@ Cardz
I read all of your "logic" the first game day. It is not correct.
The first problem is with your idealized game scenario.
You are arguing from a point of view that seems to have been contrived only to support your point. This is because in the scenario you suggest it is assumed that there are only two methodologies of killing people, the town lynch and the wolf night kill. However, one, I do not know of a game in which this is so, and two, it was a posteriori incorrect given our previous circumstances. This is because it was a game with 37 members, empirically this would make more than one method of killing people highly likely. If there were three kills every day-night cycle, the number of players would vacillate between odd and even every day night cycle, effectively flip-flopping through your "idealized" and "non-idealized" examples.
Next, your idealized game scenario completely ignores the possibility of an integral role. That is, you forget the Priest. Surely you would not say that a priest should not protect someone from death, would you? Yet that is effectively what you say should be done. Your "logic" states that it is more beneficial to gain information, as well as sustain an odd number of players once the day begins than it is to keep a townie alive. Beyond that fact, what if the priests protects someone. Would you then advocate not lynching someone in order to regain an odd number of people at the onset of the next day cycle?
The second is that your logic does not hold up even in your idealized game scenario.
Your numbers are correct. A majority in an odd numbered group is a lower proportion of the total group than a majority in an even numbered group. However what you neglect to mention is that the number of people difference cannot differ by more than one for adjacent numbers of people. Let me explain.
In number theory it is common to represent evens as numbers of the form 2k, and odds as numbers of the form 2k+1, with k being taken out of the set of integers. The fact that your numbers notice is that a strict majority of both groups is k+1, where odds are a larger group in all cases, thus the proportional difference. More specifically in the case that you criticize, evens would be of the form 2(k+1) and odds would be 2k+1 in which case a strict majority for evens would be k+2 and odds would be k+1. However as I just showed, these two value vary by a single vote. A single vote is not generally hard to get from an active player.
However there is another thing that makes your point moot, even in your idealized scenario. That is that whoever has the most votes at the end of the day is lynched. The majority rule is simply necessary for ending the day early. While it does not explicitly say this in the rules, it is the rule that this game was played by.
Lastly, what scenario can you give me in which it is more beneficial for the town to lose one kill earlier?
I ask this because it is what you are advocating in all of the cases where the first person lynched is not a townie, which is what I think statistically the majority of the cases are.