Sunday, September 02, 2007

Blood Bowl Skill Statistics: Article 1

This article is the first in a series for anyone interested in playing Bloodbowl, the fantasy football game (and that assumes that the reader knows the rules of the game). As any player knows, Bloodbowl involves a whole lot of dice rolling, and in many cases, a failed roll results in a turnover, which is never a good thing. The best way to avoid a turnover is acquiring skills for your players that are relevant to the dice rolls likely to be made for the particular player. This article (and subsequent ones) will discuss various skills in mathematical terms of success or failure including chance of success with and without the skill. Skills such as block, pass, dodge and catch are not discussed in this article because their value is obvious and many positional players start of with the skills. For the purpose of simplicity article, the “average player” will often be used as the assumed ability scores for the purpose of determining probabilities; the average player has no skills unless stated otherwise.

TABLE 1: The “average player’s” stats [equivalent to a human lineman]
MA ST AG AM Skills
6 3 3 8 none

Hopefully, this article will allow the player to better understand the value of giving players mathematically complex skills in the development of their team roster. The first skill we will discuss is the skill of champions: Diving Catch!
Diving Catch:

How good is diving catch? Diving Catch would be better described as the skill of misfits than champions. Succeeding in life despite incompetence is the objective of the diving catch skill. For teams that are bad at passing, it’s very good, but less so for teams that are already well adept at throwing. First, we need to understand what goes into a pass.

Neither the thrower, nor the catcher is in an opposing player’s tackle zone.
The pass is a short pass.
The opponent is not in position to intercept the ball.

To make a successful pass requires both a throw and catch from each player, and the probability of success can defined by this equation.

PS = ST * SC where:

PS is the probability of success
ST is the chance to successfully throw
SC is the chance to successfully catch

Throwing the ball in this case has no modifiers, and the average player needs a 4. Catching the ball is easier if the thrower is accurate, which gives a +1 to the catcher, so only needs a 3+.

PS = (3/6) * (4/6)
PS = .333

That’s not very good is it; 77.7% of time, you will fail, causing a turnover. Since several teams have throwers and catchers, it’s useful to repeat this equation assuming the thrower and catcher have the pass and catch skills, respectively. Each player can now reroll the die if they fail adding to CT and CC the probability that they will fail the first attempt times the odds that they will succeed the second attempt:

Success with a reroll:
PS = CS1 + CF1*CS2 where
CS1 is the chance to succeed the first attempt.
CF1 is the chance to fail the first attempt.
CS2 is the chance to succeed on the second attempt.

Most people know already knew how to calculate, but I mention it anyway because this mini-formula comes into to play a lot in bloodbowl. In this instance, both the thrower and catcher now use the reroll formula to boost their odds of success:

PS = [(3/6) + (3/6)*(3/6)] * [(4/6) + (2/6)*(4/6)]
PS = .75 * .889
PS = .667

Well, that’s better, but hardly an ideal scenario and we already assumed that the opponent would do nothing to stop you! This is where diving catch comes into to play by boosting the odds that the catcher may attempt to catch the ball if the thrower throws an inaccurate pass. The ball must scatter to position where it is in the catchers tackle zone (1 of 8 squares). The new equation for a successful pass looks like this:

PS = (TA * CA) + (TI * DC * CU) where
PS is the probability to succeed
TA is the chance the throwing an accurate throw.
CA is the chance of successfully catching an accurate pass.
TI is the chance of throwing an inaccurate pass.
DC is the chance the ball will scatter to square where a catch attempt can be made after an inaccurate pass by the thrower.
CU is the chance of catching an inaccurate pass.

Note that DC can only have one of two values under assumption 1 above: when the catcher does and does not have the diving catch skill. The scattering of the ball is independent of either player’s ability scores. Actually, this equation always represented PS, but without diving catch, the DC is so low that it’s not worth considering. What is the DC for player without the diving catch skill? It must scatter directly back to his square on the third, and only the third, scatter role. Look at the following illustration:

The centre dot represents the catcher, while the squares around him are empty. The squares beside him are equivalent from a mathematical perspective for our purpose, and so are the squares diagonal to the player. If the ball goes straight upwards [or to any of the other black dots], the second scatter must keep the ball exactly one square away from the player before it can scatter to him, and there is a 4/8 chance of this occurring. The ball then has 1/8 chance of going the player occupied square no matter which of the four square it went to on the second roll. If the ball goes to a white dot on the first roll, it must go to one of the two adjacent black dots on the second roll. The final calculation is:

