Variance Supplemental: Combat (Wonkish)

Introduction

In my post about variance in game design I discussed the effect of rolling multiple times for combat resolution and how this reduces variability of combat results compared to non-combat results. This was supported by some simple logic and probability. However, I was interested in getting some actual numbers to back up that hypothesis. Herein I present the results of one of those tests. Now, the variance article was written to try to present the concepts of variance to a more general audience. No such attempt has been made here. You have been warned.

The System

In order to get a more accurate idea of how skill levels determine success in combat I conducted a Monte Carlo analysis of Feng Shui. I chose to look at Feng Shui because I have a fair amount of experience with the system and one of the things we always complained about was that a point of combat skill seemed as valuable as 3 points in a non-combat skill. Thus, I had some idea of how the results would turn out and was curious as to how close our gut interpretation of the difference between combat and non-combat skills would be to the numbers.

Feng Shui Mechanics

The basic mechanics for Feng Shui are relatively simple. You have a skill. To see if you succeed in a task you roll 1d6 and add the result to your skill. This die “explodes” in that if you roll a 6 then you add 6 to your skill and roll the die again and add that number to the skill as well. If you keep rolling 6s you keep adding 6 and rolling again. In this way it is possible, though highly unlikely, to roll a +100 or higher. You also roll a second “exploding” 1d6 and subtract that result from your skill. For example, if you roll a 4 on the plus die and a 3 on the minus die and have 15 skill your result is a 16, 15 +4 –3. Obviously, the average roll in this system is +0.

To succeed at a non-combat task you need to roll equal or better than a target number. This number is usually set by the game master but may also be someone else’s skill, such as rolling your intrusion skill versus someone else’s intrusion skill to see if you get past the security system they set up. In combat you roll your combat skill against their combat skill. Only the attacker rolls, not the defender. If you equal or exceed the opponent’s combat skill you hit and go on to determine damage. You can make yourself harder to hit by giving up some of your actions to dodge, which increases the number an opponent needs to hit you by 3.

Damage is calculated by adding your damage, which is either a straight number for ranged attacks or a number plus your strength for hand-to-hand combat, to the amount you exceeded the opponent’s combat skill and then subtract the opponent’s toughness. Thus it is possible to hit someone and still do no damage. The resulting damage is subtracted from their hit points. Almost everyone has 35. When their hit points reach 0 or lower they are out.

Model Assumptions and Simplifications

In order to be able to reasonably program the model, I had to simplify the mechanics a little. In addition, I was really only interested in comparing combat to non-combat outcomes rather than exploring all possible combat outcomes. So I made certain assumptions and simplifications for the model.

First, I ignored initiative. Initiative in Feng Shui is somewhat complex in that it gives you a certain number of actions per round and different things like dodging and attacking cost different amounts. In addition, the costs can change depending on circumstance. That was just too annoying to put in the model, so I didn’t. However, if I assume that both combatants, let’s call them Pain and Suffering, have the same initiative bonus then given the multiple rounds of each fight and the fact that I’m simulating 50,000 combats initiative should really just fall out in the wash. It does have a couple of implications for the results though. Ignoring initiative allows for ties, Pain and Suffering could both get reduced to 0 or lower hit points on the same round. In play this can’t happen because one or the other would go first and the opponent wouldn’t get to carry out their attack. Assuming that they both have the same initiative bonus then in an actual game Pain would win half of the “ties” and Suffering would win the other half. It also means that when I talk about rounds I’m really talking about one exchange of attacks rather than a Feng Shui turn, which can often contain 3-4 or even more attacks.

I set the base damage, the damage done by a character minus the toughness of his opponent, at 3. Feng Shui is a combat-heavy game and people tend to have decent toughness scores so this is a not unreasonable choice. A smaller base damage would be expected to make the combat last longer, and thus have more rolls, and thus favor higher skill levels even more. A larger base damage would be expected to make combats shorter, and thus have fewer rolls, and thus favor higher skill levels somewhat less. I did run single simulations using base damages of 1 and 5 and the results were exactly as expected, though the changes in the win/ loss outcomes were actually quite small (data not shown).

