Transcript The Allocation of Shared Costs

Allocation and Social Equity
H. Paul Williams
- London School of Economics
Work with Martin Butler
University College Dublin
Allocation Problems
- Operational Research
Fairness of Allocation - Social Policy
What is Fair?
An Example
12 Grapefruit and 12 Avocados
to be split between Smith and Jones
Jones
derives 100mls of Vitamin F from each Grapefruit
none from each Avocado
Smith
derives 50mls of Vitamin F from each Grapefruit
50mls of Vitamin F from each Avocado
How should the fruit be divided?
1.
Jones 12G
Smith 12A
?
2.
Jones 9G
Smith 3G
12A
?
3.
Jones 8G
Smith 4G
12A
?
MODEL
GJ
AJ
Grapefruit to Jones
Avocados to Jones
GS Grapefruit to Smith
AS Avocados to Smith
Value to Jones
Value to Smith
=
=
100 GJ
50 GS + 50 AS
GJ + GS
AJ + AS
=
=
12
12
What Criterion Should Be
Applied?
Utilitarian
Maximise 100 GJ + 50 GS + 50 AS
Leads to GJ = 12, GS = 0, AS = 12
(Total ‘Good’ = 1800)
Egalitarian
Maximise Minimum (100GJ, 50 GS
+ 50 AS)
Leads to GJ = 8, GS = 4, AS = 12
(Total ‘Good’ = 1600)
ALLOCATION OF MEDICAL
RESOURCES
Use of QALYs (QUALITY ADJUSTED LIFE YEARS)
Allocate Resources according to greatest QALY
Cost
Utilitarian Approach
Maximise Total QALYs
subject to resource limits
Favours Young over Old
Favours Unborn over Living e.g. Fertility Treatment
Fair Approach?
Maximise Minimum shortfall of desirable QALYs
over whole population
Teacher Allocation
How to spread limited numbers of teachers over
different ability groups.
Example
Education
How should resources be allocated
fairly?
Category
Students
Desirable
Class Size
Desirable Number
of Teachers
Special Needs
1
70
3
23.33
Standard A
2
80
5
16
Standard B
3
150
10
15
Gifted
4
300
16
18.75
Very Clever
5
200
15
13.33
Total
800
86.41
How to allocate the 70 available teachers in a “fair” manner?
The negative benefit of a shortfall in a category proportional to
number in category/desirable number of teachers.
Let X i = Number of teachers allocated to
category i.
Category
Students
Desirable
Class Size
Desirable
Number of
Teachers
Actual
Number of
Teachers
1
70
3
23.33
X
1
2
80
5
16
X
2
3
150
10
15
X
3
4
300
16
18.75
X
4
5
200
15
13.33
X
5
Total
800
86.41
70
Consider Coalitions. ( Mixed ability classes )
Category
Coalition
Students
Desirable
Class Size
Desirable
Number of
Teachers
Actual
Number of
Teachers
6
1,2
150
4
37.50
X
6
7
1,2,3
300
6
50
X
7
8
2,3
230
8
28.75
X
8
9
2,3,4
530
11
48.18
X
9
10
3,4
450
14
32.14
X
10
11
3,4,5
650
15
43.33
X
11
12
4,5
500
16
31.25
X
12
Mixed Integer Optimisation Problem
Decide on possible coalitions (if at all) and
allocations of teachers within these to
Constraints
Category
Coalition
Students
Desirable
Class Size
Desirable
Number of
Teachers
1
1
70
3
23.33
X
1
Y
1
2
2
80
5
16
X
2
Y
2
3
3
150
10
15
X
3
Y
3
4
4
300
16
18.75
X
4
Y
4
5
5
200
15
13.33
X
5
Y
5
6
1,2
150
4
37.50
X
6
Y
6
7
1,2,3
300
6
50
X
7
Y
7
8
2,3
230
8
28.75
X
8
Y
8
9
2,3,4
530
11
48.18
X
9
Y
9
10
3,4
450
14
32.14
X
10
Y
10
11
3,4,5
650
15
43.33
X
11
Y
11
12
4,5
500
16
31.25
X
12
Y
12
Is the
coalition
used?
