Content
| Introduction |
| Inheritance |
| Adjusted Winner Procedure |
Chapter Objective: Use of mathematical methods
or procedures to introduce an element
of fairness for the division or allocation of objects among two or
more persons, players or groups.
What is Fairness?
1. To many fairness means equity or the equal divisions of objects among
each players
(individual to which a set of items, S is to be
allocated to) ;
The challenge here is how to divide evenly or in
equal share objects that are not easily divisible
into useful parts..
2. To some fairness is the perceived worth or value placed by each players
upon the object(s)
to be allocated.
3. To others they see fairness through the eyes or judgment of
an authority figure;
such as Mother, Manager, Arbitrator, or a Judge.
Usually this election of fairness requires trust
in the judge or a mediated person in a dispute.
4. Still others view fairness as a collective representation of a group
via a government or through laws.
Player may be powerless to challenge the group on
their enactment of fairness.
Fairness in this chapter: We will attempt
through mathematics to use each players perceived worth to
decide on fairness so that the final division or allocation of the
objects have one or more of the following properties:
Good Critera of fairness in fair-division problem:
1. Fair Share: Each players feel that they get their fair share of the object(s)
2. Envy-Free: Each player is satisfied with his or her allocation and
do not wish to have
objects or items allocated to other players
3. Pareto-optimal allocation: No other allocation would make one player
better off without
making another worse off.
Object types:
1. Continuous Items: Objects that may be easily divided into many parts: example: cake, land, money
2. Discrete Items: Objects that cannot be easily divided into many useful parts: example: House, children, a boat, a gold ring etc..
The cake cutting problem: - Division between two players
Procedure: One player cut the cake and the other player chooses.
This procedure is subject to acceptance and can be applied to all object types
Discrete Case - Two players Inheritance
Most Inheritance procedures typical follows these steps:
1. Each players present a sealed bid, a, b , etc.
2. Object(s) is awarded to the highest bidder
Example,. if b > a, then player B is awarded to object
3. Player A is then awards a fair share of player's A bid on the object(s)
Player B pays A (a/2 + (b-a)/4) or (a/2 + b/2) / 2
Example 1 : Two players one object inheritance - A house
Table 1: Player A bids $100,000 and player
B bids $150,000
| Players | A | B |
| 1. Totals bid on house
2. Fair share (1/2 of line 1.) 3.Object awarded: 4. Highest bids: 5. Remaining claim (2-4): 6. Total surplus (sum line 5): $25,000 7. Share of surplus (surplus/players): 8. Final settlement (4+7): |
100,000
50,000 None 0 50,000 - 12,500 +62,500 |
150,000
75,000 House 150,000 -75,000 - 12,500 -62,500 |
| Object allocated: | 62,500 | House and pay out $62,500 |
Final line: Sum adds to zero: more than fair share i.e., share of surplus and awarded object(s)
Table 2: Four players / many object
inheritance: Equal Share Estate: House, Cabin
and Boat
| Players | A | B | C | D |
| Bids on:
House Cabin Boat |
120,000
60,000
30,000
|
200,000
40,000
24,000
|
140,000
90,000
20,000
|
180,000
50,000
20,000
|
| 1. Sum of bids
2. Fair shares: 3. Object awarded: 4. Highest bids: 5. Remaining claims: 6. Total surplus: 76,500 7. Share of surplus: 8. Final settlements: |
210,000
52,500 Boat 30,000 22,500 - 19,125 Boat |
264,000
66,000 House 200,000 -134,000 - 19,125 House |
250,000
62,500 Cabin 90,000 -27,500 - 19,125 Cabin |
250,000
62,500 None 0 62,500 - 19,125 None |
| Objects awarded | 41,625 | -114,875 | -8,375 | 81,625 |
| Players | A
(40%) |
B
(30%) |
C
(20%) |
D
(10%) |
| Bids on:
House Cabin Boat |
120,000
60,000
30,000
|
200,000
40,000
24,000
|
140,000
90,000
20,000
|
180,000
50,000
20,000
|
| 1. Sum of bids
2. Fair shares:(% of 1) 3. Object awarded: 4. Highest bids: 5. Remaining claims: 6. Total surplus: 81,800 7. Share of surplus (% of 6) 8. Final settlements: |
210,000
84,000 Boat 30,000 54,000 - 32,720 Boat |
264,000
79,200 House 200,000 -120,800 - 24,540 House |
250,000
50,000 Cabin 90,000 -40,000 - 16,360 Cabin |
250,000
25,000 None 0 25,000 - 8,180 None |
| Objects awarded | 86,720 | -96,260 | -23,640 | 33,180 |
Purpose: Handling property settlement in a divorce or inheritance with only two players.
Example: 1991 Divorce property settlement of the Trumps (Ivana and Danold)
Steps:
Step 1: Each to assign 100 points to objects in the settlement that reflect their net worth or perceived worth to them.
Table 4: Table of Trumps Point Allocations:
| Asset | Donald | Ivana |
| Connecticut estate | 10 | 38 |
| Palm Beach mansion | 40 | 20 |
| Trump Plaza apartment | 10 | 30 |
| Trump Tower triplex | 38 | 10 |
| Cash and jewelry | 2 | 2 |
So Donald gets Trump Towers and Palm beach mansion: 38 + 40 = 78 points
And Ivana gets Connecticut estate and Trump Plaza: 38 + 30 = 68 points
Ivana gets cash and jewelry to bring her total points to 70
Step 3:. Transfer asset from player with most point to player with the least until points are equal in the following manner:
Arrange player with most points assets from left to right where leftmost
award is lowest ration of the following:
Highest player's point value of asset divided by other player's point
value for that asset:
Example. 40/20 (Palm Beach) = 2.0 38/10 (triplex) = 3.8
That is, Donald transfer asset that first that is most important to Ivana or could be of similar value to both.
Let x be equal to the portion of the asset that Donald would transfer
to Ivana that would make
both their total points be equal:
Donald: 38 + 40(x)
Ivana : 70 + 20(1-x)
Solve for x: 38 + 40x = 70 + 20 - 20x
60x = 52, x = 52/60 about 87% so Ivana gets 13% of Palm Beach
So 38 + 40(52/60) = 38 + 20(8/60) = 72.7
Actual awards of the Trumps:
Donald: Trump Tower triplex and the most part of Palm Beach mansion ...
Ivana: Connecticut estate, Trump Plaza apartment, cash and jewelry and 1 month vacation stay at at the Palm Beach mansion
The adjusted winner fulfilled: equitably, envy-free and Pareto-optimal criteria.