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 March 8th, 2010, 07:24 PM #11 Newbie   Joined: Mar 2010 Posts: 11 Thanks: 0 Re: Fair Division So funny, I just typed The Method of Sealed Bids into Google and wala, the first link was a great example of Fair Division (http://www.ctl.ua.edu/math103/FairDiv/s ... htm#SEALED BIDS EXAMPLE). Notice how Michael gets totally screwed. He gets $1,738 in cash, while Barbie gets$2,275 in items and $422 in cash. How is that fair? It sure would be nice if I could find something that spells out a scenario where everyone gets the same value. In regards to your three examples, I’m not sure if I understand: 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. Sell the two items together. Your example is like selling a house and the land separately. Who would ever structure something like that? 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). Whatever one bids on any item gets taken out of that person’s share. So sure, you could deny someone access to an item but you end up compensating them for that decision. 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). Your right, if A chooses to tip his hand and let B and C know of his intended bid, collusion could occur. Tell A to keep his mouth shut. March 9th, 2010, 05:41 AM #12 Global Moderator Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Fair Division Quote:  Originally Posted by fever So funny, I just typed The Method of Sealed Bids into Google and wala, the first link was a great example of Fair Division (http://www.ctl.ua.edu/math103/FairDiv/s ... htm#SEALED BIDS EXAMPLE). Notice how Michael gets totally screwed. He gets$1,738 in cash, while Barbie gets $2,275 in items and$422 in cash. How is that fair?
Yes, how is it fair for Barbie? Michael thinks she got $6 worth of items and$422 in cash, for a total value of $428. By his valuation, he gets over four times more than her. Quote:  Originally Posted by fever Sell the two items together. Your example is like selling a house and the land separately. Who would ever structure something like that? Yes, but that items have joint value is not always clear. For a more complicated example, look at the 2008 FCC spectrum auction. Quote:  Originally Posted by fever 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). Whatever one bids on any item gets taken out of that person’s share. So sure, you could deny someone access to an item but you end up compensating them for that decision. That means that it's possible. But consider an item which A values at only$100, but where A gains $1000 utility from denying the item to B. (Perhaps he's spiteful, or perhaps the item is noisy/garish/etc. and the two are neighbors...) If B values the item at$500 and C values the item at $300, the efficient solution is for A to pay C to place the top the top bid, and pay him$200 to $400 for it. But the mechanism doesn't make determining this joint valuation easy, nor does it determine a fair way for the surplus between A and C to be determined. Quote:  Originally Posted by fever 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). Your right, if A chooses to tip his hand and let B and C know of his intended bid, collusion could occur. Tell A to keep his mouth shut. It's frequently the case that heirs know each other well. If B or C can guess that A's bid will be$15,000 to $45,000, they can still collude by having B bid$15,000 (or $5,000). In fact, they don't even have to know that: if they trust each other, they could compare their bids and change the lower of their bids$0. If the higher bid wins, and the drop results in lowering the price of the item, they can agree to split the amount 50/50. (If they don't want to be suspected of collusion, they can reduce the bids, but by less than the full amount.)

 March 9th, 2010, 06:59 AM #13 Newbie   Joined: Mar 2010 Posts: 11 Thanks: 0 Re: Fair Division Let’s shelve the pro and con discussion of Vickeroy auctions and just say that all auctions have pros and cons. Moving on, I think I understand your response to my labeling of Michael as getting screwed. Correct me if I’m wrong, but you’re saying Michael’s realized value is actually greater than his perceived / utility value and Barbie’s realized value is less. Perhaps it’s my business background talking, but one’s perceived value should have nothing to do with an item’s market value. At the end of the day, each person should end up with an equal share of the same sized pie not different shares / different pies. Look at it this way; if all the items were sold for cash everyone’s share would be equal. Why, by bidding on items, do we suddenly give everyone different shares?
March 9th, 2010, 07:37 AM   #14
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Re: Fair Division

Quote:
 Originally Posted by fever Let’s shelve the pro and con discussion of Vickeroy auctions and just say that all auctions have pros and cons.
I don't believe we've compared the Vickrey auction to any other auction type...

Quote:
 Originally Posted by fever Perhaps it’s my business background talking, but one’s perceived value should have nothing to do with an item’s market value. At the end of the day, each person should end up with an equal share of the same sized pie not different shares / different pies.
If people value the items differently, then no matter what division system you use the people will rate the shares each person gets differently.

Quote:
 Originally Posted by fever Look at it this way; if all the items were sold for cash everyone’s share would be equal. Why, by bidding on items, do we suddenly give everyone different shares?
Because everyone values cash identically, but items differently. If each item is valued the same by all players, then everyone will get a share they agree is identical.

