Author |
Topic: A bargaining theory problem (Read 2841 times) |
|
Benny
Uberpuzzler
Gender: 
Posts: 1024
|
 |
A bargaining theory problem
« on: Jun 20th, 2009, 11:31am » |
 Quote  Modify
|
I submit to this community a bargaining theory problem :
$400 is to be divided among creditors who claim $100, $200, and $300.
They are awarded $50, $125, and $225.
Why is this consistent?
|
|
 IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: A bargaining theory problem
« Reply #1 on: Jun 20th, 2009, 2:46pm » |
Quote Modify
|
Beats me. You'd think that with A+B together having the same sized claim as C, they should be able to secure an equal share to C. And if they'd split that $200 proportionally, they'd both be better of than they are now ($67 > $50, $133 > $125), so C clearly hasn't offered either of them anything to dissuade them from cooperating.
This doesn't seem like a division they should agree to.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #2 on: Jun 21st, 2009, 12:57pm » |
Quote Modify
|
I found the explanation in this document which did surprise me.
|
|
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
Ronno
Junior Member
Gender:
Posts: 140
|
|
Re: A bargaining theory problem
« Reply #3 on: Jun 21st, 2009, 10:45pm » |
Quote Modify
|
I have read about this division elsewhere. Although it is consistent, (in the sense that any subset of the claims and awards determine the same division on application of the rules) it does not mean that it is acceptable to the creditors as towr points out.
There are more (perhaps an infinite number) equally consistent solutions to the problem (I can think of at least two other) and I don't think there is an unique "fair" division process.
|
|
IP Logged |
Sarchasm: The gulf between the author of sarcastic wit and the person who doesn't get it..
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: A bargaining theory problem
« Reply #4 on: Jun 22nd, 2009, 2:12am » |
Quote Modify
|
I can't really say I find the solution all that consistent, precisely for the reason I already stated.
A could sell his claim to B, who'd then have a $300 claim, equal to C's; and so the Talmudic solution would then also give them equal shares. Or suppose C has three claims of $100, because he gave three different loans; then he should suddenly get much less.
You will need pretty complicated rules to avoid this. It just doesn't add up (quite literally).
How many solutions can you think up there are consistent under this constraint? The only one I see is proportional division.
|
| « Last Edit: Jun 22nd, 2009, 2:23am by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: A bargaining theory problem
« Reply #5 on: Jun 22nd, 2009, 4:41am » |
Quote Modify
|
The article lays down the principle that "If A claims half and B claims all, one half is claimed by A and B and the other half is claimed only by B. Therefore A gets 1/4 and B gets 3/4."
But according to the same principle, if A claims half and B claims half, the first half is claimed by A and B and the other half is unclaimed. So A and B should get 1/4 each. Clearly, it doesn't make sense that A and B claim the same half. A and B's claims should be arranged to not overlap.
So, if now A claims 1/3, B claims 2/3 and C claims 3/3, It would be unreasonable to assume that A and B claim the same 1/3 while leaving another 1/3 undisputed.
I would say the goal of such a definitive rule is not so much fairness as to settle conflicts. The less people can argue, the faster they go on and proceed to more constructive activities.
You could argue that the proportional split isn't totally fair either. Someone who lends $200 knows that the 2nd $100 are a bit more risky than the first $100. It could be argued that he was ready to take more risks and should receive less back.
Another idea of fairness would be to pay back the debts in the order they were created.
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: A bargaining theory problem
« Reply #6 on: Jun 22nd, 2009, 5:38am » |
Quote Modify
|
on Jun 22nd, 2009, 4:41am, Grimbal wrote:| You could argue that the proportional split isn't totally fair either. Someone who lends $200 knows that the 2nd $100 are a bit more risky than the first $100. It could be argued that he was ready to take more risks and should receive less back. |
|
That does, incidentally, avoid the coalition problem, since pooling debts would decrease what you get together.
But it just implies you should use proxies to lend someone one dollar at a time. Because then those proxies have the (proportionally) highest claims, and you can recuperate more of the debt than if you lent it directly. (Granted, there's some cost involved there, like administration and of course those proxies will need some compensation as well.)
