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How Intractable is the ‘‘Invisible Hand’’: Polynomial Time Algorithms for Market Equilibria

How Intractable is the ‘‘Invisible Hand’’: Polynomial Time Algorithms for Market Equilibria. Vijay V. Vazirani Georgia Tech. Market Equilibrium. $. $$$$$$$$$. ¢. wine. bread. cheese. milk. $$$$. People want to maximize happiness Find prices s.t. market clears. Walras, 1874.

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How Intractable is the ‘‘Invisible Hand’’: Polynomial Time Algorithms for Market Equilibria

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  1. How Intractable is the ‘‘Invisible Hand’’:Polynomial Time Algorithms for Market Equilibria Vijay V. Vazirani Georgia Tech

  2. Market Equilibrium $ $$$$$$$$$ ¢ wine bread cheese milk $$$$ • People want to maximize happiness • Find prices s.t. market clears

  3. Walras, 1874 • Pioneered mathematical theory of general economic equilibrium

  4. Arrow-Debreu Theorem, 1954 • Celebrated theorem in Mathematical Economics • Shows existence of equilibrium prices using Kakutani’s fixed point theorem

  5. Arrow-Debreu Theorem is highly non-constructive • How do markets find equilibria?

  6. Arrow-Debreu Theorem is highly non-constructive • How do markets find equilibria? • “Invisible hand” of the market: Adam Smith Wealth of Nations, 1776

  7. Arrow-Debreu Theorem is highly non-constructive • How do markets find equilibria? • “Invisible hand” of the market: Adam Smith Wealth of Nations, 1776 • Scarf, 1973: approximate fixed point algorithms

  8. Arrow-Debreu Theorem is highly non-constructive • How do markets find equilibria? • “Invisible hand” of the market: Adam Smith Wealth of Nations, 1776 • Scarf, 1973: approximate fixed point algorithms • Use techniques from modern theory of algorithms

  9. Arrow-Debreu Theorem is highly non-constructive • How do markets find equilibria? • “Invisible hand” of the market: Adam Smith Wealth of Nations, 1776 • Scarf, 1973: approximate fixed point algorithms • Use techniques from modern theory of algorithms Deng, Papadimitriou & Safra, 2002: linear case in P?

  10. Market Equilibrium $ $ $ $ wine bread $ cheese milk • People want to maximize happiness • Find prices s.t. market clears

  11. History • Irving Fisher 1891 (concave functions) • Hydraulic apparatus for calculating equilibrium

  12. History • Irving Fisher 1891 (concave functions) • Hydraulic apparatus for calculating equilibrium • Eisenberg & Gale 1959 • (unique) equilibrium exists

  13. History • Irving Fisher 1891 (concave functions) • Hydraulic apparatus for calculating equilibrium • Eisenberg & Gale 1959 • (unique) equilibrium exists • Devanur, Papadimitriou, Saberi & V. 2002 • poly time alg for linear case

  14. History • Irving Fisher 1891 (concave functions) • Hydraulic apparatus for calculating equilibrium • Eisenberg & Gale 1959 • (unique) equilibrium exists • Devanur, Papadimitriou, Saberi & V. 2002 • poly time alg for linear case • V. 2002: alg for generalization of linear case

  15. Market Equilibrium • n buyers, with specified money, • m goods (unit amount) • Linear utilities: utility derived by i on obtaining one unit of j

  16. Market Equilibrium • n buyers, with specified money, • m goods (unit amount) • Linear utilities: utility derived by i on obtaining one unit of j • Find prices s.t. market clears

  17. $100 $60 $20 $140 Goods People

  18. $100 $60 $20 $140 utilities Bang per buck 10 $20 20 $40 4 $10 2 $60

  19. $100 $20 10/20 $60 $40 20/40 $20 4/10 $10 $140 $60 2/60 Bang per buck 10 20 4 2

  20. $100 $20 10/20 $60 $40 20/40 $20 4/10 $10 $140 $60 2/60 Bang per buck 10 20 4 2

  21. Bang per buck Given prices , each i picks goods to maximize her bang per buck, i.e.,

  22. $100 $20 $60 $40 $20 $10 $140 $60 Equality subgraph for all i: most desirable j’s

  23. Any goods sold in equality subgraph make agents happiest • How do we maximize sales in equality subgraph?

  24. Any goods sold in equality subgraph make agents happiest • How do we maximize sales in equality subgraph? Use max-flow!

  25. 20 100 40 60 10 20 140 60 Max flow infinite capacities

  26. 20 100 40 60 10 20 140 60 Max flow

  27. Idea of Algorithm Invariant: source edges form min-cut (agents have surplus) Want: prices s.t. sink edges also form min-cut Gradually raise prices, decrease surplus, until 0

  28. ensuring Invariant initially • Set each price to 1/n • Assume buyers’ money integral

  29. How to raise prices? • Ensure equality edges retained j i l

  30. How to raise prices? • Ensure equality edges retained j i l • Raise prices proportionately

  31. 20x 100 40x 60 10x 20 140 60x initialize: x = 1 x

  32. 20x 100 40x 60 10x 20 140 60x x = 2: another min-cut x>2: Invariant violated

  33. active 40x 100 80x 60 20 20 140 120 reinitialize: x = 1 frozen

  34. active 50 100 100 60 20 20 140 120 x = 1.25 frozen

  35. 50 100 100 60 20 20 140 120

  36. 50 100 100 60 20 20 140 120 unfreeze

  37. 50x 100 100x 60 20x 20 140 120x x = 1, x

  38. goods buyers m

  39. goods buyers equality subgraph ensure Invariant p m

  40. px m x = 1, x

  41. { } S

  42. { } S tight set freeze S

  43. { } S prices in S are market clearing

  44. frozen S active px x = 1, x

  45. frozen S active px x = 1, x

  46. frozen S active px x = 1, x

  47. frozen S active new edge enters equality subgraph

  48. frozen active unfreeze component

  49. All goods frozen => terminate • (market clears)

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