*As you stand before the ancient cogitator, its arcane mechanisms whirring with the secret knowledge of the Omnissiah, you must decide whether to pledge your sacred Fate Point in advance, thereby beseeching the machine god's divine favour to invert the digits of your result.*
*The choice, as always, lies in the balance between faith and fate, and such decisions are rarely made without consequence.*
*This, young aspirant of the Mechanicus, is why we venerate the sacred calculus of probability. Why the blessed lexmechanics of old inscribed their wisdom into the holy dataslates. And why, above all; **we have graphs**.*

## Okay, what's this all about?

__Imperium Maledictum__ is a tabletop role-playing game (TTRPG) set in __the Warhammer 40k universe__. Players take on the roles of agents working for a Patron of a powerful faction within the Imperium. Strategy, grey moral choices, and teamwork are essential for survival. And survival does not come easy.

Warhammer 40k is a __grimdark__ setting, and the TTRPG spin-offs live up to that. Things get lethal, fast. (I still can't believe that my tech priest in our original Dark Heresy game (another TTRPG 40k spinoff) survived as long as she did.)

Some people play in this universe because they enjoy the opportunities for absurd gallows humour (me). Or getting to pretend that they're a super-serial (because computers - get it? GET IT???) wizard of machinery who is also partially made of machinery. (Also me.)

Another source of enjoyment for players is wresting triumph from the gaping maw of damnation with the divine calculus of the machine god, a.k.a. mathematics. Specifically, probability.

My husband, Karl, is one of these players. And because of this, we recently had a post-dinner discussion that led to him grabbing his laptop and making some graphs while we talked. (Ideal date night TBH.)

## The mechanics of success and failure in Imperium Maledictum

Like many TTRPGs, Imperium Maledictum (IM) determines whether characters succeed or fail a challenging action via dice rolls. In IM, these are referred to as a Test. (I'll be capitalising game-specific terms throughout this post e.g. "I will test your knowledge of Tests.")

Let's briefly explore the concept of Tests by having Sabeen, a navy pilot, roll to perform certain actions.

Sabeen is searching for clues in the crashed ruin of a Chimera transport. To see whether she finds anything, the GM (Game Master) asks her player to roll a **Perception (PER)** Test. Here's what you, as Sabeen's player, need to know to do this:

**A Test is rolled using two ten-sided dice (d10s)**with one representing the "tens" column and the other representing the "ones" column, resulting in a number between 1 - 100 (rolling two zeroes is a 100 here, not a zero).**A Test is rolled against the most relevant skill**(or characteristic - I'm going to use the word "skill" solely from here on to keep things simple) for that character. In this case, it is Sabeen's capability to notice subtle details.**A successful Test requires the player to roll a lower number on their dice than the value of the associated skill.**For example, a player with a**PER**of 35 would need to roll 35 or less on their dice to succeed.

### Levels of success

Let's say our roll was a success. But by how much? This is where Success Levels (SL) come in. These are determined by the difference in the tens digit of your roll, for example:

If Sabeen's target is 35...

...and we roll a 31, Sabeen has succeeded, with 0 SL.

...and we roll a 29, that's 1 SL.

...and we roll a 12, that's 2 SLs.

In this case, the GM determined that a successful roll would reveal a locked storage box underneath the rubble, which would contain a clue vital to their investigation. If the character achieved additional levels of success, the clue may provide additional information related to Sabeen's mission. We'll explore SLs in more detail shortly.

### Fate points

Players have a number of ways they can attempt to stack the odds in their character's favour. One way to do this is to influence the outcome of dice rolls by using a limited in-game resource; Fate points.

Fate points can be spent in many ways. The two we care about in this context are:

Spending a Fate point to

**gain Advantage**for a roll.Spending a Fate point to

**increase the SL**of the roll's result.

#### Using Fate points to gain an Advantage

If a character has Advantage for their next roll, they can choose to swap the "tens" and the "ones" columns for their roll if it would deliver a better result.

For example, initially Sabeen rolls a 51 - a failure for her skill check of 35. However, as she had Advantage, we can swap those digits around to become a 15 - 2 SLs of success!

**The catch:** If you wish to use a Fate point to gain Advantage on your roll, you must declare this **before** rolling the dice. Given you have no way of knowing whether spending the Fate is worth it until after you've rolled, it's a bit of a gamble. (Mercifully, if you declare Advantage and your initial roll is better than what you'd get by swapping your digits, you can keep your original result).

#### Using Fate points to gain a Success Level

As we explored earlier, increased Success Levels can offer additional benefits to a player character. They are especially valuable for opposed Tests, and combat in general. For example...

*Thrown into unexpected combat, your character, Sabeen, grabs a chair (an Improvised Weapon) and attempts to bring it down onto the head of an unusually nimble tithe clerk.*

You roll a 63, with a Melee skill of 69, achieving 0 SL.

The dexterous tithe clerk rolls a 71 to dodge. Their dodge skill is 75, so they've also wound up with 0 SL.

In this instance, the character with the highest skill (the suspicious tithe clerk) would win, dodging the attack. However, if we spend one of Sabeen's Fate points, we can give her an additional SL. This enables Sabeen to win the contest, dealing chair-on-head-related injuries to the clerk.

Increased success levels also increase the amount of damage dealt by a successful attack. If Sabeen had already won the contest, but wanted to *really* break a (chair) leg, she could spend the Fate point to make this an extra-lethal kind of chair, splintering it into pieces in the process. Somewhere, a Manufactory Adept weeps, without knowing why.

Unlike spending a Fate Point to gain Advantage, you can choose to do this after your roll, as we did in the examples above. The only gamble is the initial dice roll itself.

