My assumption was that with a big enough sample size (as in->infinity) that the distribution of the numbers that people would guess would be equal across the range at which the guesses would likely be...so you could ignore the average and concentrate on medians..
In hindsight, the assumption is probably wrong...as evidenced by my horribly guess.
But it's a regressive game...so I didn't expect people to go with the high numbers at all...
EDIT:
Ok, not that it matters anymore...but here is what I'm thinking.
If given the instruction to choose a random number from 1 to 100...the average of everyone's number will be fifty. So, it really isn't a question about averages as much as it is a question of medians..
So the question is about the distribution of the guesses. 66 would be the highest number a person could choose to win the game...that is, even if everyone chooses 100, the average will be a hundred and two thirds of that will be 66. so, 66 is the upper limit of any rational guess.
so, what is the lower rational limit? The game is regressive..so theoretically any number from 0-66 could be the answer. So, if everyone chose zero, then the answer would be zero. If everyone chose one, the answer would be 2/3. If everyone chose 3, the answer would be two...and so on.
I'll assume that on and beyond the upper and lower rational limits, the rationality of every guess will not have a pattern. So, you get to throw out 0 and 67-100 out because they are not rational guesses and anyone guessing them won't affect the rationality of your guess. You aren't playing against irrational players..the players you need to concern yourself with are the rational ones. And those players will be found guessing something from 66-1.
So , what would a rational person, a person who knows all the above, likely guess?
You can assume, whatever the distribution is, to have a pattern. Parabolic, progressive, regressive..
I chose regressive...which means people are likely to choose lower numbers than higher ones...
Hence my guess.
It'll be wrong though, because I should've assumed that an equal amount of people will choose progressive and regressive...and gone with a parabolic distribution.
Aye, that's the variable that is seemingly too complicated to understand.
If everyone would submit their number in secrecy, and the amount of submitted numbers would be at the least a thousand, then I think my earlier answer of 44 is statistically the most likely outcome assuming most people would be influenced by the regressive nature of the problem (two third of two third). But since it is public as well, there will be people who can even calculate the current average and go from there, having a big unfair advantage over everyone that had previously submitted an answer.