In the previous article, Bunnies wrote about cost efficiencies of units, and the importance of keeping costs down. In this article, Bunnies writes about how combat results can be evaluated as a nice neat number, but cautions the reader that there’s a lot more to understanding gameplay than looking at such numbers.

## CALCULATING COMBAT RESULTS

How many combinations of ways can ten attacking infantry get three hits? n! / (k! (n-k)!), that is, 10! / (3!)((10-3)!) What is the probability of ten attacking infantry getting three hits? 10! / (3!)((10-3)!) * ((1/6)^3) * ((5/6)^7).

By using similar calculations and collating results, a player can determine the probability of different combat results. Manual calculations take a while, so it is better to use a dice simulator or a calculation program to predict results during the course of a game.

## CALCULATING EXPECTED NET IPC GAIN/LOSS

### CASE 1: ARTILLERY VS INFANTRY FOR CONTROL OF 2 IPC TERRITORY

An artillery attacks an infantry that is defending a territory worth 2 IPC. What is the expected payout?

On the first round of combat, the probability of the artillery getting a hit is 1/3; the probability of no hits is 2/3. The probability of the defending infantry getting a hit is 1/3; the probability of no hits is 2/3.

1/3 * 1/3 = 1/9 both hit. Attacker loses 4 IPC worth of units; defender loses 3 IPC worth of units. Net IPC change is (-4 + 3) = -1 IPC.

1/3 * 2/3 = 2/9 only attacker hits. Defender loses 3 IPC worth of units; attacker gains 2 IPC territory. Net IPC change is (3 + 2) = 5 IPC.

2/3 * 1/3 = 2/9 only defender hits. Attacker loses 4 IPC worth of units. Net IPC change is -4 IPC.

2/3 * 2/3 = 4/9 no hits. Combat repeats.

On the second round of combat, we take into account the chances for all rounds of combat, not just the second round. There is a 5/9 chance combat was resolved on the first round. There is a 4/9 chance that combat was not resolved on the first round (that is, that combat repeated). After the second round of combat is finished, that 4/9 chance breaks down into further fractions, some resolving the combat, and some going into a new round of combat.

On subsequent rounds of combat, the overall chance that combat has been successfully resolved on a past round approaches 100%, and the overall chance that combat goes into an even later round of combat approaches 0%.

The chance of combat repeating folds into the chance of combat resolving over time, so we can evaluate the overall chances of combat resolution by looking at the first round of combat, ignoring the chance that combat repeats, and comparing the ratio of the remaining fractions. That is a 1/9 : 2/9 : 2/9 ratio, which evaluates to a 1:2:2 ratio. That 1:2:2 ratio evaluates to 20%, 40%, and 40%.

So if we evaluate combat to its final resolution (everything destroyed, or only attacker or only defender surviving), there is a 20% chance of both attacker and defender hitting (net -1 IPC), a 40% chance of only the attacker hitting (net 5 IPC), and a 40% chance of only the defender hitting (net -4 IPC). The immediate expected overall payout of the combat is (0.2 * -1) + (0.4 * 5) + (0.4 * -4), or 0.2 IPC.

### CASE 2: INFANTRY AND ARTILLERY VS INFANTRY FOR CONTROL OF 2 IPC TERRITORY

infantry and an artillery attack an infantry that is defending a territory worth 2 IPC. What is the expected payout?

On the first round of combat, the probability of the infantry and artillery getting at least one hit is 5/9; the probability of no hits is 4/9. The probability of the defending infantry getting a hit is 1/3; the probability of no hits is 2/3.

Similar to Case 1, the chance that combat repeats with no casualties being inflicted by either side gets folded into the chance of combat resolving. However, Case 2 is a bit more complex. Now, if the defender gets a hit and the attacker does not, the same combat does not repeat; the resulting second round combat involves only the surviving artillery vs infantry for control of the 2 IPC territory.

