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Who can go the lowest?

Question

A game token costs $10 to play.  The pay out is $100.  You can purchase multiple entries if you desire.  For each entry you purchase, you must pick the lowest positive number that no one else picks.  If there are ten people, including yourself, seeking to purchase tokens, what is your strategy?

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7 answers to this question

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Spoiler

Assuming that any positive number is valid, I'd buy one token and pick:

0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

as my number.

 

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Using the same method that BobbyGo took, my number would be a...

Spoiler

...1/googolplexianth. A googolplexianth is a one followed by a googolplexian of zeros. This number can also be written out as 10^10^10^10^10^10. So, the smallest positive number I can pick is (1/googolplexianth).

 

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I guess I took this question to be something different. Rather than just "what's the smallest positive value you can think of?" for which I would go with something like: 

1-(3↑↑↑↑...↑↑3) 
or in other words: 1 over Graham's number (or simply "G")...but why stop there? why not go with
1/G↑↑↑↑↑↑↑...↑↑↑↑↑↑↑↑G

That interpretation of the OP isn't as interesting or fun because it just gets ridiculous... The way I took the original question was pick the smallest positive INTEGER that no one else picks...what is your strategy when up against 9 others doing the same (all allowed to pick however many tokens as they want)?

So, if that's the case, it becomes a much more challenging problem. Obviously it doesn't make sense to purchase more than 9 tokens, as with 10 the BEST you could hope for is breaking even...I would probably purchase the following numbers (and my rationale next to them):

  • 1 - I'm assuming I will lose this $10, but maybe everyone else will think someone else picks this and therefore no one else does!
  • 2 - Might as well try this one as well, but again, most likely will lose this $10 as well...
  • 7...ish - Some relatively low number that hopefully no one else picks...I would assume I'd lose this.
  • 10 - Let's assume everyone buys their 9 tokens and picks 1-9...10 would be the first number that no one would pick...
  • 16...ish - Again, some pretty low number, that's pretty much just a shot in the dark and you hope no one else picks it. 
  • 42 - If by some crazy chance everyone buys 9 tokens and all are duplicated, this is the lowest possible number that wouldn't be duplicated...plus isn't it really the answer to everything?

At this point, I've already bet $60 in hopes of winning $100...and really, the odds are still not in my favor...so to me, it's not worth the risk and so my ultimate strategy would be to not play and "break even" :c)

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Since the integer requirement, there haven't been any replies, so I'll go with an obvious, and not-likely-to-be-optimal one.

Spoiler

Powers: Buy a single ticket, and randomly choose X >= 1, where P(X) = 2-x. So P(1) = 1/2, P(2) = 1/4, etc. 

 

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