I've been doing a bit of number-crunching for defense analysis (also relevant to the attack on Lego). The scenario is a simplistic one - an attack on a civ with 6 cities: 3 coastal, 3 inland. The defender has 30 infantry. I assume that all coastal cities get the same number of defenders, and all inland cities get the same no. (e.g. 4 in every inland city, 6 in every coastal one).
The first attacker has 40 marines and 20 tanks. Results in table below.
Column 1 (#coastal) is no. of infantry per coastal city.
Column 2 (#inland) is no. of infantry per inland city.
Column 3 (marine) is % chance of marines taking one coastal city
Column 4 (n) is average no. of inland cities taken by the tanks if the marines land.
Column 5 (N) is the average number of cities taken in the attack (n + 1 (for coastal city) multiplied by prob. of succesful landing). This last column is slightly controversial, so bear with me.
Why is the last column controversial? you ask. Well, consider the last row. With 0 defenders in the coastal cities, there is a good chance that more than just the target city will fall. If tanks or marines can reach cities along the coast, those cities are toast. If one of the inner cities falls it may allow access to an undefended coastal city. Worst case scenario, all 3 coastal cities fall. But maybe they can't all be reached. So the N value for the final row is either 1.43, 1.43 or 3.43, depending on geography (i.e. it is very situation-dependent).
Case 1: no extra cities can be reached, and the table is accurate as it stands. Peversely, the best approach is to empty the coastal cities entirely, and stack the inland cities with defenders (really, you should count unthreatened coastal cities as inland cities for this purpose). Since you don't have enough defenders to stop the marines, realistically speaking, then your choice is essentially to let them use 40 marines to take a city killing 0 defenders, and then 20 tanks aganist all 30 infantry, or to let them use 40 marines to take a city killing say 3 defenders (without changing the odds of them taking the city), and then having 3 fewer defenders to defend against the tanks.
In case 1, the worst scenario is to put 7 defenders in each coastal city - getting the maximum number killed (with fewer to defend inland cities) without significantly affecting the odds of keeping that city anyway.
Case 2: all coastal cities can easily be reached. N for the last row is 3.43, and for the 1/9 row is probably close to that as well. The number at 5/5 is pretty much accurate - ignoring the effect of marines being able to assault a second city (we assume only tanks can do it). So we have minimum values at 10/0 and probably 4/6 (that's guesswork - it could be 5/5 or somewhere below that). We have maximum values at 8/2 and 0/10 (or 1/9 - it's hard to know).
The first attacker has 40 marines and 20 tanks. Results in table below.
Column 1 (#coastal) is no. of infantry per coastal city.
Column 2 (#inland) is no. of infantry per inland city.
Column 3 (marine) is % chance of marines taking one coastal city
Column 4 (n) is average no. of inland cities taken by the tanks if the marines land.
Column 5 (N) is the average number of cities taken in the attack (n + 1 (for coastal city) multiplied by prob. of succesful landing). This last column is slightly controversial, so bear with me.
Code:
#coastal #inland marines n N 10 0 59.1 3.00 2.36 9 1 80.7 3.00 3.23 8 2 93.9 2.99 3.75 7 3 98.9 2.67 3.63 6 4 99.9 1.95 2.95 5 5 100. 1.43 2.43 4 6 100. 1.08 2.08 3 7 100. 0.97 1.97 2 8 100. 0.87 1.87 1 9 100. 0.68 1.68 0 10 100. 0.43 1.43
Case 1: no extra cities can be reached, and the table is accurate as it stands. Peversely, the best approach is to empty the coastal cities entirely, and stack the inland cities with defenders (really, you should count unthreatened coastal cities as inland cities for this purpose). Since you don't have enough defenders to stop the marines, realistically speaking, then your choice is essentially to let them use 40 marines to take a city killing 0 defenders, and then 20 tanks aganist all 30 infantry, or to let them use 40 marines to take a city killing say 3 defenders (without changing the odds of them taking the city), and then having 3 fewer defenders to defend against the tanks.
In case 1, the worst scenario is to put 7 defenders in each coastal city - getting the maximum number killed (with fewer to defend inland cities) without significantly affecting the odds of keeping that city anyway.
Case 2: all coastal cities can easily be reached. N for the last row is 3.43, and for the 1/9 row is probably close to that as well. The number at 5/5 is pretty much accurate - ignoring the effect of marines being able to assault a second city (we assume only tanks can do it). So we have minimum values at 10/0 and probably 4/6 (that's guesswork - it could be 5/5 or somewhere below that). We have maximum values at 8/2 and 0/10 (or 1/9 - it's hard to know).
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