1. Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Georgia Tech Algorithms for the Linear Case, and Beyond …

2. Irving Fisher, 1891 Defined a fundamental market model Special case of Walras’ model

3. Several buyers with different utility functions and moneys. Find equilibrium prices!!

4. Linear Fisher Market Assume:  Buyer i’s total utility,  m i : money of buyer i.  One unit of each good j. Find market clearing prices!

5. Eisenberg-Gale Program, 1959

6. prices p j

7. Convex programs that capture market equilibria Underly all “efficient” markets (so far). Rational convex programs for many markets! Algorithms:  combinatorial, for rational (primal-dual)  continuous (ellipsoid/interior point)

8. Combinatorial algorithms for rational convex programs Natural extension of field of combinatorial optimization!

9. Combinatorial algorithms for rational convex programs Natural extension of field of combinatorial optimization! Central aspect of C.O. – efficient algorithms for solving integral LP’s, e.g. matching, flow.

10. Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 By extending the primal-dual paradigm to the setting of convex programs & KKT conditions

11. Combinatorial algorithms Yield deep structural insights. Preferable for applications.

12. Auction for Google’s TV ads N. Nisan et. al, 2009: Used market equilibrium based approach. Combinatorial algorithms for linear case provided “inspiration”.

13. Primal-Dual Paradigm 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...

18. Yin & Yang

19. An easier question Given prices p, are they equilibrium prices? If so, find equilibrium allocations.

20. An easier question Given prices p, are they equilibrium prices? If so, find equilibrium allocations. Equilibrium prices are unique!

21. At prices p, buyer i’s most desirable goods, S i = Any goods from S i worth m(i) constitute i’s optimal bundle Bang-per-buck

22. m(1) m(2) m(3) m(4) p(1) p(2) p(3) p(4) For each buyer, most desirable goods, i.e.

23. Network N(p) m(1) m(2) m(3) m(4) p(1) p(2) p(3) p(4) infinite capacities s t

24. Max flow in N(p) m(1) m(2) m(3) m(4) p(1) p(2) p(3) p(4) p: equilibrium prices iff both cuts saturated

25. Idea of algorithm “primal” variables: allocations “dual” variables: prices of goods Approach equilibrium prices from below:  start with very low prices; buyers have surplus money  iteratively keep raising prices and decreasing surplus

26. An important consideration The price of a good never exceeds its equilibrium price  Invariant: s is a min-cut

27. Invariant: s is a min-cut in N(p) m(1) m(2) m(3) m(4) p(1) p(2) p(3) p(4) p: low prices s

28. Idea of algorithm Iterations: execute primal & dual improvements Allocations Prices

29. How is primal-dual paradigm adapted to nonlinear setting?

30. Fundamental difference between LP’s and convex programs Complementary slackness conditions: involve primal or dual variables, not both. KKT conditions: involve primal and dual variables simultaneously.

31. KKT conditions

33. Primal-dual algorithms so far (i.e., LP-based) Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.)

34. Primal-dual algorithms so far Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.)  Only exception: Edmonds, 1965: algorithm for max weight matching.

35. Primal-dual algorithms so far Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.)  Only exception: Edmonds, 1965: algorithm for max weight matching. Otherwise primal objects go tight and loose. Difficult to account for these reversals -- in the running time.

36. Our algorithm Dual variables (prices) are raised greedily Yet, primal objects go tight and loose  Because of enhanced KKT conditions

37. Our algorithm Dual variables (prices) are raised greedily Yet, primal objects go tight and loose  Because of enhanced KKT conditions New algorithmic ideas needed!

38. Key Algorithmic Idea Dual variables (prices) are raised greedily Yet, primal objects go tight and loose  Because of enhanced KKT conditions Balanced Flows: For limiting no. of such events

39. Max-flow in N m p W.r.t. a max-flow f, surplus(i) = m(i) – f(i,t) i t s

40. Max-flow in N m p surplus vector = vector of surpluses w.r.t. f i

41. Obvious potential function Total surplus money = l 1 norm of surplus vector Reduce l 1 norm of surplus vector by inverse polynomial fraction in each iteration

42. Balanced flow A max-flow that minimizes l 2 norm of surplus vector. Makes surpluses as equal as possible.

43. Balanced flow A max-flow that minimizes l 2 norm of surplus vector. Makes surpluses as equal as possible. All balanced flows have same surplus vector.

44. Our algorithm Reduces l 2 norm of surplus vector by inverse polynomial fraction in each iteration.

46. Property 1 f: max-flow in N. R: residual graph w.r.t. f. If surplus (i) < surplus(j) then there is no path from i to j in R.

47. Property 1 i j R: surplus(i) < surplus(j)

48. Property 1 i surplus(i) < surplus(j) j R:

49. Property 1 i Circulation gives a more balanced flow. j R:

50. Property 1 Theorem: A max-flow is balanced iff it satisfies Property 1.

51. Construct N’(I, J) Raise prices in J New edge enters N Stop when Invariant is threatened Algorithm for an iteration

52. Network N(p) m p buyersgoods bang-per-buck edges

53. Construct N’(I, J)  Find a balanced flow in N(p) Let d = max surplus w.r.t. balanced flow  I = buyers with surplus d  J = goods desired by I Raise prices in J New edge enters N Stop when Invariant is threatened

54. Network N(p) N’(I, J) I J N - N’

55. Network N(p) N’(I, J) I J N - N’

56. Construct N’(I, J) Raise prices in J  N’ is decoupled from N - N’ New edge enters N Stop when Invariant is threatened

57. Network N(p) N’(I, J) I J N - N’

58. Network N(p) N’(I, J) I J N - N’ By Property 1, this edge did not carry any flow. Hence Invariant is not violated by its removal.