[1]
X1 + X
[2]
X1 < = 23.33 Y1
[ 13 ]
X12 < = 31.25 Y12
[ 14 ]
Y1 + Y6 + Y7 = 1
-
Category 1 only served by 1 coalition.
[ 18 ]
Y11 + Y12 = 1
-
Category 5 only served by 1 coalition.
2
+ …. X
Actual
Number of
Teachers
12
< = 70
Objective
Function
Category
Coalition
Students
Desirable
Class Size
Desirable
Number of
Teachers
1
1
70
3
23.33
X
1
Y
1
2
2
80
5
16
X
2
Y
2
3
3
150
10
15
X
3
Y
3
4
4
300
16
18.75
X
4
Y
4
5
5
200
15
13.33
X
5
Y
5
6
1,2
150
4
37.50
X
6
Y
6
7
1,2,3
300
6
50
X
7
Y
7
8
2,3
230
8
28.75
X
8
Y
8
9
2,3,4
530
11
48.18
X
9
Y
9
10
3,4
450
14
32.14
X
10
Y
10
11
3,4,5
650
15
43.33
X
11
Y
11
12
4,5
500
16
31.25
X
12
Y
12
Maximise Total Benefit :
Maximise 3X1 + 5X
2
+ …. 16 X
12
Actual
Number of
Teachers
Is the
coalition
used?
Formulation
Maximise 3X1 + 5X
2
+ …. 16 X
12
Subject to:
[1]
X1 + X
[2]
X1 < = 23.33 Y1
2
+ …. X
12
< = 70
…..
[ 13 ]
X12 < = 31.25 Y12
[ 14 ]
Y1 + Y6 + Y7 = 1
…..
[ 18 ]
Y11 + Y12 = 1
X1 , X
2
, …. X
12
Y1 , Y
2
, …. Y
12
> = 0, and integer
= {0,1}
Solution is :
Y1 = 1
X1 = 11
Y2 = 1
X2 = 16
Y11 = 1
X11 = 43
Max Benefit = 758
Solution
Coalition
Students
Desirable
Class Size
Desirable
Number of
Teachers
Actual
Number of
Teachers
Benefit
Teacher
Shortfall
Benefit
Shortfall
1
70
3
23.33
11
33
12.33
37
2
80
5
16
16
80
-
-
3
150
10
15
4
300
16
18.75
5
200
15
13.33
1,2
150
4
37.50
1,2,3
300
6
50
2,3
230
8
28.75
2,3,4
530
11
48.18
3,4
450
14
32.14
3,4,5
650
15
43.33
43
645
0.33
5
4,5
500
16
31.25
70
758
Total
The Majority Loss of Benefit Falls on Category 1. Is this fair?
42
MIN – MAX Formulation
Minimise W
Subject to:
[1]
X1 + X
[2]
X1 < = 23.33 Y1
2
+ …. X
12
< = 70
…..
[ 13 ]
X12 < = 31.25 Y12
[ 14 ]
Y1 + Y6 + Y7 = 1
…..
[ 18 ]
Y11 + Y12 = 1
[ 19 ]
W >= 70Y1 - 3X1
Solution is :
…..
[ 30 ]
W >= 500Y12 - 16X12
X1 , X
2
, …. X
12
Y1 , Y
2
, …. Y
12
Y1 = 1
X1 = 16
Y2 = 1
X2 = 12
Y11 = 1
X11 = 42
> = 0, and integer
= {0,1}
Min W = 22
Solution
Coalition
Students
Desirable
Class Size
Desirable
Number of
Teachers
Actual
Number of
Teachers
Benefit
Teacher
Shortfall
Benefit
Shortfall
1
70
3
23.33
16
48
7.33
22
2
80
5
16
12
60
4
20
3
150
10
15
4
300
16
18.75
5
200
15
13.33
1,2
150
4
37.50
1,2,3
300
6
50
2,3
230
8
28.75
2,3,4
530
11
48.18
3,4
450
14
32.14
3,4,5
650
15
43.33
42
630
1.33
20
4,5
500
16
31.25
70
738
Total
In total worse, but would seem to be a “FAIRER” solution.