## Statistically, is there a sweet spot for spending your Fate points to gain Advantage or a Success Level?

**Karl's take:** if you really, really, *really* want to succeed, it's worth spending the Fate point in advance to gain Advantage. However, if you have a very low skill level, or a crazy high skill level, then you might actually be better off waiting to see the outcome of your roll.

For an explanation of why, read on.

Here's the percentage change of a successful Test, versus the number of points a character has in the skill being tested.

The dotted line represents the base probability of success, and the solid line represents your chance of success should you choose to spend a Fate point for Advantage prior to making your roll.

How does this chart help us decide whether to add a Success Level or use Advantage? Well, we can think of a free success level as the same as making it ten points easier to succeed on the roll.

**To illustrate:**

Sabeen rolls 79 on her melee check.

Her target is 69, so that's a fail with -1 SL.

Sabeen's player spends a Fate point to increase the SL by 1.

Sabeen now passes with a result of 0 SL - the same outcome as if her target had been 79 in the first place.

So when it comes to our graph, **we can treat spending a Fate point for +1 SL as raising the dotted line by ten**. This means that when the gap between the dotted line and the solid line is less than ten, it's better to use the +1 SL after your dice roll. This also means that if the roll is *really* bad, you can skip spending the Fate point entirely, saving it for a time when it will truly make a difference.

You can see this in the table below, or if scrolling an embedded table isn't your thing, I've also created a PDF download containing the same information.

At the end of this blog post, I'll include the R code Karl wrote to generate the Advantage figures.

### Okay, but why does the Advantage line look so weird?

It *is *weird, isn't it? Adding Advantage creates a staircase-looking sort of line to our graph.

To make sense of this, we need to consider a little calculus, as a treat. (If you do not consider calculus a treat, then please instead consider this wizard pondering the graph.)

What the sloped and flat sections of our graph's "stairs" tell us is:

**"How would my Advantage rolls benefit if I spent XP to increase my skill level?"**

In the sloped section, increasing my base skill by 1 also increases my chance of success with Advantage.

In the flat sections, increasing my base skill by 1 would

*not*increase my chance of success with Advantage.

__Refer to the previous table__ to check out the exact numbers.

As I mentioned earlier: if you're interested in seeing the R code that produced these figures, I'll be including that at the end of the blog post.

### But why does the graph behave this way?

**A hint:** the point where the graph stops being flat is at double digits. For example:

If my skill is, say, 64, then my chance of success with Advantage is the same as if my skill were 65 (due to the '65' becoming a '56').

If my skill was already 65, then with or without Advantage, 65 is also a success.

This means that **increasing my skill level to 65 does not increase the size of my set of successful outcomes **(or change my successful outcome space, modify my Venn diagram, or any other mathematically equivalent term you prefer).

*However*… increasing my skill level to 66 *does* increase the size of my set of successful outcomes - 66 was a failure, and even with Advantage (at skill level 65) it is still a failure! **But if I increase my base skill to 66, then it's now a success.** There's no other dice roll I can get that would change to 66 via Advantage, so it only increases my set of successful dice rolls by 1.

Now, you may have noticed - if you look *really* closely - that there is actually a slightly smaller slope at that change. It goes: flat bit, shallow slope bit, then steep slope bit.

To get your head around this, consider now that we go from a skill value of 66 to 67. Now we're in the steep slope.

If my skill is increased from 66 to 67, then I am adding **two **successful outcomes when declaring Advantage: firstly, if I roll 67, and secondly, if I roll 76.

So now I have increased my set's cardinality (the number of possible success outcomes) by **two** - hence the steeper slope in that section.

**In short:**

If the value in the "ones" column for my skill is in the range between zero and double digits (e.g. 50 - 54) then increasing my skill does not increase my chance of success when using Advantage.

If the value in the "ones" column for my skill is one less than double digits, then increasing my skill by one point (e.g. 54 to 55) increases my chance of success with Advantage by one.

If the value in the "ones" column for my skill is double digits or higher (e.g. 55 - 59) then increasing my skill by one increases my chance of success by two.

Here's a few more examples to help bring this concept home. Use __the lookup table__ as you follow along.

**If Sabeen's Rapport skill level goes from 53 to 54**

Her chance of success was with Advantage was 74.

Her chance of success with Advantage remains 74.

This is because 54 was already a successful roll via Advantage even at skill level 53.

So while your base chance of success improves by increasing your skill level, your chance of success via Advantage is unchanged between these two values.

**If Sabeen's Rapport skill level goes from 54 to 55**

Her chance of success was 74.

It now increases by one point, to 75.

This is because now, 55 is a successful roll.

**If Sabeen's Rapport skill level goes from 55 to 56**

Her chance of success with Advantage was 75.

Now, it increases by two points, to 77.

This is because now, both 56 and 65 will be successful rolls.

Can you guess at what point the slope will become flat again?

**One thing we've not addressed in text, but is present in the graph**

A roll of 96 to 00 is always a failure in Imperium Maledictum (although Advantage may still work there until you hit 99 - 00).

## Bonus: the R code to generate this dataset

For anyone interested in how my husband generated the set of numbers used for this article, or looking to replicate results, here's the code he wrote in R, with his commentary:

Unfortunately this particular time-saving blog CMS I am using does not allow me to add a plugin to auto-colour code snippets. (While this system does offer some nice features, I'm very strongly tempted to switch over my website to being powered by something visually simple but powerful under the hood, like __Obsidian__... (yes, it can be used to publish sites, not just store notes as markdown!))

Another flaw I've discovered with this blog system is I cannot credit multiple authors for a single post. While I wrote the post itself, it was only by talking through the concepts with Karl that I was able to put it together, and the code, graph and initial insights were his. So, credit where credit is due.

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