Thanks for looking at it.

Probability is on a per player basis (ie: each turn, a player has a chance p for detecting each undetected player). I'll edit this so it's more clear.

For LK (as well as LR and L_P), the term outside the summation is essentially (probability the player survives the whole game) * (game length). It's necessary since the game is of fixed length, and the summation is adding (probability of dying on turn x) * (turn x). Consider if the probability of detection is zero, and players will never die - without the term outside the summation, the expected lifetime calculation will return zero.