Philosophy Talk and the Paradoxical Facebook Contest

So we’re approaching  1,000 fans on our  Philosophy Talk Facebook Page and we say to ourselves,  “We ought to have a contest.  Maybe we could give the 1,000th fan and the person, if any,  who invited the 1000th fan to become a fan some really cool philosophy related prizes.”

Sounded like a fun idea at first.   But after we thought about it for, oh,  a few seconds, we realized that there’s a catch! Facebook pages publicly display the number  of fans who have subscribed to that page.  That means that if we were to announce that we were holding such contest   anybody who wanted to win the contest would, in all likelihood,   refrain from signing up or inviting a friend to sign up, until the the fan count was, say,  999.   If everybody thought this way,  we’d be in serious trouble because then nobody would sign up or invite another to sign up.

Maybe  we could get around that perverse incentive by making the prize so valuable that 94 people — fans 907 – 10000 — might be incentivized to share the prize.  But believe me,  we’re not likely to come up with a prize such that 1/94 share of it would be of significant value. 
 
Or maybe we could just give out a prize and not announce that we intend to do so.  But then,  where’s the increased incentive for new fans to sign up?  
 
So what are we to do?   We  want to have a contest.  And we want the contest to incentivize more people to sign up on our facebook page as Philosophy Talk fans.  And we want to make it known that we are holding such a contest. 
 
So here’s the question,   how do we run a contest that satisfies al those constraints?   We came up with two quick  ideas.
 
 The first is to hold a drawing that all fans — not just the 1000th fan — have a chance of winning.    The trick is that the drawing is  triggered by the 1000th fan to sign up.  The announcement  would run something like this:    Become the 1000th fan of Philosophy Talk on Facebook and trigger a drawing that you have a 1/1000  chance of winning.   
 
The obvious problem with that,  it seems to me, is that it’s not much of an incentive  — unless the prize is really good.   Plus something in me says it’s not quite fair to the 1000th fan.  Shouldn’t  the triggering  get something special, something that nobody else gets?   Maybe two chances to everybody else’s one, perhaps? 
 
Another thought is that we could do a sort of “surprise exam” type contest.  We announce that there is a contest and announce that for  some number n,  the nth fan will win a bunch of cool prizes, where n is greater than or equal to 1000.  But what we don’t do is announce what n is in advance.  
 
I think I like that one.   But it’s probably got some hidden downside too. 
 
Anyway, there will be a contest.   And it will be announce and it will incentivize people to sign up.  
I’d love to hear your ideas — serious, tongue in check,  whatever — about how we should do it.