Even More Bell Curve Bollocks

 I'm sure everyone is familiar with the remark attributed to a senior French civil servant after something had been explained to him: "That's all very well in practice," he said "but will it work in theory?". I feel somewhat the same about the domino method of determining initiative for games of Piquet. I'd previously been very happy with it, and then I asked myself what theoretical underpinnings it had. I should have let it lie.

The winner's probability distribution is indeed, as I predicted, right-skewed and the mean (9.5) is higher than the mode (7). There is a long tail of high initiatives, which individually have a low probability, but collectively add up to quite a lot. One third of the time the winner will get 12 or more.

The graph of the loser's initiative isn't particularly illuminating. The loser gets a mean of 3.5, with a mode of 4. The system this replaced was opposed D20s with the winner getting the difference. That produces a mean initiative of 7 to the winner, with 0 to the loser (*). Therefore there's a net benefit of 2.5 to the winner and 3.5 to the loser.

Other points to note: 

• 7% of the time the loser still won't get any initiative at all.
• The mode of opposed D20 rolls is of course 1, so there's clearly less overhead involved in drawing dominos.
• The loser can get as much as 10 initiative. That sounds good until you couple it with the fact that it can only happen if the winner gets 20, 21 or 22. Because of the way the initiative interacts with the card decks the chances are quite high that the winner will end the turn with that much initiative and so winning an initiative of 10 is definitely not the same as getting to use an initiative of 10.
• D20s only give 20% chance of getting 12 initiative or more. 

So, my feeling that dominos give big swings to the initiative winner more often is (probably) correct. However, on average it's better for the loser, which was always the point. I still think I'd be inclined to take out all dominos with a six on, but basically, however unthought through the theory behind it was, it mostly works OK in practice.

 

* All the analysis excludes either the same domino or the same result on the dice, both of which indicate the end of a turn.