For instance, say days 1-99 are treated thus (all prisoners initially hold one conceptual token):
If the light is off:
1. And it is your first time in the room, remove your token and leave the light off.
2. And it is your second time in the room, designate yourself the counter, and count Days-1 as your initial count. Turn the light on.
If the light is on do nothing.
On day 100:
If the light is off:
1. And it is your first time in the room, then everyone has been in the room. Demand freedom.
2. And it is your second time in the room, designate yourself the counter, and count Days-1 (=99) as your initial count. Leave the light off in this instance since it is day 100.
If the light is on:
Turn it off.
That method gives you a head-start. On average the initial count is 12, however it is not an optimal head-start. For instance there is no reason to dedicate the first 100 days to it, in fact the first 20 days would be more than enough. It is so improbably that your initial count will be more than 20 you may as well save the other 60 days for better use.
Like I said, that concept I didn't think of initially myself. However my concept is that the more people you have on zero tokens the faster you will be able to transfer all the tokens to the counter. If you were to designate the first 20 days to this step you would instead have:
For instance, say days 1-19 are treated thus (all prisoners initially hold one conceptual token):
If the light is off:
1. And it is your first time in the room, remove your token and leave the light off.
2. And it is your second time in the room, designate yourself the counter, and count Days-1 as your initial count. Turn the light on.
If the light is on do nothing.
On day 20:
If the light is off:
1. And it is your first time in the room, then designate yourself the counter and start the count at 20. Leave the light off.
2. And it is your second time in the room, designate yourself the counter, and start the count at 19. Leave the light off.
That works much better, you can also improve your chances by having a special case for days 2 and 3 - that is, there is a 1% chance that the same person will be chosen twice in a row, if this happens you could insert the rules for days 2 and 3:
Day 2:
If it is your second time in the room, switch the light on but do not designate yourself the role of counter.
Day 3:
If the light is on, and it is your third time in the room (0.1% chance) then designate yourself the role of counter and start the count at 1.
If the light is on, and it is your second time in the room then designate yourself the role of counter and start the count at 2.
You get away with this because according to the rules for this period if on the third day the light is on, you know with certainty only one person had been in the room before you. On the forum which discussed this riddle over 3 years, no one ever mentioned that (at least I don't think so, I didn't go through all of it). What no one figured out, on the forum that discussed this riddle (in over 3 years), is that you can give yourself a further head-start by repeating this step! I couldn't believe it either. I know the average for repeating it goes down and down, but you only need to designate a certain number of days to it.
I would designate 20 more days for repeating it (possibly 1 or 2 days less). Thus:
The rules for days 21-39:
If the light is off:
And you are not the counter, and you have one token then remove your token and leave the light off.
And you are not the counter, and you have no tokens, then turn the light on and your new token count = days - 21.
And you are the counter then switch the light on and amend your count days - 21. Turn the light on.
I do not yet know the optimal number of times to repeat this, but I imagine it might be 3 or 4 times. The whole idea is to get as many people onto zero tokens as quickly as possible. Yes, you will have people with counts of any possible number, but they can be transferred to the counter in larger clumps at later times.
People on zero tokens are passive, that means when they enter the room they don't have to get rid of any tokens by switching the light on, and they don't have to collect any tokens by turning the light off. This is a huge advantage. by the end of 80 or 100 days, I imagine you could have average it to ~20 people on zero tokens (that is just an estimate at this stage). Compare that to running it once with 100 days allocated to it - which has an average of just 11 people on zero tokens. You've got a huge head-start.
Anyhow, that's not as far as I've got, but that's as far as hints from me come. I'm sure you can easily take this in the right direction to find the optimal plan now (something that apparantly no one from the original site could find).