1. Combinatorial Algorithms for Market Equilibria Vijay V. Vazirani

2. Arrow-Debreu Theorem: Equilibria exist.

3. Do markets operate at equilibria?

4. Arrow-Debreu Theorem: Equilibria exist. Do markets operate at equilibria? Can equilibria be computed efficiently?

5. Arrow-Debreu is highly non-constructive

6. “Invisible hand” of the market: Adam Smith Scarf, 1973: approximate fixed point algs. Convex programs:  Fisher: Eisenberg & Gale, 1957  Arrow-Debreu: Newman and Primak, 1992

7. Used for deciding tax policies, price of new products etc. New markets on the Internet

8. Algorithmic Game Theory Use powerful techniques from modern algorithmic theory and notions from game theory to address issues raised by Internet. Combinatorial algorithms for finding market equilibria.

9. Two Fundamental Models Fisher’s model Arrow-Debreu model, also known as exchange model

10. Combinatorial Algorithms Primal-dual schema based algorithms  Devanur, Papadimitriou, Saberi & V., 2002 Combinatorial algorithm for Fisher’s model Auction-based algorithms  Garg & Kapoor, 2004 Approximation algorithms.

11. Approximation Find prices s.t. all goods clear Each buyer get goods providing at least optimal utility.

12. Primal-Dual Schema Highly successful algorithm design technique from exact and approximation algorithms

13. Exact Algorithms for Cornerstone Problems in P: Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching

14. Approximation Algorithms set cover facility location Steiner tree k-median Steiner network multicut k-MST feedback vertex set scheduling...

15. Main new idea Previous: problems captured via linear programs DPSV algorithm: problem captured via a nonlinear convex program

16. Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i,

17. Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i, Find prices s.t. market clears

18. Eisenberg-Gale Program, 1959

19. DPSV Algorithm “primal” variables: allocations of goods “dual” variables: prices algorithm: primal & dual improvements Allocations Prices

20.  Buyer i’s optimization program:  Global Constraint: Market Equilibrium

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

22. Prices and utilities $100 $60 $20 $140 $20 $40 $10 $60 10 20 4 2 utilities

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

24. Bang per buck Utility of $1 worth of goods Buyers will only buy goods providing maximum bang per buck

25. Equality subgraph $100 $60 $20 $140 $20 $40 $10 $60 10 20 4 2 10/20 20/40 4/10 2/60

26. Equality subgraph $100 $60 $20 $140 $20 $40 $10 $60 Most desirable goods for each buyer

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

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

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

30. Idea of Algorithm Invariant: source edges form min-cut (agents have surplus) Iterations: gradually raise prices, decrease surplus Terminate: when surplus = 0, i.e., sink edges also form a min-cut

31. Ensuring Invariant initially Set each price to 1/n Assume buyers’ money integral

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

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

34. 100 60 20 140 20x 40x 10x 60x initialize: x = 1 x

35. 100 60 20 140 20x 40x 10x 60x x = 2: another min-cut x>2: Invariant violated

36. 100 60 20 140 40x 80x 20 120 active frozen reinitialize: x = 1

37. 100 60 20 140 50 100 20 120 active frozen x = 1.25

38. 100 60 20 140 50 100 20 120

39. 100 60 20 140 50 100 20 120 unfreeze

40. 100 60 20 140 50x 100x 20x 120x x = 1, x

41. m buyers goods

42. m p buyers goods ensure Invariant

43. m p buyers goods equality subgraph ensure Invariant

44. m pxpx x = 1, x

45. } { S

46. } { S freeze S tight set

47. } { S prices in S are market clearing

48. x = 1, x S active frozen pxpx

49. x = 1, x S active frozen pxpx

50. x = 1, x S active frozen pxpx

51. new edge enters equality subgraph S active frozen

52. unfreeze component active frozen

53. All goods frozen => terminate (market clears)

54. All goods frozen => terminate (market clears) When does a new set go tight? Solve as parametric cut problem

55. Termination Prices in S* have denominators Terminates in max-flows.

56. Polynomial time? Problem: very little price increase between freezings

57. Polynomial time? Problem: very little price increase between freezings Solution: work with buyers having large surplus

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

59. 100 60 20 140 20 40 10 60 20 0 10 60 40 0 Max flow

60. surplus(i) = m(i) – f(i) 100 60 20 140 20 40 10 60 20 0 10 60 40 0 60 20 70

61. surplus(i) = m(i) – f(i) 100 60 20 140 20 40 10 60 20 0 10 60 40 0 60 20 70 Surplus vector = (40, 60, 20, 70)

62. Balanced flow A max-flow that minimizes l 2 norm of surplus vector  tries to make surpluses as equal as possible

63. Algorithm Compute balanced flow

64. active frozen Active subgraph: Buyers with maximum surplus

65. active frozen x = 1, x pxpx

66. active frozen new edge enters equality subgraph

67. active frozen Unfreeze buyers having residual path to active subgraph

68. active frozen Unfreeze buyers having residual path to active subgraph Do they have large surplus?

69. f: balanced flow R(f): residual graph Theorem: If R(f) has a path from i to j then surplus(i) > surplus(j)

70. active frozen New set tight

71. active frozen New set tight: freeze

72. Theorem: After each freezing, l 2 norm of surplus vector drops by (1 - 1/n 2 ) factor. Two reasons:  total surplus decreases  flow becomes more balanced

73. Idea of Algorithm algorithm: primal & dual improvements measure of progress: l 2 -norm of surplus vector Allocations Prices

74. Polynomial time Theorem: max-flow computations suffice.

75. Weak gross substitutability Increasing price of one good cannot decrease demand for another good.

76. Weak gross substitutability Increasing price of one good cannot decrease demand for another good. => never need to decrease prices (dual variables).

77. Weak gross substitutability Increasing price of one good cannot decrease demand for another good. => never need to decrease prices (dual variables). Almost all primal-dual algs work this way.

78. Arrow-Debreu Model Approximate equilibrium algorithms:  Jain, Mahdian & Saberi, 2003: Use DPSV as black box.  Devanur & V., 2003: More efficient, by opening DPSV.

79. Garg & Kapoor, 2004 Auction-based algorithm Start with very low prices Keep increasing price of good that is in demand B has excess money. Favorite good: g Currently at price p and owned by B’ B outbids B’

80. Outbid

81. Auction-based algorithm Go in rounds: In each round, total surplus decreases by factor Hence iterations suffice, total money M= total money

82. Arrow-Debreu Model Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money

83. Arrow-Debreu Model Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money Any good with price >1 is fully sold

84. Arrow-Debreu Model Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money Any good with price >1 is fully sold Eventually every good will have price >1

85. Garg, Kapoor & V., 2004: Auction-based algorithms for additively separable concave utilities satisfying weak gross substitutability

86. Kapoor, Mehta & V., 2005: Auction-based algorithm for a (restricted) production model

87. Q: Distributed algorithm for equilibria? Appropriate model? Primal-dual schema operates via local improvements