Algorithmic Game Theory and Internet Computing

1. The “Invisible Hand of the Market”: Algorithmic Ratification and the Digital Economy Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

2. Adam Smith • The Wealth of Nations, 1776. “It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard for their own interest.” Each participant in a competitive economy is “led by an invisible hand to promote an end which was no part of his intention.”

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

4. How are scarce resources assigned to alternative uses?

6. How are scarce resources assigned to alternative uses? Prices!

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

9. How are scarce resources assigned to alternative uses? Prices Parity between demand and supplyequilibrium prices

10. Leon Walras, 1874 • Pioneered general equilibrium theory

11. General Equilibrium TheoryOccupied center stage in MathematicalEconomics for over a century Mathematical ratification!

12. Central tenet • Markets should operate at equilibrium

13. Central tenet • Markets should operate at equilibrium i.e., prices s.t. Parity between supply and demand

14. Do markets even admitequilibrium prices?

15. Easy if only one good! Do markets even admitequilibrium prices?

16. Supply-demand curves

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

18. Irving Fisher, 1891 • Defined a fundamental market model • Special case of Walras’ model

21. amount ofj Concave utility function (Of buyer i for good j) utility

22. total utility

23. For given prices,find optimal bundle of goods

24. Several buyers with different utility functions and moneys.

25. Several buyers with different utility functions and moneys.Equilibrium prices

26. Several buyers with different utility functions and moneys.Show equilibrium prices exist.

27. 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.

28. First Welfare Theorem • Competitive equilibrium => Pareto optimal allocation of resources • Pareto optimal = impossible to make an agent better off without making some other agent worse off

29. Second Welfare Theorem • Every Pareto optimal allocation of resources comes from a competitive equilibrium (after redistribution of initial endowments).

30. Kenneth Arrow • Nobel Prize, 1972

31. Gerard Debreu • Nobel Prize, 1983

32. 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!

33. Leon Walras • Tatonnement process: Price adjustment process to arrive at equilibrium • Deficient goods: raise prices • Excess goods: lower prices

34. Leon Walras • Tatonnement process: Price adjustment process to arrive at equilibrium • Deficient goods: raise prices • Excess goods: lower prices • Does it converge to equilibrium?

35. GETTING TO ECONOMIC EQUILIBRIUM: A PROBLEM AND ITS HISTORY For the third International Workshop on Internet and Network Economics Kenneth J. Arrow

36. OUTLINE • BEFORE THE FORMULATION OF GENERAL EQUILIBRIUM THEORY • PARTIAL EQUILIBRIUM • WALRAS, PARETO, AND HICKS • SOCIALISM AND DECENTRALIZATION • SAMUELSON AND SUCCESSORS • THE END OF THE PROGRAM

37. Part VI: THE END OF THE PROGRAM • Scarf’s example • Saari-Simon Theorem: For any dynamic system depending on first-order information (z) only, there is a set of excess demand functions for which stability fails. (In fact, theorem is stronger). • Uzawa: Existence of general equilibrium is equivalent to fixed-point theorem • Assumptions on individual demand functions do not constrain aggregate demand function (Sonnenschein, Debreu, Mantel)

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

39. The new face of computing

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

41. Standard sufficient conditions on utility functions (in Arrow-Debreu Theorem): • Continuous, quasiconcave, satisfying non-satiation.

42. Complexity-theoretic question • For “reasonable” utility fns., can market equilibrium be computed in P? • If not, what is its complexity?

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

45. “Stock prices have reached what looks likea permanently high plateau”

46. “Stock prices have reached what looks likea permanently high plateau” Irving Fisher, October 1929

49. Herbert Scarf, 1973 Approximate fixed point algorithms for market equilibria. Not polynomial time.

50. Curtis Eaves, 1984 Polynomial time algorithm for Cobb-Douglas utilities.