1. Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Georgia Tech Combinatorial Algorithms for Convex Programs (Capturing Market Equilibria and Nash Bargaining Solutions)

2. What is Economics? ‘‘Economics is the study of the use of scarce resources which have alternative uses.’’ Lionel Robbins (1898 – 1984)

3. How are scarce resources assigned to alternative uses?

5. Prices!

7. How are scarce resources assigned to alternative uses? Prices Parity between demand and supply

8. How are scarce resources assigned to alternative uses? Prices Parity between demand and supply equilibrium prices

9. Do markets even admit equilibrium prices?

10. General Equilibrium Theory Occupied center stage in Mathematical Economics for over a century Do markets even admit equilibrium prices?

11. Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem.

12. Do markets even admit equilibrium prices?

13. Easy if only one good!

14. Supply-demand curves

15. Do markets even admit equilibrium prices? What if there are multiple goods and multiple buyers with diverse desires and different buying power?

16. Irving Fisher, 1891 Defined a fundamental market model

19. linear utilities

20. For given prices, find optimal bundle of goods

21. Several buyers with different utility functions and moneys.

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

24. Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem. Highly non-constructive!

25. An almost entirely non-algorithmic theory! General Equilibrium Theory

26. The new face of computing

27. New markets defined by Internet companies, e.g.,  Google  eBay  Yahoo!  Amazon Massive computing power available. Need an inherenltly-algorithmic theory of markets and market equilibria. Today’s reality

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

29. Combinatorial algorithm Conducts an efficient search over a discrete space. E.g., for LP: simplex algorithm vs ellipsoid algorithm or interior point algorithms.

30. Combinatorial algorithm Conducts an efficient search over a discrete space. E.g., for LP: simplex algorithm vs ellipsoid algorithm or interior point algorithms. Yields deep insights into structure.

31. No LP’s known for capturing equilibrium allocations for Fisher’s model

32. No LP’s known for capturing equilibrium allocations for Fisher’s model Eisenberg-Gale convex program, 1959

33. Eisenberg-Gale Program, 1959

34. prices p j

35. No LP’s known for capturing equilibrium allocations for Fisher’s model Eisenberg-Gale convex program, 1959 Extended primal-dual paradigm to solving a nonlinear convex program

36. Theorem If all parameters are rational, Eisenberg-Gale convex program has a rational solution!  Polynomially many bits in size of instance

37. Theorem If all parameters are rational, Eisenberg-Gale convex program has a rational solution!  Polynomially many bits in size of instance Combinatorial polynomial time algorithm for finding it.

38. Theorem If all parameters are rational, Eisenberg-Gale convex program has a rational solution!  Polynomially many bits in size of instance Combinatorial polynomial time algorithm for finding it. Discrete space

39. Idea of algorithm primal variables: allocations dual variables: prices of goods iterations: execute primal & dual improvements Allocations Prices (Money)

40. How are scarce resources assigned to alternative uses? Prices Parity between demand and supply

43. Yin & Yang

45. 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. Complete information game.

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

47. Utilities derived jointly : convex + compact feasible set

48. Disagreement point = status quo utilities Disagreement point =

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

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

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

52. Thm: Unique solution satisfying 4 axioms

53. Generalizes to n-players Theorem: Unique solution

54. Linear Nash Bargaining (LNB) Feasible set is a polytope defined by linear packing constraints Nash bargaining solution is optimal solution to convex program:

55. Q: Compute solution combinatorially in polynomial time?

57. How should they exchange their goods?

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

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

60. Convex program for ADNB

61. Theorem (V., 2008) If all parameters are rational, solution to ADNB is rational!  Polynomially many bits in size of instance

62. Theorem (V., 2008) If all parameters are rational, solution to ADNB is rational!  Polynomially many bits in size of instance Combinatorial polynomial time algorithm for finding it.

63. Flexible budget markets Natural variant of linear Fisher markets ADNB flexible budget markets Primal-dual algorithm for finding an equilibrium

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

65. 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.

66. KKT conditions

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

69. 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.

70. 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.

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

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

73. Nonlinear programs with rational solutions! Open

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

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

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

77. 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

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

79. 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

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 Algorithmic Game Theory (New Millennium): Rational solutions to nonlinear convex programs Approximation algorithms for convex programs?!

83. Convex program for ADNB

84. Eisenberg-Gale Program, 1959

85. Common generalization

86. Is it meaningful? Can it be solved via a combinatorial, polynomial time algorithm?

87. Common generalization Is it meaningful? Nonsymmetric ADNB Kalai, 1975: Nonsymmetric bargaining games  w i : clout of player i.

88. Common generalization Is it meaningful? Nonsymmetric ADNB Kalai, 1975: Nonsymmetric bargaining games  w i : clout of player i. Algorithm