miker49,
take a look at this. this was my first attempt to setup a KGF 4x4 shuck with USA in 3 rounds:
http://www.axisandallies.org/forums/index.php?topic=13076.0
you can modify it to clear africa, to include a carrier purchase on USA1, or to get some armor going. but, whatever you do, try and get a 4x4 going by USA4.
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
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
… and if Japan takes India quickly-securing its AA gun
most UK players move the india AA gun to persia and then caucus. they don’t give it to you.
(even on the Caucuses because any roll over 4 still takes four IPC’s and 4 is greater than the average roll of 3.5)
no, even if you play with the box rules, if you roll [1,2,3,4,5,6], the damage would be [1,2,3,4,4,4], which is an average of 3.0 per rocket strike.
the Axis can have 4 rocket attacks and two bomber attacks on USSR. Thus the Axis should be able to bring a 17 to 21 IPC loss the vast majority of the time resulting in a near incapacity for the USSR to build new units by the third turn.
i mostly agree with the analysis. with a rocket attack on moscow (3.5) + rocket attack on cauc (3) + BMR raid on moscow over 1 AA gun (2.9) = average damage of 9.4 IPCs. if you do this with japan, then another 9.4 IPCs of average damage. an average of 18.8 IPCs, about what you said. i agree that ~19 IPCs of damage would be backbreaking. but…
I believe the odds are overwhelmingly in favor of this strategy succeeding. Of course it is relying on a dice roll, but so is every action in this game.
maybe i can convince you otherwise.
what is your replacement bomber policy? if the german bomber gets shot down, do you buy a replacement? do you buy a replacment japanese bomber? if so, yeah, you do 3.5 IPCs of damage when you don’t get shot down, but you have to buy a new bomber 1/6th of the time, for an average damage of 2.5 IPCs to your economy. plus the allies can position their AA guns so that at least one of the two bombers will be fired at at least twice to make a run on moscow.
what exactly is your research policy going to be? spend 1 research die a round per power (16.6%)? 2 (30.5%)? 3 (42.1%)? more? the axis only starts with a 70 IPC income, and if they spend 30 IPCs on research, then they spent almost 1/2 their income on something other than boots and ships. if you fail to get rockets the first try (likely) with either power, do you just keep trying until you do? i assume so.
plus, if japan moves its original AA gun to wake, japan has to buy 2 more AA guns to fire on moscow (10 IPCs). plus it looks like you plan to build an IC (15 IPCs) on the mainland to produce them. then you have to build them and move them into position. that’s 25 IPCs more that doesn’t do boots/ships.
it appears to me that your J1 plan is to build 1 IC on the mainland and buy 3 research die. J2 you would buy an AA there, maybe buy more research. on J3 you would move the AA into position, and it could finally fire the first time on J4. in short–you spend the first few turns doing nothing but setting up for a rocket attack. you are leaving yourself wide open for USA to sweep in and take away your islands and kill your fleet.
germany will have to move 1 AA gun to eastern europe, and the other to balkins for this plan. the allied fleets can threaten both your AA guns here, and russia might stack ukraine since germany will be troop light.
the necessary investment to complete the strategy is relatively minor compared to its potential payoff: Domination over USSR very quickly!
what? minor? even if you get the tech on the 1st round of 15IPCs for both axis powers. adding in the IC and 2 AA guns, you’ve spent 55 IPCs to invest in rockets. that’s a lot. not counting the other IPCs you lose by forgetting to build troops to defend/exchange your territories, it will take you ~3 rounds of rocketing/bombing to do enough damage to get paid back. plus both axis powers will be on their heels trying to stay alive long enough for their rocket investment to payoff–the US and UK pretty much have free reign.
My first round purchase is usually 5 tanks and and IC to be placed in Eastern Europe. Round 2 I purchase 8 tanks. 5 in Germany and 3 in Eastern Europe.
ok, this has really been bugging me.
plan 1) you’re going to spend $15 in G1 to buy an IC in EE. then spend $15 on G2 to build tanks there. on G3 you can finally use them.
plan 2) spend that same $15 on G1 to buy 3 tanks and place them in germany. spend $15 in G2 to build 3 more tanks in germany.
(2) is superior to (1) in every case.
*in (2) you can use the G1 tanks in G2 in kar/blk/ukr, or just park in them in EE, but in (1) they aren’t even on the board yet.
*the G1 tanks in (2) are 1 space ahead of the G2 tanks in (1)
*in (2) you have 3 more tanks on the ground compared to (1) at any point in time.
*in (1) you have to worry about defending EE. once UK gets transports running, he can just unload in EE and kill your 3 tanks. so this IC is no longer an asset (and likely becomes a liability). in (2) you do not have to worry about defending EE.
Thanks bigchris, what if we assume that the dice rolls of the attacker and defenders are two separate events?
