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Faragdar the Wise
Commander
3440 Posts
    
Albuquerque, NM, USA
 | | 09/24/2006 11:47 AM |
| I imagine this has been done before, but I started looking at probability to get a handle on this elusive thing folks call the "spawn curve". Since I didn't find a previous analysis, I went ahead and did it myself. I wanted to get an idea of the probability spread for spawn totals, so this is what I've found:
On the first turn, since 1s and ties are rerolled, you can only get a result from 5 to 11 spawn points. The probability of each result is:
5 - .1
6 - .1
7 - .2
8 - .2
9 - .2
10 - .1
11 - .1
Average result is 8, high probability of getting 7 - 9. But still, the probability of 7 - 9 is only a little higher than a 5,6,10 or 11 result. Though you want to skew toward the middle, the probability of a 5 or an 11 is significant enough that it should be planned for.
In the second turn, ties will be rerolled. With the abilities of the minis from the first set, it is impossible for either player to win the first turn. (Whirlwind Dervish is too expensive to be spawned and immediately Teleport Self in the first turn.) It's also impossible for any creatures or locations to be in place to generate extra spawn points in the 2nd turn, so you can count on a "straight-up" spawn curve. I could look at all possible initiative results, but the only things that matter are a) the probability there will be a spawn phase, and b) the total of the two dice if there's a spawn phase.
In turn 2 (or any turn where ties must be rerolled and a turn without spawn phase is possible), there is a 1/3 chance of no spawn phase. Assuming there is a spawn phase, you have the same probability curve as in the first turn--8 is average and 7, 8 and 9 are equally likely results.
There are two more probability curves to look at. In the 3rd and subsequent turn, ties may be allowed. Also, if the previous turn had no spawn phase (assuming tournament rules), then 1s will be rerolled.
Ties allowed, 1s are not rerolled:
In this case, the chance of having no spawn phase is a teensie bit less than one in three. It drops to 11/36, which is still roughly 1/3. Assuming there is a spawn phase, the curve looks like this:
4 - 1/25
5 - 2/25
6 - 3/25
7 - 4/25
8 - 1/5
9 - 4/25
10 - 3/25
11 - 2/25
12 - 1/25
Note that 8 is still the center of the curve. In fact, it is now the highest probability result. In the last case, where ties are allowed but ones must be rerolled, the curve looks exactly the same.
People familiar with games of probability like craps are generally programmed to know that 7 is the most probable result when you sum two d6s, but in Dreamblade, you have to remember that the total of the 2 initiative dice only matters when there's a spawn phase. If you throw out all the 1s, then 8 is the most probable spawn total result.
Last note: If it weren't for the tournament rule than you can't have 2 no-spawn turns in a row, you could say that, on average, 1/3 of your turns after the first would have no spawn phase. With the tournament rule, looking at the long haul, it got a little harder to analyze the different paths, but to the 4th turn, I figure there's roughly 1/3 chance you get a 4th spawn phase. It's roughly 5/9 chance for 3 of your 4 turns have a spawn phase, and about 1/9 that you'll have had just two spawn phases (though that automatically means your 5 turn has a spawn phase). Maybe I'll look out several more turns, but based on that, it seems like about 4/5 of your turns should have a spawn phase. I'm not sure what that means for planning a strategy, though. | |
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Prince o the Raven Banner
Sergeant
606 Posts
| | 09/24/2006 4:11 PM |
| Locally, we've switched to using tournament rules for virtually all of our games, even the casual ones. If this is being done in a widespread fashion, then we could assume that your 4/5 number is a reliable guide to real play.
Thank you for writing this up, I looked and couldn't find it either. Strange that it is so widely referred to. | |
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Orc Mage
Skirmisher
23 Posts
| | 09/24/2006 10:40 PM |
| I suspect that since the game is sold as a tournament based game, we are all playing tournament rules in order to improve our gameplay and really get to grips with it.
We've been playing with your stats in mind from pretty much the beginning, it's nice to have our thoughts confirmed and nice to see that we're all playing from the same page.
Thanks for the effort. | |
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XAos
Underboss
2358 Posts
London
| | 09/25/2006 3:03 AM |
| The only time I have seen the game played by the rulebook on spawn-rolls. It was a couple of players who had not seen the DCI floor rules. As Orc Mage said, most players find it more convenient to using tournament rules for all games.
On the probability of getting a spawn phase, it occilates between 3 values {25/36, 20/30 & Guarenteed}. Depending on the two issues of did the current turn have a spawn phase. & is the score currently a tie.
Overall the average is about 7/9ths of the turns will have a spawn phase. The average cannot be precisely calculated. because you can't predetermine how many of the turns will have a tied score. However the maximum increase above 7/9 is only 2%
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Faragdar the Wise
Commander
3440 Posts
Albuquerque, NM, USA
| | 09/25/2006 10:14 AM |
| | Of course, that 7/9 is for turns after the first. That's probably a better stat to keep in mind, since it's also impossible to predict the number of turns in a game--a number that would skew any statistic that included the fact that the first turn always has a spawn phase. | |
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DMG
Sneak
107 Posts
Dallas
| | 09/26/2006 4:10 AM |
| hmm...
I always thought "spawn curve" referred to the collective spawn costs of the miniatures you chose to include in a warband...
