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Most players of Axis and Allies understand intuitively that the original heavy bombers tech "unbalances" the game by being disproportionately good. My own research confirms this by valuing the two additional rolls as 30 IPCs (per bomber), meaning that the tech pays for its 30 IPC research costs in one turn.

Some players allow heavy bombers only two rolls instead of three. You basically get two bombers for the price of one, +15 IPCs. That's still higher than other techs (6-10 IPCs), but less of an outlier than before.

My own preference is for a 50% increment in the bomber's original powers. That is two rolls for heavy bombers but only THREE numbers hit (instead of four), six numbers instead of four or eight. For strategic bombing, take the die roll +2 for a range of 3-8 instead of 1-6 (average of 5.5 hits instead of 3.5). This gives heavy bombers an incremental value of about 8, squarely in line with other techs'.

Is there a heavy bomber tech, either the "two roll" rule, or my 50% increment version, or some other suggestion that allows people to use heavy bombers without throwing the game out of whack?

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3 Answers 3

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Over at this question about A&A variants I suggesting changing heavy bombers such that each bomber is worth two dice, choose the higher value. For attacking this means the probability of successful attack increases 33 percent: from 0.67 (4/6) to 0.89 (1 - 2/6^2). For strategic bombing the expected value of IPCs would goes up by 28 percent.

So my rule makes bombers about 33 percent better than they were before, which using Tom's methodology is worth about 5 IPCs, since each bomber is costs 15. This is a 6-turn pay-back with 1 bomber, or two turns with 3--not accounting for the time-value of IPCs. This seems much more in line with the increases in the other techs. (Also see the footnote at the bottom explaining how this rule reduces the likelihood of bad rolls in addition to increasing the average. This is additional value not included in the 5 IPC figure.)


Derivation of the increase in expected strategic bombing value

For 1-die the value is 3.5. For rolling two dice and picking the higher one, the math is much more complicated. From this paper on dice probabilities, the probability that the highest number rolled with n dice is k is given by (sorry, no tex formatting on the board game site):

(k^n - (k - 1)^n) / 6^n

which for n = 2 reduces to

(k^2 - (k - 1)^2) / 36

Now we calculate the probability that our two rolls will give us each value from 1 - 6.

Value  Probability Equation
1       1/36       (1^2 - (1-1)^2)/36 
2       3/36       (2^2 - (2-1)^2)/36
3       5/36       (3^2 - (3-1)^2)/36 
4       7/36       (4^2 - (4-1)^2)/36 
5       9/36       (5^2 - (5-1)^2)/36 
6      11/36       (6^2 - (6-1)^2)/36 
       36/36  <--- just checking the sum

Now to calculate the expected value we have to weight these probabilities by the likelihood (i.e. multiple the value and probability columns and sum), which results in 4.47--a 28 percent increase over 3.5.

This table also makes clear that rolls are now going to be biased on the high side. Where as before the chances of getting a 1 were 1/6 = 17 percent, now they are 3 percent. And the probability of getting a 6 has increased from 17 percent to 30 percent. So the average has moved up and moved up in a way that makes it very unlikely you will get a bad roll, removing some of the luck from strategic bombing, further increasing their value above the postulated value of 5 IPCs per bomber.

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Your version of heavy bombers is more or less in line with other techs, right? –  Tom Au Jun 4 '11 at 19:01
    
@Tom: I think so, but I haven't played with it so long I honestly don't remember. From a non-empirical first-principles perspective I think it is. –  Adam Wuerl Jun 4 '11 at 19:07

Good solutions. In our group in MI we have a percentile tech table cross indexed by nationality. We give a free roll for a 6 each turn (2 for GE) and charge 4$ for extra rolls. There are SO MANY techs that the odds of getting heavy bombers is low (and there are also improved bombers that roll 2 d6, and the nuke that lets you for 10$ make a bomb that one of your bombers can use to devestate an enemy area) so if you get them you deserve it (though we do limit the 3d6 to economic damage, not single handedly destroying whole fleets).

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The rule I have used is very simple, "No Heavy Bombers", if you roll HB's you get your choice of the others.

An option I think might be workable is that instead of 2 or 3 dice, roll one die and it costs your opponent IPC's as normal unless you roll a 1, on a roll of 1 the factory is eliminated.

In WWII Allied Bombers were a key factor to victory, which makes for a good war and bad game. But with a board game you don't have to worry about the terrible consequences to humanity of a NAZI victory.

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@cos: I don't like to "suspend" game rules, would rather "fix" them instead. By my calculations, the original heavy bomber was worth 30 IPCs per turn, too much. Basically you get two "incomes" instead of one. Adam Wuerl and I found ways of "crippling" them so they were worth only 5-8 IPCs per turn. That's 1/4 to 1/6 of the original, and in line with other techs (6-10 IPCs). But thanks for writing. –  Tom Au Jun 10 '11 at 14:26
    
@Tom, what (in your perceptions) is the difference between changing (or fixing) a rule you don't like and suspending a rule you don't like? You are, indeed, right that heavy bombers tend to 'spoil' the game. I simply point out that one way to tame them is to just take them out of play. –  Cos Callis Jun 12 '11 at 10:32
    
@cos: the game designers intended six techs, and then "went berserk" in giving so much power to one. I enjoy the challenge of "reverse engineering" ALL techs, and using that knowledge to bring heavy bombers in line. Using my die roll +2 rule, a heavy bomber does 3-8 of damage versus a rocket's 1-6. But the bomber costs 10IPCs more and can be shot down. Now you have a real dilemma. –  Tom Au Jun 12 '11 at 13:02
    
@Tom, you simply restate the dilemma of the OP. I am not contradicting that the r+2 rule is a reasonable alternative (amongst many) way to "Tame Heavy Bombers". My most recent question is "How is changing a rule somehow different from suspending a rule. In your response to my suggestion you imply that "suspending" a rule was not as good as "fix"ing them. (whatever the rule, whatever the fix). –  Cos Callis Jun 12 '11 at 13:59
    
@cos: If you fix a rule, you have SOME version of the rule, potentially fulfilling the intentions of the game designer (perhaps better than he did himself). If you suspend a rule, you destroy part of the framework/fabric of the game. –  Tom Au Jun 12 '11 at 21:30

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