@a44bigdog:

Bombers are bought to bomb with. There will be losses to AA and there will be rounds where they do not do much damage. That is just part of it, however with regard to the lost bombers that is what they are being bought for.

good read, a44bigdog. i also agree my “break even” idea is not best. i’d like to propose 2 statistics for reasoning about SBR: bomber lifetime, and damage over lifetime.

it appears that most of the reasoning is over probability of being shot down (1/6), and expected bomber damage given the bomber survives AA (3.5 IPCs), but those are not as useful numbers for reasoning about an SBR campaign.

these are more reasonable metrics IMO since you are buying bombers for the primary purpose of bombing. yes, as jen points out, you can use the bombers for other purposes other than SBR. you can also use them to win land or navy battles (direct damage). you can also use them to forcing US/UK to buy a carrier or other capital ship (indirect strategical effects). you can also use them to prevent unescorted transports to the karela sea zone (indirect tactical effects). these are hard to quantify, so let’s just consider that flexibility a bonus.

**assumptions**

- each bomber will endure 1 and exactly 1 AA shot on each SBR. if the allies position AA guns so a bomber would take 2 or more shots, the the bomber does something else other than SBR.
- an axis power shall have 3 bombers for SBR. 2 bombers shall SBR moscow or UK, with a max combined damage of 8. 1 bomber shall SBR caucus with a max damage of 4. the axis powers could station a single SBR base in EE to hit all 3 targets.

**bomber lifetime**

let the lifetime of a bomber be the random variable N. from assumption (1), the probability of a bomber being shot down on exactly the Nth SBR is

p(N=n) = (1/6) * (5/6)^(n-1) for n>=1

so

n p(N=n)

1 0.1667

2 0.1389

3 0.1157

4 0.0965

5 0.0804

6 0.0670

7 0.0558

8 0.0465

9 0.0388

10 0.0323

11 0.0269

12 0.0224

13 0.0187

or, to phrase the distribution a different way, here’s the probability of a bomber successfully completing n or more SBRs before being shot down.

P(N>=n) = (5/6)^n n>=1

n p(N>=n)

1 0.8333

2 0.6944

3 0.5787

4 0.4823

5 0.4019

6 0.3349

7 0.2791

8 0.2326

9 0.1938

10 0.1615

11 0.1346

12 0.1122

13 0.0935

i’m only interested in the first moment (mean) of this distribution, and it can be shown that the expected value is E[N] = 6. so each bomber is shot down on the 6th SBR on average, so it makes 5 successful SBRs on average. yes, i know it can be shot down in the first round, but it is equally likely that the bomber is shot down on the 11th or higher round. so i’m sticking with 5 successful SBR per bomber.

**SBR damage over lifetime**

let’s consider caucus first, since that’s easy. let the damage to caucus given a successful SBR be C. it can be shown that the average result E[C] = 3.0.

c P(C=c)

1 0.1667

2 0.1667

3 0.1667

4 0.5000

now let’s consider the damage moscow/UK, which is will be the random variable W. if we send only 1 successful bomber, then E[W] = 3.5.

if we send 2 successful bombers, then E[W] = 6.44

w P(W=w)

2 0.0278

3 0.0556

4 0.0833

5 0.1111

6 0.1389

7 0.1667

8 0.4167

**conclusion**

the caucus bomber will do an average damage of 3/round and will survive for 5 rounds. therefore, the caucus bomber is expected to deliver 3*5=15 IPCs worth of damage over it’s lifetime. that means for each bomber purchased to SBR caucus, the russians should expect to lose 15 IPCs. a $ for $ trade is advantageous.

the moscow/UK bombers will do an average damage of 3.27/round each and will also survive for 5 rounds. therefore, the moscow/UK bombers are expected to deliver 5*3.27 = 16.34 IPCs worth of damage over their lifetimes. that means for each bomber purchased to SBR moscow/UK, the allies should expect to lose 16.34 IPCs. this is better than $ for $.

so this seems like a perfectly viable long term strategy. if you couple this with the bonuses mentioned earlier, this becomes very tough for the allies.

**defense**

IMO, the best defense for the allies would be to work out a 2+ AA route against one of the bomber bases. we’ll have 4 AA defending russian territories–2 under R control and 2 under UK control. can’t afford more than 4. and 4 can really help slow the bleeding.

move the india AA gun to caucus, and move the UK gun to moscow. the UK AA gun can be replaced by the EUS AA gun. the russians would then position their AA pair to take away at least one japanese bomber base.

for the japs the 3 most likely bomber bases are bury, china, india. evenki/novo defends against bury. kaz/novo defends against china. kaz defends against india. if the japs spread out and have 2+ bomber bases, then the only thing russia can do is just block the base with the most bombers. can’t block everything without a 3rd AA, and russia can’t afford it. for the germans, the most likely bomber base is EE. arch/WR defends against EE.

if the russians get into the position where they are trading these territories (evenki/novo/kaz/arch/WR), then they’re screwed. they can’t afford to leave an AA gun there, so there will be at least 1 bomber base that does not have a 2+ AA defense. the russians are probably screwed anyways if they are trading territories next to moscow.

-c