Did I misunderstand? | | | |
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Faragdar the Wise
Commander
3440 Posts
Albuquerque, NM, USA
| | 09/26/2006 5:55 AM |
| I believe "spawn curve" refers to the probability curve for various spawn total results when rolling initiative dice. You want to build your warband so that you can take maximum advantage of that curve. That means you should have creatures with spawn cost (or spawn + aspect cost when considering early rounds) such that no matter what spawn total is rolled, you can take full advantage of it, but you need to build the majority of your band to take full advantage of the most probable values in the curve.
XAos, is that 7/9 right? That assumes equal occurrence of the 3 probability values, but I'm not sure that's the case (having trouble wrapping my head around the probability on this one). I have no problem assuming 2/3 = 25/36, since the difference is minor. The issue is that the "guarranteed" result occurs exactly once each time you get a "no spawn phase" result. That makes me wonder if it's 5/6: 2/3 chance of rolling a spawn phase (call that weight 1) plus 1/3 chance of no spawn phase this turn but guarranteed next turn (call that weight 1/2). | |
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XAos
Underboss
2358 Posts
London
| | 09/26/2006 9:48 AM |
| | On the subject of what "spawn-curve" means. I use it to describe both the binomial distribution of the initiative rolls (as modified by the rules for re-rolling & energise abilities)
And a specific warbands spread of spawn costs for it's 16 units. A well designed warband will have a spawn-curve which closely matches the distribution of the spawn rolls and can tolerate most probably variations in those rolls for specific games. The objective is to minimise the number of spawn points you waste because a roll does not match the minis in your reserve.
[Quote] Posted By Faragdar the Wise on 09/26/2006 5:55 AM
XAos, is that 7/9 right? That assumes equal occurrence of the 3 probability values, but I'm not sure that's the case (having trouble wrapping my head around the probability on this one). I have no problem assuming 2/3 = 25/36, since the difference is minor. The issue is that the "guarranteed" result occurs exactly once each time you get a "no spawn phase" result. That makes me wonder if it's 5/6: 2/3 chance of rolling a spawn phase (call that weight 1) plus 1/3 chance of no spawn phase this turn but guarranteed next turn (call that weight 1/2).
7/9 is approximatly correct over multiple game turns.
Clearly for 1 turn ahead, the actual odds are one of the 3 distinct values. e.g. if the current turn did not have a spawn phase. The next turn certainly (100%) will.
The lowest value for the overall avearge occures if you always reroll ties i.e. if the score is tied every turn (Very unlikly, that would require that neither player ever wins a turn);
= 7/9 = 0.7777777...
The maximum value for the average occurs if the scores are never tied (and in some games the score is tied for the first 2 turns & never ties again); = (11*36+25*25)/1296 = 0.787808642...
The difference between those extremes is only 0.010030864...
So if you use an average of 7/9 for designing warbands. Thats a good enough approximation. The variance in the dice rolls for any specific game. Will vastly exceed the minor error in ignoreing the extra +1%. | |
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gss_000
Commander
3204 Posts
Baltimore, MD
| | 09/26/2006 4:07 PM |
| Very nicely done. I really appreciate the accompanying analysis as well.
One question: I don't know all of the figures abilities by heart, but is there any figure that will change this analysis? | |
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Faragdar the Wise
Commander
3440 Posts
Albuquerque, NM, USA
| | 09/26/2006 6:18 PM |
| | There are a number of figures that move the curve up by adding to spawn points, but no figure, yet, actually skews the curve. In later sets, we might see a figure that can actually allow a player to modify an initiative result, though that would have to be carefully done or be horribly unbalancing. If you use a Greenlife Dryad, Heartsblood Temple or Rainforest Shaman (all passion), then you'll want to factor in the possibility of bonus spawn points when building your warband to take full advantage of the curve. | |
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XAos
Underboss
2358 Posts
London
| | 09/27/2006 1:38 AM |
| Faragdar, I missed a factor in the maths for calculating the average number of spawn phases.
With a recursive definition of a pair of rolls being forbidden, you remove the odds of that roll occuring from the equation.
Correct formulaes are;
For tied score= (2*2 + 1*2) / (9 - 1) = 0.75
For non tied score = (25*25 + 11*25) / (6^4 - 11*11) = 0.765957...
So the easy number to remember is Three-Quarters.
[Quote]then you'll want to factor in the possibility of bonus spawn points when building your warband to take full advantage of the curve.[/Quote]
Agreed, thats a factor I forgot to list when describeing the meaaning of "Spawn-curve".
I suspect the simplest mathematical model is to assume the bell curve is multiple overlapping curves; for the various number of destroyed units since the last spawn phase.
With a "guess" at the weighting for how often you get a specific number of destroyed units.
Note; This factor changes significantly depending on your turn-1 spawning strategy.
e.g.;
a) If you spawn 3x Spellbound Scissors on turn 1, you won't get bonus spawn points from destroyed units for several turns.
b) If you spawn 3xCannibal Pariahs, you can expect 2 - 6 extra spawn points fairly rapidly.
c) If you spawn a single creature on turn-1, you will certainly loose vp's at turn-2 and you can expect to not gain spawn points for a while.
So you need to modify the spawn-curve of your warband accordingly.
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Don't worry about the current metagame. It doesn't matter if it's ugly, bad, or the best ever. In 2 years time, set rotation will ban everything. | |
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