Heavy Tanks - a new technology

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.

I am starting to believe we’re doing something wrong.

Don’t feel bad, it is a really complex game that takes hours and hours to learn. When I first started we spent hours to get just 1 turn through as well! But it’s worth it! We’re rooting for you, feel free to ask any questions (well I don’t have any specific advice, just wanted to give you a pep talk  :lol:)

ONLY if you have the Combined Bombardment technology.

Otherwise, ONLY the Battleship fires, and then only once for the entire battle, and then only if it did not engage in combat the same turn as the amphibious landing.

I made two new rules to that can help change strategy and make the game more historically accurate.  They are air transports/paradrops.  You need bombers for this.  For air transports, you can carry anything you can on a transport on a bomber and land it in another friendly territory where it drops off all units in the air transport, but aa rules apply if you fly over an enemy aa gun.  Also, once the bomber lands it stays in that territory for the rest of the turn.  In paradrops, you take a bomber and use it to transport 1 or 2 infantry to either an enemy or friendly territory and then fly it back to a friendly territory.  Again, aa rules apply.  Also, when paradroping into enemy territory the bomber can’t be used both to paradrop and bomb, you either send in two troops or use it to bomb, not both.  Also, if an aa shoots down a bomber with units on it, the bomber as well as the units are lost, and if attacking an enemy territory with an aa gun with paratroopers, the aa gun fires at the bomber before the paratroopers are drooped.  One more thing is that you cannot use bombers used in an attack, airdrop, or bombing raid during the combat phase as an air transport or to paradrop into friendly territory in the noncombat phase.
  Another new rule I made is neutral armies and territory value.  With this, you give neutral territories armies and value so that it becomes legal to take these territories, but you have to fight a neutral army first.  Also, as soon as you attack a neutral territory, the territory and troops in that territory are turned over to one of your enemy nations until you take it over.

No bid, with equally skilled players and no major mistakes, and no bid…

Axis win from 30% to 40% of games.

after you launch the program select “load new game” look for the file revised.xml, select it and click  open.  now try starting a game, it should give you the revised version.

@Revelade:

Okay, two more questions…

After you attack a territory with an air unit and you kill all of their units, must the air unit that you used, land back to any friendly territory within one space or does it go back to any territory within its movement range?

They can move to land in any friendly territory within the limit of their movement

@Revelade:

If air units can not capture land territories, what’s the point of leaving men in territories like Yakut SSR?  Wouldn’t the game just be about stationing all men at the borders?

You can try to do this, and many folks do.  The risk is, once a break-through has been made in your front lines, you might not have any reserves to seal the breach.  Also, no nation (except Russia) has an IC on the frontier, thus all builds have to work their way to the front from an IC.