1. Markets and the Primal-Dual Paradigm Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani

2. Markets

3. Stock Markets

4. Internet

5. Revolution in definition of markets

6. Revolution in definition of markets • New markets defined by • Google • Amazon • Yahoo! • Ebay

7. Revolution in definition of markets • Massive computational power available for running these markets in a centralized or distributed manner

8. Revolution in definition of markets • Massive computational power available for running these markets in a centralized or distributed manner • Important to find good models and algorithms for these markets

9. Theory of Algorithms • Powerful tools and techniques developed over last 4 decades.

10. Theory of Algorithms • Powerful tools and techniques developed over last 4 decades. • Recent study of markets has contributed handsomely to this theory as well!

11. AdWords Market • Created by search engine companies • Google • Yahoo! • MSN • Multi-billion dollar market – and still growing! • Totally revolutionized advertising, especially by small companies.

14. Historically, the study of markets • has been of central importance, especially in the West

15. Historically, the study of markets • has been of central importance, especially in the West General Equilibrium TheoryOccupied center stage in MathematicalEconomics for over a century

16. Leon Walras, 1874 • Pioneered general equilibrium theory

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

18. Kenneth Arrow • Nobel Prize, 1972

19. Gerard Debreu • Nobel Prize, 1983

20. General Equilibrium Theory • Also gave us some algorithmic results • Convex programs, whose optimal solutions capture equilibrium allocations, e.g., Eisenberg & Gale, 1959 Nenakov & Primak, 1983 • Cottle and Eaves, 1960’s: Linear complimentarity • Scarf, 1973: Algorithms for approximately computing fixed points

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

22. What is needed today? • An inherently algorithmictheory of market equilibrium • New models that capture new markets and are easier to use than traditional models

23. Beginnings of such a theory, within Algorithmic Game Theory • Started with combinatorial algorithms for traditional market models • New market models emerging

24. A central tenet • Prices are such that demand equals supply, i.e., equilibrium prices.

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

26. Supply-demand curves

27. Irving Fisher, 1891 • Defined a fundamental market model

30. Utility function utility amount ofmilk

31. Utility function utility amount ofbread

32. Utility function utility amount ofcheese

33. Total utility of a bundle of goods = Sum of utilities of individual goods

34. For given prices,

35. For given prices,find optimal bundle of goods

36. Fisher market • Several goods, fixed amount of each good • Several buyers, with individual money and utilities • Find equilibrium prices of goods, i.e., prices s.t., • Each buyer gets an optimal bundle • No deficiency or surplus of any good

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

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

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

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

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

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

45. No LP’s known for capturing equilibrium allocations for Fisher’s model • Eisenberg-Gale convex program, 1959 • DPSV:Extended primal-dual schema to solving a nonlinear convex program

46. Fisher’s Model • n buyers, money m(i) for buyer i • k goods (unit amount of each good) • : utility derived by i on obtaining one unit of j • Total utility of i,

47. Fisher’s Model • n buyers, money m(i) for buyer i • k goods (unit amount of each good) • : utility derived by i on obtaining one unit of j • Total utility of i, • Find market clearing prices

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

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

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