59. Raise prices in J proportionately, so that edges in N’ don’t change. p. x, for each p in J  initialize x = 1  raise x

60. Construct N’(I, J) Raise prices in J New edge enters N Stop when Invariant is threatened

61. Network N(p) N’(I, J) I J N - N’

62. Construct N’(I, J) Raise prices in J New edge enters N  Recompute balanced flow  Buyers in N - N’ having residual paths to N’  Move to N’ Stop when Invariant is threatened

63. Network N(p) N’(I, J) I J N - N’

64. Network N(p) N’(I, J) I J N - N’

65. Network N(p) N’(I, J) I J N - N’

66. Construct N’(I, J) Raise prices in J New edge enters N  Recompute balanced flow  Buyers moved to N’ will have sufficiently large surplus Stop when Invariant is threatened

67. Construct N’(I, J) Raise prices in J New edge enters N Stop when Invariant is threatened Algorithm for an iteration

68. Tight set: p(S) = m(T) N’(I, J) T S N - N’

69. Surplus of buyers in T drops to 0

70. l 1 norm of surplus vector drops by 1/n fraction after the iteration.

71. Assume k sub-iterations. Let d 0 = d. At the end of l th sub-iteration, d l = min {surplus(i) | i is in I}. So, d k = 0.

72. Network N(p) N’(I, J) I J N - N’

73. Some i in old I will achieve minimum. Its surplus must drop by at least (d l-1 – d l ). Therefore, decrease in l 1 norm in sub-iteration l is at least (d l-1 – d l ) Therefore, decrease in iteration is at least d

74. Network N(p) N’(I, J) I J N - N’

75. Assume k sub-iterations. Let d 0 = d. At the end of l th sub-iteration, d l = min {surplus(i) | i is in I}. So, d k = 0. Decrease in l 1 norm in sub-iteration l is at least (d l-1 – d l ) Decrease in l 2 2 norm in sub-iteration l is at least (d l-1 – d l ) 2

76. Our algorithm Reduces l 2 norm of surplus vector by 1/n 2 fraction in each iteration

77. Open question Can define balanced flow without l 2 norm  Balanced flow = lexicographically smallest flow Q: Can we dispense with l 2 norm in proof?

78. Open question Can define balanced flow without l 2 norm  Balanced flow = lexicographically smallest flow Q: Can we dispense with l 2 norm in proof? V, 2008: Family of examples s.t. l 1 norm of surplus vector decreases by inverse exponential fraction in an iteration!

80. KKT conditions were relaxed e(i): money currently spent by i w.r.t. a balanced flow in N surplus money of i

81. Relaxed KKT conditions e(i)

82. Potential function Algorithm drops potential by an inverse polynomial factor in each iteration (strongly polynomial time).

83. Potential function Algorithm drops potential by an inverse polynomial factor in each iteration (strongly polynomial time).

84. Second point of departure KKT conditions are satisfied via a continuous process Normally: in discrete steps

86. utility Piecewise linear, concave amount of j Additively separable over goods

87. Long-standing open problem Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, plc utilities?

88. How do we build on solution to the linear case?

89. utility amount of j Generalize EG program to piecewise-linear, concave utilities? utility/unit of j

90. Generalization of EG program

92. V. & Yannakakis, 2007: Equilibrium is rational for Fisher and Arrow-Debreu models under separable, plc utilities. Given prices p, are they equilibrium prices? Build on combinatorial insights

93. utility amount of j Case 1 fully allocated partially allocated

94. utility amount of j Case 2: no p.a. segment fully allocated

95. p full & partial segments Network N(p) Theorem: p equilibrium prices iff max-flow in N(p) = unspent money.

96. Network N(p) partially allocated segments s t

97. LP for max-flow in N(p); variables = f e ’s Next, let p be variables! “Guess” full & partial segments – gives N(p) Write max-flow LP -- it is still linear!  variables = f e ’s & p j ’s

98. Rationality proof If “guess” is correct, at optimality, p j ’s are equilibrium prices. Hence rational!

99. Rationality proof If “guess” is correct, at optimality, p j ’s are equilibrium prices. Hence rational! In P??

100. NP-hardness does not apply Megiddo, 1988:  Equilibrium NP-hard => NP = co-NP Papadimitriou, 1991: PPAD  2-player Nash equilibrium is PPAD-complete  Rational Etessami & Yannakakis, 2007: FIXP  3-player Nash equilibrium is FIXP-complete  Irrational

101. Markets with piecewise-linear, concave utilities Chen, Dai, Du, Teng, 2009:  PPAD-hardness for Arrow-Debreu model

102. Markets with piecewise-linear, concave utilities Chen, Dai, Du, Teng, 2009:  PPAD-hardness for Arrow-Debreu model Chen & Teng, 2009:  PPAD-hardness for Fisher’s model V. & Yannakakis, 2009:  PPAD-hardness for Fisher’s model

103. Markets with piecewise-linear, concave utilities V, & Yannakakis, 2009: Membership in PPAD for both models,

104. How do we salvage the situation?? Algorithmic ratification of the “invisible hand of the market”

105. Is PPAD really hard?? What is the “right” model??

106. Open Can Fisher’s linear case be captured via an LP?