Combinatorial Approximation Algorithms for Convex Programs

1. Combinatorial Approximation Algorithms for Convex Programs?! Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

2. Rational convex program • Always has a rational solution, using polynomially many bits, if all parameters are rational. • Some important problems in mathematical economics and game theory are captured by rational (nonlinear) convex programs.

3. A recent development • Combinatorial exact algorithms for these problems and hence for optimally solving their convex programs.

4. General equilibrium theory

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

6. Supply-demand curves

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

10. amount ofj Concave utility function (for good j) utility

11. total utility

12. For given prices,find optimal bundle of goods

13. Several buyers with different utility functions and moneys.

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

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

16. Combinatorial Algorithm for Linear Case of Fisher’s Model • Devanur, Papadimitriou, Saberi & V., 2002 Using primal-dual paradigm Solves Eisenberg-Gale convex program

17. Eisenberg-Gale Program, 1959

18. Eisenberg-Gale Program, 1959 prices pj

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

20. Why seek combinatorial algorithms?

21. Why seek combinatorial algorithms? • Structural insights • Have led to progress on related problems • Better understanding of solution concept • Useful in applications

22. Auction for Google’s TV ads N. Nisan et. al, 2009: • Used market equilibrium based approach. • Combinatorial algorithms for linear case provided “inspiration”.

24. Piecewise linear, concave utility Additively separable over goods amount ofj

25. Long-standing open problem • Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, plc utilities?

26. How do we build on solution to linear case?

27. amount ofj Generalize EG program to piecewise-linear, concave utilities? utility/unit of j utility

28. Generalization of EG program

29. Generalization of EG program

30. Long-standing open problem • Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, plc utilities? • 2009: Both PPAD-complete (using combinatorial insights from [DPSV]) • Chen, Dai, Du, Teng • Chen, Teng • V., Yannakakis

31. Piecewise linear, concave utility Additively separable over goods amount ofj

32. What makes linear utilities easy? • Weak gross substitutability: Increasing price of one good cannot decrease demand of another. • Piecewise-linear, concave utilities do not satisfy this.

33. rate = utility/unit amount of j rate amount ofj Differentiate

34. rate = utility/unit amount of j rate amount ofj money spent on j

35. Spending constraint utility function rate = utility/unit amount of j rate $20 $40 $60 money spent onj

36. Theorem (V., 2002): Spending constraint utilities: 1). Satisfy weak gross substitutability 2). Polynomial time algorithm for computing equilibrium.

37. An unexpected fallout!! • Has applications to Google’s AdWords Market!

38. $20 $40 $60 Application to Adwords market rate = utility/click rate money spent on keywordj

39. Is there a convex program for this model? • “We believe the answer to this question should be ‘yes’. In our experience, non-trivial polynomial time algorithms for problems are rare and happen for a good reason – a deep mathematical structure intimately connected to the problem.”

40. Devanur’s program for linear Fisher

41. C. P. for spending constraint!

42. Spending constraint market Fisher market with plc utilities EG convex program = Devanur’s program

44. Price discrimination markets • Business charges different prices from different customers for essentially same goods or services. • Goel & V., 2009: Perfect price discrimination market. Business charges each consumer what they are willing and able to pay.

45. plc utilities

46. Middleman buys all goods and sells to buyers, charging according to utility accrued. • Given p, there is a well defined rate for each buyer.

47. Middleman buys all goods and sells to buyers, charging according to utility accrued. • Given p, there is a well defined rate for each buyer. • Equilibrium is captured by a convex program • Efficient algorithm for equilibrium

48. Middleman buys all goods and sells to buyers, charging according to utility accrued. • Given p, there is a well defined rate for each buyer. • Equilibrium is captured by a convex program • Efficient algorithm for equilibrium • Market satisfies both welfare theorems!

49. Generalization of EG program works!

50. Spending constraint market Price discrimination market (plc utilities) EG convex program = Devanur’s program