DC = (4/8)*(4/8)*(1/8) + (4/8)*(2/8)*(1/8)
DC = 0.031 + 0.016
DC = 0.047

There is slightly less that a 1/20 chance of the catcher being able to catch an inaccurate pass, but what about when the player. Determining the DC for a player with diving catch involves a similar process, working out the allowable rolls one by one, but this time, the 8 dots themselves are also squares where the ball can land in. Actually, if the ball scatter directly back to the catcher on the second roll, the catcher will be able to attempt a catch no matter what the third roll! I will not spell out the process a second time, but the odds of being able to attempt a catch are:

Black dots on first roll

(4/8)*(2/8)*(5/8) = 0.078
(4/8)*(2/8)*(3/8) = 0.047
(4/8)*(2/8)*(2/8) = 0.031
(4/8)*(1/8)*(3/8) = 0.023
Total: = 0.179

White dots on first roll

(4/8)*(2/8)*(5/8) = 0.078
(4/8)*(2/8)*(3/8) = 0.047
(4/8)*(2/8)*(2/8) = 0.031
(4/8)*(1/8)*(1/8) = 0.008
Total: = 0.164
Note: this table omits the special case where the second roll is directly back to the catcher, which can happen no matter what the 1st or 3rd roll is.

DC = (8/8)*(1/8)*(8/8) + 0.170 + 0.164
DC = 0.459

The DC improves dramatically when the catcher has a diving catch skill, increasing 41.2%! The real question is how much does this increase the likelihood of a successful pass? Let’s recalculate PS for both skilled and unskilled players using the new equation, and assuming the catcher has the diving catch skill.

PS = (TA * CA) + (TI * DC * CU)
PS = (3/6)*(4/6) + (3/6)*0.459*(3/6)
PS = 0.333 + 0.115
PS = 0.448

Two average players have about a 50/50 shot at making a short pass under these conditions, and for skilled players:

PS = (TA * CA) + (TI * DC * CU)
PS = [(3/6) + (3/6)*(3/6)] * [(4/6) + (2/6)*(4/6)] + [((3/6)*(3/6))*0.459*((3/6) + (3/6)*(3/6))]
PS = (0.75 * 0.889) + (0.25*0.459*0.75)
PS = 0.667 + 0.086
PS = 0.753

TABLE 2: the statistical benefit to making a completion with diving catch skill
Players Catcher doesn’t have Diving Catch skill Catcher does have Diving Catch skill
Skilled 0.333 0.448
unskilled 0.667 0.753

Finally, our players now have a decent chance of making the pass, succeeding ¾ of the time. How good is the diving catch skill? Well, it depends on the TI really; the more likely the thrower is to fail, the more likely the diving catch skill is to be useful. The hail merry pass is often the best combination for someone with diving catch as it is always inaccurate. The odds of any thrower successfully making a hail merry pass are always (5/6), which is represented by the constant, HM, in the following equation:

PS = hm * DC * CU
PS = (5/6) * 0.459 * (3/6)
PS = 0.191

A favorite of bloodbowl teams with no positional players, the hail merry pass allows the coach a 1/5 chance of making any pass to anywhere on the field, including one end zone to the other despite having an extremely inept thrower. Would a wood elf coach ever give this skill one of his catchers? Well, as the range of the pass increases to long bomb, the TI also increases, so let’s assume both the thrower with agility 4 has pass, the catcher has agility 4, catch, and diving catch, and the throw is a long bomb.

PS = (TA * CA) + (TI * DC * CU)
PS = [(2/6) + (4/6)*(2/6))*((5/6) + (1/6)*(5/6)] + [((2/6)*(2/6))*0.459*((4/6)*(2/6)*(4/6))]

PS = 0.889*0.972 + 0.111*.0459*0.889
PS = 0.864 + 0.045
PS = 0.909

Obviously, such a catcher probably wouldn’t bother with skill unless they already had several other skills, but none the less diving catch did help the coach increase his confidence 4.5%, which is better than nothing. The decision is yours, but hopefully this article was of some interest by providing some solid numbers for strategic decision making. Let me know if you any skills you want to see analyzed for the next article.


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