Quote:| Another idea of fairness would be to pay back the debts in the order they were created. |
|
That requires good administration though, otherwise it just opens up another line of argument.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Ronno
Junior Member
Gender:
Posts: 140
|
|
Re: A bargaining theory problem
« Reply #7 on: Jun 22nd, 2009, 6:08am » |
Quote Modify
|
How about this division:
You keep distributing the money equally until the one with lowest claim is fully credited. Then the rest of the money goes to the others in the same way.
For example, for A, B, C, A gets 100, B & C get 150 each.
|
|
IP Logged |
Sarchasm: The gulf between the author of sarcastic wit and the person who doesn't get it..
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: A bargaining theory problem
« Reply #8 on: Jun 22nd, 2009, 6:14am » |
Quote Modify
|
Honestly, I also prefer the proportional division.
|
|
IP Logged |
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #9 on: Jun 22nd, 2009, 11:28am » |
Quote Modify
|
on Jun 22nd, 2009, 2:12am, towr wrote: ......
The only one I see is proportional division. |
|
on Jun 22nd, 2009, 6:14am, Grimbal wrote:| Honestly, I also prefer the proportional division. |
|
And thank you Ronno for your input.
Fair division is a concept that depends as much on logic as it does on social custom ... There might not be one correct answer.
To see why, consider the following 3 situations that present very different solutions:
Suppose I owe debts of $100, $200, and $300 to you guys
Ronno -----> $100,
Grimbal ---> $200
Towr ------> $300
But I have less than $600, say $400
(1) Suppose that we are all related, we could be brothers or cousins ... and I don't have
$600 to pay my debts to you
(2) Suppose we are partners and running for example, a limited partnership
(3) Suppose we go out for a dinner and each of us order food items on the menu at the
restaurant with promises to pay, and then ... here comes the stinkin' diaper situation:
how are we going to split the bill?
In each case, we could have a different scheme to split the money.
I don't think there's just one way, or a right way to approach these types of problems.
That's why we end up with conflicts, litigations, lawsuits everyday ... Lawyers must be very happy people ... and I must be in the wrong racket.
Disputes, conflicts are a matter of perspectives.
See other similar problems :
- The disputed garment problem
- Sharing the Cost of a Runway
- The Marriage Contract Problem
- The Bankruptcy Game
- Some Lawyer’s Arguments
- Nash’s Bargaining Game
|
| « Last Edit: Jun 22nd, 2009, 11:45am by Benny » |
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
Obob
Senior Riddler
Gender:
Posts: 489
|
|
Re: A bargaining theory problem
« Reply #10 on: Jun 22nd, 2009, 1:42pm » |
Quote Modify
|
In current financial matters, these types of problems have come up. For instance, in the collapse of the Madoff pyramid scheme, I seem to recall investors were repaid up to the first million of their investment, and then further claims were divided proportionally from any remaining funds.
|
|
IP Logged |
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #11 on: Jun 22nd, 2009, 2:16pm » |
Quote Modify
|
on Jun 22nd, 2009, 1:42pm, Obob wrote:| In current financial matters, these types of problems have come up. For instance, in the collapse of the Madoff pyramid scheme, I seem to recall investors were repaid up to the first million of their investment, and then further claims were divided proportionally from any remaining funds. |
|
Do you have the details?
I thought that Madoff's direct customers would be covered by Securities Investor Protection Corp. , which typically covers up to $500,000 in losses.
Here's a Physical Interpretation in part #4
http://dept.econ.yorku.ca/~jros/docs/AumannGame.pdf
|
| « Last Edit: Jun 22nd, 2009, 2:59pm by Benny » |
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: A bargaining theory problem
« Reply #12 on: Jun 22nd, 2009, 10:58pm » |
Quote Modify
|
on Jun 22nd, 2009, 2:16pm, BenVitale wrote:That's the same thing as in that other article, except it's drawn differently.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #13 on: Jun 23rd, 2009, 3:46pm » |
Quote Modify
|
on Jun 22nd, 2009, 10:58pm, towr wrote:
That's the same thing as in that other article, except it's drawn differently. |
|
Oh, yeah ... sorry.
|
|
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #14 on: Jun 24th, 2009, 9:14am » |
Quote Modify
|
Another example that I enjoy is the problem of splitting up rent.
