Wallet Game, All Welcome To Play

[Jazza]

there was a post a little while ago about a game called wallet, which got me thinking about the game and the best way to play it

so here are the rules: you and an opponent choose how much money to put in your wallet, when you both have done this you put your wallets on the table, and the person who's wallet has the most money wins all the money in both wallets

you both have infinite bankrolls, but have to choose a finite amount of money

so what strategy can you use to guarantee a minimum EV of 0?

also, i will create a contest for those who wish to enter:

everyone PM's me their prefered strategy (in the form of a probability distribution function), and after say 3 days i will see who's strategy is best overall, by calculating the EV of all the match-ups possible, and then averaging all the EV's so each person has one overall EV, highest overall EV wins

one thing to note is that the best strategy for the contest may not be the strategy that guarantees a minimum EV of 0

and btw i haven't tried to work out the answer yet

[kyro]

Maybe I'm missing something crucial, but there's seems to be an obvious way to guarantee an EV of 0.

[Jazza]

good point, i hereby change the rules:

the amount of money you put in your wallet has to be greater than 0, and finite

[mostsmooth]

there needs to be more to this game , no?
one could just keep putting money into(or writing a total)the wallet forever to guarantee he has the most. wouldnt that guarantee >0ev?what person is going to put $100? in the wallet?
you need to incorporate some maximum difference or something i would think

[Jazza]

[ QUOTE ]
one could just keep putting money into(or writing a total)the wallet forever

[/ QUOTE ]

forever seems to imply infinite, which is not within my rules

and yes sometimes i would put $100 on my wallet, but not always

[mostsmooth]

forever is an infinite amount of time, not money
youd put more than $100 if it would prevent you from losing it

[joshman1204]

forever is finite but you could say that I spent 80 years putting money in my wallet at which point I had 80,000,000,000,000,000,000,000 so why would you ever stop putting money in? even if the time is infinite the amount of money could still be finite correct? I think you must have an upper bounds to make this work.

[mannika]

[ QUOTE ]
I think you must have an upper bounds to make this work.

[/ QUOTE ]

That's what you would think, but in fact, having an upper bound makes this game not work. If the upper bound is set at $1000, putting $1000 in your wallet guarantees a win or a tie. I think there must be an inherant contradiction or paradox or something in the rules of the game.

[TomCollins]

There is no contradiction. It doesn't have to be money, it could be "pick a positive integer. I will pick a positive ineteger". There is no contradiction without specifying an upper bound.

[mostsmooth]

[ QUOTE ]
There is no contradiction. It doesn't have to be money, it could be "pick a positive integer. I will pick a positive ineteger". There is no contradiction without specifying an upper bound.

[/ QUOTE ]
then its just stupid
what would be the challenge? basically whoever thinks of the biggest number/dollar amount wins. theres no need for any strategy as the OP was looking for.

[joshman1204]

Exactly what I was thinking. If I have an infinite amount of anything to choose from and all I have to do is choose the highest number then there is no possible strategy to this game. Am I missing something here?

[riverdance]

i'll put 100 million billion google trillion dollars in my wallet.

ok, i put 100 million billion google trillion and one dollars in my wallet, i win.

easy game

[mostsmooth]

[ QUOTE ]
i'll put 100 million billion google trillion dollars in my wallet.

ok, i put 100 million billion google trillion and one dollars in my wallet, i win.

easy game

[/ QUOTE ]
a better answer is 999 trillion trillion trillion trillion googleplexes^trillion trillion trillion trillion googleplexes
but it would probably still lose

[pzhon]

[ QUOTE ]

so what strategy can you use to guarantee a minimum EV of 0?

[/ QUOTE ]
This game is not compact. You can't assume there is a strategy that has an EV of 0.

[ QUOTE ]

everyone PM's me their prefered strategy (in the form of a probability distribution function), and after say 3 days i will see who's strategy is best overall,


[/ QUOTE ]
Three days may not be enough time to figure out what I would send you unless you have studied that type of mathematics, but it may be enough to determine it is larger than anything anyone else sent.

Here is a start. (http://www.sci.wsu.edu/math/faculty/hudelson/moser.html)

[PairTheBoard]

It seems to me that using probabilty distributions doesn't change the game that much from just trying to choose the largest number. You could still do this anyway by choosing a delta distribution.

Also, I think that every Distribution will beat infinitely many and lose to infinitely many, possibly without finite expectation if both distributions are not tight. Unless you define some finite measure on the space of Distributions I don't see how you can hope to compute some kind of overall expectation for a strategy, or a measure for how it does overall against the others. And whatever measure you might be able to define would be arbitrary. I'm not even sure if the space of unrestricted Distributions is measurable, although I suppose I should know that. I'm afraid I've forgotten more than I remember.

PairTheBoard

[PairTheBoard]

With a little more thought, I think it's easy to see that no Distribution insures at least 0 EV against all others.

proof: Let the probabilty measure on the reals be P. Define P*[0,1]=0 and for all b>a>1 define P*[a,b]=P[a-1,b-1].

PairTheBoard

[Jazza]

woops, i apologize to everyone, i goofed, this game is indeed stupid /images/graemlins/blush.gif