Playing the odds in Battle of the Bulge’s Bastogne conundrum

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A long time ago, strategy articles about boardgames went through an extensive process to get to you. First, they had to be typed. Diagrams had to be mocked up. They then had to be mailed to an editor, who put them in a magazine, which had to be printed and mailed. Eventually, they appeared in your mailbox.

Those days are gone, and I’m a little sad about it. Now, people apparently watch gameplay videos. That’s okay, I guess. There is a set of gameplay videos up for Battle of the Bulge which I haven’t watched yet. Videos are great for watching, but less great for savoring. You can bet that’s what I did with every morsel of strategy advice I ever read for Afrika Korps. Maybe games are more disposable now, or maybe they always were and I just didn’t know it. But there is something to examining a game methodically, and turning it over and over until you have a better appreciation of what it offers. Even if it doesn’t come in a cool magazine.

After the jump, first we take Bastogne

One of the staples of those magazines was the “perfect plan” that subsequently got dissected and refuted and argued over for months or years. I’m not crazy enough to write a perfect plan for Battle of the Bulge, first of all because I don’t think there is one, and second of all because that would just tell you, my potential opponent, how to win. And that’s crazy. But I was curious enough to do a deeper analysis of the game, and I found some interesting things that made me love it even more.*

Battle of the Bulge is a fascinating study in the cumulative effects of tiny variations in results. The Germans may succeed a little more here, a little less there. But those results make enough of a differencethat it’s hard to make reliable predictions about what the map will look like in a few days. But the general feeling seems to be that the Allies need to preserve their forces and avoid giving away any cheap victory points.

That’s not bad advice, but I think it discourages players from trying to disrupt the German advance when the opportunity arises, giving the Axis room to roam with the only pressure being the clock. A good Allied player should give them something to think about as well.

The Allies have an interesting dilemma at the outset: what to do about Bastogne. Because the Germans have the first impulse on Dec. 17, if there are no defenders in Bastogne when dusk falls on Dec. 16, the Germans can have the city for free. If the Allies don’t move Combat Command A of the 9th Armored Division out of Beaufort to Bastogne on the first impulse of Dec. 16, the Germans can execute what I call the “Longvilly Lunge” and send one infantry division from Clervaux to Lullange to attack the 9th Armored Division’s Combat Command Reserve, and one to Longvilly to cut the road. (This depends on clearing Clervaux with one of the surprise attacks, which will happen 96.5% of the time.) If the Germans manage to destroy the CCR/9 Arm with the Lullange attack (an 8.4% chance), the Allies will be forced to either divert CCB/9 Arm from the north or lose Bastogne without a fight. Moving CCA/9 Arm to Bastogne on the first Allied impulse is a pretty standard opening Allied move.

But even assuming the Allies cover the town on their first impulse, what happens next at Bastogne is an interesting problem. The Germans can still attack CCR/9 Arm in Lullange, but they’re limited to one division since the attack is across a river. As I mentioned before, destroying the armor is an unlikely result, although it can happen. See the table below.

But wait, you say, the table shows a 26.5% chance of getting two hits. That U.S. armor unit has two pips. Aren’t my chances of destroying it much better than 8.4%? No, because any unit that takes exactly as many hits as its strength will attempt to retreat, which will reduce its hits by one. In this case, CCR/9 Arm can retreat right back to Bastogne, to form the scratch force that will have to hold that town until 10th Armored Division arrives on the 17th.

So that chart says that CCR/9 Arm only has a 24% chance of taking no hits from the German infantry. Wouldn’t it make more sense to move it back to Bastogne before it gets attacked? I’m glad you asked, because I was wondering that as well. So I did a little number-crunching.
If CCR/9 Arm moves back to Bastogne before it is attacked, it allows both German infantry in Clervaux to cross the bridge into Lullange, and they’ll presumably be joined before too long by 2 Pz. That’s 13 strength points that can attack the puny American garrison the next day.

