21 September 2009

Math of 2DX Dice Systems

Hi, this is Kit from Orloff Military Academy here to do a guest post. I got the idea from my time playing BattleTech, but it didn't seem quite specialized enough to post on my blog. Luckily I have been kindly permitted to post it on GBR, so I would like to start out saying thanks for the opportunity.

This is an article I have been wanting to do for awhile and involves the statistics of systems that roll two identical sized dice. The one I actually know best is 2D6 systems because of my hobby playing BattleTech, however the math remains pretty much the same for all 2DX systems. Because of this I decided to assume 2D10 is being used to make the examples and graphs a bit more clean.

Simple Distribution

The first thing I feel is worth talking about involves the simple distribution that arise out of these systems just as a matter of course when you add the result of both dice together.

One of the easiest to spot is the % chance that any particular number will be rolled when using a 2DX system. The lowest and highest value will only have one possible dice combination (2 and 20 for the 2D10 example). The number of combinations lineally increases at a 1-to-1 ratio as you move toward the value of [N/2]+1 where N is the maximum sum of the two dice (in the following examples N=X+X=10+10=20). This is actually a very simple distribution as you can see from the graph:

Chance To Roll Better

From that simple distribution we get a very nice probability curve for attempting to roll equal or better than a target number. This type of roll is actually quite typical of games where some sort of roll must be made in order to successfully complete actions that can range anywhere from combat to convincing a merchant to lower the price of goods.

Take a look at the graph to the above. Notice that the chance of success is not linear at all, but rather curves more steeply toward the center, while leveling off near the ends. This is a direct result of the probability distribution I talked about above and it has a very significant impact on game play: a change of 1 to the difficulty of a roll will have a more profound impact on success toward the center of the possible number range than at either end.

For example, the difference between a target number of 11 and a target number of 12 is a 10% loss in success rate (from 55% to 45%). In contrast the change from 19 to 20 is only 2% difference (1% down from 3%). Likewise there is only a 2% impact in a change from 2 to 3.

Impact: Opposed Rolls

What I call an opposed roll is when two different rolls are made: one by whatever is trying to take the action (often a player or an NPC) and another by whatever is opposing them. In effect the second dice roll is setting the target number to beat on the spot.

In this type of play players need to be very aware of bonuses and penalties because so many rolls fall into the central portion of the graph and a penalty or bonus of 1 can have a profound impact. For example, 70% of all rolls in a 2D10 system will fall into the 7-15 range and 58% actually fall into the 8-14 range. This is in the steepest portion of the probability graph.

The result of this is that any type of action that poses a penalty upon the opposing roll without penalizing the player equally (or more), or one which gives the player a bonus without providing an equally helpful bonus to the opposed roll is amazingly powerful. This results in a 7% impact on the result from just a change of 1 to the dice roll. That is quite the significant difference. However conversely penalties and bonuses also reach a point of diminishing returns - which means that after a +3 or +4 bonus (or -3 or -4 penalty) is rarely worth going for more it if there are costs associated with doing so even if higher bonuses/penalties are possible. Dan was actually kind enough to make the following table:

Impact: Combat Systems

Another thing to consider is the style of combat that BattleTech uses: where factors such as range, movement, and so forth set the difficulty to make a shot. The target number doesn't change, however often any action you take to make your target easier to hit will also make it easier to hit you. By the same token, anything you do to make yourself more difficult to hit makes it more difficult for you to hit something else.

Now this means that when taking an action that will make your target more easy to hit you need to ask if the cost is worth it since typically you will be easier to hit as well. The way to do this is, thankfully, quite easy. You calculate how much your chance to hit the enemy will change and divide that by how much their chance to hit you will change. The larger the number the more the change favors you, while the smaller the number the more it favors the enemy (a result of 1 means things are perfectly even).

Doing this it quickly becomes apparent that if the enemy's target number is at either extreme there is usually no reason to not make both them and you easier to hit - if they are always hitting anyway you are best served to improve your hitting chances as well, or if they are always missing because of high target numbers any change is not too significant. This holds especially true if your target numbers fall toward the middle of the target range. Of course this works the other way as well and the logic is almost exactly the same when it comes to raising target numbers.

If anyone spots a flaw in my numbers/reasoning or has something worth pointing out please feel free to chime in! There is actually a lot more that can be talked about regarding these types of dice systems so don't hesitate to share your thoughts.

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Christian Lindke said...

This is a wonderful thought provoking piece and one that I would love to see expanded with examples.

Would it be possible to see your Battletech relative advantage calculation used in play? It is an easy concept to understand, but I would love to see it in action.

Dan Eastwood said...

I'll take a stab at this Battletech example. Suppose that you are moving last and have two choices for your move:

1) If you stand still you will have a Target Number (2d6) of 9 or better to hit your opponent (15 in 36 chance), and your opponent has a TN of 6 (26 in 36).

2) If you walk your mech 3 hexes, gaining a +1 defensive modifier and a +1 offensive modifier, you will have a TN of 10 (6 in 36) and your opponent will have TN 7 (21 in 36).

If you choose to walk, your probability of hitting will decrease by 4 in 36 (4/36) chance (relative to standing), and your opponents chances will decrease by 5/36. Your opponent loses more in absolute probability of hitting (5), but GAINS in relative probability of hitting (opponent/you): 26/15 ~= 1.7 versus 21/6 = 3.5

So far walking doesn't look like a great choice, but let's add in range:

3) If you walk your mech 3 hexes AND decrease your range bracket from long to medium (reducing your range penalty), gaining a +1 defensive modifier and a -1 offensive modifier (relative to standing!), you will have a TN of 8 (15 in 36) and your opponent will have TN 7 (21 in 36).

Now the relative probability of hitting is: 26/15 ~= 1.7 versus 21/15 = 1.3, a change in your favor.

The lesson is to try to move so that it gives you a relative advantage. In these examples your opponent has the absolute advantage (lower TN) and won't be motivated to move unless forced. You will want to take away that advantage as quickly as possible, probably by moving closer.