62
Fixed Cost Allocation
Examples:
• How should cost of an airport runway be spread
among different sizes of aircraft?
• How should cost of a dam be spread among
different beneficiaries?
(hydro generators, water sports, irrigation)
• How should cost of an ATM be spread among
different credit card companies?
Co-operative Game Theory
Not fair to charge users within a coalition more, in
total, than the coalition would be charged (core
solutions)
Nucleolus Solution:
Minimise Maximum (i.e. try to equalise) savings of
each coalition from forming coalition
Example:
Cost of Computer Provision in a University (in 100k)
Veterinary Science
Medicine
Architecture
Engineering
Arts
Commerce
Agriculture
Science
Social Science
6
7
2
10
18
30
11
29
7
Cost of Coalitions
Veterinary Science, Medicine
Architecture, Engineering
Arts, Social Science
Agriculture, Science
Veterinary Science, Medicine, Agriculture, Science
Arts, Commerce, Social Science
Cost of Central Provision
11
14
22
37
46
50
96
What is a Fair division of the central provision?
Cost of Computer Provision (in £100k)
Independent A Core
Cost
Cost
Veterinary Science
Medicine
Architecture
Engineering
Arts
Commerce
Agriculture
Science
Social Science
6
7
2
10
18
30
11
29
7
6
3
2
0
11
30
8
29
7
96
Nucleolus
Cost
Weighted
Nucleolus Cost
4
1
0
8
15
28
8
27
5
96
1.83
5
0
7.5
16
24.67
9
27
5
96
Facility Location
Customer A
requires 1 of Facilities 1 or 2 or 3
and 1 of Facilities 4 or 5 or 6
and has a benefit of 8
Customer B
requires 1 of Facilities 1 or 4
and 1 of Facilities 2 or 5
and has a Benefit of 11
Customer C
requires 1 of Facilities 1 or 5
and 1 of Facilities 3 or 6
and has a Benefit of 19
Fixed Costs of Facilities (1 to 6) 8, 7, 8, 9, 11, 10
How do we split fixed costs of Facilities among
Customers who use them?
Optimal Solution (Maximum Benefit – Cost) is to
build Facilities 1, 2, 6 and supply all Customers.
There is no satisfactory cost allocation which
will lead to this.
Find Optimal Solution (Integer Programming)
and then allocate costs.
Possible Allocation
Surpluses Customers
Facilities
8
8
2
7
7
3
0
4
0
5
0
6
10
B
19
1
8
A
11
4
1
8
C
10
Allocation from Minimising Maximum
Surpluses
8
4 /3
1
A
1/
2
7
3
0
4
0
5
0
6
10
3
11
B
8
32/3
3
41/3
1
3
1/
3
4 2/ 3
19
4 /3
1
C
10
Allocation from Minimising Weighted Maximum
Surpluses
8
2.74
A
11
3.75
1
5.26
0.24
8
2
7
3
0
4
0
5
0
6
10
7
7.76
B
19
6.5
C
4.75
References
M. Butler & H.P. Williams, Fairness versus Efficiency in
Charging for the Use of Common Facilities, Journal of the
Operational Research Society, 53 (2002)
M. Butler & H.P. Williams, The Allocation of Shared Fixed Costs,
European Journal of Operational Research, 170 (2006)
J. Broome, Good, Fairness and QALYS, Philosophy and Medical
Welfare, 3 (1988)
J. Rawls, A Theory of Justice, Oxford University Press, 1971
J. Rawls & E. Kelly Justice as Fairness: A Restatement Harvard
University Press, 2001
M. Yaari & M. Bar-Hillel, On Dividing Justly, Social Choice
Welfare 1, 1984