The Paradox of Sharing: should you share a contest if you get extra entries?
findingscience.com
findingscience.com
So, for a given n, your chances of winning is (n+1)/(2n+1) . The limit of (n+1)/(2n+1) as n->infinity isn't 1, as the graph implies; it's 1/2. (http://www.wolframalpha.com/input/?i=limit%20%28n%2B1%29%2F%...). So the more people you invite, the closer your chances of winning get to 50%.
If none of your friends accepts your invite, you have one vote, and your friends have no votes. 100% chance of winning.
If one of your friends accepts your invite, you have two votes (your original, plus your bonus), and your friend has one vote. 2/3 chance of winning. For two friends, you're at 3/5; for three, 4/7, etc. Note that we're above 50%, so the delta in your chance of winning is actually negative for each friend you invite.
The author is also confused whether you get an initial vote or not, but that's less important. The author didn't seem to count the friends' votes anywhere: "If two friends sign up, then you have a 2/3 chance of winning, then 3/4, then 4/5, and so on." If two friends sign up, there are either four or five votes total, neither of which is divisible by three, as that sentence states.
But the author's final conclusion is correct -- at least, it's close enough; you want to share with your friends such that they have enough time to enter the contest, but not enough time to share withe their friends. The appropriate time, of course, varies per-friend.
Assuming you are the only one in the competition, you have 100% chance to win. Invite one friend and you now have a 2/3 chance to win (2 entries for you, one for him). Invite another friend and you now have 3/5 chance (3 entries for you, 2 for your friends). Invite another and you have a 4/7 chance to win (4 entries for you, 3 for your friends). The general equation is (n +1)/(2n + 1), which approaches 50%.
Am I wrong with that?
I'm no statistician though so I suspect I'm missing something.
First, the case in which you are the only entrant is irrelevant. Your chances are already 100%... why share?
For all other cases, think about this:
-Getting one extra entry DOUBLES your chances of winning.
-Supposedly, the network effect of inviting friends offsets the above. Think about what that means. The number of people you are able to get to join through invites has to be MORE THAN HALF of the total number of final entrants (if there are 10,000 entrants already, you have to cause at least 10,000 more to join to offset the doubling above).
The author is essentially stating that by the end of the contest the majority of the entrants will have been caused by you. There is no universe in which this makes sense.
If you were in a competition with 10k (or any high number) other entrants, inviting a friend will almost double your chance of success.
Example: let's assume there 10,000 entrants including you, maintain the assumption that "a new person will sign up each day based on your initial invitation", and give it 30 days. Therefore, you only get 1 extra entry and 30 of your extended friends have signed up (each getting an extra entry except the last). Your chances have gone from:
1/10,000
to
2/(10,000+30+29)
You've increased your odds by 98.8%! And you could do even better by waiting until right before the contest deadline and inviting as many people as possible.
The break even point is ridiculously low too: only 60 entrants under these assumptions. So even with just "hundreds of other entrants", you should share.
Keep looking for that science, bro.
If I were really to invite 5 of my friends, their most likely response is:
a) not to join at all
b) not to share (that's extra work)
c) or being generous: invite one or two people
But let's roll with the post's bizarro assumptions that each accepted invite results in 2 more and see what happens if there are just 6 other entrants initially (this problem favors your argument with less entrants, not the other way around).
Your chances without sharing (remember, those other entrants are going to be able to bring in a billion persons each):
1/(1 + 2 * 6 * 2^30)
Your chances with sharing (you get two extra entries):
(1 + 2)/(2 * 7 * 2^30)
That's right. Every human being alive has joined. Only 6 other people started in the contest and you still improve your chances 157% by sharing.
The absolute only case in which it makes sense not to share is when you're the only person in the contest. And I would hope most people know not to try to improve on 100% odds.