Monday, 18 February 2013

The Renaissance Diceman Cometh - Shooting

On Sunday I played a game of Field of Glory Renaissance (FOG R) using my TYW Danish army. It was a practice game for the Northern League this coming Sunday.



Graham, my regular opponent, fielded an army with a lot of bow power and afterwards we discussed his choice of bow power over fire power. He said he'd faced it a few times last year and wanted to try it out.

I was intrigued. The whole point seems to be to cause your opponent to take as many Cohesion Tests as possible. This despite the fact that FOG R seems to me to be designed to model unit degradation through base removal from fire power and melee rather than the steady decline of unit cohesion seen in the Ancient & Medieval variant.

I decided to spend some time looking at the maths behind this choice.

Initially, let's look at a battlegroup of four bases shooting at another battlegroup of four bases. This is a very common occurrence in FOG but really I've chosen it because it neatly illustrates all the factors involved; even if it is something a special case because of the way the maths works.

The shooting player rolls 4 dice needing 4 or more to hit the target: 4 is the standard score in FOG assuming no Points of Advantage (POA) apply. The probability of hitting the target twice is 37.50%. That's the easy bit over with. Things fragment as the size of the target and the nature of the weapons shooting come into play. To explain:
  1. Depending on battlegroup size, shooting can effect the target by making the target battlegroup take a Cohesion test and failing the test weakens the battlegroup especially if it happens repeatedly. When shot at battlegroups need only take a Cohesion Test when they have received at least one hit for every three basses (rounded up). So two hits would cause battlegroups of up to six bases to test.
  2. In addition to making the target take a Cohesion Test, the number of hits from shooting can also affect the outcome of any test. If a battlegroup has to test, and has suffered one hit for every two bases, it adds 1 to the score required to pass the test. This is usually 7 on 2d6. So 2 hits on four bases would require 7 + 1 = 8 to pass.
  3. The type of weapons shooting and the nature of the target further affect the Cohesion test. For certain targets, like cavalry, simply being shot at by firearms adds 1 to the score required to pass the test. So 2 hits involving firearms on four bases would require 7 + 1 + 1 = 9 to pass.
  4. In addition to a drop in Cohesion there's the possibility of base removal via "death rolls" (in Field of Glory speak). This too is weapon dependent. For firearms death rolls result in base removal when the target fails to roll more that the number of hits on 1d6. For bows the number of hits is reduced by 2 meaning base removal is impossible with only two hits.
So with these points in mind let's continue with the comparison.

A target battlegroup of four bases receiving 2 hits from bow fire must take a Cohesion Test needing a score of 8 on 2d6 to pass. The target has a 58.33% chance of failing this test but there's no possibility of the bow fire resulting in the removal of a base.

A target battlegroup of four bases receiving 2 hits from firearms also needs to take a Cohesion Test. This time a score of 8 or 9 on 2d6 is needed to pass depending on the target. There's either a 58.33 or a 72.22% chance of failing this test but more importantly there is 33.33% chance of the target failing its death roll by rolling 1 or 2 on 1d6.

Now let's combine the chance of scoring two hits by shooting with the chances of passing the resultant Cohesion Test. For a bow armed battlegroup of four bases shooting, the chance of inflicting two hits and causing a drop in its target's cohesion level is 21.87% with zero chance of killing a base. For a similar battlegroup using firearms, there's either a 21.87 or a 27.08% chance of two hits causing a drop in its target's cohesion level and a 12.50% chance of removing a base.

Interestingly, battlegroups with firearms retain the 12.50% chance of killing a base with two hits irrespective of the size of the target whilst the impact of two hits on the Cohesion Test itself declines with the increasing size of the target.

The above is a simple situation where no shooting POA apply. However, there is often at least one negative POA in play meaning a 5 or 6 is required to hit. In the above example this would reduce the probability of two hits to 29.63% affecting the overall outcome of bow fire (17.43%) and firearms (17.43 or 21.40%) equally. The chance of removing a base with two hits from firearms drops to 9.78%.

Using this analysis you'd choose troops with firearms every time: they are more effective at eroding a target's cohesion and always offer the possibility of removing a base. So why are some players choosing to use bows in FOG R? As always the devil is in the detail.

Firstly, things are never that straightforward. The shooting POA for bows and firearms are very different and so things get more convoluted. Bow fire doesn't suffer a negative POA against most mounted whilst firearms do. Against foot it's slightly better as only pistols and carbines suffer a negative POA.

So the most meaningful comparison is really between bows needing 4-6 to hit and firearms needing 5-6 to hit. Bow fire retains its 21.87% overall chance of causing a drop in its target's cohesion level with two hits but firearms' chance of achieving the same result drops to 17.43 or 21.40%, depending on the target, with a 9.78% chance of removing a base: a much closer set of probabilities.

Secondly, bows are cheap. They are 2 points per base less than muskets and 1 point less than arquebus, pistols or carbines. So it costs less to increase the size of bow armed battlegroups. For example: 8 MF bow costs 40 points cf. 6 MF musket at 42 or 6 MF arquebus at 36 points.

Larger battlegroups roll more shooting dice resulting in a higher number of hits (on average) with a consequent increase in their impact on their target's cohesion and they may even result in base removal if they score more than two hits.

So there you have it. Bows offer a more cost effective chance of eroding a units cohesion compared to firearms. The added benefit of firearms, in always offering a chance of base removal, is quite low at around 1 in 10 and is counterbalanced by the fact bows are cheaper allowing for bigger battlegroups for the same number of points.

Overall, I think the choice is finely balanced and I suppose that's why players can field, and be effective with, massed bow armies.

Notes:

As before I've used the excellent www.anydice.com to calculate the odds. If you're interested the functions used were:

output [count {4,5,6} in 4d6] named "Rolling >4 on 4d6"
output 2d6 named "Cohesion test results"
output [count {5,6} in 4d6] named "Rolling >5 on 4d6"


You can use the results to work out the probabilities for 3 and 4 hits and larger battlegroups if you fancy a challenge.

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