16 rolls for attacker, 7 rolls for defender.
sure, that’s fine. as long as you are separating the rolls based on some arbitrary labeling and not based on their values, you can test any group/sub-group you like.
Ho: the dice are uniformly distributed
Ha: the dice are not uniformly distributed
Attacker: 6,5,6,5,4,6,5,6,6,4,3,5,4,6,3,6
histogram for entire sequence (16 rolls total)
6s: 6666666
5s: 5555
4s: 444
3s: 33
2s:
1s:
pearson chi-square test:
with 16 rolls, we expect 16/6=2.67 rolls in each bin according to Ho.
(7 - 2.67)^2 / 2.67 +
(4 - 2.67)^2 / 2.67 +
(3 - 2.67)^2 / 2.67 +
(2 - 2.67)^2 / 2.67 +
(0 - 2.67)^2 / 2.67 +
(0 - 2.67)^2 / 2.67 = 13.25
chi-square value = 13.25
degrees-of-freedom = 5
Probability (One-Tailed): 0.0211
that is, for the attacker, if Ho is in fact true, the probability of getting results as or more extreme than these is 0.0211. These data are sufficient evidence (at the 5% level) to reject Ho and adopt Ha.
Defender: 2,2,1,3,4,2,1
histogram for entire sequence (7 rolls total)
6s:
5s:
4s: 4
3s: 3
2s: 222
1s: 11
pearson chi-square test:
with 7 rolls, we expect 7/6=1.17 rolls in each bin according to Ho.
(0 - 1.17)^2 / 1.17 +
(0 - 1.17)^2 / 1.17 +
(1 - 1.17)^2 / 1.17 +
(1 - 1.17)^2 / 1.17 +
(3 - 1.17)^2 / 1.17 +
(2 - 1.17)^2 / 1.17 = 5.857
chi-square value = 5.857
degrees-of-freedom = 5
Probability (One-Tailed): 0.3204
that is, for the defender, if Ho is in fact true, the probability of getting results as or more extreme than these is 0.3204. this is insufficient evidence to reject Ho. therefore, we do not reject Ho.
Exact dice rolls:
Round 1:
Attack: 6,5,6,5,4,6,5,6,6 (no hits)
Defense: 2,2,1 (3 hits)Round 2:
Attack: 4,3,5,4 (1 hit)
Defense: 3,4,2 (1 hit)Round 3:
Attack: 6,3,6 (no hits)
Defense: 1 (1 hit)
histogram for entire sequence (23 rolls total)
6s: 6666666
5s: 5555
4s: 4444
3s: 333
2s: 222
1s: 11
Ho: the dice are uniformly distributed
Ha: the dice are not uniformly distributed
pearson chi-square test:
with 23 rolls, we expect 23/6=3.83 rolls in each bin according to Ho.
(7 - 3.83)^2 / 3.83 +
(4 - 3.83)^2 / 3.83 +
(4 - 3.83)^2 / 3.83 +
(3 - 3.83)^2 / 3.83 +
(3 - 3.83)^2 / 3.83 +
(2 - 3.83)^2 / 3.83 = 3.870
chi-square value = 3.870
degrees-of-freedom = 5
Probability (One-Tailed): 0.5684
you would need a chi-square value of 11 before you would adopt Ha (at the 5% level), and you are only at 3.870.
that is, if the dice are in fact uniformly distributed (Ho), the probability of getting results as or more extreme than these is 0.5684. this is insufficient evidence to reject Ho. therefore, we do not reject Ho.
now, the a priori probability of your friend winning that battle was likely greater than 99.5%, but that’s another issue.
whenever I make a move late at night usually around or close to midnight, I have noticed that the dice rolls turn out poorly.
Example:
2 inf, 1 art, 1 arm, 2 ftr, 1 bmb, 2 bb shot vs. 3 inf
Result: held, 1 ftr, 1 bmb retreats.
gnasape, be careful here that you are not falling victim to a statistical/psychological phenomenon known as selection bias. here, the emotional pain you (and us all) feel when you get hosed by the dice makes the event so memorable that it appears to occur more often than it actually does.
in your example, you’ve cited all of the units from a battle and the exact outcome. i bet you couldn’t do that with a battle that was very favorable for YOU during that same game. could you? the painful battles are just more memorable.
also in your example, you aren’t giving the actual dice results which could be analyzed for statistical significance, just that you got hosed. and big time. yeah, sure, your rolls in the example were all misses, but the opponents were all hits. so the pseudo-random number generator was probably producing samples in it’s normal distribution, but just not sorted in an order that you liked. it happens.
i imagine what happened is that once you got hosed, you looked at the clock for some reason and noticed it was around midnight. probably the next night it happened again at midnight, and your brain decided these samples were statistically significant (selection bias) and created an completely illusory artifact that “i get hosed at midnight”.
