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Subject: Statistical Studies #3: How Valuable are Shield Carriers? rss

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Joe Gola
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Aaron and Balthazar are finishing creating their teams. Aaron's turn is before Balthazar's, and he completes a four-swordsman team adjacent to Balthazar's unfinished team, which currently holds three swordsmen. At the completion of this homogeneous team, Aaron brashly vows to attack Balthazar's adjacent team with his first move, no matter what.

Q: Will Balthazar improve his chances of scoring a kill against Aaron if he includes a shield carrier with his team?

First we'll look at what happens if Balthazar defends with five swordsmen. Aaron attacks first with five dice, and there are 7,776 possible combinations on five six-sided dice. Below is a table showing the possible outcomes (ignoring wounds), the number of combinations associated with those outcomes, and then the probability of rolling each particular outcome.

0-1 stars, 0 kills: 1053/7776 (13.5%)
2-3 stars, 1 kill: 3285/7776 (42.2%)
4-5 stars, 2 kills: 2642/7776 (34.0%)
6-7 stars, 3 kills: 730/7776 (9.4%)
8-10 stars, 4 kills: 66/7776 (0.8%)

To what degree does a shield carrier reduce these losses? To answer the question we have to find out the number of instances in which the shield carrier will prevent a second, third or fourth kill, since a shield carrier that absorbs a single kill is, from an offensive standpoint, functionally the same as a dead swordsman.

Here is the breakdown of possible outcomes of Aaron's five dice:

0 stars: 243/7776 (3.1%)
1 star: 810/7776 (10.4%)
2 stars: total 1485/7776 (19.1%)
....2 singles, 3 blanks: 1080/7776
....2 doubles: 405/7776
3 stars: total 1800/7776 (23.1%)
....3 singles, 2 blanks: 720/7776
....1 double, 1 single, 3 blanks: 1080/7776
4 stars: total 1590/7776 (20.4%)
....* 4 singles, 1 blank: 240/7776
....* 1 double, 2 singles, 1 blank: 1080/7776
....2 doubles, 3 blank: 270/7776
5 stars: total 1052/7776 (13.5% )
....5 singles: 32/7776
....1 double, 3 singles, 1 blank: 480/7776
....2 doubles, 1 single, 1 blank: 540/7776
6 stars: total 530/7776 (6.8%)
....* 1 double, 4 singles: 80/7776
....* 2 doubles, 2 singles, 1 blank: 360/7776
....3 doubles, 2 blanks: 90/7776
7 stars: total 200/7776 (2.6%)
....2 doubles, 3 singles: 80/7776
....3 doubles, 1 single, 1 blank: 120/7776
8 stars: total 55/7776 (0.7%)
....* 3 doubles, 2 singles: 40/7776
....4 doubles, 1 blank: 15/7776
9 stars: 10/7776 (0.1%)
....4 doubles, 1 single
10 stars: 1/7776 (0.01%)
....5 doubles: 1/7776

Adding up the totals of all the outcomes with asterisks—the outcomes for which the shield will prevent the death of the second, third or fourth unit—we arrive at 1,800 of 7,776 instances, or 23.1% of rolls. Overall the probabilities are as follows:

0 kills: 2133/7776 (27.4%)
1 kill: 3525/7776 (45.3%)
2 kills: 1762/7776 (22.7%)
3 kills: 330/7776 (4.2%)
4 kills: 26/7776 (0.3%)

Compare that to the first table and you will see that there is a significant improvement defensively.

The next question is whether the saved lives will translate into an offensive advantage despite the disadvantage of only having four dice to roll in the cases when the attacker whiffs entirely.

In both the four-swordsman scenario and the three-swordsmen-one-shield scenario, we can map out all the possible outcomes of both rolls (always assuming that the first kill would be assigned to the shield carrier) and then calculate the probability of each particular outcome by multiplying the probability of Aaron killing X number of Balthazar's units by the probability of Balthazar's return fire with the number of dice left to him after removing losses. For example, the chances of Aaron scoring two kills on five dice when facing a four-swordsman team are 2,642 in 7,776, or 34%; the chances of Balthazar scoring one kill on the three dice left to him are 107 in 216, or 49.5%; multiply those two together and you arrive at 16.8%. If one then adds all of Balthazar's matching results together, regardless of how many dice he rolled to arrive at those results, one can know the probabilities of his counterattack even before Aaron picks up his dice.

First, the table for the shield-carrier scenario:

0 kills: 28.0%
1 kill: 48.6%
2 kills: 20.8%
3 kills: 2.6%
4 kills: 0.06%
I had miscalculated this table originally; this is the corrected version. The odds are significantly improved!

Compare that to the table of probabilities had Balthazar instead included a fourth swordsman:

0 kills: 30.6%
1 kill: 46.9%
2 kills: 19.5%
3 kills: 2.9%
4 kills: 0.1%

One can see from the foregoing that in this case there is no offensive disadvantage to the inclusion of the shield bearer. If anything, one could argue that there is an overall improvement of the odds.

Additionally, it should be mentioned that we are taking a rather narrow view of the shield carriers insomuch as we are only considering them from a short-term offensive standpoint. They will also be preventing a wound in 3,872 of 7,776 or 49.8% of cases when attacked by five swordsmen. A wound does not affect the immediate turn, of course, but, as the smell of blood draws in the sharks, a limping team offers an attractive target for all the other players. It's also worth considering whether simple longevity has any long-term offensive advantage for a team; conceivably in the latter half of the game a defense-oriented team may find itself to be the strongest offensive force when compared to a field which has spent the entire game hacking itself into bits.

Next: What's the Deal with the Net Casters?
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Chester
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Re: Statistical Studies #3: How Valuable are Shield Carriers
My son has made teams with just 4 net-casters before (he was 4 at the time) and done pretty well with them. Seems kind of silly to me, but they do about even head to head against my scientifically-created optimum forces. It always makes me smile.
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Chester
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After playing again tonight....the long strategic dissection after the game revealed a sentiment that the optimum team is:

3 swords and 1 shield-bearer

Your rebuttal, Mr Gola?
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Alex Sorbello
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NO NO NO No,
It's 3 net guys and a sword guy.
Has the upper hand against any other team except tied against a all swords team.
And bad against animals in the end not to mention.

My conclusion after much deliberation is that it's still based on dice, thus the Ultimate and unbeaten strategy is to get the best rolls.
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Joe Gola
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I counter with two swordsmen, one prong bearer, and one net caster! Your shield bearer is neutralized! The prongings are exquisite! The power of exclamation points is revealed!
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Jeff Wiles
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Neil
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20 Plastic Ninjas
There was a Batman Begins game that had 20, yes, 20 plastic ninjas.
I think that would be a cool re-theme and the animals could be dragon creatures.
 
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