Win Free Stuff at Work: Game Theory Strategies

[Diffy]

In the office I work, there is a popular game when someone wants to give something away. When a person has something to give away, they send out an email to 'n' people. The email directs the recipients to try and guess the lowest positive integer they can that is unique amoung all the responses. So the email would look like:

"You need to send a guess which is a positive integer (1, 2, 3…) The person who guesses the lowest unique number wins."

I was wondering, based on the number of people 'n' is there a best guess?

For 1 person, 1 is the best guess for obvious reasons.
For 2 people I think 1 is still the best guess. If the other person also guess is 1, its a tie, if not you win.

For 3 people I am not sure.

I am looking for people who are smart then me to help me devise a strategy to win baseball tickets, software licenses, old hardware, booze, and other misc giveaways at my work.

 

[Gerenuk]

I tried googling:
http://www.tinbergen.nl/discussionpapers/08049.pdf
http://mathoverflow.net/questions/27004/lowest-unique-bid

Let me know what it say :)
I guess some exponential distribution of probabilities is best. Obviously the best strategy cannot be just one number, as everyone wants to use the best strategy, however if that was the best, then surely someone would want to depart.
The Nash equilibrium means the best strategy can be adopted by everyone and no-one would like to depart if everyone uses it.

 

[Office_Shredder]

Initial observation: the best number has to be between 1 and (n/2+1). If any of those numbers are untaken by everyone else, then your best choice is to take that number or to guess lower. If all those numbers are taken, at least one of them has only one person guessing it and you're screwed.

I noticed that the mathoverflow post and one of the papers cited there don't take this into account it seems (the paper restricts picks to be between 1 and n). It might be that picking numbers larger than n/2+1 is a good idea to minimize your chance of overlapping with someone else, so it brings up a question: If nobody has a unique guess what happens?

Also observe that if you can get half the people in the office into an alliance, you can guarantee your alliance wins all the prizes by just inputting those numbers. How you then divvy up the winnings is up to you of course (you can play this game again amongst yourselves. And then see how far down the rabbit hole you can go with it. Note: This is unlikely to make you many friends, but if everyone is sworn to the proper level of secrecy nobody will know that they're still one of the chumps).

 

[Hurkyl]

Initial observation: the best number has to be between 1 and (n/2+1).

While intuitively appealing, it is not true for the standard game.

For example, consider a three-player game where you know your opponents will choose according to the distribution:

1. 1/2
2. 1/4
3. 1/4

In the standard game, your odds of winning for each choice of number is

1. 4/16
2. 5/16
3. 5/16
4. 6/16

In the variant of the opening post, the odds are

1. 1/3
2. 1/3
3. 1/3
4. 3/8