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

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

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

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

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

15. 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

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

17. 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

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

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

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

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

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

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

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

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

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

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

28. 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

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

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

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

32. 100 60 20 140 20x 40x 10x 60x initialize: x = 1 x

33. 100 60 20 140 20x 40x 10x 60x x = 2: another min-cut x>2: Invariant violated

34. 100 60 20 140 40x 80x 20 120 active frozen reinitialize: x = 1

35. 100 60 20 140 50 100 20 120 active frozen x = 1.25

36. 100 60 20 140 50 100 20 120

37. 100 60 20 140 50 100 20 120 unfreeze

38. 100 60 20 140 50x 100x 20x 120x x = 1, x

39. m buyers goods

40. m p buyers goods equality subgraph ensure Invariant

41. m pxpx x = 1, x

42. } { S

43. } { S freeze S tight set

44. } { S prices in S are market clearing

45. x = 1, x S active frozen pxpx

46. x = 1, x S active frozen pxpx

47. x = 1, x S active frozen pxpx

48. new edge enters equality subgraph S active frozen

49. unfreeze component active frozen

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

51. All goods frozen => terminate (market clears) When does a new set go tight?

52. } { S

53. Try S = A (all goods) Let Clearly, If s is min-cut, x=x* and S* = A.

54. m pxpx Otherwise, x > x*. s S*

55. Sufficient to recurse on smaller graph s S*

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

57. Polynomial time Pre-emptively freeze sets that have small surplus (at most ).

58. m s S*

59. m s

60. m s } freeze

61. m add to prices and find new min-cut s S*

62. Next freezing: prices must increase Problem: at end, surplus

63. Next freezing: prices must increase Problem: at end, surplus But, surplus

64. initial surplus M

65. ε

66. final surplus

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

71. Question Is main algorithm, i.e., without pre-emptive freezing, polynomial time?

72. Question Is main algorithm, i.e., without pre-emptive freezing, polynomial time? Strongly polynomial?

73. Post-mortem

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

75. Central Idea Behind Primal-Dual Schema Two processes making local improvements (relative to each other) and achieving global objective

76. Post-mortem “primal” variables: flow in equality subgraph “dual” variables: prices algorithm: primal & dual improvements

77. Post-mortem “primal” variables: flow in equality subgraph “dual” variables: prices algorithm: primal & dual improvements nonzero flow from j to i

78. Post-mortem “primal” variables: flow in equality subgraph “dual” variables: prices algorithm: primal & dual improvements nonzero flow from j to i “complementary slackness condition”

79. Post-mortem “primal” variables: flow in equality subgraph “dual” variables: prices algorithm: primal & dual improvements algorithm inspired by Kuhn’s primal-dual algorithm for bipartite matching

80. ‘‘Primal-Dual-Type’’ Algorithms Formal mathematical setting? Analogous setting for primal-dual algorithms: LP-Duality Theory

81. Concave utilities Buyers get satiated by goods Fix prices => each buyer has unique optimal bundle

82. Concave utilities Buyers get satiated by goods Fix prices => each buyer has unique optimal bundle Economy of communication! Distributed market clearing

83. utility Concave utility function amount of j

84. utility Piece-wise linear, concave amount of j

85. utility PTAS for concave fn. amount of j

86. utility Piece-wise linear, concave amount of j

87. rate rate = utility/amount of j amount of j Differentiate

88. $20$70$100 for buyer i, good j rate rate = utility/unit amount of j

89. Fix price of j, then utility/$ given by utility derived,

90. $20$70$100 utility fix price of j piecewise-linear, concave

91. V. 2002: Rate at which i derives happiness depends on fraction of budget spent on j. Spending constraint model

92. Theorem: Equilibrium price is unique, and can be computed in polynomial time

93. Theorem: Equilibrium price is unique, and can be computed in polynomial time (Not unique in traditional model, even for piece-wise linear case)

94. $20$40$100 Can generalize notion of ‘‘market clearing’’ -- assume that buyers have utility for money. rate

95. Does the spending constraint model measure up to traditional theory?

96. $100 for buyer i, good j rate continuous, decreasing rate function

97. utility derived, Strictly concave function. Each buyer has unique optimal bundle.

98. Devanur & V., 2003 Equilibrium exists for cts., decreasing rate fns. (Proof uses Brauwer’s fixed point theorem),

99. Devanur & V., 2003 Equilibrium exists for cts., decreasing rate fns. (Proof uses Brauwer’s fixed point theorem), and prices are unique.

100. Devanur & V., 2003 Equilibrium exists for cts., decreasing rate fns. (Proof uses Brauwer’s fixed point theorem), and prices are unique. PTAS for computing equilibrium.

101. Devanur & V., 2003 Extend model to Arrow-Debreu setting.

102. Devanur & V., 2003 Extend model to Arrow-Debreu setting. Equilibrium exists. (Proof uses Kakutani’s fixed point theorem.)

103. Algorithmic Game Theory Mechanism design: find equilibria that ensure truthful, fair functioning of agents, and are efficiently computable. Approximations: deal with NP-hardness, stringent game-theoretic notions

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

105. Global optimality via local improvement Exploit in distributed setting Kelly, Low, Lapsley, Doyle, Paganini …

106. TCP congestion control primal process: packet rates at sources dual process: packet drop at links AIMD + RED solves utility maximization problem in limit

107. Kelly, `97: charging, rate control and routing for elastic traffic Kelly & V. 2002: It is essentially a market equilibrium question, and generalizes Fisher’s problem!

108. Kelly, `97: charging, rate control and routing for elastic traffic Kelly & V. 2002: It is essentially a market equilibrium question, and generalizes Fisher’s problem! Q: Polynomial time alg?

109. Develop an algorithmic theory of market equilibria, via polynomial time exact and approximation algorithms

110. w.r.t. prices p, i sorts segments according to utility/$, and partitions into classes

111. w.r.t. prices p, i sorts segments according to utility/$, and partitions into classes $50$30$40 $60

112. w.r.t. prices p, i sorts segments according to utility/$, and partitions into classes $50$30$40 $60 Assume i has $100

113. w.r.t. prices p, i sorts segments according to utility/$, and partitions into classes $50$30$40 $60 forced flexible undesirable

114. Invariant 1: Approach eq. from below Invariant 2: Forced allocs follow sorted order

115. Invariant 1: Approach eq. from below Invariant 2: Forced allocs follow sorted order –simultaneously, for all i

116. Invariant 1: Approach eq. from below Invariant 2: Forced allocs follow sorted order –simultaneously, for all i –as prices change, allocs may become undesirable

117. Invariant 1: Approach eq. from below Invariant 2: Forced allocs follow sorted order –simultaneously, for all i –as prices change, allocs may become undesirable Deallocate

118. Invariant 1: Approach eq. from below Invariant 2: Forced allocs follow sorted order –simultaneously, for all i –as prices change, allocs may become undesirable Deallocate - exponential time??

119. Invariant 1: Approach eq. from below Invariant 2: Forced allocs follow sorted order –simultaneously, for all i –as prices change, allocs may become undesirable Deallocate - exponential time?? Reduce prices

120. Invariant 1: Approach eq. from below Invariant 2: Forced allocs follow sorted order –simultaneously, for all i –as prices change, allocs may become undesirable Deallocate - exponential time?? Reduce prices - measure of progress??

121. Maintains both Invariants + deallocations + monotonicity of prices Algorithm

122. flexibleforcedundesirable

123. flexibleforcedundesirable

124. flexibleforcedundesirable Can ensure prices