T.R | Title | User | Personal Name | Date | Lines |
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605.1 | 3rd parties are not all bad | MODEL::YARBROUGH | | Thu Oct 30 1986 11:32 | 27 |
| >Cost matrix:
>
> Plan1 Plan2
>me c1 c2
>you c3 c4
>
> We'd like to arrange to choose the scheme that yields the lowest global
>cost (i.e. choose plan1 if c1 + c3 < c2 + c4.)
>
> Let's say that we choose plan1.
>
> Unfortunately, it may work out that, for the chosen plan, you wind up
>paying more than me. To be fair, it seems that I should pay you a certain
>amount of money to make it worth your while to go along with plan1.
>
> But if I have to pay you enough that my cost rises above c4 [I think you
>mean c2; c4 is the other guy's cost], then it no longer makes sense for me
>to cooperate with plan1. Also, neither one of us likes to pay more than we
>have to.
But if you agree on plan 2, then an equitable cost division is going to be
still worse. In other words, if your adjusted cost for plan 1 is greater
than c2, then the other guy's c4 must be way out of line, so that an
equitable adjustment for the c2-c4 cost will be more expensive for both
of you than the plan 1 adjustment.
Sounds like it may be worth the money to agree to hire an arbiter.
|
605.2 | | CLT::GILBERT | eager like a child | Thu Oct 30 1986 21:11 | 23 |
| Right. If the payments between the two parties are considered in
the cost to each party, the plan with smallest (summed) cost should
be chosen.
Suppose there are multiple parties and multiple plans, and each plan
has different benefits to the various parties. For example:
Plan1 Plan2
PartyA cA1 bA1 cA2 bA2
PartyB cB1 bB1 cB2 bB2
Here, too, the plan that maximizes the sum of (Benefits-Costs) should
be chosen, and cross-player payments can redistribute the costs, possibly
in proportion to each player's benefits.
This reminds me of an old problem from the classic "Mathematical Snapshots".
Three brothers are to divide a rectangular plot of ground between themselves,
with vertical (North/South?) dividers. However, the land varies in quality,
so some sections may be preferred over others, and the brothers may disagree
in their assesments of the land. How can this be arbitrated so each brother
will be satisfied that he has received at least his 1/3 share of the land?
|
605.3 | Why No Math? | CLOSUS::TAVARES | John--Stay low, keep moving | Fri Oct 31 1986 13:26 | 6 |
| When I first saw this one, I thought you folks would come up with
a generalized solution using linear programming. Is the reason
why linear programming won't work because the shares of each
participant can't be put in equation form? That is, the choice
of "quality" is a judgement call, and as such subject only to
negotiation? Hope I'm making my question clear.
|
605.4 | | CLT::GILBERT | eager like a child | Fri Oct 31 1986 14:13 | 57 |
605.5 | got any more "Snapshots"? | VINO::JMUNZER | | Fri Oct 31 1986 14:26 | 23 |
| Re .2, last paragraph:
Require each brother to set boundaries of "thirds" that seem fair to him
(is this requirement permissible?); e.g.
|AAAAAAAA-AAAAAA-AAAAAAA|
West |BBBBBBB-BBBBBB-BBBBBBBB| East
|CCCCCC-CCCCCCCC-CCCCCCC|
Give the Western piece to the brother who picked the Westernmost
Western/middle boundary (C).
Then give the middle piece to the remaining brother who picked the
Westernmost Eastern/middle boundary (B).
Then give the Eastern piece to the remaining brother (A):
West |CCCCCC-BBBBBBB-AAAAAAAA| East
(This extends to more than three siblings.)
John
|
605.6 | Another division problem from Mathematical Snapshots | CLT::GILBERT | eager like a child | Fri Oct 31 1986 16:50 | 35 |
| From "Mathematical Snapshots", 1960 edition, page 68:
"There is another problem of division encountered in economic life: the
division of indivisible objects like houses, domestic animals, pieces of
furniture, cars, and works of art. If for instance, an inheritance composed of
a house, a mill, and a car has to be divided among four inheritors A, B, C, D
participating in equal shares, the division is generally made by a sworn
appraiser who deetrmines the values of the objects so that the inheritors can
choose the objects, and, if they agree, in principle, satisfy by payments in
cash the mutual claims arising from the differences in value.
This procedure has many inconveniences connected with the determination of
the objective value of things by an official appraiser or by a court of
justice. It is possible to make a fair division without appealing to them:
An umpire, who has to act only as a sort of automaton to keep records and
make computations, summons the inheritors to write down their estimates of the
objects. They are not supposed to discuss the matter among themselves but
every one of them is allowed to be helped by friends and experienced persons.
Thus a table of values is put down by the umpire:
A B C D
House $6,000 $10,000 $7,000 $9,000
Mill 3,000 2,000 4,000 2,000
Car 1,500 1,200 1,000 1,000
..."
[here, I've omitted the solution to the problem]
"Thus everybody will finally get more than his due share of the inheritance,
the value of the total and of the objects given to him being estimated being
estimated according to his own valuation."
Alright, how did the umpire do it?
|
605.7 | | BEING::POSTPISCHIL | Always mount a scratch monkey. | Fri Oct 31 1986 19:26 | 17 |
| There is a fundamental difference between the division problems and the
cooperation problem. In the division problems, the people own equal
shares of the property being divided or otherwise have some reason for
getting equal shares -- "fairness" is already defined.
In the cooperation problem, there is no ownership or other way to
define "fairness". If one party is stubborn and holds out for more
money, they may be able to get more -- and there is no reason stated
why they should not do that. In the real world, and assuming people
were rational and intelligent, the willingess of each person to pay or
receive various amounts for various plans would actually depend upon
external factors. For example, somebody with another source of income
would be more willing to take a risk of losing money if the expected
gain is large enough.
-- edp
|
605.8 | Simpler than I thought at first glance | EAGLE1::BEST | R D Best, Systems architecture, I/O | Fri Nov 07 1986 13:35 | 10 |
|
re .1-.4:
I see. This problem is much simpler than I realised at first glance.
The best strategy is always to choose the minimum global cost plan and
split the difference. I had also forgotten to factor in the effect of
relative benefit to each party. Equal benefit seems a reasonable assumption
for a lot of situations. If the benefits are not equal, we should use a
different matrix where the elements are ( benefit[i,j] - cost[i,j] ).
Thanks !
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