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Algorithmic Game Theory and Internet Computing

Markets and the Primal-Dual Paradigm . Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani. Markets. Stock Markets. Internet. Revolution in definition of markets. Revolution in definition of markets New markets defined by Google Amazon Yahoo! Ebay .

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Algorithmic Game Theory and Internet Computing

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  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.

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

  13. 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

  14. Leon Walras, 1874 • Pioneered general equilibrium theory

  15. 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.

  16. Kenneth Arrow • Nobel Prize, 1972

  17. Gerard Debreu • Nobel Prize, 1983

  18. 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

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

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

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

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

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

  24. Supply-demand curves

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

  26. Utility function utility amount ofmilk

  27. Utility function utility amount ofbread

  28. Utility function utility amount ofcheese

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

  30. For given prices,

  31. For given prices,find optimal bundle of goods

  32. 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

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

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

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

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

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

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

  39. 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

  40. 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,

  41. 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

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

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

  44. 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

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