It's the old roommate/rent Dilemma
Quote:
Two friends — a merchandising analyst and a law student — and I are attempting to split up rent of a three-bedroom apartment with two common bathrooms. All rooms have their pros and cons, with the major differentiators being closet space and sheer square footage:
|
|
Room # 1 ..... 15 ft. x 15 ft. = 225 sq.ft.
Room # 2 ..... 12 ft. x 12 ft. = 144 sq.ft.
Room # 3 ..... 20 ft. x 8 ft. = 160 sq.ft.
(Room # 1) - (Room # 2) = 225 - 144 = 81 sq.ft.
(Room # 1) - (Room # 3) = 225 - 160 = 65 sq.ft.
Rent is $2,200 per month and the apartment is approximately 2,200 square feet.
Simple math shows one would pay $1 per sq.ft.
That goes out the window with the ranking intangibles, and the fact that no one necessarily
wants the big room.
The roommates threw out these prices:
Room # 1: $800/month
Room # 2: $710/month
Room # 3: $690/month
over the course of a year
Room # 1: $800/month X 12 = $9,600
Room # 2: $710/month X 12 = $8,520
Room # 3: $690/month X 12 = $8,280
Differences show
9,600 - 8,520 = $1,080
9,600 - 8,280 = $1,320
How do you recommend solving this situation?
[proposed suggestions are at the bottom of the document]
|
|
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: A bargaining theory problem
« Reply #15 on: Jun 24th, 2009, 3:50pm » |
Quote Modify
|
on Jun 24th, 2009, 9:14am, BenVitale wrote:
How do you recommend solving this situation? |
|
Maybe turn it into a bidding problem instead.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #16 on: Jun 24th, 2009, 10:23pm » |
Quote Modify
|
on Jun 24th, 2009, 3:50pm, towr wrote:
Maybe turn it into a bidding problem instead. |
|
how we can model this situation as a game?
Do you agree with these prices:
Room # 1: $800/month
Room # 2: $710/month
Room # 3: $690/month
Would you like to start the opening bid for the better room? In your opinion, which room is the better room? How to allocate the rights to joint furniture purchases? groceries?
Would you propose a bid on the good room with chores -- meaning the person who is willing to do the most domestic chores to compensate wins the auction and the room?
Bidding may look like as a fair system, but it presents a problem : The winner's curse
Winner’s curse was originally studied in the laboratory by Bazerman and Samuelson (1983) using “the jar game”, in which participants bid on a jar full of change.
Bazerman and Samuelson's experiment : bid on a jar of nickels
The winner's curse is frequently observed in auctions: The person who bids the most and wins
the auction may ultimately regret the bid since it often exceeds the value of the object being auctioned.
It's when we get the feeling in our stomach when we realize that we paid much more for something that it is actually worth. This is a bad enough situation when it happens every once in a while.
The trick is to recognize this problem and avoid it.
Read about The Winner’s Curse in The bidding game
Quote:
Why did the oil companies -— which on average are pretty good at guessing how much oil lies buried in a tract -— seem so often to pay more than the tract turned out to be worth?
As an analogy, imagine that a jar of nickels is being sold in a sealed first-price auction. The jar holds $10 in nickels, but none of the bidders know that; all they can see is how big the jar is. The players independently estimate how much the jar is worth. Maybe Alice guesses right, while Bob and Charlie guess the jar holds $8 and $12, respectively. Diane and Ethel are farther off, putting the value at $6 and $14, respectively.
If all the bidders bid what they think the jar is worth, Ethel will win, but she’ll pay $14 for $10 in nickels—what economists call the “winner’s curse.” Even if the jar is sold in a second-price auction, she will still overpay. Although on average the bidders are correct about how much money is in the jar, the winner is far from correct; she is the one who has overestimated the value the most ...
|
|
|
| « Last Edit: Jun 24th, 2009, 10:39pm by Benny » |
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: A bargaining theory problem
« Reply #17 on: Jun 25th, 2009, 12:25am » |
Quote Modify
|
on Jun 24th, 2009, 10:23pm, BenVitale wrote:That only occurs in some bidding schemes, you can easily avoid it. I forgot what it's called, but if you use a closed bidding system, and the winner pays the second highest bid, then people should not overbid.