However, if CCR/9 Arm retreats out of Lullange after combat with one hit, it will have kept the other German infantry out of Lullange altogether. (The other German infantry can’t just move in later, as it had to be activated in Clervaux when the first one attacked, and presumably moved to Longvilly or Vianden.) So a reduced three-strength American force in Bastogne would only face a 9-strength attack on the 17th, from 2 Pz and one infantry division. In fact, the Americans have a 75% chance of inflicting at least one hit on the German infantry, which would take the assaulting force down to 8 strength or even 7. So the Germans will likely not be fighting at full strength.

That gives us three likely scenarios:

(1) The Americans withdraw to Bastogne and are attacked by 2nd Panzer, 6th Fallschirmjager, and 26 Volksgrenadier divisions, for a total of 13 strength points.

(2) The Americans hold in Lullange, are attacked, and either retreat to Bastogne with one hit of damage, or withdraw later after the attack.

(3) Same as (2) but the Germans suffer damage to their infantry.

Note that there are other scenarios, such as the Americans inflicting more damage on the German infantry, but I’ve kept it fairly simple here. And here’s the chart of results.

“Hits” just means the number of pips destroyed. Probability is the odds of rolling exactly that many hits. And the “cumulative” column shows the chances of rolling that many hits or fewer.

So in the left column, the 13-strength attack could theoretically inflict up to 13 hits, although the chances of rolling the highest numbers on the table are vanishingly small. Two undamaged American armor units defending in Bastogne will have four strength and benefit from both the woods and town terrain. That means it will take six hits to take Bastogne. And for the German maximum attack, the odds of rolling five or fewer hits is 88.4%. That means this attack succeeds just 11.6% of the time. No wonder Bastogne holds out so often.

The second column lists the odds for a 9-strength attack by 2nd Panzer and one infantry division. Under this assumption, the Germans attacked Lullange and forced a retreat, inflicting one hit of damage. That means taking Bastogne will require inflicting just five hits. But according to the chart, the Germans will roll four or fewer hits 89.3% of the time, making their success rate 10.7%.

And what if the German infantry is damaged in that first attack? The chances of inflicting four or fewer hits go up to 92.6%, meaning Bastogne falls only 7.4% of the time.

But what about that 24% chance of the German infantry will not inflict any damage on CCR/9 Arm? That would leave Bastogne with four strength points plus terrain reductions against just a 9-strength attack or less. And according to the chart, the chances of that succeeding are around 2%.

These are the small shifts in probability that add up over the course of the game. But what I like most about this chart is that it shows that these decisions make at best a few percentage points of difference in the results against Bastogne. There isn’t a clear “best move” based on these odds, as there would be if this were some cheap puzzle game. Rather, the most valuable information to take away is that if you want to defend forward, the odds back you up. And if you want to play it safe and preserve your troops, that’s a reasonable option as well. And there are some long-term implications to consider as well.

Because there is actually another possibility that I didn’t mention, and that’s the Allied armor being forced to retreat to Longvilly. If the Germans move their second infantry division to Vianden instead of Longvilly, CCR/9 Arm may end up not retreating to Bastogne at all! If that happens, the Allies will have only two strength points in Bastogne, and will therefore only be eligible for one terrain reduction. With the Germans needing only three hits instead of five or six, Bastogne’s odds of holding out drop to less than 50%.

But by not moving to Longvilly, the Germans give up the positional advantage of being able to work around the Allies’ southern flank. While historically the “southern shoulder” of the Bulge was comparatively quiet due to the lack of German armor commitment, the iPad version sees the Allies off balance, trying to keep the Germans from taking their reinforcement entry zones. This is an important part of German strategy and I’ve seen players even feint toward Bastogne and then send 2nd Panzer south through Vianden, trying to permanently delay Patton’s Third Army’s arrival by capturing all the spaces with roads off the southern map edge.** Without a unit in Longvilly, the Germans will be behind schedule. Is it worth passing it up to give CCR/9 Arm a place to retreat, knowing it may not end up going there, even if it has to retreat after combat?

