I'm impressed by both the OP and Paul's reasoning. But judging from the OP, for me to calculate my average multiplier, I would have to know my win rate. So if I required to keep track of my win rate, why won't I just keep track my average multiplier instead, which isn't that much harder.
Yup, that's a great point. I figured this out because I'm trying to make a tool for people to use to figure out how much money they'll make or lose based on the equipment they're wearing. While it's true that if you want exact values of either win rate or average multiplier, neither is particularly preferable since the same recorded data would tell you both.
However, I was more thinking along the lines of someone who was guesstimating, and it's much more intuitive for a player to estimate their win rate than their average multiplier. Most people won't know their average multiplier off the top of their head, but they can reason out a win rate.
Eg. I'm an ok player, but I'm in a top notch clan that lets me stack teams in battle, so I probably win about 60-65% of the time.
I'm a pretty good player, but I'm not in a clan, and I mostly play siege with a lot of players, so my individual impact is small, so I probably win about 50-55% of the time.
I'm not that great a player, but I'm a natural leader in siege, so I probably win about 60% of the time.
Playing battle you have a 50% winrate when you play an infinite number of rounds.
Playing siege is a whole different matter though. I rarely get past x3 on a siege server UNLESS there's a clan steamrolling the server and I'm lucky enough to be in their team for some rounds.
Some clans can hold a an x5 for hours on end by playing well together.
Overall, I make much less gold/exp on siege compared to battle.
Actually, siege is a closer approximation to my calculations than battle, for the reasons you stated. You might have an overall win rate of 50% on both battle and siege, but still get a higher average multiplier on battle. The reason for this is that the probabilities across rounds are not independent in battle, especially with heavy banner balance effects. That is to say, when you get your x2, usually you have a higher than 50% chance of continuing the trend and going up to x3, x4, and x5.
Siege team swaps like no other, so your probabilities are more likely to be independent between rounds. Of course, I don't actually have any empirical evidence that battle gives a higher average multiplier than siege, it may well be that the probability linkage balances out multiplier sprees of x2, x3, x4, x5 with bouts of x1, x1, x1, x1. Without meticulous data gathering, the impression that battle gives more gold/exp than siege for the same win rate can easily be chalked up to psychological bias, and a tendency to make a note of x5 sprees, given their salience.
