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Friday, March 30, 2012

Flames of War Statistics 101 (pt 2)

Post by Eric Riha
Hello Again Everyone,
I’m back again to continue our conversation on FoW Statistics. Last time we talked about Expected Values in Flames of War, how to calculate them, and how these values can help us make better tactical decisions on the battlefield. Today I’m going to talk about the lesser known, but often better ‘remembered’ Potential Value.

If you need a quick refresher on Expected Value, you can find the previous article here.
Much like Expected Value, Potential Value is exactly what it sounds like: the ‘potential’ number of successes or failures out of a set of conditions. Potential Value helps us make tactical decisions in two ways: proving certain shooting options are better than others, and providing the foundation for ‘tactical pessimism’.

The first point relates to certain shooting/tactical options one usually has with American Tanks (Stabilizers) or any other situation where you have the option to fire two shots at 6’s to hit or 1 shot at 5’s to hit.

People in this position often make a random decision – usually based on ‘how well’ they’ve been rolling that game. However, by looking at Potential Values, we can see that one option is statistically better every time.

Using our Expected Value method, let’s take a look at the second part of that same Sherman 76 vs StuG G scenario. In that scenario, we had a platoon of Sherman 76’s moving into short range on a platoon of concealed, Veteran StuG G’s. Our previous equation for that was:
This was a simple ‘move n shoot’ scenario, with no special rules thrown into the set-up. At moving RoF (1 shot per tank), we saw how this was used to find a final result of .74074 dead StuG’s (given a platoon of 5 Sherman 76’s).

Now let’s bring in that famous FoW special rule: the U.S. Stabilizer. In this scenario, our To Hit number increases from 5+ to 6 (Vet 4+, concealed +1, Stabilizers +1), but we fire at our stationary RoF (2 shots per tank). Now our equation looks like this:
When we multiply our ‘per shot’ value by our platoon’s RoF (10 shots on the move w/Stabilizers), we get an Expected Value of .74074 dead StuG G’s – exactly the same Expected Value as our scenario without using the Stabilizers. If we only looked at Expected Value to compare our tactical options, we could assume that both maneuvers would achieve the exact same tactical results. However, we can see that only one option is truly best once we calculate the Potential Value of each scenario.

Think of Potential Value as the best possible result you can achieve out of whatever scenario you are looking at. Assume all of the dice rolls go in your favor – you roll a 6 on every die and your opponent rolls a 1 on every die. While this is a very improbable scenario, it is certainly still a statistically possible one. I’m sure you know somebody in your local group that’s run into that luck before; although usually the stories are told from the opposite point of view!

When we look at our Sherman 76 vs StuG G scenario, a roll of a 6 to hit, a 1 for an armor save, and a 6 for firepower will result in exactly 1 dead StuG G. This result will occur with or without the use of Stabilizers. However, without Stabilizers, we can only achieve a Potential Value of 5 dead StuG G’s. By enabling the Stabilizers, we achieve a Potential Value of 10 dead StuG G’s.

While Potential Value calculates the most extreme positive result, it also calculates the most unlikely result. However, it does provide us with a gauge of ‘how much better’ an above-average result might be.
If we simply look at “to-hit” results for a single Sherman tank in our scenario, we have the option of firing 1 shot at a 5+ to hit or 2 shots at a 6 to hit. You have the same statistical chance to roll a 5+ on 1 die, or a single 6 on two dice. Once again, the Stabilizer option is better, because it is the only option that gives you the possibility of hitting twice.

The same results can be seen when firing weapons at unarmoured teams – firing full RoF at long range vs. moving to close range at reduced RoF. Two shots at 6’s to hit will always be a better option - if the only tactical result of movement is a reduced To Hit number. Unlike Expected Values, which can produce a wide range of results and ‘rules’ regarding how to interpret a variety of tactical situations, the Potential Value gives us a hard rule across all given scenarios – it is always better to roll more dice, if the Expected Value for each option are equal.

Potential Values also provide us with a reason to employ what I call ‘Tactical Pessimism”. Since Potential Values can also swing in your opponent’s favor, it is not wise to rely completely on Expected Values. I’m sure we can all think of a time when we had a ‘game in the bag’ and our opponent pulled some crazy shit by rolling four 6’s to hit and we rolled 1’s for every save – handing the game to our opponent on a silver platter.

Our tendency is to ‘blame the dice’ in these scenarios; however, there is no such thing as ‘bad dice’ – only poorly calculated maneuvers that failed to account for Potential Values.
I will illustrate my point with a ‘real-world’ example. At the 2011 Masters event, Bill Willcox’s only loss came at the hands of Phillip Messier; mainly because he failed to properly account for his opponent’s Potential Value during a particularly important shooting step.