These are questions of style rather than probability. Staying in Lullange to defend against the infantry assault may make sense if you’re trying to inflict as many points of damage on the Germans as possible, because those early casualties often add up to victory points when those same units are worn down to nothing as the campaign progresses. But you need to be able to adapt if that backfires and Bastogne is left with one armor unit on Dec. 17.

I took a similar look at what happens in Elsenborn, another space certain to be attacked by a strong German force on Dec. 16. In real life, the “northern shoulder” of the Bulge formed because of the stubborn resistance put up by the American troops on Elsenborn Ridge, which was never captured by the Germans. In the iPad version, Elsenborn is a speed bump for the Sixth Panzer Army and the formidable German force which starts in Gemund. It’s essential for the area to be cleared on Dec. 16th, because on the 17th those same units should press straight for Eupen with the hope of clearing that and threatening Verviers. Facing a 10-strength German force which includes the (non-elite) 12th SS Panzer Division, the Allies have only the understrength 99th Infantry. However, one space north in Monschau is the elite 2nd Infantry as well as the 102 Cav. The former can’t move on the 16th, but the latter can. Does it make sense to move it to Elsenborn to help defend against the onrushing Axis?

You can decide for yourself by looking at that chart. The 99th Infantry has three strength points, and can take advantage of both the woods and the town, for an effective strength of five. The Germans have only a 24% chance of getting four hits or fewer, so 76% of the time, Elsenborn is cleared on the 16th and the 17th will see a battle in Eupen. With the 102 Cav added to the defense, the odds of keeping the space Allied rise to almost 50%. One strength point doubles the Allies’ chances.

Of course, that means one fewer strength point in Eupen, which the German may exploit on the first impulse if 99th Infantry wasn’t able to retreat from Elsenborn. But keeping Elsenborn throws a serious wrench into the German first-day timetable, and a 50-50 shot is probably worth it. If the Germans use their first impulse of Dec. 17 against Bastogne, the elite 2nd Infantry in Monschau can move to Eupen and provide a reasonable defense.

The last opportunity to muck up the panzer schwerpunkt only happens 23% of the time, when 14 Cav in Losheim escapes the surprise attack and retreats to Malmedy to join CCB/9 Arm (11% of the time, 14 Cav will be unscathed in Losheim). This is just begging for a first Allied impulse activation of Malmedy and a move into St. Vith.

The terrain rules in Battle of the Bulge penalize low-strength defenders. There is probably a solid historical role-playing simulation reason for this, like a small force cannot take advantage of terrain to cover all the approaches. Whatever. What it means is that a force with two or fewer strength pips cannot take advantage of more than one terrain hit reduction, while a force with three strength cannot use more than two. St. Vith happens to have two features available: woods and town. If you try to cover St. Vith with just the two-strength CCB/9 Arm, you can only use one of those to reduce your hits. Your effective strength is then only three. But if 14 Cav survives and retreats to Malmedy, and you move them both to St. Vith, your three-strength force can now use both the woods and town reductions. Now your effective strength is five. And then take a look at the chart below***:

The chance of CCB/9 Arm surviving more than doubles. And if you count a successful result as anything that keeps an American unit alive, that chance jumps to a whopping 92%!

So what, right? I mean, what good does it do to have a unit in St. Vith? Won’t it get crushed? Well, yes. But because armor cannot exploit after combat on Dec. 16th, any German units attacking St. Vith will get stuck there, even if they clear the space. Even with no interference, there is no way for any German unit except 116 Pz to get to Vielsalm on the 16th. With St. Vith blocked, no German unit can reach Manhay on the 17th, and the only additional unit able to attack Bastogne on the 17th will be Panzer Lehr, if it has a chance to get to Clervaux. If not, the Germans are stuck with just 2 Pz and the infantry that made it to Lullange, and their chances of taking Bastogne if it’s defended by three or four strength points of American armor, is … well, we just did that math.

