Friday, 17 January 2020

Linear versus exponential in BKC4

I have been asked to expand on my statement yesterday that "the benefits of being in cover increase in an exponential manner while the benefits of better quality or greater quantity are linear". Before I do so, let me clarify a couple of points:

  • The sentence in question is constructed using the rhetorical form of antithesis largely for literary effect. It's really only the cover bit that bothers me. Even then it's the least of the reasons that I don't like Blitzkrieg Commander; it got on the list because it features a lot in the Sidi Rezegh scenario.
  • I haven't actually read the latest version of the rules. I have read BKC2, but for BKC4 we are relying on James, who is umpiring. If there is some error in the way we are playing then I will be the first to hold my hand up and acknowledge that it's all his fault, although obviously I will not be angry with him, just disappointed.
  • I am well versed in the higher mathematics, but clearly can't be arsed to do any heavy number crunching, so have therefore limited myself to first order approximations; specifically I have chosen to ignore the impact of saving throws. Should anyone wish to take me on in the area of probability theory I feel obliged to remind you that my first degree was in Mathematical Sciences and to warn you that in doing so you will be entering a world of pain.
  • As a contrast I claim no particular knowledge of Second World War tactics, equipment, combat or anything else battlefield related really; I'm more of a grand strategy sort of guy. Some things don't look right to me - which is why I'm going to mention them - but if you tell me they are OK then I won't argue.

Blitzkrieg Commander is a D6 based game. Units have a number of attack dice and a range, either or both of which may be different for anti-tank and anti-personnel firing. There are slightly different rules for off-table artillery, but they don't change the overall point being made. As the quality of units increases (e.g. better types of tanks) they gain extra dice and/or longer range. Any unit firing within half of its range gains an extra dice. If more than one unit fires at the same target they add the number of dice together. All this is recognisable, perfectly sensible and, I believe, fits being described as linear.

When a unit or more fires on another the dice are rolled. In the open any 4,5 or 6 counts as a hit. If insufficient hits are made to eliminate the target (hits are only cumulative within the firing players turn; they are removed before the owning player's turn begins) then a number of dice equal to the number of hits made is rolled and any 4,5 or 6 causes the target to be suppressed i.e. it can't activate on its next turn. (Suppression markers are not removed until then end of the owning player's turn.) It is, in essence, a game of suppression; make the enemy keep their heads down and then carry out your own objectives. We can see that, in the open, we would expect to suppress a unit once for every four dice rolled against it. Four dice would, on average, result in two hits, and two hits would, on average, result in one suppression.

When in cover two things happen: the to-hit throw required becomes harder and there may be a saving throw (actually armoured vehicles in the open get a saving throw as well). I'm going to ignore saving throws in the calculation because they complicate matters and actually work to skew the thing even more anyway. So in the next type of cover the roll required is a 5 or 6 to hit, and also subsequently to suppress. The same logic as above shows us that we expect to suppress the target for every nine dice rolled against it. In the hardest cover we have to roll a 6 to hit and afterwards to suppress. We therefore expect one suppression for every thirty six dice. It is my contention that the progression 4, 9, 36 is exponential in the everyday sense that the increase is becoming more and more rapid.

It is also my contention that something isn't right. An infantry unit firing at close range at another infantry unit rolls four dice. The target requires six hits to destroy it. So if you get caught in the open you would expect, on average, to get suppressed if the opposing unit activates and fires once and destroyed if it activates and fires three times. If the target is in hard cover then in order to expect to suppress it on the first activation (I use the term 'expect' in the sense of probability theory) you would have to fire at it with nine units. Now, clearly one would anticipate using a larger number of units, but does nine seem right to you? That point is to some extent moot, because the real problem is how many turns you would expect it take you the eliminate the target. The answer is, of course, also one. Having thrown the thirty six dice one would expect six sixes. In other words firing against infantry in hard cover it is as easy to kill them as to make them keep their heads down; or, if you prefer, it is as hard to make infantry in trenches keep their heads down as it is to kill them. Does that make sense?

In reality I am well aware that if you repeated this unlikely 9 vs 1 scenario many times you would on occasion suppress some units without killing them because the other side of the distribution curve contains a whole number of units killed without being first suppressed. To which I reply, so what? 


  1. First, let's get the preliminaries out of the way...

    "In this corner"

    My undergrad degree was in applied math (actuarial science) with advanced degree in economics and statistics. Still, I know when not to pick fight! I agree that the progression of suppression as terrain hardens looks exponential, for sure.

    Thank you for the BK rules' explanation. Based on that explanation and your examples, you make clear and valid points. I also find it interesting (that is, if I understand your examples correctly) that it takes throwing 36 dice in one turn to have an expectation (in the probability sense) of destroying a unit NO MATTER THE DEFENDER'S TERRAIN!

    Something seems akilter here in the rules in both killing and suppression.

    Next Question: Are mathematicians inherently drawn to wargaming or are wargamers drawn to mathematics?

    1. Pass friend, and welcome to the inner circle. I think actuarial techniques are exactly what we need here.

      I shall answer your questions in detail in another blog post, but no, that's not what I was trying to say. Apologies for the confusion which I seem to be sowing.