight adjustments and adjustments with an increased risk attached.
The Increased Risk adjustment
These represent an increased capability in exchange for a reduction in likelihood to occur. In this case, I applied them to those time periods when nocturnal variations of day creatures were most likely to be operating, and to when diurnal variations of traditionally nocturnal creatures would be most likely. I also took into account the increased competition and danger that would be faced during the time periods when the greatest number of the “normal creatures of opposing type” would be active – that’s why there is no pre-dawn adjustment when one would otherwise be expected.
What do these values mean? Well, I wouldn’t write this whole table in my modifiers table; just the conditions and the final scores. Those that got a -5% adjustment for increased risk would be marked with an asterisk.
If it’s, say, Midmorning when I make my encounter roll, then the base chance of an encounter is down 7%, but if an encounter does result, the creature has +1EL relative to whatever the encounter table says. When I select from my list of appropriate encounters, if the creature is normally diurnal, I can choose to use that to “flip” their orientation; if not, or I choose to have them at a disadvantage in respect of the time of day, I can use it for something else.
That’s how you can take a specific condition and turn it into a general one so that there’s no need to differentiate between them in your encounter lists.
How Often should checks be made?
It’s important to realize that if you wait long enough, any given level of encounter will eventually take place. The risks being assessed are the risks in a specific time frame. In roleplaying terms, this should be the risk of whatever time unit is next highest compared to the frequency with which you are going to be making die rolls to check against the risk, on average. If your checks are to be daily, then your base levels should be the risk of such an encounter in a week; if hourly, then in a day; and so on.
This means that you can produce a different base chance chart for each of the major time-spans that you might want to hand-wave. One table – the one I showed earlier – would be reasonable for a week’s, or perhaps a month’s, travels. That’s why there’s such a high risk shown for the “Extremely High” habitation level; daily, it would probably be only half or even a third of that amount, despite the population “density”.
Most of the time, nothing happens.
Other Modifiers to consider
ANY condition that might impact the chance of an encounter can be taken into account.
Are the PCs carrying torches, advertising their presence and potentially attracting diurnal creatures?
Or are they using some form of Infravision to get around without so advertising their presence, which reduces the opportunity for other creatures to flee a potential danger?
The first would increase the risk of encountering high-level creatures who have enough confidence that they aren’t frightened off, while reducing the risk of encountering low-level creatures who don’t have that confidence.
The latter increases the likelihood of low-level encounters and removes the opportunity for high-level creatures to be attracted to the PCs – by the time the relative adjustment is made, the high-level encounter likelihood is effectively reduced. So this would be a case of a modifier with a bias to the encounter chart. In some cases, it would be a reduction in encounter chance, in others it would be a reduction in encounter chance and a bias toward low results on the encounter table – perhaps a -10% or a -20%.
Use your imagination a little. The PCs have a lot of magic with them? Then they are more likely to encounter beings that can detect that magic. They have captured some rare artifact? Then they are more likely to encounter beings that have an interest in that artifact. Both of these suggest an increase in the encounter rating. The PCs are currently engaging in, or have just had, a battle? The noise generated must increase the chance of another encounter, but it will reduce the risk of a low-level encounter (which is more likely to run) and increase the risk of a high-level encounter (which feels secure enough to investigate) – in other words, it would be an increase in the chance of an encounter and a bias high on the encounter table.
Check Frequency Revisited
It gets even more entertaining when the time comes to use the table.
First, vary the frequency of die rolls according to the risk the party are taking. One roll in camp on the way to an encounter per night is enough; maybe 2 a night on the way back. Perhaps you might increase the risk of a low-level encounter if they have no campfire, or the risk of a high-level encounter if they light one.
Instead of a roll every hour when they are exploring the dungeon, make one every 3 hours – but make an extra roll at the start of combat to see if an unexpected encounter will take place during, and another one afterwards to see if one is drawn by the sounds of battle.
By doing this, you let the circumstances help you to determine the nature of the encounter – (What type of encounter would be brave enough to show up in the middle of the fight? Or perhaps the extra is cowering in a hidden lair within the room – until the fight crashes through into their living room. After the battle, looters and carrion eaters are more likely) – and use the history of what the PCs have been and are currently doing to compile modifiers as you need them and as you think of them.
Remember, because of the “Base Chance” system, any factor you haven’t specifically included a modifier for is automatically taken into account. All of them.