if you like, record a bunch of samples (~100) and post them here. i’d be happy to statistical significance test for you.
it is well known that the expected value for damage on an SBR is 3.5 IPCs (given the bomber survives AA fire and given the territory is 6 IPCs or more). there are a few posts taking a stab at the distribution for heavy bombers in LHTR, but no one has formulated it correctly. the work below shows the average SBR damage is 5.472 IPCs per heavy bomber.
in LHTR, heavy bombers roll 2 dice, take the max, add one, limit to the territory value, and that’s the IPC damage inflicted.
let A and B be independent random variables corresponding to the two dice rolled. they are distributed as:
P(A=a) ={ 1/6 a=1,2,3,4,5,6
{ 0 otherwise
P(B=b) ={ 1/6 b=1,2,3,4,5,6
{ 0 otherwise
let C be the derived random variable defined as:
C = max(A,B) + 1
Theorem: C is distributed as:
P(C=c) ={ 1/36 c=2
{ 3/36 c=3
{ 5/36 c=4
{ 7/36 c=5
{ 9/36 c=6
{ 11/36 c=7
{ 0 otherwise
Proof:
note: <= is the less-than-or-equal-to operator
note: E is the summation operator
P( C<=c ) = P( max(A,B)+1 <= c )
6
= E P( max(A,b)+1 <= c | B=b ) P(B=b)
b=1
6
= E P( max(A,b) <= c-1 ) P(B=b)
b=1
c-1
= 1/6 * E P( A <= c-1 ) c<=7
b=1
c-1 c-1
= 1/6 * E E P(A=a) c<=7
b=1 a=1
c-1 c-1
= 1/36 * E E 1 c<=7
b=1 a=1
= 1/36 * (c-1)^2 c<=7
and
P(C=c) = P(C<=c) - P(C<=c-1)
={ 1/36 c=2
{ 3/36 c=3
{ 5/36 c=4
{ 7/36 c=5
{ 9/36 c=6
{ 11/36 c=7
{ 0 otherwise
Analysis:
note: E is the expectation operator
so, if one were heavy SBR’ing a 7 (or more) territory, the expected damage is:
E© = 21/36 + 33/36 + 45/36 + 57/36 + 69/36 + 711/36 = 197/36 = 5.472 IPC
so is the extra ~2 IPC per bomber worth the investment? i’d like to know what other analysts think now that i’ve quantified the difference.
glad to see someone putting that idea to good use! i’ve also found that 5 INF, 2 ARM, 1 ART = 8 units for 29 IPCs is more effective than 4 INF, 4 ART = 8 units for 28 IPCs (if you can get the extra $).
1. Germany plays defensive from turn one and buy almost only infantry. This is the hardest situation where I often lose. Let’s say I have 32 inf. 32 art + 5 ftrs in Eastern Europe (16 inf. and 16 art. from UK and US each). Then Germany have about 50 inf. 6 fighters and some tanks and Japan is sending about 5-6 ftrs to support Ger. When running a combat simulator I have no chance to win. What shall I do then? Wait for more reinforcements? Try to attack Balkans and then Southern Europe to establish a base?
so, G is reduced down to only 3 territories (G, WE, SE) worth only 22 IPCs? meanwhile both UK and USA are cashing in around 40 IPCs each. this position is an allied win. if you keep G contained to these 3 territories, G can produce only 7 inf a round, and that can’t keep up with the allies’ 16 units a round. you will eventually have enough to take G, but it might take 10 more rounds. as long as (1) the allied stack is too large for G to kill, (2) you outproduce G, and (3) the allies occasionally help russia with japan–you won’t lose. maybe expand USA’s supply line to a 5x5 or 6x6.
2. What shall I do when Germany leaves France open? Let’s say I have 8 troops to Norway and I’m ready to shuck 8 more to Karelia. Shall I start to invade France or is it best to ignore it and stick to the plan to attack Karelia with everything.
WE is awesome! 6 IPCs! if you go for WE, make sure that you can continue retaking it each round. this really screws G, since she will have to spend at least 1/2 her income trading this territory with the allies. G can’t afford to leave WE in allied hands, since WE threatens both SE and G. if you do go for WE, the question is: do you reinforce WE with USA or does USA continue the Nor/Kar/EE build up? either way puts G in a tough spot.
the thing i like about tripleA is that i have a history in a saved game file. after entering all the moves (and editing), i can go back R1, step through each move and purchase and study the position. once i see the clever moves in round 4-5, i can go back and see how they were setup.
i pretty much have to do it this way since many of the allies moves aren’t obvious at first glance. it takes a few turns of the allies coordinating together (for me) to see why things were done several turns ago and why certain things were not done.
can abattlemap do that for me? or will i just have to use my memory?