

The Paradox of Sharing: should you share a contest if you get extra entries? - dremmettbrown
http://findingscience.com/statistics/2012/06/04/playing-for-3k-share-or-no-share.html

======
zck
The first graph here is wrong. Recall the constraint that only you and people
you invite can enter. If you invite _n_ friends, you have _n+1_ votes. Since
each of your invites also gets a vote, there are _(n+1) + n_ votes total, or
_2n + 1_.

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%...](http://www.wolframalpha.com/input/?i=limit%20%28n%2B1%29%2F%282n%2B1%29%20as%20n%20goes%20to%20infinity)).
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.

------
doomslice
I don't quite understand the conclusion formed in the first half... that your
odds increase as more friends join. I would think it would actually decrease
for every friend that joined with an asymptote at 50%.

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?

~~~
petercooper
Haha, I just posted the very same comment on the blog. The only thing I could
think of is that you win _if one of your referrals wins too_ (like AppSumo
does with their MacBook Air contests).. but then in the hypothetical "only you
+ friends" scenario, you'd have a 100% chance of winning(!)

I'm no statistician though so I suspect I'm missing something.

~~~
doomslice
I think the author messed up. Logically I was thinking "wait... it should go
down", and then when I started to do the math I _started_ doing the same
mistake that the author made in the calculations until I realized I wasn't
adding 2 entrants to the total pool for every friend invite (the bottom term).
Easy mistake to make, but I fear the article's conclusion is wrong. Instead,
the answer should be a resounding no (assuming you don't gain some marginal
utility for seeing your friend win instead of some random stranger).

------
jere
The final conclusion is incorrect. I've gone into detail responding to heeton,
but I now thought of a simpler way to explain it.

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.

------
heeton
This completely ignores the fact that in a real competition, you are NOT the
only person entering.

If you were in a competition with 10k (or any high number) other entrants,
inviting a friend will almost double your chance of success.

~~~
jere
Exactly. The comment at the end "Now imagine there are hundreds of other
entrants, even further decreasing your odds" makes no sense. Those hundreds
(actually, more like you said: tens of thousands) of other entrants make the
negative effect of inviting your own friends negligible.

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.

~~~
dremmettbrown
That ignores the effect of your friends sharing with their friends and so on.
I added the final graph to illustrate this point - yes, having 2 entries is
better than having 1, but the network effect of your friends pulling in their
friends eventually kills your odds.

~~~
jere
Your "network effect" assumptions are rather silly. After all, by the end of
your last graph you've caused a _billion_ extra people to join the contest
(2^30). Give me a break.

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.

------
dremmettbrown
zck is right about failing to account for the friends votes - which does mean
an asymptotic approach to 0.5 rather than 1. I updated the graph (and the
example fractions). That does, however, change the conclusion, that you
shouldn't share with friends at all.

------
planetguy
Is this a "should" as in "is it in my own selfish interests" question, or a
"should" as in "is it right to harass my friends in order to marginally
promote my own interests as part of yet another goddamn evil viral marketing
scheme for yet another goddamn internet startup" question?

~~~
davefp
Read the article and find out?

