vicawoo
Chieftain
- Joined
- Feb 12, 2007
- Messages
- 3,226
With more time, this will become an article. But for now it's a work in progress.
Flanking is used to maximize value for weakening high strength units. The downside is that your attrition units are weaker and will be less effective and win less often. Should your war chariot go flanking 2 or combat 2 versus archers? Should knights go flanking versus longbows? Answers lie ahead.
In another post, somebody calculated the increased effectiveness of getting flanking bonuses. So a 50% flanking bonus buys you another unit (once saved and healed) 50% of the time, right? However, what if you withdraw multiple times? There's a 50% chance of chance of withdrawing once, but a 25% chance of withdrawing twice, and so on. However, one might argue that you'll only withdraw and heal 3 times before your unit becomes outdated, perhaps 5 times for later units.
I. Increase in value from flanking
The formula is simply p+p^2+p^3+..., or sum(p^n,i=0..infinity), which is a telescoping sum equalling p/(1-p). The formula for up to 3 withdrawals is p+p^2+p^3
Flanking Value increase Value w/3 wd's Value w/5 wd's
0.00 0.00 0.00 0.00
0.10 0.11 0.11 0.11
0.15 0.18 0.18 0.18
0.20 0.25 0.25 0.25
0.25 0.33 0.33 0.33
0.30 0.43 0.42 0.43
0.35 0.54 0.52 0.54
0.40 0.67 0.62 0.66
0.45 0.82 0.74 0.80
0.50 1.00 0.88 0.97
0.55 1.22 1.02 1.16
0.60 1.50 1.18 1.38
0.65 1.86 1.35 1.64
0.70 2.33 1.53 1.94
0.75 3.00 1.73 2.29
0.80 4.00 1.95 2.69
0.85 5.67 2.19 3.15
0.90 9.00 2.44 3.69
As you can see, a 25% chance is a noticable 1/3 increase in value to a 0 flanking unit. Of course, this doesn't mean you have that extra unit at hand, but that you don't have to rebuild it. Also, we note that limiting the number of withdrawals to realistic numbers doesn't cause problems until higher levels.
Flanking 2 cavalry then have a 150% increase in value, right? But even combat 2 cavalry have an intrinsic flanking. So let's calculate the increase in value with each flanking upgrade. The formula is (1+(p+.1)/(1-p-.1))/(1+p/(1-p)) and (1+(p+.3)/(1-p-.3))/(1+p/(1-p)). Sorry, I'm not bothering with great general yet.
Base F F1 value F2 Value (F2 value/F1 value)
0.00 1.11 1.43 1.29
0.10 1.13 1.50 1.33
0.15 1.13 1.55 1.36
0.20 1.14 1.60 1.40
0.25 1.15 1.67 1.44
0.30 1.17 1.75 1.50
0.35 1.18 1.86 1.57
0.40 1.20 2.00 1.67
0.45 1.22 2.20 1.80
0.50 1.25 2.50 2.00
So Flanking 1 increasing a unit without any withdrawal chances by 11%, flanking 2 proportionally increases it by 29% to a total of 43% over a non-flanking unit. Flanking 2 also has very large value bonuses, from 43% to 150%
Limited withdrawals, the same tables are below
Base F (F1+,3w) (F2+,3w) (F2+/F1+,3w) (F1+,5w) (F2+, 5w) (F2+/F1+, 5w)
0.00 1.11 1.42 1.28 1.11 1.43 1.28
0.10 1.12 1.46 1.30 1.12 1.49 1.33
0.15 1.13 1.48 1.31 1.13 1.53 1.35
0.20 1.14 1.50 1.32 1.14 1.58 1.38
0.25 1.14 1.52 1.33 1.15 1.62 1.41
0.30 1.15 1.54 1.34 1.16 1.67 1.44
0.35 1.15 1.55 1.35 1.17 1.72 1.47
0.40 1.15 1.56 1.35 1.19 1.77 1.49
0.45 1.16 1.57 1.35 1.20 1.82 1.52
0.50 1.16 1.57 1.36 1.21 1.87 1.55
Note that flanking bonuses increases as withdrawal chance goes up. Even with a cap of 3 withdrawals, flanking 2 goes from 42% to a 57% increase.
