To dice or not to dice?
Firstly, JJMicromegs (Canada) commented:
“When it comes to wargaming I purposely try to not get into the stochastic side of dice roles, even though sometimes it's hard to not think about it.I thought the last point was both apposite and close to my own view:
The reason is I want to take the 'game' elements out of it and want to be play as if I were a general. When I get start calculating probabilities it breaks the immersion.”
“I started looking at the maths because the way the dice work in FOG is different from DBM and much harder to estimate at a tactical level. The maths has shown me why and I thought the analysis would help others.I particularly like the phrase “When I start calculating probabilities it breaks the immersion” it’s such a clean encapsulation of the fun of wargaming.
More importantly, it has created a framework for me. With it I hope to stop undermining my generalship with false tactical hopes and fears based on dodgy estimates of success or failure.”
Glass half full or empty?
The next comment came from John Acar (USA) who wrote:
“When you analyze one outcome, you are correct in that you will see more wild swings with smaller amounts of dice.
When you test for the same outcome over and over again, your luck will tend to balance itself out as you are adding more and more dice to the equation.
Example: you have 3 units of light horse trying to hold a line that might not work out so well for you. [Later specified as three separate] light horse units [against three cavalry units with], for argument's sake, a 30% chance of victory for each individual unit. No unit influences the next as one outcome is independent of [any] other. However, if you look at it in terms of [the] chances that all three will win, then it [is] 0.3 x 0.3 x 0.3 or about a 2.7% … significantly less than the 30%.
While it is good to know the statistics, the larger question should always be, "How much damage will be caused should I fail?""This was a very thought provoking comment for three reasons. One is the luck balancing out point; the next is the three unit argument and the final one is the glass half full or empty approach. It’s a bit tricky to paraphrase my replies but here are the main points. Firstly, the luck balancing out issue;
“[3 units of light horse] against 3 units of cavalry [is] just three versions of the 4 v 2 combat at the same odds and nothing changes. Three attempts at (4 vs 2) is not the same as one at (12 vs 6).
When you repeatedly test something the odds for each test remain the same whilst the rolling average of all the results combined approaches the statistical average as previous high and low scores counteract one another. However you can't combine the rolls from different combats [in this way] for three reasons; two maths and one FOG related.
 The FOG reason is that high and low scores coupled with failed cohesion tests for either side will change the number of dice between each round of combat. So you can't repeat the test (or combat) ad infinitum.
 The maths reason #1 is that, even if things stay the same, each round of combat is an independent event and previous results have no impact on its outcome. The average changes; not the odds for each combat.
 The maths reason #2 is that the approach to the average is governed by the "Law of Large Numbers". The numbers of dice involved in this far exceeds those rolled in one FOG combat. Have a play with "Law of Large Numbers" demo to see this in action. Be sure to show the mean and roll totals; the latter continue to vary whilst the mean converges on the average.”Next is the three unit observation:
“There's potential [logic] trap here: a 2.7% chance of the light horse losing all three does not mean there's a 97.3% chance of the cavalry always winning.
If the odds of the cavalry winning are 66% which include 33% winning by one and 33% by 2 or more (in rounded odds). The chance of the cavalry winning all three by one, or all three by 2 or more, is the same as the light horse winning: 0.33 x 0.33 x 0.33 = 0.036 [or 3.6%]. [In contrast] the chance of the cavalry winning all three by any score is much better at 0.66 x 0.66 x 0.66 = 0.29 [or 29%].
So the great majority of outcomes lie in between and include partial wins for both sides. [Again] in FOG these only have a negative outcome if you also fail the cohesion test and any death rolls ...”This example illustrates one of the main points of the original article: extreme results with low numbers of dice are quite common and that it’s quite easy to kid yourself that things are worse than they actually are. On reflection I think the high odds of partial wins or draws (some 60%) strongly suggests that the initial conditions aren’t likely to persist much beyond the first two rounds of combat but that’s only an estimate. Finally, there’s the the glass half empty / half full view of options to which I replied:
"“How often will I lose and what happens if I do?" are equally important questions [not just “How often will I lose”]. Or even better "How often will I win and what happens when I do?".John then replied:
My FOG experience has been that the initial failure is quickly followed by further failure and so you're far better off fully understanding how often you're likely to win/draw/lose or you'll be unnecessarily timid.”
“I am most concerned with the negative aspects of any outcome. There is little in the way of positive aspects that would prevent me from performing an attack! Other options maybe but no positive aspects!”A marked contrast to my approach. Feel free to add yours below. I’d like to know if you are a glass half empty or a glass half full wargamer?