The Net Effect
There are obvious advantages to this approach. Not only are you assessing the risks of encounters based on the behavior of the party, encouraging clever game play, but you are adding realism. You are also increasing the risks when the party have more to lose (and hence making the party more interested in the outcome) but you are increasing the risks when they are freshly-weakened by battle. In short, you are making your random encounters more interesting, more relevant, and more appropriate.
Application II: Setbacks And Plot Complications
When originally conceiving of this article, the Random Encounters portion was as far as I intended to go. But in the course of writing it, I began to catch glimpses of other ways of utilizing the same basic approach.
For example, scenario generation – plot twists always happen about 2/3 of the way through the story or later, right? That gives the players time to understand the situation, to have wasted efforts and resources going down the wrong path, to be committed to “X” when they suddenly find that “Y” is what they should have been trying to achieve all along, and still have time to reverse course and achieve a last-minute opportunity to turn things around.
But setbacks and plot complications can occur ANYWHERE in the story, provided they are not resolved until the end. You can work out a table of likelihood of a complication occurring at each given point in a scenario in exactly the same way as I have done with encounters, and roll for them according to how quickly the party are getting through the scenario. If your campaign has strong continuity, you don’t even have to explain them in the course of the scenario – you can just leave the events a mystery for a later occasion.
The PCs are receiving their mission briefing when their car explodes in the parking lot (a medium level setback).
The PCs are about to board their flight to the adventure location when they discover a mistake with their tickets (another medium-level setback).
The PCs break into the vault to recover the stolen crown jewels only to discover that they have tracked down the crown jewels of a completely different Kingdom – which may or may not have been stolen (a high-level setback).
Another favorite trick (if not overused) is to make the triviality of the encounter proportional to the level of paranoia of the party. I once had a group obsessing for hours about a flower, much to my entertainment; a young woman approached a member of the party, took a flower out from a bunch of them that she happened to have with her, and pinned it to the lapel of one of the PCs, simply saying this would look better on you, and then walking away, casually tossing the rest of the flowers in the garbage… (she had just broken up with her fiancé, who had given her the flowers. The PC reminded her for a moment of her boyfriend) – (a low-level setback).
The more of this sort of thing that you can come up with on-the-spot and leave to be explained later, the more dynamic your plots will be as actually played. Some of your best ideas will be spur-of-the-moment and I’ll-explain-it-later. If YOU can’t predict when these things will come up, neither can your PCs – and they will never be completely sure of the significance of any encounter. Your players are more than capable of complicating the lives of their characters with no help from you – given enough rope – and this helps massively when it comes to avoiding plot trains.
Application III: Weather
Another area where this sort of risk assessment approach is useful is in the determination of the weather. Assume an average day and then roll for deviations from it. Here, the risks are unlikely to be modified too much by what the PCs are doing or have done; the basis of the risk is the combination of local geography and preceding weather conditions.
As a trivial news item within one campaign, I once mentioned that a Weather Wizard had escaped from custody while the PCs were on their last adventure, and a large reward had been posted. By sheer chance, every time they passed through a certain small town thereafter, it began to rain. After 4 or 5 such occasions of sudden shifts in the weather, they became convinced that the Weather Wizard was hiding there. He wasn’t, but an economic war was being waged in the vicinity using weather magic, true enough. Why? Because when they went looking for the Wizard, I decided that they should find SOMETHING for their troubles. Meanwhile, the Weather Wizard was safely tucked away in his glacier, putting the finishing touches on his plans to trigger a new ice age, and carefully NOT doing anything to give away his location….
These basic examples show how the principles of risk assessment, as used by insurance companies the world over, can be applied to RPGs in a number of interesting and beneficial ways. But these only scratch the surface. Tactical situations can be described in terms of minimizing risks, for example. There has been no real effort to treat different kinds of risks separately. Even combat can be considered in terms of the level of risk of taking damage given the conditions, defenses, and armaments involved.
And if a really fun idea occurs to you, as GM, you can always fudge the dice. The risk of a powerful artifact disguised as something trivial that everyone wants to get their hands on being mistakenly sold to the PCs for a silver piece must be pretty low… “But he gave all the right recognition signals and code phrases!”
This is the last post at Campaign Mastery for 2016. I had hoped to generate Christmas/New Years Greeting and schedule publication in advance, but time (and Christmas planning) simply won’t permit it. For the first time in about 10 years, I’m taking the Christmas/New Year period off (aside from ongoing maintenance of the site, of course). So let me take this opportunity to wish every reader a Very Merry Christmas and a Happy New Year! See you in 2017…