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Allocation and Social Equity H. Paul Williams -London School of Economics Work with Martin Butler University College Dublin

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Allocation Problems - Operational Research Fairness of Allocation - Social Policy

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What is Fair?

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12 Grapefruit and 12 Avocados to be split between Smith and Jones An Example

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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

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How should the fruit be divided? 1.Jones 12GSmith 12A? 2.Jones 9GSmith 3G12A? 3.Jones 8GSmith 4G12A?

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MODEL GJGrapefruit to JonesGS Grapefruit to Smith AJAvocados to JonesAS Avocados to Smith Value to Jones =100 GJ Value to Smith = 50 GS + 50 AS GJ + GS=12 AJ + AS=12

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What Criterion Should Be Applied?

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UtilitarianMaximise 100 GJ + 50 GS + 50 AS Leads to GJ = 12, GS = 0, AS = 12 (Total ‘Good’ = 1800) EgalitarianMaximise Minimum (100GJ, 50 GS + 50 AS) Leads to GJ = 8, GS = 4, AS = 12 (Total ‘Good’ = 1600)

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ALLOCATION OF MEDICAL RESOURCES Use of QALYs (QUALITY ADJUSTED LIFE YEARS) Allocate Resources according to greatest QALY Cost

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Utilitarian Approach Maximise Total QALYs subject to resource limits Favours Young over Old Favours Unborn over Living e.g. Fertility Treatment

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Fair Approach? Maximise Minimum shortfall of desirable QALYs over whole population

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Teacher Allocation How to spread limited numbers of teachers over different ability groups.

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Example Education CategoryStudentsDesirable Class Size Desirable Number of Teachers Special Needs 170323.33 Standard A 280516 Standard B 31501015 Gifted 43001618.75 Very Clever 52001513.33 Total80086.41 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. How should resources be allocated fairly?

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Let X i = Number of teachers allocated to category i. CategoryStudentsDesirable Class Size Desirable Number of Teachers Actual Number of Teachers 170323.33 X 1 280516 X 2 31501015 X 3 43001618.75 X 4 52001513.33 X 5 Total80086.4170

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Consider Coalitions. ( Mixed ability classes ) CategoryCoalitionStudentsDesirable Class Size Desirable Number of Teachers Actual Number of Teachers 61,2150437.50 X 6 71,2,3300650 X 7 82,3230828.75 X 8 92,3,45301148.18 X 9 103,44501432.14 X 10 113,4,56501543.33 X 11 124,55001631.25 X 12

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Mixed Integer Optimisation Problem Decide on possible coalitions (if at all) and allocations of teachers within these to

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Constraints CategoryCoalitionStudentsDesirable Class Size Desirable Number of Teachers Actual Number of Teachers Is the coalition used? 1170323.33 X 1 Y 1 2280516 X 2 Y 2 331501015 X 3 Y 3 443001618.75 X 4 Y 4 552001513.33 X 5 Y 5 61,2150437.50 X 6 Y 6 71,2,3300650 X 7 Y 7 82,3230828.75 X 8 Y 8 92,3,45301148.18 X 9 Y 9 103,44501432.14 X 10 Y 10 113,4,56501543.33 X 11 Y 11 124,55001631.25 X 12 Y 12 [ 1 ] X 1 + X 2 + …. X 12 < = 70 [ 2 ]X 1 < = 23.33 Y 1 [ 13 ]X 12 < = 31.25 Y 12 [ 14 ]Y 1 + Y 6 + Y 7 = 1 - Category 1 only served by 1 coalition. [ 18 ]Y 11 + Y 12 = 1 - Category 5 only served by 1 coalition.

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Objective Function CategoryCoalitionStudentsDesirable Class Size Desirable Number of Teachers Actual Number of Teachers Is the coalition used? 1170323.33 X 1 Y 1 2280516 X 2 Y 2 331501015 X 3 Y 3 443001618.75 X 4 Y 4 552001513.33 X 5 Y 5 61,2150437.50 X 6 Y 6 71,2,3300650 X 7 Y 7 82,3230828.75 X 8 Y 8 92,3,45301148.18 X 9 Y 9 103,44501432.14 X 10 Y 10 113,4,56501543.33 X 11 Y 11 124,55001631.25 X 12 Y 12 Maximise Total Benefit : Maximise 3X 1 + 5X 2 + …. 16 X 12

