Thursday, 17 November 2022

Bell Curve

The refight of Salamanca ended with a pretty overwhelming British victory. I came away thinking, as I often do after wargames, about probability theory. The ongoing rule changes to which I frequently refer are, as I hope you have realised all along, a feature rather than a bug. They are part of our (well James's really) Sisyphean attempt to develop the 'perfect' rules for any given period, usually but not always based on one of the Piquet family; a process which I rather enjoy and think the others do too. There are wargamers out there who indulge in a bit of Free Kriegspiel or in Matrix Games, but we in the Lower Wharfe Valley stick to the turning of cards, rolling of dice, drawing of dominos etc to establish the outcome of events. These randomisers usually involve probabilities which approximate a normal distribution (*). What we and other wargames developers are trying to do - although of course we wouldn't express it in this way - is to make sure that the centre of the bell curve is where it should be (**) on the x-axis, and that the outcomes which occur way along in the extremities of the tails don't ruin the game.

The second of those issues typically manifested itself in the original version of Piquet in the occasional very large swings of initiative leaving one set of players twiddling their thumbs while the other side got to move and fire all their units repeatedly. The second version of Piquet (i.e. FoB) got round this by giving each side the same initiative, while we adopted a domino draw mechanism that, for the most part, suits us by providing different, but mostly acceptable, initiative amounts for each side. It can occasionally go wrong though, and last night it did. The draw which one would most want is to get double six whilst one's opponent draws the six five. The chances of a particular side doing this are 1 in 784. Last night the British did it twice in an evening, say a dozen draws in total. I'll leave you to calculate how unlikely that is for yourselves (***). It did rather skew things. If I had been called in as a consultant by James, and I think you should assume that I haven't, I'd suggest taking out all the dominos with sixes on. As well as reducing the maximum initiative swing from 23:6 to 19:5 it would increase the odds of a turn ending early to 1 in 21, or virtually the same as in the original Piquet rules.

As for the other issue: is the centre of the curve for combat resolution where it ought to be? No, but it is of course still a work in progress. 

*     i.e. are normal in the limit

**   Given both our interpretation of history and our desire for a decent game

*** I believe that the number of times that particular draw of two dominos would be repeated in any given set of twelve draws would itself follow a binomial distribution, but I really can't be arsed to work it out.

1 comment:

  1. It's a perennial problem, Alan. Once, in a WSS game on our (used to be, pre Covid) annual wargaming week away, I took eight hits and had to role eight saving dice. I cannot remember the specifics and they are home brewed rules anyway, but all I needed was anything but a 1....and of course, I duly rolled EIGHT 1's....I don't suppose you can be arsed working out the odds against that happening either...I certainly can't! The one thing to be said for such an incident always remember where you were and what game it happened in ūüėÄ