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Thursday, May 17, 2012

Why the Pie Hates You

The Math Behind Artillery and Air
Soviet forces come under fire from a German armored train
By now, I'm sure you've all read Eric Riha's excellent article on artillery math in Flames of War.  (If not, go read it here.)  You'll notice he has a whole bunch of complicated charts calculating the chance of killing a single tank under an artillery template. You'll also notice those numbers are pretty small.

I want to talk about why they're so small and what that means for using your artillery on the battlefield.

Imagine the chances of killing a single target under a template as a pie.  If it is automatically hit and killed, you'd get the full pie.  Each die roll after the first slices a little bit more out of the pie.  Artillery always needs at least three rolls to hit and kill.  Once to range in, once to hit, and one save.  If the situation requires a firepower test, that's four rolls.

2+ = 83.33%

2+ 2+ = 69.44%
2+ 2+ 2+ =57.87%

2+ 2+ 2+ 2+ = 48.23%

A US AOP calls in artillery fire
That may not sound like much, but it adds up quickly.  Look at the charts above.  The first pie is automatic success, of course.  The second pie shows the reduction from a 2+ roll, the second, two 2+ rolls, then three.  By the time you have made four 2+ rolls, the overall chance of success is less than 50%. 

So in practical terms, imagine there was a Tiger in the open, and all you had to do to kill it was not roll a 1 on four dice.  You'd fail more than half the time.

What's the moral of this?  Artillery makes a lousy tank sniper.  Even the best artillery is terrible at killing any single target.  I know we've all felt the temptation to aim the big guns at the enemy's heaviest armor and hope for the best -- but that's usually a bad idea.

Here is my own version of Eric's charts.  They don't take into account re-rolls or the chance to kill.  All I'm calculating is the chance to range in and to hit:

To range To hit Range EV Hits EV
6 6 0.166667 0.027778
6 5 0.166667 0.055556
6 4 0.166667 0.083333
6 3 0.166667 0.111111

5 6 0.444444 0.055556
5 5 0.444444 0.12963
5 4 0.444444 0.203704
5 3 0.444444 0.277778

4 6 0.722222 0.083333
4 5 0.722222 0.194444
4 4 0.722222 0.314815
4 3 0.722222 0.435185

3 6 0.888889 0.111111
3 5 0.888889 0.25
3 4 0.888889 0.398148
3 3 0.888889 0.546296

2 6 0.972222 0.138889
2 5 0.972222 0.296296
2 4 0.972222 0.458333
2 3 0.972222 0.62037

Notice how rapidly the EV to diminishes even under favorable circumstances.

But this is for a single target under the template.  If you can put more than one team under the template, you can multiply your EV by the number of targets.  So you are (surprise, surprise) better off aiming at a cluster of targets than a single target.  A wary opponent will keep his tanks spread out, but that's a harder game to play with infantry.

The pie also favors pinning over killing.  Merely hitting a target requires fewer die rolls than killing it.  That's one or two slices of the pie you get to keep. 
Italian cannons fire over open sights in a night battle

So the mathematics of artillery dictate an optimal use -- aim at a large cluster of teams, preferably infantry or guns, and expect to pin more than you kill.  About what you'd expect from the historical use of artillery.   (You'd think this was a simulation or something!)  If you do have to use your artillery as a tank sniper, don't expect it to do jack-all. Know that you are trying to beat the odds.

Air is a bit trickier.  Air in v3 automatically ranges in on a target in the open.  So it has (potentially) one fewer roll to make than artillery.  This should make it easier to use.  On the other hand, there are other rolls air must make -- and these slice the pie just as much as a ranging test.  There are rolls to see if your air arrives, whether it is intercepted, how many planes arrive, and whether they are shot down by AA.  All in all, air remains probably less reliable than artillery.

The dilemma for air lies in the armament for planes.  Many of them are specialist tank hunters with good FP and great AT.  It seems like a waste to fire them at infantry or guns; but (as we've seen) the chance of killing a single tank is low.  If you do go tank-hunting, you can take comfort in the fact that many of the most critical rolls are made before you place the template.  By the time you begin aiming, you will already know how many planes you have and where the enemy AA guns are located.

Just remember, every roll is a slice off the pie.  Every slice off the pie is bad.  The pie hates you.  Plan accordingly.

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