The Correlation Effect

The Warhammer Fantasy Roleplaying Game 3rd edition (from now on WFR3) employs an unusual resolution mechanic with specialized dice. The results are very non-linear and the resulting probabilities are not intuitive. As a result there have been a number of good statistical analyses of the new mechanic. So why am I doing another one? One aspect of the mechanics is generally left out of these analyses because it is such a pain to deal with. So, in my fine tradition of dealing with stupidly complicated game mechanics for no good reason, I decided to see just how important this aspect really is to the game.

The Question

So what is this aspect? That takes a bit of explaining. Thanks to the different specialized symbols rather than just numbers like an ordinary die WFR3 has several axes for resolution on any roll.

What do I mean by that? The classic d20 system only has one axis. Roll a d20 and add a number to it. The result can be higher or lower but it just produces one number. This is why in d20 systems everything uses a separate roll. In contrast the One Roll Engine, as the name indicates, resolves entire actions from initiative to hit location with a single roll. Its dicing mechanic has two axes. A player rolls a number of 10 sided dice and looks for matching numbers. For example if 3 of the dice showed the number 2 this would be 3 2’s, while 2 dice showing 8’s would be 2 8’s. The number of matching dice and the number on the die give two different results to use to resolve an action. This is why I don’t care for the One Roll Engine. It only has two axes. However, those two results are used to determine every aspect of a roll leading to strange links. So an attack will have initiative, damage, to hit, and target location, four different things but using only two numbers for them. Thus the earlier an attack goes in the round the more damage it does and it is easier to avoid an attack that might hit your feet than it is to avoid one that might hit your head. 

WFR3 has five different axes. It has success, boons (basically side effects, negative side effects are called banes but are really just negative boons), a special bad effect called a Chaos Star, a wild card called a Comet, and the possibility of a separate penalty. For those that know the game a roll can produce a delay penalty or an exertion penalty. While these are different only one type is possible on any roll so there is only one extra penalty possible and not two. Having this many axes allows the game to make task resolution as complicated as they wish with many possible outcomes including succeeding but with problems or failing but with some positive outcome.

Now to the question. When conducting statistical analyses of the rolls these different axes are treated as independent. If a roll has a 50% chance to succeed and a 50% chance of getting a positive side effect this is treated as 25% chance of succeeding with a positive side effect. However, they are not independent. If the good die that can roll successes or positive side effects rolls successes then it didn’t roll positive side effects. Likewise a bad die that rolled negative side effects wouldn’t have rolled negative successes. As a result rolls with large numbers of success should tend to have worse side effects while rolls that fail should tend to have positive side effects.

This would be irrelevant if the side effects didn’t have anything directly to do with success or failure. However, interpreting the results from the die roll is done based on the action the character is taking. Talking to someone has different possible outcomes than swinging a sword. For many of these actions the positive side effects amplify the results of a success. This means that positive side effects are often only useful if the task succeeds. So if positive side effects are skewed to occur when the task fails this could significantly change the distribution of the final results.

Correlation of Success and Boons

In order to test how boons (side effects) correlate with success I chose to look at a dice pool of 2 blue, 2 red, and 2 purple dice. This is just gibberish to those unfamiliar with the game. In brief, blue dice are good, purple dice are bad, and red dice basically good but with some bad results on them. Relatively small numbers of dice were chosen to try to maximize the correlation. Larger numbers of dice will tend to even out across the outcomes as it becomes less and less likely that all of the dice will roll the same result and so the results will behave closer to true independence.

In order to model the exclusionary effects of actually rolling the dice I constructed a Monte Carlo model of WFR3 dice rolls. 1,000,000 rolls of the dice pool were generated.

(As a side note: the number of distinct results generated by the Monte Carlo program given the five different axes, well four really as I didn’t use any dice that give Comets, was 390. This gives some idea of just how complicated they could make results using this system. Fortunately, most cards only have 3 different levels of success and often only 4 or 5 levels of boons so rolls only yield dozens rather than hundreds of different results.)

The basic distribution of boon results, ignoring successes for the moment, was compared to calculated results. This provided a test of the Monte Carlo model. As can be seen in Figure 1 the Monte Carlo results are effectively the same as the calculated results.

Figure 1. Boon Distribution from Calculated and Monte Carlo

control
To see how the boons are affected by success I broke the results down by type of success. In this case I chose three categories, failure, success, and three or more successes. This is how most actions are broken down. Some use two or more successes, or four or more instead of three, but three is the most common break point for succeeding very well on a roll. Figure 2 plots the distribution of boons within each category along with the distribution if they really were independent variables.

Figure 2. Distribution of Boons

boons
There is a very pronounced skew in the results. On rolls that fail the roll is less likely to score 0 or fewer boons than would be the case if the results were independent but more likely to score 1 or more boons. At the level of 3 or more successes the chance of also scoring 3 or more boons is around a tenth of the independent result.

Correlation of Success and Exhaustion

The extra penalty associated with red dice is exhaustion. What’s interesting from a correlation point of view is that the exhaustion result is linked to a success result. This means that rolls that result in exhaustion should be more common for successful rolls compared to rolls that fail. In addition, because the red dice also have a 2 success roll, which wouldn’t be rolled if the 1 success plus exhaustion side came up, there may be a skew against rolling exhaustion in the high success results. Breaking the results down as above, Figure 3 shows how exhaustion correlates with success.

Figure 3. Distribution of Exhaustion

exhaust

As can be seen, failed results are less likely to result in exhaustion. The effect is very low when it comes to 3+ successes but there is a tiny reduction compared to a simple success. Of course, this doesn’t mean all that much. Exhaustion is a cumulative penalty that only causes problems over time and so it is irrelevant whether exhaustion occurs with failed or successful rolls. While the results may not be independent, from a game stand point they might as well be.

Does Any of this Matter?

As mentioned above, even if a skew exists it may not matter much from a game mechanical point of view. In order to test the results of an action I applied the rolls to the action Trollfeller Strike. For those unfamiliar with the game this is an attack developed by Dwarfs to kill big monsters like Trolls. It is also a commonly used action example. It has two levels of success, base damage +1 and for 3+ successes base damage +3. Since all successful results include the base damage and the overall chance of success is identical between the independent and Monte Carlo models the actual level of the base damage is not that important so I’ll indicate it with a D. Trollfeller also has three levels of positive boons, +1 damage and ignore armor, +3 damage and 1 critical hit, and +4 damage, ignore armor, and 1 critical hit. Table 1 shows the percentage chance to obtain each of these results for both models as well as the average bonus damage per attack, average critical hits per attack, and percent of hits that ignore armor.

Table 1. Trollfeller Results

Result Percent
Independent Monte Carlo
D+1 26.1 27.2
D+2 no armor 7.6 7.6
D+4 critical 4.3 4.1
D+5 no armor critical 2.5 1.7
D+3 11.8 14.7
D+4 no armor 3.5 2.7
D+6 critical 2.0 0.9
D+7 no armor critical 1.1 0.1
Bonus damage 1.40 1.28
Criticals 0.10 0.07
Ignore armor percent 25 21

There certainly is a difference when non-independence is taken into account. However, that difference is pretty small (0.12 points of damage). If D were 10, pretty common for this attack, then the average damages would be 7.29 and 7.17 respectively, a difference of less than 2 percent. If ignoring armor were worth 2 extra damage done, it would barely break 2 percent. The effect is present but even in an example chosen to maximize the difference it’s just not large enough to matter.

This is good news for modelers since assuming independence makes calculations so much easier.

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