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        riddles >> general problem-solving / chatting / whatever >> To buy or not to buy? And how to share?
        
(Message started by: BenVitale on Jul 18^th, 2009, 2:51pm)

Title: To buy or not to buy? And how to share?
Post by BenVitale on Jul 18^th, 2009, 2:51pm

Two roomates are condering purchasing an espresso machine.
The machine costs $50.
Each has a maximum willingness to pay, but each knows only his own willingness to pay and not
the other's.

It is efficient to buy the machine if and only if the sum exceeds $50.

They have to decide two things: whether or not to buy the machine and how to share the cost.

Title: Re: To buy or not to buy? And how to share?
Post by pex on Jul 18^th, 2009, 2:58pm

- If both are willing to spend at least $25, just buy the thing and split the cost evenly.
- If not, but one of them is willing to spend $50, he buys it and does not allow the other to use it.
- Otherwise, there will be no espresso machine.

Okay, it's not game-theoretically solid, but that's how it works in practice.

Title: Re: To buy or not to buy? And how to share?
Post by BenVitale on Jul 18^th, 2009, 5:02pm

A natural proposal is to share the cost equally... but this can be inefficient.

Game:

Two roomates are considering buying an espresso machine.
Cost = $50.

The Roomates: You and your buddy.
Each of you has a maximum willingness to pay. You know your own willingness but don't know his and vice versa.

Suppose that one of you has a high willingness to pay, say $40, but the other has a low willingness to pay, say $20.

How do we define a "high willingness"?

An amount higher than $25.

The sum ($40 + $20) exceeds $50

In conclusion, one of will reject splitting the cost

So now we may ask, "how can they improve the mechanism?"

Title: Re: To buy or not to buy? And how to share?
Post by Grimbal on Jul 26^th, 2009, 8:29pm

Well, they put what they are willing to pay in an envelope each.
They go to the shop get the machine. As payment they give the envelopes. If the salesman complains it is not enough, they take back the envelopes. If the salesman returns money, they share what he returns.

Title: Re: To buy or not to buy? And how to share?
Post by BenVitale on Jul 27^th, 2009, 1:31am

Before playing this game, you and your roomate need to set the rules of the game.

You know that your roomate has a willingness to pay v₂ but you don’t know what it is.

Your values are zero when you do not purchase the machine.

You and your roomate are choosing among the following alternatives.
(1) no machine, no monetary payments.
(2) espressso machine, any transfer scheme t = t₁, t₂ such that t₁ + t₂ = -50

The utilitiarian policy is to purchase the espresso machine if
v₁ + v₂ is $50 or more

and not to purchase the machine if v₁ + v₂ < 50.

Suppose you agree to split the cost.
When would you be willing to do it?
Only when v₁ is $25 or more.
And the same is true of your roomate.
You both reach the agreement that the salesman -- as you mentioned -- become the impartial judge.

- You and your roomate simultaneously pledge a contribution (some number.)

- If the contributions add up to at least 50 then the espresso machine is bought (and the surplus divided proportionally)
- Otherwise not.

Title: Re: To buy or not to buy? And how to share?
Post by raven on Sep 15^th, 2009, 8:48pm

This is great for one of the roommates if they intend to contribute half or less; but if both must contribute half or more to the purchase, there is no game (unless I'm missing something here).

What I propose -- still using the sales clerk as the impartial arbitrator -- is each roommate puts one dollar in the envelope for each day of the month they intend to be allowed to make espresso; if the total is greater than $50 when they get to the counter they can buy some beans too; if the total is less they can go back and pick out a coffee maker instead.

;D

Title: Re: To buy or not to buy? And how to share?
Post by BenVitale on Sep 16^th, 2009, 12:32pm

on 09/15/09 at 20:48:17, raven wrote:

This is great for one of the roommates if they intend to contribute half or less; but if both must contribute half or more to the purchase, there is no game (unless I'm missing something here).

Could u please tell me whether you are familiar with Game Theory? If you are not, then I will attempt a different approach.

Title: Re: To buy or not to buy?
Post by Noke Lieu on Sep 17^th, 2009, 3:28am

hey, I wouldn't discount it because he doesn't have formal training. Insight is insight.

i really like Grimbals tweak on it. If the game is to maximise the amount of money you get.
I'd further twist it to if there's not enough m oney, then they get thrown out of the shop - sans money as a penalty fro wasiting the shopkeep's time.

Title: Re: To buy or not to buy? And how to share?
Post by raven on Sep 18^th, 2009, 5:34pm

on 09/16/09 at 12:32:08, BenVitale wrote:

Could u please tell me whether you are familiar with Game Theory? If you are not, then I will attempt a different approach.

Hi BenVitale,

I just Wikipedia'd it -- but formally, no :-/

I just hack away at things with the body of knowledge I do have -- but I'm not adverse to learning...

Feel free to educate me or not -- up to you. :)

Title: Re: To buy or not to buy? And how to share?
Post by BenVitale on Sep 18^th, 2009, 11:40pm

Okay, let's start with ....

This game is a Vickrey-Clarke-Groves auction (http://en.wikipedia.org/wiki/Vickrey-Clarke-Groves_auction)