In Axis and Allies, Can Heavy Bombers Be "Tamed"?

Question

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?

Answer 1

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.)

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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.

Answer 2

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.

Answer 3

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).