Algorithmic Game Theory and Internet Computing

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

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?

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

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

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

7. Do markets even admitequilibrium prices?

8. General Equilibrium TheoryOccupied center stage in MathematicalEconomics for over a century Do markets even admitequilibrium prices?

9. 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.

10. Do markets even admitequilibrium prices?

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

12. Supply-demand curves

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

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

17. linear utilities

18. For given prices,find optimal bundle of goods

19. Several buyers with different utility functions and moneys.

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

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

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

26. 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!

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

28. The new face of computing

29. Today’s reality • 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.

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

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

32. 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.

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

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

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. Linear Fisher Market • B = n buyers, money mifor buyer i • G = g goods, w.l.o.g. unit amount of each good • : utility derived by i on obtaining one unit of j • Total utility of i, • Find market clearing prices.

37. Eisenberg-Gale Program, 1959

38. Eisenberg-Gale Program, 1959 prices pj

39. Why remarkable? • Equilibrium simultaneously optimizes for all agents. • How is this done via a single objective function?

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

41. 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.

42. 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

43. AllocationsPrices (Money) Idea of algorithm • primal variables: allocations • dual variables: prices of goods • iterations: execute primal & dual improvements

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

47. Yin & Yang

49. 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.

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