Milliways Bar Bet

I arrange to meet you in a bar of our mutual choosing. [Perhaps at Milliways, so we needn't be concerned about having the time we'll need to do this.] This is not a magic trick, so I'm going to request 3:1 odds from you at the outset.

I offer you one of my many "special decks" of twenty-seven playing cards whose "face up" surface has been bleached blank, and ask you to write any single number you wish on each card. You are free to use real numbers (positive, negative, fraction, irrational), large and small, provided that they are all different, and that we can both agree easily on their relative ranking (e.g. low to high). So I suggest you avoid formulas whose value might be difficult to evaluate.

Once you've marked all of the cards, they are shuffled thoroughly by a third party (with no ulterior motive), and placed face down in front of me.

The Challenge:

I'm allowed to draw cards, one at a time, from the top of the deck and view their number. At some point, I will stop drawing and declare that the current card I'm holding contains the highest number present in the entire deck.

Q1: Over enough of these rounds, can I expect to make a profit from you? If so, how much profit can I expect to make over, say 10,000 rounds, if I'm putting up $1 against your $3 on every round?

Q2: What is my "winning" strategy?

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"That's hot!" --Paris Hilton (who recently tried to trademark the expression)

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Can you add a possibility in the poll?
"Déjà vu"

This known as the "perfect secretary" problem in decision theory.

Quote: kubikulann

Can you add a possibility in the poll?
"Déjà vu"

This known as the "perfect secretary" problem in decision theory.

I've always heard it referred to as the marriage problem or perfect suitor problem something along those lines. But yeah nice interesting little problem. Though now I wonder if regional and national difference lead to different namings of these types of problems.

http://en.wikipedia.org/wiki/Secretary_problem

Optimal strategy selects the winning card 37% of the time, so your bet has positive EV.

Reject the first 27/e cards (rounds up to 10), then select the drawn card if it beats all other previously drawn cards.

Quote: kubikulann

Can you add a possibility in the poll?
"Déjà vu"

This known as the "perfect secretary" problem in decision theory.

Let me guess, you can only interview one girl at a time, and since Einstein has proven that time=money,
you want to maximize the talent you hire with respect to how much time you spend interviewing?

I think I my do just that. I'm having trouble choosing between the "oldies but goodies" versus
less well known problems. I guess some of that is a free choice of how elitist one wants to be.
You and I both know that some "classic" problems are new, and I hope educational to many
members, and likewise boring to other members who've seem them dozens of times with only
superficial changes.

But at least, the "Déjà vu" option you suggest would allow me to see where I'm actually operating
along that spectrum. Noted and appreciated.

Quote: BleedingChipsSlowly

Reject the first 27/e cards (rounds up to 10), then select the drawn card if it beats all other previously drawn cards.

Correct.

But please use a spoiler tag in the future so others have a chance to play! Maybe something like:

<open bracket>spoiler=Who's Your Daddy, Now?<close bracket><Your reply><open bracket>/spoiler<close bracket>

Finally, does Wikipedia explain why this is optimal? If so, a simple "yes" is good enough for me--no need to paste derivation or link to derivation here.

Yes. I will use a spoiler tag in the future.