Monday, October 5, 2009

Gear Probabilities

So Seth Jaffee from over on Cumbersome Blog gave Gear Burn a try last night.  I really appreciate him trying it out, and he had some great feedback on it.

One of the big things he mentioned was how rough the gear probabilities were.  Right now, I just have a player roll a number of d6's (Six-Sided Dice) equal to their gear.  So 2d6 in second, 3d6 in third etc.

Then a player looks at the largest value showing on a single die; that is their speed.
Then a player looks at the smallest value showing on a single die; that is their maneuver.

This works out really well on the margin.  Each time you shift into a higher gear your speed will increase and your maneuver will decrease on average.  But the question is of course are there enough dice rolls to average out?  The answer is.. kind of.

The biggest glitch is probably first gear.  A uniform distribution from 1-6 has a dang high variance for a race game.  It can really suck to be 5 spaces behind after the first turn :(

I'm going to define a new stat too, the difference between the average speed and maneuver in a given gear shall be the "drift".  This is how much closer to the wall a car will veer.  It doesn't need to appear in the rules, but it does need to be addressed when designing the math behind them.  Anyway, the drift is always zero for a single die.  But that drift jumps fast when moving from 1d6 to 2d6.  The drift goes up 2 units for only 1 unit increase in speed.

So, Seth played once with the just Nd6 gears, but then tried replacing them with a more interesting distribution of dice.  I'll outline it below, but the most important differences were:

1. The drift and speed increased at about the same amount.
2. The speed kept increasing at at least +1 for each gear.

The downside to Seth's?  Not everyone has a d3.  Also, the jump from 1st to 2nd was weak compared to the rest of the gear jumps.

I thought I'd just go ahead and outline a lot of different gear possibilities and calc the math on them.  The results are in this chart:

So, what does the above chart say?

Original - Yea, it looks pretty crappy when shown in this light.  I'm definitely moving away from that.  1st gear was highly variable and 3rd vs 4th gear didn't matter at all.

Seth - Pretty solid really.  I'd like to see the margin smoothed out a little more.  The drift per speed is still pretty flat between 3rd and 4th.

Fixed Pairs - Rather oddly, the Drift doesn't increase nearly as fast as the speed.  I'm afraid this would come down to who could roll double 10s in 4th.  Also, the drift per speed is just flat.  Might as well run top gear.

Simple Additive - Here we start with what I call "additive" styles.  This is because each consecutive gear just adds one die to the prior gear.  Would be easy to make and hardly require any dice at all.  Simple additive actually has some very nice characteristics. Except for 4th gear, it keeps roughly the same absolute drift numbers from the original design.  The drift per speed is pretty close too; hopefully that means the maps wouldn't need much tweaking.

Big Base Additive - Starts with 2d4 and then builds.  Really harsh drift numbers, even drift per speed numbers.  Probably couldn't go very fast on this one.  The speed/drift margins don't look much better than the original :(

Slow Additive - Here is probably the most promising to me.  It's got steadily increasing marginal speed boosts.  3rd and 4th gear should still feel very different, and 4th should bring more danger.

Anyway, this is way too much information for a simple dice game.  But hey, the math is fun to some of us!


  1. This is really intereating, although I admit that at first blush I don't really understand your "drift" value or how it's useful.

    I like the additive version, requiring only 1 of each die in the game - it's elegant and keeps components down. HOWEVER, I'm not sure it's better than my version for this reason:

    A player upshifts because they want to haul ass. If the chances of actually moving 7+ spaces aren't that good, then why take the risk of upshifting to 4th? That's kinda what happened with 3d6 vs 4d6 in the original when Iplayed it - the MAX speed was 6, and I was already fairly likely to get that in 3rd gear. I seldom felt inclined to upshift to 4th to help guarantee a 6 at the cost of likely less maneuverability.

    WAY more fun to upshift to bigger dice, and much better chances of GOING REAL FAST, as well as more chances of not maneuverig well.

    So I'd kinda like to see not just the avg values, but the distribution curves for each of these setups. Because if I wanna go fast, I think I'd rather roll 4d8 than 1d4, 1d6, 1d8, and 1d10.

    - Seth

    P.S. 1d3 is easy, just roll 1d6 and devide by 2.

  2. Oh yeah, and of course the main reason I started fiddling with different dice was this: I thought it was lame that you could go as fast (indeed, faster) in 1st gear than you could in 3rd.

  3. Trust me, you're not the only one to mention that it's possible to go faster in 1st than 3rd. It's true that on *average* 3rd is faster, but not always.

    The "Drift" is important because it's sort of what controls crashing. Actually, Drift Per Speed would be the critical stat. Drift Per Speed would be how many spaces you are forced towards the wall for each space forward. The higher that stat, the riskier the gear.

    And yes, I agree that originally, there was very little difference between 3rd and 4th gears.

    I'll just have to try out a couple of the above ones. I think I'll probably lean hard on one of the additive ones for the same reasons you mentioned: it keeps the components down :)

    However, I also like your comment about how nice it was to just haul ass in 4th when you really wanted to.

    Hrm hrm hrm.

  4. (I found this via Seth's post.)

    This is very interesting. There is one thing I want to mention: in your chart, you're looking at the difference in speed as a measure of how much you gain from one gear to the next. And you commented, for example, that in Seth's version, 2nd gear doesn't gain you much.

    I think that's looking at it wrong. Your absolute speed doesn't matter; the number you're trying to optimize is something like turns to destination. Say you have a destination 100 spaces away. If your (average) speed is 2 and the next gear's speed is 3, shifting will gain you 17 turns. If the next gear's speed is then 5, shifting again gains you only 13 turns. The gain from going to second gear is greater even though the absolute speed increase is half the size. From your comments, it sounds like you are looking more at the absolute increase (at least in your theoretical analyses) and would thus be undervaluing speed increases in the low gears and overvaluing speed increases in the high gears.

  5. Indeed, that is a good point Sean. The current track is about 85 spaces long. I should probably judge how much of the track each gear eats up.


  6. Aaand now that I think about it you probably care even more about how many turns you're gaining *per turn*, rather than how many turns you're gaining over the whole game.

    In my example, that would be 17/33 for 2nd over 1st gear, and 13/20 for 3rd over second.

    More simply, it's (New Speed)/(Previous Speed) - 1, and it means that if you want the gear differences to have approximately equal speed benefits, you want the ratios to be similar. So for example average speeds of 2, 3, 4.5, 6.75... Not that it's clear that you want the speed gains to be equal. The middle speeds also have the advantage of being more quickly adjustable to other speeds, and the dangers of different speeds probably don't change linearly either. But anyway... I think my observation about you underrating the early speed gains is still accurate. :)


Behave. Your mamma could read this.