What if the Germans then throw Panzer Lehr into St. Vith? Well, their chances of clearing it get much larger. By my calculation, the chances of a combination of 116 Panzer and Panzer Lehr clearing St. Vith is just a hair better than 47%. But if they don’t, they suddenly have a big traffic jam.

And if the Germans throw in an infantry division in a desperate attempt to salvage the situation? Depending on how the previous attacks went, this might not even be possible. Based on what could have happened before, the overall chance of an infantry division fixing what the panzers couldn’t is just 3.5%. Almost not worth diverting the infantry, which is needed in Malmedy to back up the move on Spa.

So the question is, are the odds of surviving with at least one American unit in St. Vith into Dec. 17th worth (a) the weakening of the front in the north, and (b) the loss of the units themselves? That’s a really good question, and something the numbers don’t tell you. But imagine the situation: by now we have 116 Panzer, Panzer Lehr, and maybe an infantry division in St. Vith. If there is even a single Allied strength point in St. Vith at dusk on the 16th, I submit that this is a fantastic result for the Allies. There will be no one to cover Spa, but if the Germans use their first impulse to take it, then Bastogne will be saved for sure. And the push across the Ourthe will be significantly delayed, possibly allowing the Allies to shift some of their reinforcements northward. Plus the Germans will have been denied a victory point. Even if St. Vith is German-controlled, the diversion of 116 Panzer is probably worth it. And those St. Vith defenders will be firing across a river, increasing the chances of damage to the precious German panzer units.

As you can see, the choices offered by the survival of 14th Cav in Losheim open up a cascade of move and countermove. If the Allies don’t use their little delaying force to take St. Vith on their very first impulse, the Germans should immediately activate Bleialf and move 116 Pz to Vielsalm and 560 Volksgrenadier Division to St. Vith. If the Allies pursue the St. Vith stopgap defense, that means they have left CCA/9 Arm in Beaufort. The Germans then can make the Longvilly Lunge and deal with St. Vith on a subsequent impulse. After all, it isn’t going anywhere.

I think the numbers show that an aggressive defense as the Allies has a pretty good chance of slowing the Germans down. Because the German player is under such time pressure, throwing the timetable off in just a couple areas can snowball into real problems later on. In the next article, we’ll talk about some strategy beyond the first day.

If you used to read magazines back when those existed, this would have been about ten pages of text and diagrams. Now it is just one long screenful, however big that is. Infinite, I think. But it does show you how much things have changed since people first started writing these kinds of analysis articles. Except now, instead of having to write a letter to the editor and hope it got printed, you can now just post your disagreements in the comments.

*I know I promised a series replay earlier, which is still going to happen, but I may be able to do it against a live opponent rather than the AI. And that’s worth waiting for, I think. So you get a strategy article in the meantime.
**Technically you only have to capture Arlon, Herbeumont and Sedan, as these reinforcements cannot enter at Beaufort.
***Sample calculation for 116 Pz vs St. Vith: To hit chance is 60% (elite armor = 50% + 10% surprise + 10% artillery – 10% defender includes armor). Chance to get zero hits on five dice is 0.4^5. Chance to get one hit is 0.6*0.4^4*5 (roll one hit and four misses, and there are five different ways to get that result). Chance to get two hits is 0.6^2*0.4^3*10 (roll two hits and three misses, and there are ten ways to get that result). Overall chance of a two-strength unit surviving an attack in terrain is the chances of getting zero, one, and two hits added together (one hit is reduced to zero and two hits are reduced to one by terrain). Chances of remaining in the attacked hex are the chances of getting zero and one hit added together, and the chance of getting two hits is multiplied by 30% and then added, because a two-strength unit sustaining two hits in an objective hex has a 70% chance of being forced to retreat (die roll). Now you can do all your own calculations!