I also assumed no dodging. The dodging mechanic is tied in with initiative and so can’t be modeled exactly without it. I did run one simulation assuming that both opponents would dodge all attacks. As expected, the combat lasted much longer and the higher skill level character was slightly more successful than without the dodge (data not shown).

I left out the various combat stunts that characters might have, they would be often difficult to model and there are just too many of them and they vary too much between characters. Needless to say, a character with combat stunts will have an advantage over a character without such abilities, but examining this is beyond the scope of this analysis.

I also left out Fortune points. Fortune points are a limited supply of points that can be spent to add one, non-exploding, d6 to a skill roll or active dodge. Most characters have few, if any, Fortune points and in our games we tended to roll many combat rolls. So overall the effects of Fortune points were minimal for most characters. However, some characters have a fair number of Fortune points, and if they know that this is the last combat of the session they might as well use them fast and furiously. In addition, while most combat stunts just amplify normal combat parameters, i.e. you do more damage or attack more often, some stunts can very seriously inconvenience an opponent. For such stunts Fortune points can become quite powerful. For an example of a Feng Shui combat with stunts and Fortune points see Sophie’s write up.

Feng Shui Results

In order to test the model I set both combatants, Pain and Suffering, to the same combat skill level and ran 50,000 simulated fights until either Pain, Suffering, or both collapsed. The results are shown in Figure 1.

fengpaine

If the model is working, then two combatants with equal levels of skill should have equal numbers of wins as well as a few ties. As you can see, this is the case. It can also be seen that the typical combat length is around 8 to 10 rounds. This is 8 to 10 attacks and with Feng Shui turns being about 3 to 4 attacks per turn, the combat should last about 2 to 3 turns. This is in keeping with my personal experience with the system and made me more confident of the model.

To test the effects of skill difference on combat outcomes, I ran tests setting Pain’s skill at 1 to 3 points lower and 1 to 3 points higher than Suffering’s. Unless something is wrong with the model, the results should be symmetrical about the even skill result. The results for Pain at one skill lower than Suffering should be the same as the results for Suffering at one skill lower than Pain. This was the case. Also, the 2 points lower result was intermediate between the 1 point lower and 3 points lower results. So Figure 2 only shows the results for Pain at 1 and 3 points lower than Suffering.

fengpain1a3

As expected, with a lower skill level Pain is much less likely to win the combats than Suffering and that chance decreases as Pain’s skill is reduced. The combats also tend to be a little shorter, typically about 7 to 9 rounds for one lower skill and 6 to 8 rounds for three lower skill.

But the real question is how this compares to the chance of success with non-combat skill. This poses a slight problem. Non-combat success is obtained by rolling equal to or greater than the target number. Basically, ties go to the player character. However, combat would not normally have ties, but it does because I ignored initiative. Assuming equal initiative bonuses, the player would only win half of those ties in an actual game. However, in order to make the non-combat, player wins ties, more comparable to the combat results I added together both Pain wins and ties. This is the equivalent of the player winning all ties. This isn’t quite accurate, but seems a fair way to compare the two. Additionally, the number of ties is relatively small and this decision does not change the final numbers by that much. The results are presented in Table 1.

Table 1: Feng Shui Combat Versus Non-Combat Results

Pain’s Skill

relative to difficulty

Non-Combat Combat
% chance of success % chance of win or tie
3 84.3 99.6
2 77.6 97.9
1 68.6 87.9
0 57.1 53.8
-1 42.9 16.2
-2 31.4 3.2
-3 22.4 0.7

You will notice that even being generous to Pain by granting him all ties, the success rate falls off quite dramatically as Pain’s skill gets lower, proving the hypothesis that combat results are more consistent than non-combat results. Another interesting result is the chance of success at 1 skill higher for combat is about equal the chance of success for 3 points higher on a non-combat skill, my Feng Shui group’s gut reaction was pretty accurate!

Conclusion

I feel satisfied that combat results really are more consistent than non-combat rolls.

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2 Responses

  1. […] Game Design article. While there are some new observations, it covers much the same ground as the first supplement but for a different system, Spirit of the Century. So if you haven’t read the first supplement […]

  2. […] than the Feng Shui fights, tending to be around 16 to 19 rounds long rather than 8 to 10 (see supplement 1). After all, it takes only 9 minimum damage hits to take someone out in the SotC example while it […]

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