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Allocation and Social Equity

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

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Allocation and Social Equity

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

  2. Allocation Problems - Operational Research Fairness of Allocation - Social Policy

  3. What is Fair?

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

  5. 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

  6. How should the fruit be divided? 1. Jones 12G Smith 12A ? 2. Jones 9G Smith 3G 12A ? 3. Jones 8G Smith 4G 12A ?

  7. MODEL GJ Grapefruit to Jones GS Grapefruit to Smith AJ Avocados to Jones AS Avocados to Smith Value to Jones = 100 GJ Value to Smith = 50 GS + 50 AS GJ + GS = 12 AJ + AS = 12

  8. What Criterion Should Be Applied?

  9. 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)

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

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

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

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

  14. Example Education How should resources be allocated fairly? 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.

  15. Let X i = Number of teachers allocated to category i.

  16. Consider Coalitions. ( Mixed ability classes )

  17. MixedInteger Optimisation Problem Decide on possible coalitions (if at all) and allocations of teachers within these to

  18. Constraints [ 1 ] X1 + X 2 + …. X 12 < = 70 [ 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.

  19. Objective Function Maximise Total Benefit : Maximise 3X1 + 5X 2 + …. 16X 12

  20. Formulation Maximise 3X1 + 5X 2 + …. 16X 12 Subject to: [ 1 ] X1 + X 2 + …. X 12 < = 70 [ 2 ] X1 < = 23.33 Y1 ….. [ 13 ] X12 < = 31.25 Y12 [ 14 ] Y1 + Y6 + Y7 = 1 ….. [ 18 ] Y11 + Y12 = 1 X1 , X 2 , …. X 12 > = 0, and integer Y1 , Y 2 , …. Y 12 = {0,1} Solution is : Y1 = 1 X1 = 11 Y2 = 1 X2 = 16 Y11 = 1 X11 = 43 Max Benefit = 758

  21. Solution The Majority Loss of Benefit Falls on Category 1. Is this fair?

  22. MIN – MAX Formulation Minimise W Subject to: [ 1 ] X1 + X 2 + …. X 12 < = 70 [ 2 ] X1 < = 23.33 Y1 ….. [ 13 ] X12 < = 31.25 Y12 [ 14 ] Y1 + Y6 + Y7 = 1 ….. [ 18 ] Y11 + Y12 = 1 [ 19 ] W >= 70Y1 - 3X1 ….. [ 30 ] W >= 500Y12 - 16X12 X1 , X 2 , …. X 12 > = 0, and integer Y1 , Y 2 , …. Y 12 = {0,1} Solution is : Y1 = 1 X1 = 16 Y2 = 1 X2 = 12 Y11 = 1 X11 = 42 Min W = 22

  23. Solution In total worse, but would seem to be a “FAIRER” solution.

  24. 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?

  25. 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

  26. Example: Cost of Computer Provision in a University (in 100k)

  27. Cost of Coalitions What is a Fair division of the central provision?

  28. Cost of Computer Provision (in £100k)

  29. 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

  30. 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.

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

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

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

  34. 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 RestatementHarvard University Press, 2001 M. Yaari & M. Bar-Hillel, On Dividing Justly, Social Choice Welfare 1, 1984

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