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

 

[CellCoree]

I would approach this game from a mathematical perspective using principles of game theory. Game theory is a branch of mathematics that studies decision-making in competitive situations. In this case, the goal is to come up with a strategy that maximizes your chances of winning the prize.

Firstly, it is important to note that the best guess for 1 person is indeed 1, as there are no other options. For 2 people, the best strategy would be to guess 1, as you mentioned. However, for 3 people, the strategy changes.

Assuming all players are rational and trying to win, the key to winning in this game is to think about what other players might guess. If you think about it, the lowest unique number will always be the smallest number that no one else has guessed. So, for 3 people, the best strategy would be to guess 2. This way, if the other two players guess 1, you win. If one of them guesses 2 as well, it's a tie. And if both of them guess higher numbers, you still have a chance to win.

This same logic can be applied for any number of players. For example, if there are 4 players, the best guess would be 3. And for 5 players, it would be 4. In general, the best guess for 'n' players would be 'n-1'.

However, it's important to keep in mind that this strategy assumes that all players are rational and trying to win. In reality, there may be other factors at play such as personal preferences or biases that can affect their guessing behavior.

In addition, it's also worth considering the potential consequences of this strategy. By always guessing one number lower than the total number of players, you may be perceived as trying too hard to win and could potentially create tension or animosity among your colleagues.

In conclusion, while there may be a mathematical strategy to increase your chances of winning in this game, it's important to consider the social dynamics and potential consequences as well. It's always best to approach these types of situations with good sportsmanship and not let winning be the sole focus.