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March 6th, 2010, 08:58 PM   #1
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Fair Division

I'm taking a discrete math course and the subject of fair division was presented by the teacher as a mechanism to resolve estate disputes. The method makes no sense to me; I believe in a sealed-bid auction format for resolving estate disputes. A format where the highest bidder essentially buys the item and that dollar amount is both taken out of his share and helps to build the total estate value. But I can't find anything in writing/official to support that concept.

An example of the format I would use is as follows:

Dad dies leaving the family quilt and $1,000,000 in his estate. In a sealed bid format son #1 bids $200,00 for the quilt and son#2 bids $100,000.

Value of Estate $1,200,000

Son #1 gets the quilt and $400,000
Son #2 gets $600,000
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March 7th, 2010, 01:29 PM   #2
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Re: Fair Division

I'm not sure what the question is.

I would usually prefer a second-price (Vickrey) auction. If the bids were the same as in your example -- typically, bids in a second-price auction are higher -- then the value of the estate would be $1,100,000 so first son would get $450,000 plus the quilt.
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March 7th, 2010, 08:31 PM   #3
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Re: Fair Division

My question is: how would a mathematician resolve division of assets amongst heirs of an estate? Is there a math concept like "Fair Division" that I could reference?

Thanks for the Vickrey auction tip.
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March 7th, 2010, 10:02 PM   #4
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Re: Fair Division

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Originally Posted by fever
My question is: how would a mathematician resolve division of assets amongst heirs of an estate? Is there a math concept like "Fair Division" that I could reference?
Yes, it's actually called "fair division" (so I assumed you already knew about it). It's considered a part of (mathematical) economics (specifically, game theory), so I moved this discussion to Math & Economics.

Your method is a good one for estates with cash and atoms (indivisible goods). Deciding exactly how to value the atoms is tricky; I said I would prefer a Vickrey auction, but modifications might be useful in some situations. If there are few heirs and the goods are valuable, having the items appraised and using that as a 'floor' value might be worthwhile. If there are many identical items, a VCG auction could have value... but it's hard to work with. (Fortunately, its theoretical shortcomings aren't don't really hurt it in this setting, though, so it's a quite good method.) Etc.
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March 8th, 2010, 04:56 AM   #5
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Re: Fair Division

Yes I am familiar with Fair Division and I think itís an incredibly flawed approach. However, I am familiar with no other ďnamedĒ mathematical approach which accomplishes the same thing as Fair Division. Sure there are lots of different auction methodologies but none of them goes the extra step of mathematically dividing assets ďfairlyĒ. Keep in mind, Fair Division doesnít require market values or appraisals.
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March 8th, 2010, 07:36 AM   #6
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Re: Fair Division

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Originally Posted by fever
Yes I am familiar with Fair Division and I think itís an incredibly flawed approach.
What, to you, is "fair division"? To me it's a field with hundreds of different techniques: "you cut, I choose", the various moving-knife procedures, combinatorial auctions, etc. There are also many various 'fairness' criteria: envy-freeness, exact division, proportional divisions, Pareto efficiency, Kaldor-Hicks efficiency, ... also techniques can be discrete, moving knife, exact or inexact [in a different sense from "exact division"], NP-hard, weakly additive, etc.

Also possibly relevant: does the procedure have a Nash equilibrium? Do the results lie in the core, the nucleolus, etc.? Are there issues of joint welfare or spite?
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March 8th, 2010, 08:56 AM   #7
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Re: Fair Division

When capitalized, Fair Division is something I donít agree with. However, I agree, fair division consists of hundreds of techniques. The only technique Iíve ever seen spelled out in detail is Fair Division. For example, pretend Dad dies and leaves three equal beneficiaries the car, the house, the business and the family quilt. Fair Division provides a detailed, albeit flawed, approach on how to resolve splitting these items amongst the heirs. I know, this question is probably better fielded by a bunch of lawyers than mathematicians, but I sure would love to know how a mathematician would tackle the problem.
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March 8th, 2010, 09:31 AM   #8
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Re: Fair Division

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Originally Posted by fever
When capitalized, Fair Division is something I donít agree with.
What does "Fair Division" mean to you (with capitals)? Until you tell me I can't agree or disagree about it being flawed.

Also, can you tell me anything about the sorts of estate disputes you're talking about?
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March 8th, 2010, 11:19 AM   #9
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Re: Fair Division

Short of copying and scanning a bunch of pages from my textbook, it will be difficult to explain it. But Iím not looking for another opinion/analysis; Iím looking for an alternative methodology to the one presented in my textbook (Fair Division). Together, Iím sure we could come up with something that involves an auction but my professor asked me to find something concrete. Something that might exist in another textbook or lawyerís periodical. Something named.

ďAlso, can you tell me anything about the sorts of estate disputes you're talking about?Ē
Imagine someone who wants to leave his estate to his three children and have it split 33%/33%/33%. How would everyone resolve who gets what?
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March 8th, 2010, 12:23 PM   #10
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Re: Fair Division

Your description doesn't really tell me much about the kinds of problems you wanted to apply the method to. Are there always three players? Does the estate consist of discrete items, continuous items, or both? Are the continuous items, if any, homogeneous? Need the pieces be continuous? Is there a trusted arbiter? Which of the fairness criteria are considered important?

So here are some general methods:
  • The Divider-Chooser Method[/*:m:1kr5o7oq]
  • The Lone Divider Method[/*:m:1kr5o7oq]
  • The Lone Chooser Method[/*:m:1kr5o7oq]
  • The Last Diminisher Method[/*:m:1kr5o7oq]
  • The Method of Sealed Bids[/*:m:1kr5o7oq]
  • The Method of Markers[/*:m:1kr5o7oq]
http://www.colorado.edu/education/DMP/f ... ision.html

The method I suggested is what Peressini calls "The Method of Sealed Bids". It has potential issues, though, depending on the situation; if players (heirs) have nonlinear valuations on the items. There is also possibility for collusion for those with insider knowledge of those outside the coalition.

Example: The estate has a time machine (without power source), a time machine power source, and $1 billion. A player values the time machine at $1 million, the time machine power source at $1 million, and the combination of the two at $1 billion.
Example: Two heirs are rivals, and gain positive utility from denying the other access to desired items (by taking them or causing them to go to another heir).
Example: A painting is valued by A at $30,000, by B at $60,000, and by C at $90,000. The honest Vickrey valuation of the painting is $60,000. But if B and C know that A values the painting at $30,000 then B can dishonestly bid $30,000 for the painting and have C give him a side-payment between $10,000 and $30,000 (thus defrauding A).
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