Algorithmic Game Theory and Internet Computing

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

2. The new face of computing

3. A paradigm shift inthe notion of a “market”!

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

8. 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

9. 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

10. 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 • Scarf, 1973: Algorithms for approximately computing fixed points

11. Today’s reality • New markets defined by Internet companies, e.g., • Google • Yahoo! • Amazon • eBay • Massive computing power available for running markets in a distributed or centralized manner • A deep theory of algorithms with many powerful techniques

12. What is needed today? • An inherently-algorithmic theory of markets and market equilibria

13. What is needed today? • An inherently-algorithmic theory of markets and market equilibria • Beginnings of such a theory, within Algorithmic Game Theory

14. What is needed today? • An inherently-algorithmic theory of markets and market equilibria • Beginnings of such a theory, within Algorithmic Game Theory • Natural starting point: algorithms for traditional market models

15. What is needed today? • An inherently-algorithmic theory of markets and market equilibria • Beginnings of such a theory, within Algorithmic Game Theory • Natural starting point: algorithms for traditional market models • New market models emerging!

16. Theory of algorithms • Interestingly enough, recent study of markets has contributed handsomely to this theory!

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

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

19. Supply-demand curves

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

23. Utility function utility amount ofmilk

24. Utility function utility amount ofbread

25. Utility function utility amount ofcheese

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

27. For given prices,

28. For given prices,find optimal bundle of goods

29. 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

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

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

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

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

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

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

38. 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

39. 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,

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, • Find market clearing prices

41. 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

42. A convex program • whose optimal solution is equilibrium allocations.

43. A convex program • whose optimal solution is equilibrium allocations. • Constraints: packing constraints on the xij’s

44. A convex program • whose optimal solution is equilibrium allocations. • Constraints: packing constraints on the xij’s • Objective fn: max utilities derived.

45. A convex program • whose optimal solution is equilibrium allocations. • Constraints: packing constraints on the xij’s • Objective fn: max utilities derived. Must satisfy • If utilities of a buyer are scaled by a constant, optimal allocations remain unchanged • If money of buyer b is split among two new buyers, whose utility fns same as b, then union of optimal allocations to new buyers = optimal allocation for b

46. Money-weighed geometric mean of utilities

47. Eisenberg-Gale Program, 1959

48. KKT conditions

49. Therefore, buyer i buys from only, i.e., gets an optimal bundle

50. Therefore, buyer i buys from only, i.e., gets an optimal bundle • Can prove that equilibrium prices are unique!