II. Bonus efficiency of combat promotions
Here comes the complicated part. How much more valuable is a combat upgraded unit than a non-combat (in this case, flanking) upgraded unit? Is it really more than 30% for chariots, 50% for horse archers, and 54% for cavalry? Suppose we are throwing our unit at the strongest defender.
Further suppose they are at full health, so we get to use Arathorn's guide.
If you are trying to attrition the defending unit, the most critical breakpoint is dying in 4 hits instead of 5. 20*(3+R)/(1+3R)=25 is solved for R=7/11 ~ 0.636. A worse one is at 20*(3+R)/(1+3R)=34, R=13/41 ~ 0.32. So definitely don't get below 3 to 1 odds, people.
I'll post those breakpoints for war chariot with no, combat 1, or combat 2 upgrades versus fortified archers in a city with the possibilities of being on a hill, 20% culture bonus, 40% culture bonus, city garrison 1/guerrilla 1, city garrison 2, but I retrieve the calculations.
Next, the mathematically more difficult part, is solving when the chariot takes 4 hits and the archer takes 6 hits, what's the chance of winning and by how much, only generally speaking. I don't know off hand how to do this in the general case. For the mathematically inclined, you can't solve this through the typical martingale solution to the gambler's dilemma. You have to solve the discrete difference equation P(x,y)=p*P(x-1,y)+(1-p)*P(x,y-1), through an extension of something called Lanchester's equations, which are used to model military models. There's a paper by Robert H. Brown on how to solve those in discrete situations, but I don't have access to the journal. So if someone has the solution to the equation or access to the paper, let me know.
Then from there, I'll calculate how much damage you need to do to get "good" odds versus the archer, based on modified jump points, and the chance of getting those with various mounted upgrades, and from that how worse off flanking chariots are than combat 2 or combat shock.
Later on, I'll tackle how many axemen you'll need to take archers in cities in certain situations, swordsmen versus archers. I'll also do macemen versus longbows, and knights versus longbows on a much simpler set of conditions, thanks to catapults.
And as an aside, first strike seems by far the best way to do attrition damage if you're not losing by too much (that is, you still die in 5 hits).
Flanking is used to maximize value for weakening high strength units. The downside is that your attrition units are weaker and will be less effective and win less often. Should your war chariot go flanking 2 or combat 2 versus archers? Should knights go flanking versus longbows? Answers lie ahead.
In another post, somebody calculated the increased effectiveness of getting flanking bonuses. So a 50% flanking bonus buys you another unit (once saved and healed) 50% of the time, right? However, what if you withdraw multiple times? There's a 50% chance of chance of withdrawing once, but a 25% chance of withdrawing twice, and so on. However, one might argue that you'll only withdraw and heal 3 times before your unit becomes outdated, perhaps 5 times for later units.
I. Increase in value from flanking
The formula is simply p+p^2+p^3+..., or sum(p^n,i=0..infinity), which is a telescoping sum equalling p/(1-p). The formula for up to 3 withdrawals is p+p^2+p^3
Flanking Value increase Value w/3 wd's Value w/5 wd's
0.00 0.00 0.00 0.00
0.10 0.11 0.11 0.11
0.15 0.18 0.18 0.18
0.20 0.25 0.25 0.25
0.25 0.33 0.33 0.33
0.30 0.43 0.42 0.43
0.35 0.54 0.52 0.54
0.40 0.67 0.62 0.66
0.45 0.82 0.74 0.80
0.50 1.00 0.88 0.97
0.55 1.22 1.02 1.16
0.60 1.50 1.18 1.38
0.65 1.86 1.35 1.64
0.70 2.33 1.53 1.94
0.75 3.00 1.73 2.29
0.80 4.00 1.95 2.69
0.85 5.67 2.19 3.15
0.90 9.00 2.44 3.69
As you can see, a 25% chance is a noticable 1/3 increase in value to a 0 flanking unit. Of course, this doesn't mean you have that extra unit at hand, but that you don't have to rebuild it. Also, we note that limiting the number of withdrawals to realistic numbers doesn't cause problems until higher levels.