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Subject to: [ 1 ] X 1 + X 2 + …. X 12 < = 70 [ 2 ]X 1 < = 23.33 Y 1 ….. [ 13 ]X 12 < = 31.25 Y 12 [ 14 ]Y 1 + Y 6 + Y 7 = 1 ….. [ 18 ]Y 11 + Y 12 = 1 X 1, X 2, …. X 12 > = 0, and integer Y 1, Y 2, …. Y 12 = {0,1} Formulation Solution is : Y 1 = 1 X 1 = 11 Y 2 = 1 X 2 = 16 Y 11 = 1 X 11 = 43 Max Benefit = 758

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Solution CoalitionStudentsDesirable Class Size Desirable Number of Teachers Actual Number of Teachers BenefitTeacher Shortfall Benefit Shortfall 170323.33113312.3337 280516 80-- 31501015 43001618.75 52001513.33 1,2150437.50 1,2,3300650 2,3230828.75 2,3,45301148.18 3,44501432.14 3,4,56501543.33436450.335 4,55001631.25 Total7075842 The Majority Loss of Benefit Falls on Category 1. Is this fair?

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Minimise W Subject to: [ 1 ] X 1 + X 2 + …. X 12 < = 70 [ 2 ]X 1 < = 23.33 Y 1 ….. [ 13 ]X 12 < = 31.25 Y 12 [ 14 ]Y 1 + Y 6 + Y 7 = 1 ….. [ 18 ]Y 11 + Y 12 = 1 [ 19 ]W >= 70Y 1 - 3X 1 ….. [ 30 ]W >= 500Y 12 - 16X 12 X 1, X 2, …. X 12 > = 0, and integer Y 1, Y 2, …. Y 12 = {0,1} MIN – MAX Formulation Solution is : Y 1 = 1 X 1 = 16 Y 2 = 1 X 2 = 12 Y 11 = 1 X 11 = 42 Min W = 22

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Solution CoalitionStudentsDesirable Class Size Desirable Number of Teachers Actual Number of Teachers BenefitTeacher Shortfall Benefit Shortfall 170323.3316487.3322 2805161260420 31501015 43001618.75 52001513.33 1,2150437.50 1,2,3300650 2,3230828.75 2,3,45301148.18 3,44501432.14 3,4,56501543.33426301.3320 4,55001631.25 Total7073862 In total worse, but would seem to be a “FAIRER” solution.

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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?

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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

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Veterinary Science 6 Medicine 7 Architecture 2 Engineering 10 Arts 18 Commerce 30 Agriculture 11 Science 29 Social Science 7 Example: Cost of Computer Provision in a University (in 100k)

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Cost of Coalitions What is a Fair division of the central provision? Veterinary Science, Medicine 11 Architecture, Engineering 14 Arts, Social Science 22 Agriculture, Science 37 Veterinary Science, Medicine, Agriculture, Science 46 Arts, Commerce, Social Science 50 Cost of Central Provision 96

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Cost of Computer Provision (in £100k) Independent Cost A Core Cost Nucleolus Cost Weighted Nucleolus Cost Veterinary Science 6641.83 Medicine 7315 Architecture 2200 Engineering 10087.5 Arts 18111516 Commerce 30 2824.67 Agriculture 11889 Science 29 27 Social Science 7755 96

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Facility Location Customer Arequires 1 of Facilities 1 or 2 or 3 and 1 of Facilities 4 or 5 or 6 and has a benefit of 8 Customer Brequires 1 of Facilities 1 or 4 and 1 of Facilities 2 or 5 and has a Benefit of 11 Customer Crequires 1 of Facilities 1 or 5 and 1 of Facilities 3 or 6 and has a Benefit of 19

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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.

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Possible Allocation A Surpluses Customers Facilities 8 B 6 5 4 3 2 8 4 1 11 19 8 7 0 0 0 10 7 8 C 1

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Allocation from Minimising Maximum Surpluses A 41/341/3 1 B 6 5 4 3 2 41/341/3 11 19 8 7 0 0 10 C 0 31/331/3 42/342/3 8 32/332/3 31/331/3 41/341/3

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Allocation from Minimising Weighted Maximum Surpluses 8 A 2.74 1 B 6 5 4 3 2 6.5 11 19 7 0 0 10 4.75 C 0 7 7.76 8 5.26 0.24 3.75

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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

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