Quote:how we can model this situation as a game?
Do you agree with these prices:
Room # 1: $800/month
Room # 2: $710/month
Room # 3: $690/month |
|
Of course not, it depends on how much people value the rooms.
Quote:| Would you like to start the opening bid for the better room? |
|
I'd use a closed bidding system, so there isn't an opening bid.
I haven't really thought about the specific scheme. I suspect it would work best if everyone presents a closed bid for all the rooms at once, i.e. how much they're willing to pay to end up in each room.
One problem, however, is that the solution might not necessarily add up to the total rent; depending on the details. As I said, I haven't really thought it out yet.
Quote:| In your opinion, which room is the better room? |
|
I'd have to see them. But the apartment is probably way too expensive. The rooms are twice as expensive as the one I have, and I have everything (utilities, maintenance etc) included.
Quote:| How to allocate the rights to joint furniture purchases? |
|
"Rights to joint furniture purchases"? You mean like how long they can sit on the couch? WTF?
Quote:If they eat together, I'd say split it, or alternate. If someone has special dietary needs/habits, then it could reasonably be separate. And if someone wants to have a claim on an item of food, obviously they need to pay for it themselves and label it accordingly.
[quote]Would you propose a bid on the good room with chores -- meaning the person who is willing to do the most domestic chores to compensate wins the auction and the room? [quote]I wouldn't. But it's an alternative if the people involved want it.
You can use bidding to divide chores, so each gets those chores best suited to him. One person might like cooking more than vacuuming, the other might like vacuuming more than cooking. So you can maximize satisfaction in domestic life by arranging the chores accordingly.
|
| « Last Edit: Jun 25th, 2009, 12:44am by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #18 on: Jun 25th, 2009, 10:20am » |
Quote Modify
|
Yes, the winner's curse does not apply here, because it's a private bid ... independent of the market value.
The rent that must be paid is $2,200 per month ... that's a fixed price, a market value.
When the 3 roomates threw out the following prices
Room # 1: $800/month
Room # 2: $710/month
Room # 3: $690/month
That was their first bid.
|
|
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #19 on: Jun 25th, 2009, 12:39pm » |
Quote Modify
|
It also depends on your roommates ... say you and your roommates are all math major students ... and all of you are big fans of Game Theory. And you guys get together discussing this problem before getting an apartment or a 3-bedroom house.
The rent is $2,200 per month not including utilities and such.
A 3-way split (= $2,200/3) would be unfair since the rooms have different sizes.
As for utilities and other matters, they are divided up individually.
How do you like the following scheme:
Rent is $2,200 per month and the apartment is approximately 2,200 square feet. That means one would pay $1 per sq.ft.
We know that:
Room # 1 ..... 15 ft. x 15 ft. = 225 sq.ft.
Room # 2 ..... 12 ft. x 12 ft. = 144 sq.ft.
Room # 3 ..... 20 ft. x 8 ft. = 160 sq.ft.
Total area = 529 sq.ft.
Remaning area = 2200 - 529 = 1671 sq.ft.
Split in 3: 1671/3 = 557
Rent for room #1: (225 sq.ft. * $1) + (557 sq.ft. * $1) = $782
Rent for room #2: (144 sq.ft. * $1) + (557 sq.ft. * $1) = $701
Rent for room #3: (160 sq.ft. * $1) + (557 sq.ft. * $1) = $717
|
| « Last Edit: Jun 25th, 2009, 12:41pm by Benny » |
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: A bargaining theory problem
« Reply #20 on: Jun 25th, 2009, 12:51pm » |
Quote Modify
|
Do you think a room of 1 ft x 300 ft would be more desirable than one that's 10 ft x 10 ft, just because it has three times the surface?
And what about view? How close they are to the bathroom? How easily you can get to a fire escape? How resilient the door is against zombie attacks? People have different priorities and can value the same thing differently. I would think that's one thing you'd need to find out first.