Flanking 2 cavalry then have a 150% increase in value, right? But even combat 2 cavalry have an intrinsic flanking. So let's calculate the increase in value with each flanking upgrade. The formula is (1+(p+.1)/(1-p-.1))/(1+p/(1-p)) and (1+(p+.3)/(1-p-.3))/(1+p/(1-p)). Sorry, I'm not bothering with great general yet.
Base F F1 value F2 Value (F2 value/F1 value)
0.00 1.11 1.43 1.29
0.10 1.13 1.50 1.33
0.15 1.13 1.55 1.36
0.20 1.14 1.60 1.40
0.25 1.15 1.67 1.44
0.30 1.17 1.75 1.50
0.35 1.18 1.86 1.57
0.40 1.20 2.00 1.67
0.45 1.22 2.20 1.80
0.50 1.25 2.50 2.00
So Flanking 1 increasing a unit without any withdrawal chances by 11%, flanking 2 proportionally increases it by 29% to a total of 43% over a non-flanking unit. Flanking 2 also has very large value bonuses, from 43% to 150%
Limited withdrawals, the same tables are below
Base F (F1+,3w) (F2+,3w) (F2+/F1+,3w) (F1+,5w) (F2+, 5w) (F2+/F1+, 5w)
0.00 1.11 1.42 1.28 1.11 1.43 1.28
0.10 1.12 1.46 1.30 1.12 1.49 1.33
0.15 1.13 1.48 1.31 1.13 1.53 1.35
0.20 1.14 1.50 1.32 1.14 1.58 1.38
0.25 1.14 1.52 1.33 1.15 1.62 1.41
0.30 1.15 1.54 1.34 1.16 1.67 1.44
0.35 1.15 1.55 1.35 1.17 1.72 1.47
0.40 1.15 1.56 1.35 1.19 1.77 1.49
0.45 1.16 1.57 1.35 1.20 1.82 1.52
0.50 1.16 1.57 1.36 1.21 1.87 1.55
Note that flanking bonuses increases as withdrawal chance goes up. Even with a cap of 3 withdrawals, flanking 2 goes from 42% to a 57% increase.
II. Bonus efficiency of combat promotions
Here comes the complicated part. How much more valuable is a combat upgraded unit than a non-combat (in this case, flanking) upgraded unit? Is it really more than 30% for chariots, 50% for horse archers, and 54% for cavalry? Suppose we are throwing our unit at the strongest defender.
Further suppose they are at full health, so we get to use Arathorn's guide.
If you are trying to attrition the defending unit, the most critical breakpoint is dying in 4 hits instead of 5. 20*(3+R)/(1+3R)=25 is solved for R=7/11 ~ 0.636. A worse one is at 20*(3+R)/(1+3R)=34, R=13/41 ~ 0.32. So definitely don't get below 3 to 1 odds, people.
I'll post those breakpoints for war chariot with no, combat 1, or combat 2 upgrades versus fortified archers in a city with the possibilities of being on a hill, 20% culture bonus, 40% culture bonus, city garrison 1/guerrilla 1, city garrison 2, but I retrieve the calculations.
Next, the mathematically more difficult part, is solving when the chariot takes 4 hits and the archer takes 6 hits, what's the chance of winning and by how much, only generally speaking. I don't know off hand how to do this in the general case. For the mathematically inclined, you can't solve this through the typical martingale solution to the gambler's dilemma. You have to solve the discrete difference equation P(x,y)=p*P(x-1,y)+(1-p)*P(x,y-1), through an extension of something called Lanchester's equations, which are used to model military models. There's a paper by Robert H. Brown on how to solve those in discrete situations, but I don't have access to the journal. So if someone has the solution to the equation or access to the paper, let me know.
Then from there, I'll calculate how much damage you need to do to get "good" odds versus the archer, based on modified jump points, and the chance of getting those with various mounted upgrades, and from that how worse off flanking chariots are than combat 2 or combat shock.
Later on, I'll tackle how many axemen you'll need to take archers in cities in certain situations, swordsmen versus archers. I'll also do macemen versus longbows, and knights versus longbows on a much simpler set of conditions, thanks to catapults.
And as an aside, first strike seems by far the best way to do attrition damage if you're not losing by too much (that is, you still die in 5 hits).
