1. Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Georgia Tech Combinatorial Approximation Algorithms for Convex Programs?!

2. Rational convex program Always has a rational solution, using polynomially many bits, if all parameters are rational. Some important problems in mathematical economics and game theory are captured by rational (nonlinear) convex programs.

3. A recent development Combinatorial exact algorithms for these problems and hence for optimally solving their convex programs.

4. General equilibrium theory

5. A central tenet Prices are such that demand equals supply, i.e., equilibrium prices. Easy if only one good

6. Supply-demand curves

7. Irving Fisher, 1891 Defined a fundamental market model

10. utility Concave utility function (for good j) amount of j

11. total utility

12. For given prices, find optimal bundle of goods

13. Several buyers with different utility functions and moneys.

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

15. Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 Using primal-dual paradigm

16. Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 Using primal-dual paradigm Solves Eisenberg-Gale convex program

17. Eisenberg-Gale Program, 1959

18. prices p j

19. Why remarkable? Equilibrium simultaneously optimizes for all agents. How is this done via a single objective function?

20. Why seek combinatorial algorithms?

21. Structural insights  Have led to progress on related problems  Better understanding of solution concept Useful in applications

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

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

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

26. How do we build on solution to linear case?

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

28. Generalization of EG program

30. Long-standing open problem Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, plc utilities? 2009: Both PPAD-complete (using combinatorial insights from [DPSV])  Chen, Dai, Du, Teng  Chen, Teng  V., Yannakakis

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

32. What makes linear utilities easy? Weak gross substitutability: Increasing price of one good cannot decrease demand of another. Piecewise-linear, concave utilities do not satisfy this.

33. rate rate = utility/unit amount of j amount of j Differentiate

34. rate amount of j rate = utility/unit amount of j money spent on j

35. rate rate = utility/unit amount of j money spent on j Spending constraint utility function $20$40 $60

36. Theorem (V., 2002): Spending constraint utilities: 1). Satisfy weak gross substitutability 2). Polynomial time algorithm for computing equilibrium.

37. An unexpected fallout!! Has applications to Google’s AdWords Market!

38. rate rate = utility/click money spent on keyword j Application to Adwords market $20$40 $60

39. Is there a convex program for this model? “We believe the answer to this question should be ‘yes’. In our experience, non-trivial polynomial time algorithms for problems are rare and happen for a good reason – a deep mathematical structure intimately connected to the problem.”

40. Devanur’s program for linear Fisher

41. C. P. for spending constraint!

42. EG convex program = Devanur’s program Fisher market with plc utilities Spending constraint market

44. Price discrimination markets Business charges different prices from different customers for essentially same goods or services. Goel & V., 2009: Perfect price discrimination market. Business charges each consumer what they are willing and able to pay.

45. plc utilities

46. Middleman buys all goods and sells to buyers, charging according to utility accrued.  Given p, there is a well defined rate for each buyer.

47. Middleman buys all goods and sells to buyers, charging according to utility accrued.  Given p, there is a well defined rate for each buyer. Equilibrium is captured by a convex program  Efficient algorithm for equilibrium

48. Middleman buys all goods and sells to buyers, charging according to utility accrued.  Given p, there is a well defined rate for each buyer. Equilibrium is captured by a convex program  Efficient algorithm for equilibrium Market satisfies both welfare theorems!

49. Generalization of EG program works!

50. EG convex program = Devanur’s program Price discrimination market (plc utilities) Spending constraint market

52. Nash bargaining game, 1950 Captures the main idea that both players gain if they agree on a solution. Else, they go back to status quo.

53. Example Two players, 1 and 2, have vacation homes:  1: in the mountains  2: on the beach Consider all possible ways of sharing.

54. Utilities derived jointly : convex + compact feasible set

55. Disagreement point = status quo utilities Disagreement point =

56. Nash bargaining problem = (S, c) Disagreement point =

57. Nash bargaining Q: Which solution is the “right” one?

58. Solution must satisfy 4 axioms: Pareto optimality Invariance under affine transforms Symmetry Independence of irrelevant alternatives

59. Thm: Unique solution satisfying 4 axioms

60. Generalizes to n-players Theorem: Unique solution

61. Nash bargaining solution is optimal solution to convex program:

62. Nash bargaining solution is optimal solution to convex program: Polynomial time separation oracle

63. Q: Compute solution combinatorially in polynomial time?

65. How should they exchange their goods?

66. State as a Nash bargaining game S = utility vectors obtained by distributing goods among players

67. Special case: linear utility functions S = utility vectors obtained by distributing goods among players

68. ADNB B: n players with disagreement points, c i G: g goods, unit amount each S = utility vectors obtained by distributing goods among players

69. Convex program for ADNB

70. Theorem (V., 2008) Nash bargaining program is rational.

71. Theorem (V., 2008) Nash bargaining solution is rational. Combinatorial polynomial time algorithm for finding it.

72. Game-theoretic properties of NB games -- “stress tests” Chakrabarty, Goel, V., Wang & Yu, 2008:  Efficiency (Price of bargaining)  Fairness  Full competitiveness

73. An application (Lucent) “fair” throughput problem on a wireless channel.

74. EG convex program = Devanur’s program Price disc. market Spending constraint market ADNB

75. EG convex program = Devanur’s program Price disc. market Spending constraint market Kelly, 1997: proportional fairness Jain & V., 2007: Eisenberg-Gale markets ADNB

76. A new development Orlin, 2009: Strongly polynomial algorithm for Fisher’s linear case. Open: For rest.

77. AGT’s gift to theory of algorithms! New complexity classes: PPAD, FIXP  Study complexity of total problems A new algorithmic direction  Combinatorial algorithms for convex programs

78. Nonlinear programs with rational solutions! Open

79. Nonlinear programs with rational solutions! Solvable combinatorially!! Open

80. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s

81. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s Approximation Algorithms (1990’s): Near-optimal integral solutions to LP’s

82. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s Approximation Algorithms (1990’s): Near-optimal integral solutions to LP’s Algorithmic Game Theory (New Millennium): Rational solutions to nonlinear convex programs

83. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s Approximation Algorithms (1990’s): Near-optimal integral solutions to LP’s Algorithmic Game Theory (New Millennium): Rational solutions to nonlinear convex programs Approximation algorithms for convex programs?!

85. Extending primal-dual paradigm to framework of convex programs and KKT conditions

86. Eisenberg-Gale Program, 1959

87. Main point of departure Complementary slackness conditions: involve primal or dual variables, not both. KKT conditions: involve primal and dual variables simultaneously.

88. KKT conditions

89. Primal-dual algorithms so far Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.)

90. 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 weight matching.

91. 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 weight matching. Otherwise primal objects go tight and loose. Difficult to account for these reversals in the running time.

92. Our algorithms Dual variables (prices) are raised greedily

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

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