Example: 5 people, 7 items that everyone agrees are worth $1000, plus$3000 in cash; two people get two items and the other three get 1 item and $1000. Example: As above, except that one person values the items at$500. He would be just as happy with $1000/$1500 as the above split (depending on which share he gets). To him, his fair share is (7 * $500 +$3000)/5 = $1300, or$800 plus an item, or $300 plus two items. By choosing the first of these, you can give him more than his fair share (by his standards) and give the others more than their fair share (by their standards). How should the surplus be shared? That's what the method has to decide. (Same) example: 7 items are worth$500 on the open market, but four of the five place special value on the items; they would be willing to pay $1000 for them. If the items were all sold, each person would get$1300. Then the four could each try to buy the items back; depending on how this happens and whether they collude, they'll pay between $500 and$1000 for the items. (If I value an item at $500 but someone who values it at$1000 wants to buy it from me, I may very well sell for more than $500. If the items were auctioned, surely some of the items would go for more than$500.) Let's say that the average buyback price is $600.$400 was lost to outside sellers; this method is a way to keep that $400 internal and so give the players a Pareto improvement. March 9th, 2010, 07:40 AM #15 Global Moderator Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Fair Division Quote:  Originally Posted by fever At the end of the day, each person should end up with an equal share of the same sized pie not different shares / different pies. Let's do this differently. Take the example you posted and divide the goods by some method you prefer. Let's then compare the results.  March 9th, 2010, 08:25 AM #16 Newbie Joined: Mar 2010 Posts: 11 Thanks: 0 Re: Fair Division Item Michael Barbie Jamie Who gets it Watch$1 $200$100 Barbie for $100 Camera$890 $500$2500 Jamie for $890 Clock$5 $2000$1000 Barbie for $1000 Sunglassess$0 $75$55 Barbie for $55 Jacket$50 $50$450 Jamie for $50 Jet Ski$1 $1000$3000 Jamie for $1000 Total$0 $1,155$1,940 $3,095 Fair Share$1032 $1032$1032 $3,095 Settlement Receives$1032 Pays $123 Pays$908 All heirs receive the equivalent of $1,032 The above would be how I would split it, if conducting a Vickeroy Auction. I have no idea why Michael would bid$1 for a jet ski both in my system and the previous system.
 March 9th, 2010, 08:29 AM #17 Newbie   Joined: Mar 2010 Posts: 11 Thanks: 0 Re: Fair Division That didn't post very well. How about this one. Item, Michael, Barbie, Jamie, Who gets it Watch, $1,$200, $100, Barbie for$100 Camera, $890,$500, $2500, Jamie for$890 Clock , $5,$2000, $1000, Barbie for$1000 Sunglassess, $0,$75, $55, Barbie for$55 Jacket, $50,$50, $450, Jamie for$50 Jet Ski, $1,$1000, $3000, Jamie for$1000 Total, $0,$1,155, $1,940,$3,095 Fair Share, $1032,$1032, $1032,$3,095 Settlement Michael Receives $1032 Barbie Pays$123 Jamie Pays $908 All heirs receive the equivalent of$1,032 The above would be how I would split it, if conducting a Vickeroy Auction. I have no idea why Michael would bid $1 for a jet ski both in my system and the previous system.  March 9th, 2010, 09:57 AM #18 Global Moderator Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Fair Division I see. I've used that method before myself, so I can't say I object to it. It's hard to say which is better; both methods give out goods to the player who wants them the most [assuming honest bids], so they're equivalent in a Coasian sense. It's not clear to me which method, if either, has claim to being the 'right' one -- nor even that such a method exists. Are you familiar with Arrow's General Possibility Theorem, or better yet the similar theorem of Gibbard & Satterthwaite?  March 9th, 2010, 06:16 PM #19 Newbie Joined: Mar 2010 Posts: 11 Thanks: 0 Re: Fair Division I still can’t rationalize Fair Division as being very fair. Choose whatever auction methodology you like, at the end of the bidding, the estate value should be based on the bids and split evenly amongst the bidders. Keep in mind, the focus really isn’t on the winning bidder in either methodology; it’s on how we’re calculating their respective shares/pie. I just did some research on Arrow’ and Gibbard and I fail to see their application in this discussion but admittedly I’m not well versed in either theory. March 10th, 2010, 07:50 AM #20 Global Moderator Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Fair Division Quote:  Originally Posted by fever I still can’t rationalize Fair Division as being very fair. Choose whatever auction methodology you like, at the end of the bidding, the estate value should be based on the bids and split evenly amongst the bidders. I don't agree with that (even if I might be convinced to pick 'your' auction vs. 'their' auction). That division may result in the players being strictly less well-off than a different method; I think we should remain open to other possibilities. Quote:  Originally Posted by fever Keep in mind, the focus really isn’t on the winning bidder in either methodology; it’s on how we’re calculating their respective shares/pie. Sensibly, since the auction methods don't change who wins from auction to auction. Quote:  Originally Posted by fever I just did some research on Arrow’ and Gibbard and I fail to see their application in this discussion but admittedly I’m not well versed in either theory. Arrow's theorem isn't that relevant, but it's better known than the other so I thought I'd suggest it. Gibbard & Satterthwaite is actually extremely relevant, though the exact statement of the theorem likely does not apply. Gloss of the theorem: Take a 'reasonable' voting method. If there's no way to manipulate the system, then it is a dictatorship. It seems extremely likely that there are similar results that can be/have been proved relating to this division problem: that there are ways to manipulate the system by submitting dishonest/sophisticated bids. ("Bid" is dual to "vote", in this comparison.) I've already given examples of how the players can manipulate various systems. So in light of that manipulation possibility, any conclusions you draw on the assumption that the bids are honest is questionable. Take, for example, your assertion that Quote:  the estate value should be based on the bids and split evenly amongst the bidders If the bids are honest, this may be a desirable principle (though I'm not convinced; see above). But when the votes are known to be dishonest, there's no reason to stick to that. Say the estate has just one item. A values it at$90, B at $600, and C at$1200. First of all, it's not obvious that the Vickrey price of $600 is the right one -- any price from$600 to $1200 might lay claim to that. But what if B dishonestly bids$900? He knows that this will increase his share of the estate by $100, so it's in his interests. So just because we *want* to give everyone$200 doesn't mean that we should actually give everyone one-third of the second-highest bid, because that might not reflect reality.

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