Fair Division of an estate

This page describes a fair division of an estate between 5 people. The process is described in the "method of Sealed Bids" section. It that requires each participant has money they can use to pay off the others. I think this is a fairly serious drawback as not everyone has loads of money. Also in the case of an estate division it seems slightly galling that the person with more money gets more things, possibly of only sentimental value, then others.

The division works by
"# Step 1. Bidding. Each player produces a sealed bid in which he or she attaches a dollar value to each item in S. A player's fair share is 1/N of his total assessment.

# Step 2. Allocation. Each item in S goes to the highest bidder for that item. If her/his assessed value of the items received exceeds her/his fair share, she/he must pay the difference. If the assessed value of the items received falls short of a fair share, then she/he is paid out of money that others have had to pay.

# Step 3. Dividing the Surplus There is almost always a surplus of cash that is divided equally among the players. "
given results in people getting

Al  BMW, Saab, Miro 
Ben  House  
Cal  Cottage 
Don  Klee
Ed  Yacht

And the minimum receive is $62,500 + $52,100+$18,340=132 940 out of a possible $679,000 or 19% of their desire.

If instead of assuming each player has money to buy stuff off the others I assumed each layer had equal claim to happiness. So I give each persons 100% desire is worth $679000 and those who claimed less then that use the remainder of their allocation as cash.

So this gives a table of desire of this

data;
param m := 5;
param n := 1;
 param o := 7;
 
 /* n divisible items, o non divisible items*/

/*Divorce between Donald and Ivana Trump described in Win-Win solution.
Items to be divided are 
4 divisible items:cash
2 non divisible items: House cottage, bmw,saab,yacht,miro, klee
Item   Al   Ben   Cal   Don   Ed
House  29.46 31.66 28.71 25.77 30.19
Cottage  8.84 7.22 9.2 8.76 8.1
BMW  4.27 3.6 3.68 4.06 4.06
Saab  3.68 2.8 3.33 3.6 2.87
Yacht  17.67 18.4 17.53 19 19.51
Miro  13.99 13.13 7.36 11.05 9.57
Klee  22.09 19.88 14.58 26.36 21.94
cash 0 3.31 15.61 1.4 3.76
*/

param d :  1  2  3  4 5 6 7 :=
      1   29.46 8.84 4.27 3.68 17.67 13.99 22.09
      2   31.66 7.22 3.6 2.8 18.4 13.13 19.88
      3   28.71 9.2 3.68 3.33 17.53 7.36 14.58
      4  25.77 8.76 4.06 3.6 19 11.05 26.36
      5   30.19 8.1 4.06 2.87 19.51 9.57 21.94; 
      
param c :  1 :=
      1   0
      2   3.31
      3   15.61
      4   1.4
      5   3.76; 
      
end;

Running this on the linear programming fair division calculator program(you need to add s.t. wor5{i in I}: ((sum{j in J} c[5,j] * x[5,j])+(sum{k in K} d[5,k] * y[5,k])) >= worst;). gives a the person who gets least 20.5388%

Al BMW Saab Miro 
Ben House 
Cal  0.726381 of the cash  Cottage  
Don Klee
Ed Yacht 0.273619 of the cash

which is an slight improvement of over on the initial allocation. Everyone gets the same items as in the initial allocation but the money is divided amongst those who receive the least rather then everyone.