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Folks,

Say that you attack (or you must hold) two bases: one is worth 100 points for you and 10 points for the enemy; the other is worth 10 points for you and 100 points for the enemy. Which option will give the best victory ratio?

In summary:

(a) Go for the bases that are worth a lot for the enemy if your victory points plus the base value is more than the enemy victory points (that is, if you're winning)
(b) Otherwise go for the bases that are worth a lot for you (that is, if you're losing)

The recommendations above can pretty much be explained with common sense: if you have a lot of victory points, some more won't make difference. If the enemy has only a few victory points, some less will make a lot of difference.

Of course, a ton of other considerations apply: which base is easier to conquer, defend, supply, have greater strategic importance, etc... But the analysis above give some indications if I have to defend the Aleutians at all (value a lot for the enemy, little to me, so I can forget about them and go for more strategic targets if I'm losing from the AI; otherwise I should stick to them.

You can either believe on me on go through the mathematical proof below.

Regards,
F.

One base is valued “k.a” victory points for me
Sme base is valued “(1-k).a” victory points for the enemy

I have a total of VM victory points
Enemy has a total of VE victory points

If I conquer the base the victory ratio becomes (NR = New Victory Ratio):

NR = (VM + k.a) / (VE – (1-k).a)
NR = (VM + k.a) / (VE – a + k.a)

First derivative

dNR / da = [ (VE – a + k.a) . k – (VM + k.a).k ] / (VE – a + k.a)^2

Equating to zero

dNR / da = 0 => (VE – a + k.a).k = (VM + k.a).k
VE – a + k.a = VM + k.a
VE – a = VM

Therefore, there is local maximum or minimum that doesn’t depend on “k”; if that precise condition is not met, there is no inflexion point, and either k=0 is the maximum or k=1 is the maximum.

At k=0, NR(k=0) = VM / (VE – a)
At k=1, NR(k=1) = (VM+a) / VE

NR(k=0) > NR(k=1) if VM / (VE-a) > (VM+a) / VE

Or VM.VE > (VM+a)(VE-a)
VM.VE > VM.VE – VM.a + VE.a –a^2
VM.a – VE.a + a^2 > 0
VM + a > VE

Therefore, going for a base that is worth little to me and a lot for the enemy is a good idea if the I have more victory points than the enemy and the difference of victory points is greater than the added value of the base. Otherwise, I should go for bases that are worth a lot to me and little to the enemy

Example: I have 5,100 points and the enemy has 4,900 points. There are two bases. Base “A” is worth 100 points to me and 10 points to the enemy; base “B” is worth 10 points to me and 100 points to the enemy. The added value of both bases is 100 + 10 = 110 points. As 5,100 + 110 = 5,210 and is greater than 4,900, I should go for base “B” (value is little to me and much to the enemy). Verifying that:

Verification: if I go for base “A”, the points will be 5,200 for me and 4,890 for the enemy (victory ratio is 1.0633). If I had gone for base “B”, the points woul be 5,110 for me and 4,800 for the enemy (ratio is 1.0646). Therefore, base “B” is the best option.

Same situation, if I had 4,000 points and the enemy has 4,900 points. I have less points, so I should go for base “A”.

Verification: if I go for base “A”, the points will be 4,100 and 4,890 (ratio is 0.8384). If I go for base “B”, the points will be 4,010 and 4,800 (ratio is 0.8354). Therefore, base “A” is the best option.

It's a combination package. What you aim for is denial of high value bases for the enemy coupled with capturing high value bases for your cause. As the score is is calculated as a ratio of points between the two sides, this plays on both sides of the ratio, lowering one and boosting the other.