Math Nerds Needed, Probability, curves, comparisons, oh my! Options

[noonesshowmonkey]

I am putzing about with a game design that I'd like to use in a Birthright-esque game. The core mechanic is based on a Risk themed roll & compare.

The core mechanic is Roll + Keep + Compare. Players Roll a number of control dice generated from their statistics, Keeping a certain number of their best dice as determined by their statistics, and then Compare these best dice against their opponent's. Control dice can be of different weights, d6 through d12.

For example, a player has 6d6k3, and rolls 6 six-sided dice, keeping his best three. His opponent has 5d6k3. The two roll off in a test.

Player A: 6 - 6 - 5 - 3 - 2 - 2
Player B: 6 - 5 - 4 - 3 - 3 - -

The two 6s tie, Player A's 6 beats B's 5, and Player A's 5 beats player B's 4. The total is two net success for Player A.

Now, here is my problem: I need to make sure that I can generate broad, reliable probability curves that scale by die number and by die weight. I need to make sure that it isn't wildly, obviously, painfully better to have 6d6 than 4d8 or some such. Or that any other such optimizations are limited in both quantity and quality. The system would ideally use dice pools ranging from 4 to 10 dice, as any fewer make for easy statistical aberrance and any more are unwieldy to pick up and use. Most tests would keep between two and six dice.

I am not really sure how to generate numbers to describe exactly how beneficial keeping an extra die is in a given test with a given number of dice, with a given weight, against a different number of dice of a different weight.

Halp?

[Aaron]

Do you have the opportunity to take a statistics course?

EDIT: There's a nifty site called AnyDice that is pretty cool.

[thorya]

Do you have access to matlab, because I can write you a program to do this for you if you do. (I could probably do it in visual too, but I'm really rusty, so it might not be worth the time)

Anydice is pretty good, but getting it to work for some things can be hard. How does the compare work if people are keeping different numbers of dice? Is it an automatic success?

Are you looking for a spreadsheet or a program? Because with the three factors and varying levels of success, it will be hard to represent this concisely.

[thorya]

Another note: For quick consideration, remember the huge role that the dice size will have in this. With a d6 vs. d8, the d8 wins at least 25% of the time regardless of the dice pool size, simply because 25% of the time it will roll something that the d6 can't beat no matter how well it rolls. At the more extreme, d12 vs. d6, the d12 will win at least 50% of the time regardless of the dice pool size.

So if the choice is between increasing the dice pool size and increasing the dice size, it's almost always better to increase the dice size.

As far as increasing dice pool, you pretty quickly hit diminishing returns for adding more dice because you're taking the highest. It's not quite as bad as some other cases because you're comparing several dice, but it still rapidly approaches the maximum roll on the dice. Which means that for the first comparison, you're very likely going to be comparing the maximum numbers each time.
Think about it this way: If I'm rolling one die I have equal probability for any number. If I'm rolling two dice and keeping the highest, I'm going to roll something in the top half of the die's range 75% of the time (4,5,6 for a d6). (because 75% of the possible combinations have one dice in the top half of the range). If I'm rolling three dice and keeping the highest, I'm going to roll something in the top half of the die's range 87.5% of the time. If I'm rolling four dice and keeping the highest, I'm going to roll something in the top half of the die's range 94% of the time. For 5 it's 97% of the time. For 6 it's 98% of the time. The chances of getting a six as your highest is: 1 die= 16.7%, 2 dice= 30%, 3 dice= 42%, 4 dice= 51%, 5 dice= 60%, 6 dice= 66%.
Which means for 4 dice on a d12, I'm going to beat a d6 at least 94% of the time, even if they are rolling infd6. (assuming we're keeping the same number of dice)

That might work for your system, not entirely sure what sort of breakdown you're looking for in terms of odds.

Also, consider your example case of 6d6 versus 4d8, I imagine you want them to be somewhat similar, but just my gut on the odds tells me that the 4d8 will be better. If we are keeping just 1, we can quickly calculate that 6d6 only wins 11% of the time, while the 4d8 wins 75% of the time (tying the other 13% of the time). The odds get slightly better as you keep more dice, but the larger dice still has a distinct advantage. I would worry someone less mathematically minded would look at this and say, 6+6=4+8=12, so they should be the same! I'm rolling 6 dice how is he beating me?

If keeping more than your opponent is an automatic success, it becomes the most important factor. Because it gives automatic successes that an opponent can't beat regardless of how many or what kind of dice they roll. If I can get to keep significantly more than an opponent, they have no hope of winning.

Edit: To more clearly indicated the relative differences each makes, a number of cases are studied.
Odds for 6d6 vs. 4d8, keep 2 (from a quick monte carlo simulation with 4000 cases, so expect +/- 2% on these values): 4d8 wins 63% of the time, 6d6 wins 17% of the time
Odds for 6d6 vs 4d8, keep 3 (again 4000 trials): 4d8 wins 58% of the time, 6d6 wins 30% of the time.
Increasing to 7d6 makes it:
Odds for 7d6 vs. 4d8, keep 2: 4d8 wins 61% of the time, 7d6 wins 18% of the time.
Odds for 7d6 vs. 4d8, keep 3: 4d8 wins 55% of the time, 7d6 wins 33% of the time
Increasing to 4d10:
Odds for 6d6 vs. 4d10, keep 2: 4d10 wins 80% of the time, 6d6 wins 8% of the time
Odds for 6d6 vs. 4d10, keep 3: 4d10 wins 76%, 6d6 wins 17%
Now the situation when you have both dice pool and dice size advantage:
Odds for 4d6 vs. 6d8, keep 2: 4d6 wins 5% of the time, 6d8 wins 88% of the time
Odds for 4d6 vs. 6d8, keep 3: 4d6 wins 6% of the time, 6d8 wins 90% of the time
And when one has a large advantage in both dice size and dice pool:
4d6 vs. 7d10, keep 2: 4d6 wins 1% of the time, 7d10 wins 97% of the time
4d6 vs. 7d10, keep 3: 4d6 wins 1% of the time, 7d10 wins 97% of the time
Finally, changing dice pools with the same size dice:
Odds for 6d6 vs. 4d6, keep 2: 4d6 wins 26% of the time, 6d6 wins 56% of the time.
Odds for 6d6 vs. 4d6, keep 3: 4d6 wins 24% of the time, 6d6 wins 63% of the time.

Conclusions:
Dice pools matter a lot for the same dice size and matter a lot less when the dice sizes differ. Which is actually workable, if you reflect the more significant role of dice size in your description. I.E. Maybe a factor that's really important changes dice size and a secondary benefit changes the pool size. When you have very neck and neck competitions, the little factors shine, but they're unlikely to be enough to win the day when completely outclassed. Against a similarly skilled opponent, your personalized sword that you practice with and has your favorite grip is a slight edge. But your personalized sword grip doesn't really help very much against a world class fencer if you usually compete at a local level.
The keeping more dice tends to benefit the larger pool, but doesn't change the outcome drastically.
There is still a significant chance that someone with only a slight edge in dice pool size or a slight edge in dice size but a disadvantage in pool size will lose. Someone with a small advantage in both has very little chance of losing, though it's still likely to occur in play occasionally.
Possible problems: There are a lot of ties, which can lead to a lot of indecision or possibly a lot of rerolling. Possible solution: Keep 2 and use the 3rd as a tie breaker (and 4th if necessary, etc.)