1. 2) Combinatorial Algorithms for Traditional Market Models 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

7. Arrow-Debreu is highly non-constructive “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

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

9. 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.

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

11. 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.

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

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

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

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

16. Main new idea Previous: problems captured via linear programs DPSV: nonlinear convex program Eisenberg-Gale Convex Program, 1959

17. 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,

18. 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

19. Can equilibrium allocations be captured via an LP? Set of feasible allocations:

20. Does equilibrium optimize a global objective function? Guess 1: Maximize sum of utilities, i.e., Problem: and are equivalent utility functions.

21. However,

22. Guess 2: Product of utilities.

23. However, suppose a buyer with $200 is split into two buyers with $100 each And same utility function. Clearly, equilibrium should not change.

24. However,

25. Money of buyers is relevant. Assume a utility function is written on each dollar in market

26. Guess 3: Product of utilities over all dollars

27. Eisenberg-Gale Program, 1959

28. Via KKT Conditions can establish: Optimal solution gives equilibrium allocations Lagrange variables give prices of goods

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

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

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

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

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

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

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

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

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

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

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

40. 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

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

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

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

44. 100 60 20 140 20x 40x 10x 60x initialize: x = 1 x

45. 100 60 20 140 20x 40x 10x 60x x = 2: another min-cut x>2: Invariant violated

46. 100 60 20 140 40x 80x 20 120 active frozen reinitialize: x = 1

47. 100 60 20 140 50 100 20 120 active frozen x = 1.25

48. 100 60 20 140 50 100 20 120

49. 100 60 20 140 50 100 20 120 unfreeze

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

51. m buyers goods

52. m p buyers goods ensure Invariant

53. m p buyers goods equality subgraph ensure Invariant

54. m pxpx x = 1, x

55. } { S

56. } { S freeze S tight set

57. } { S prices in S are market clearing

58. x = 1, x S active frozen pxpx

59. x = 1, x S active frozen pxpx

60. x = 1, x S active frozen pxpx

61. new edge enters equality subgraph S active frozen

62. unfreeze component active frozen

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

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

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

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

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

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

69. 100 60 20 140 20 40 10 60 20 0 10 60 40 0 Max flow

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

71. 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)

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

73. Algorithm Compute balanced flow

74. active frozen Active subgraph: Buyers with maximum surplus

75. active frozen x = 1, x pxpx

76. active frozen new edge enters equality subgraph

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

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

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

80. active frozen New set tight

81. active frozen New set tight: freeze

82. 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

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

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

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

86. 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.

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

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

89. 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’

90. Outbid

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

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

93. 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

94. 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

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

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

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