I'm not going to like any division that doesn't take people's individual valuation into account.
|
| « Last Edit: Jun 25th, 2009, 12:53pm by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: A bargaining theory problem
« Reply #21 on: Jun 25th, 2009, 12:58pm » |
Quote Modify
|
You should take advantage of the fact that different people will value a room differently. If everybody agrees on how much each room is worth there is no problem. Otherwise, each person makes a list of how much the rooms are worth to them, so that the sum of their values is equal to the total rent. Then assign the rooms to maximize the sum of the values chosen. This will be strictly larger than the rent due, so you can scale back the payments and everybody wins.
|
|
IP Logged |
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #22 on: Jun 26th, 2009, 11:14am » |
Quote Modify
|
on Jun 25th, 2009, 12:51pm, towr wrote:Do you think a room of 1 ft x 300 ft would be more desirable than one that's 10 ft x 10 ft, just because it has three times the surface?
And what about view? How close they are to the bathroom? How easily you can get to a fire escape? How resilient the door is against zombie attacks? People have different priorities and can value the same thing differently. I would think that's one thing you'd need to find out first.
I'm not going to like any division that doesn't take people's individual valuation into account. |
|
I agree with you ... you're right to raise these questions ... we don't have much info on the location of this apartment, about these rooms ... we only know about the rent and the surface areas of these rooms. I've worked with what we got.
|
| « Last Edit: Jun 26th, 2009, 11:36am by Benny » |
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
Benny
Uberpuzzler
Gender:
Posts: 1024
|
|
Re: A bargaining theory problem
« Reply #23 on: Jun 26th, 2009, 11:25am » |
Quote Modify
|
on Jun 25th, 2009, 12:58pm, Eigenray wrote:| You should take advantage of the fact that different people will value a room differently. If everybody agrees on how much each room is worth there is no problem. Otherwise, each person makes a list of how much the rooms are worth to them, so that the sum of their values is equal to the total rent. Then assign the rooms to maximize the sum of the values chosen. This will be strictly larger than the rent due, so you can scale back the payments and everybody wins. |
|
A bid was offer in the story. So, I offered a bid ... naturally, my offer is open for negotiations.
We see these problems all too often to the delight of mathematicians, and math students (myself included) ... attaching numerical values to human feelings is the first obstacle whenever we need to divide a finite resource in a fair, envy-free manner. The second obstacle is to find a workable algorithm, formula to produce such a division.
Problems such the one that started this thread ... and other problems, namely:
(1) Cake-cutting problem, see in Wolfram and in wiki
Here we ask: Can a cake be cut into 3 pieces and allocated so that every person believes he or she
receives the most desirable piece?
(2) How to split a shared cab ride? Very carefully, say economists
Here, the story is: 3 economists get into a cab. They're each getting off at different places along the route. How should they split the bill?
I see a parallel between all these wonderful problems.
|
| « Last Edit: Jun 26th, 2009, 11:34am by Benny » |
IP Logged |
If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: A bargaining theory problem
« Reply #24 on: Jun 26th, 2009, 2:31pm » |
Quote Modify
|
on Jun 26th, 2009, 11:25am, BenVitale wrote:Here we ask: Can a cake be cut into 3 pieces and allocated so that every person believes he or she
receives the most desirable piece? |
|
The answer is yes, and interestingly it only gets easier as they value the various elements of the cake more differently.
Quote:| Here, the story is: 3 economists get into a cab. They're each getting off at different places along the route. How should they split the bill? |
|
I think not looking like a penny-pinching ass might be a good investment in future social relations; so you could do worst than paying more than your "fair share".
I mean, really, a $10 cab ride is not exactly going to drive you into bankruptcy as a well-paid economist. I'd happily let A ride for free if I was B or C, and as A I'd happily pay the entire fee up to that point if they hadn't suggested otherwise. As B, again, I'd happily pay the fee up to my stop; and as C as well, presuming the cab ride wasn't absurdly much longer; and even then. I'd be happy to have good company on the way; and if they're not good company I'd take another cab.
|
| « Last Edit: Jun 26th, 2009, 2:36pm by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print