Luck is the sum over all of your attack rolls of the difference between what you lost and what, on average, you should have lost for that attack roll.  So I had naively thought that "luck is how many more troops you have than you should expect to have given your attacks."

I was quite wrong.  And the following disastrous start to a game shows why.  (Be prepared to see a lot of figures.  The td;lr is that when you're unlucky you both have bad rolls and more rolls.  More rolls means that your actual excess losses are much higher than the luck stat indicates.  In this case about 82% higher.)

Here is what happened.  I had 25 troops.  Attacked 10 in a fair battle.  I lost 20 (hence 20 deaths), neutral lost 10, and I won a victory that surely will doom me.  I cannot see a history for this game, but I know enough stats to figure out what happened.  I tie out the number of deaths and the luck score exactly if in the first 14 dice rolls I lost 19 troops, neutral 9, and then I lost the next dice roll, and won the following.  My luck score to greater precision is -6.42669753086418.  (Incidentally the standard deviations work out to -2.06769920352304 and so according to a normal approximation the odds of my having that initial combat go so horribly were about 1.99%.  Yes, I only seem to do these calculations when my luck is horrible...)

But am I only down about 6.43 troops from where I expect?  No, I'm not!  I did a single attack 25 vs 10.  My average losses in that case work out to 8.3023101151994.  I'm actually down about 11.70 troops!  (While I'm at it, my exact odds of my having done so badly in that attack were about 1.84%.  That's a pretty good convergence to the normal for being so far out in the tail and just have 16 rolls.)  That's a discrepancy of about 5.27 troops.

Why the discrepancy?

The discrepancy on luck stats is that bad luck in my initial rolls not only meant that I had higher than expected losses, but I also faced more rolls that I could expect losses over.  In this case I had roughly 5.7 more rolls than I should have, each of which cost me.  When you work it out, that's about 82% more losses than the luck stat by itself would have lead me to think.

Hopefully this analysis proves of interest to someone else.

you are wrinkling my brain.

This is "Expected losses for a continuous attack" versus "Expected losses rolling the dice as many times as I rolled", right?

They are indeed different. I like the continuous attack analysis for some things. If you compare a 2-4 defense versus a 3-3 defense, continuous attack expected losses is a reasonable thing to compute for comparison. And... the math is more fun than roll expectation :)

Hugh wrote:

This is "Expected losses for a continuous attack" versus "Expected losses rolling the dice as many times as I rolled", right?

They are indeed different. I like the continuous attack analysis for some things. If you compare a 2-4 defense versus a 3-3 defense, continuous attack expected losses is a reasonable thing to compute for comparison. And... the math is more fun than roll expectation :)

Exactly.  I was caught by surprise at how different.

As for the math, what's a little dynamic programming between friends?

And to Ozyman, be glad I didn't share code...

Thank you for this. I was very perplexed by the luck stat but it makes sense now.