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Chapter Thirty-One

Chapter Thirty-One. Welfare. Social Choice. Different economic states will be preferred by different individuals. How can individual preferences be “aggregated” into a social preference over all possible economic states?. Aggregating Preferences.

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Chapter Thirty-One

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  1. Chapter Thirty-One Welfare

  2. Social Choice • Different economic states will be preferred by different individuals. • How can individual preferences be “aggregated” into a social preference over all possible economic states?

  3. Aggregating Preferences • x, y, z denote different economic states. • 3 agents; Bill, Bertha and Bob. • Use simple majority voting to decide a state?

  4. Aggregating Preferences More preferred Less preferred

  5. Aggregating Preferences Majority Vote Results x beats y

  6. Aggregating Preferences Majority Vote Results x beats y y beats z

  7. Aggregating Preferences Majority Vote Results x beats y y beats z z beats x

  8. Aggregating Preferences Majority Vote Results No socially best alternative! x beats y y beats z z beats x

  9. Aggregating Preferences Majority Vote Results No socially best alternative! x beats y y beats z z beats x Majority voting does not always aggregate transitive individual preferences into a transitive social preference.

  10. Aggregating Preferences

  11. Aggregating Preferences Rank-order vote results (low score wins).

  12. Aggregating Preferences Rank-order vote results (low score wins). x-score = 6

  13. Aggregating Preferences Rank-order vote results (low score wins). x-score = 6 y-score = 6

  14. Aggregating Preferences Rank-order vote results (low score wins). x-score = 6 y-score = 6 z-score = 6

  15. Aggregating Preferences Rank-order vote results (low score wins). No state is selected! x-score = 6 y-score = 6 z-score = 6

  16. Aggregating Preferences Rank-order vote results (low score wins). No state is selected! x-score = 6 y-score = 6 z-score = 6 Rank-order voting is indecisive in this case.

  17. Manipulating Preferences • As well, most voting schemes are manipulable. • I.e. one individual can cast an “untruthful” vote to improve the social outcome for himself. • Again consider rank-order voting.

  18. Manipulating Preferences These are truthful preferences.

  19. Manipulating Preferences These are truthful preferences. Bob introduces a new alternative

  20. Manipulating Preferences These are truthful preferences. Bob introduces a new alternative

  21. Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies.

  22. Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8

  23. Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 y-score = 7

  24. Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 y-score = 7 z-score = 6

  25. Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 y-score = 7 z-score = 6 -score = 9 z wins!!

  26. Desirable Voting Rule Properties • 1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. • 2. If all individuals rank x before y then so should the voting rule. • 3. Social preference between x and y should depend on individuals’ preferences between x and y only.

  27. Desirable Voting Rule Properties • Kenneth Arrow’s Impossibility Theorem: The only voting rule with all of properties 1, 2 and 3 is dictatorial.

  28. Desirable Voting Rule Properties • Kenneth Arrow’s Impossibility Theorem: The only voting rule with all of properties 1, 2 and 3 is dictatorial. • Implication is that a nondictatorial voting rule requires giving up at least one of properties 1, 2 or 3.

  29. Social Welfare Functions • 1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. • 2. If all individuals rank x before y then so should the voting rule. • 3. Social preference between x and y should depend on individuals’ preferences between x and y only.

  30. Social Welfare Functions • 1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. • 2. If all individuals rank x before y then so should the voting rule. • 3. Social preference between x and y should depend on individuals’ preferences between x and y only. Give up which one of these?

  31. Social Welfare Functions • 1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. • 2. If all individuals rank x before y then so should the voting rule. • 3. Social preference between x and y should depend on individuals’ preferences between x and y only. Give up which one of these?

  32. Social Welfare Functions • 1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. • 2. If all individuals rank x before y then so should the voting rule. There is a variety of voting procedures with both properties 1 and 2.

  33. Social Welfare Functions • ui(x) is individual i’s utility from overall allocation x.

  34. Social Welfare Functions • ui(x) is individual i’s utility from overall allocation x. • Utilitarian:

  35. Social Welfare Functions • ui(x) is individual i’s utility from overall allocation x. • Utilitarian: • Weighted-sum:

  36. Social Welfare Functions • ui(x) is individual i’s utility from overall allocation x. • Utilitarian: • Weighted-sum: • Minimax:

  37. Social Welfare Functions • Suppose social welfare depends only on individuals’ own allocations, instead of overall allocations. • I.e. individual utility is ui(xi), rather than ui(x). • Then social welfare iswhere is an increasing function.

  38. Social Optima & Efficiency • Any social optimal allocation must be Pareto optimal. • Why?

  39. Social Optima & Efficiency • Any social optimal allocation must be Pareto optimal. • Why? • If not, then somebody’s utility can be increased without reducing anyone else’s utility; i.e. social suboptimality  inefficiency.

  40. Utility Possibilities OB 0 0 OA

  41. Utility Possibilities OB 0 0 OA

  42. Utility Possibilities OB 0 0 OA

  43. Utility Possibilities OB 0 0 OA

  44. Utility Possibilities OB 0 0 OA

  45. Utility Possibilities OB 0 0 OA

  46. Utility Possibilities Utility possibility frontier (upf) OB 0 0 OA

  47. Utility Possibilities Utility possibility frontier (upf) OB 0 0 Utility possibility set OA

  48. Social Optima & Efficiency Upf is the set of efficient utility pairs.

  49. Social Optima & Efficiency Upf is the set of efficient utility pairs. Social indifference curves

  50. Social Optima & Efficiency Upf is the set of efficient utility pairs. Higher social welfare Social indifference curves

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