5/9 * 1/3 = 5/27 both sides hit. Attacker loses 3 IPC worth of units; defender loses 3 IPC worth of units; attacker gains 2 IPC territory. Net IPC change is (-3 + 3 + 2) = 2 IPC.

5/9 * 2/3=10/27 only attacker hits. Defender loses 3 IPC worth of units; attacker gains 2 IPC territory. Net IPC change is (3 + 2) = 5 IPC.

4/9 * 1/3 = 4/27 only defender hits. A new combat results, with artillery vs infantry for control of 2 IPC territory. Net IPC change is 0.2 IPC. (This new combat’s results have been evaluated in Case 1.)

4/9 * 2/3 = 8/27 no hits. Combat repeats.

Resolving these figures as in case 1, we get the ratios 5/27 : 10/27 : 4/27, or 5/19 : 10/19 : 4/19. If we resolve the final payouts, we have ((5/19) * 2) + ((10/19) * 5) + ((4/19) * 0.2) = 60.8 / 19, or 3.2 IPC.

## BEYOND IMMEDIATE NET IPC CHANGE

### COUNTERATTACK

Assume the best result for the attacker in case 2. That is, suppose the attacker destroyed the defending infantry and both the attacking infantry and artillery survived to take control of the 2 IPC territory. The immediate net IPC change for the attacker is 5 IPC. But that does not account for what may happen on the counterattack. The attacker committed a valuable infantry and artillery to taking the territory. That’s 7 IPCs worth of units that can be destroyed by counterattack.

Suppose the defender had an overwhelming counter that used infantry fodder. The chance of the counterattack destroying the infantry and artillery on the first round of counterattack fire would be close to 100%. The chance of the infantry and artillery inflicting casaualties would be 1/3 each. The value of an inflicted casualty would be 3 IPC (one of the counterattack’s infantry fodder).

On the defender’s counterattack, the original attacker would lose 7 IPCs of units (the infantry and artillery) for an expected average 2 IPCs worth of enemy units, or a net -5 IPC.

That loss would completely wipe out the immediate net IPC gain from the attacker’s best case scenario.

### OPPORTUNITY COST ON ATTACK

In case 2, the attacker committed an infantry and an artillery to a battle for a 2 IPC territory, but those units may have been more useful for defense or another attack. The presence of one or two additional units can make a difference of 5%, 10%, or even 20% or more of success in a critical battle, and can make a difference of more than one or two units in the survivors of a battle.

Suppose the attacker had another attack that turn that involved 5 infantry 5 tanks against 10 infantry. The odds for that battle favor the attacker with around a 66% chance of the attacker winning, usually with two or three attacking tanks left. But if the attacker added an infantry and artillery to attack with 6 infantry 1 artillery 5 tanks, the attacker’s odds jump to about 92%, usually with about five attacking tanks surviving. About two and a half additional tanks are preserved by adding two more units to the attack, plus a sizable safety margin in the percentage of success. The value of the tanks alone is 12.5 IPC, which is greater than any of the potential payouts given in Case 2.

### OPPORTUNITY COST ON DEFENSE

Suppose Germany controls Eastern Europe with 6 infantry 1 artillery, that Russia controls Karelia with 3 infantry 2 tanks 2 UK infantry, and that Russia controls Belorussia with 1 infantry. Suppose Case 2 were Germany attacking Belorussia, and ignore the rest of the map.

If Germany attacks Belorussia, it weakens Eastern Europe to the point that Russia has good odds for an attack. On the other hand, if Germany doesn’t attack anywhere, it has good odds of holding on to Eastern Europe in case of Russian attack. It is better here for Germany not to attack at all, and to keep the units it would otherwise send to attack in Case 2 on defense.

In the counterattack, the problem is Germany losing the units it sent to attack. Here, Germany instead loses other units entirely; units that could have been preserved if Germany kept the units it sent to attack on defense.

## IN OUR NEXT EXCITING EPISODE

Bunnies describes some of the mistakes players make when they think about probability as it applies to Axis and Allies Spring 1942.