The story as Bill retells it, is that he had several tanks capable of firing upon a variety of targets, and a platoon of 5 Hetzers was his main target. After firing about half of his tanks, the platoon of Hetzers had 1 dead tank and 4 bailed tanks. If we were to calculate the string of events necessary for Phillip to use his Hetzers on the following turn to kill one of Bill’s Shermans, it would be the following:
Interpreted, each remaining Heter tank had about an 8% chance to kill one of Bill’s Shermans in return fire the next round. Given these odds (I’m sure Bill didn’t calculate these on the fly, but ‘reasoned’ something very close), Bill used the remainder of his tanks to fire on other enemy units – something he admits was the crucial mistake that put the game soundly in Phillip’s hands. (This is not to say Phillip is a sub-par player and Bill ‘gave’ him the game. Rather, at the highest levels of competition, a great opponent will wring every advantage possible out of a mistake of this magnitude). Phil passed his platoon morale check, remounted all 4 Hetzers, and proceeded to blow the snot out of an entire Armoured platoon in one round. While unlikely the result of his choice was unlikely, Bill should have (again, under his own admission) ‘confirmed the kill’ and used his remaining tanks to destroy the remnents of the Hetzer platoon.

Now obviously, one does not play a game of Flames of War where every shot bounces, every opposing infantry team saves, and every opposing air strike destroys an entire platoon (even if it seems like that now and again!). If one only followed Potential Values, they would play a ‘cowardly’ game of Flames of War – never engaging, always retreating – fearing that his units would be so easily destroyed. One must use these tools – Expected Values and Potential Values – to build a set of tactical ‘rules’ for one’s self; something that allows you to properly account for risk on the field of battle and mitigate it as much as possible.

In my next article, I will go into some tactical theories in detail where we use modified versions of the Expected Value to produce scenarios we can plan and account for during a game of Flames of War. These scenarios allow us to account for some effects of our opponent’s potential value, but without hamstringing our offensive capabilities.

Eric Riha
FoW Mathamagician


Whiskey51 said...

My brain exploded.

WEBGriffin said...

Keep 'em coming!

But then you are sharing "secrets" with the masses...

The Riha said...

I do want to make one addendum to the Stabilizer's rule, based on a suggestion from the previous article.

If one looks at the ACTUAL probability (Binomial) of using a stabilizer to hit a target that is already 5+ to hit (making it 6+ w/Stabilizers), you will actually find that the percentage chance to score one hit is LOWER than that of rolling one die by itself.

The true % chance to score 1 or more hits using Stabilizers to go from 5+ to 6+ to hit is actually 30.55% - slightly lower than the 33% to hit when rolling 1 die at 5+.

While I still agree that it is best to roll with Stabilizers in this situation (the ability to roll 2 hits instead of 1 outweighs the slightly lower chance to hit), there may be particular situations where you want to avoid using stabilizers.

meads594 said...

Very nice posts! I do have one minor quibble - the number you give in the last example (hetzers gonna hetz on Bill Wilcox!) is wrong. The platoon morale roll is for the whole platoon, not each individual vehicle. The rule for "summing the individual probabilities" (which you are implicitly using by multiplying your calculated per-vehicle value by the number of vehicles) only applies for uncorrelated probabilities. The platoon morale is correlated across all the vehicles.

A minor point, and probably isn't going to change the ultimate conclusion anyway. I would have to think a bit on the "correct" way to do it (and since it is Friday afternoon I'm trying to avoid thinking).

The Riha said...

Agreed on the quibble, though I did not sum by the number of vehicles, only their individual RoF.

"each remaining Heter tank had about an 8% chance to kill one of Bill’s Shermans in return fire"

That being said, I purposely avoided summing by the number of vehicles because I could not readily figure out how to approach that equation.

If you could find the 'correct' way to do it, I would be all ears.

The Riha said...

Actually, now that I think about it, it may not matter.

If you calculated the probability for return fire without the morale check for the whole platoon, then multiplied that number by the % chance to pass a morale check, you would find the correct result, no?

If that is the case, then it would not matter where the fraction was multiplied in the equation, no?

Sorry, Dirty Jon - I know, not supposed to post-counter-post here :P

Nathan said...

One thing that I think should also be added to the mix is that the more dice you roll the closer to the statistical average your results will be. So rolling 10 dice vs 5 in the stabilizer example also improves the chances that you will get the expected results.

strelkovaya said...

From Phillip Messier's page on the FoW Website:
"I’m also hoping to get a chance to go up against Bill Wilcox again. My only loss at the 2010 nationals was to him and the game came down to a single reluctant platoon morale roll, which he made and thus won the game. There’d be a certain irony if I ended up defeating him in a similar kind of circumstance. If that happened it’d certainly make my Masters experience."


Izimu said...

I really like this article series. As much as I like the other tactics articles, I really appreciate getting down to the bone and gristles of FOW mechanics like this.

Outcome in FOW is ultimately driven by a series of dice rolls. Tactics within the game often comes down to how you stack the odds in your favor. Some tactics advice I've seen are abstract to the point being vague and almost a bit fanciful. Almost like a zen koan. So it's nice to see an article that gets down to concrete mechanics.

Max M. said...

One day Riha... someone WILL nerf you!!

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