Algorithmic Game Theory and Internet Computing

1. Market Equilibrium: The Quest for the “Right” Model Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

2. Arrow-Debreu Theorem, 1954 • Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem. • Provides a mathematical ratification of Adam Smith’s “invisible hand of the market”.

3. Need algorithmic ratification!!

4. The new face of computing

5. 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.

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

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

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

9. Several buyers with different utility functions and moneys.

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

11. Linear Fisher Market • DPSV, 2002: First polynomial time algorithm • Assume: • Buyer i’s total utility, • mi: money of buyer i. • One unit of each good j.

12. Eisenberg-Gale Program, 1959

13. Eisenberg-Gale Program, 1959 prices pj

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

15. Rational convex program • Always has a rational solution, using polynomially many bits, if all parameters are rational. • Eisenberg-Gale program is rational.

16. Combinatorial Algorithm for Linear Case of Fisher’s Model • Devanur, Papadimitriou, Saberi & V., 2002 By extending the primal-dual paradigm to the setting of convex programs & KKT conditions

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

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

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

20. How do we build on solution to the linear case?

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

22. Generalization of EG program

23. Generalization of EG program

24. Build on combinatorial insights • V. & Yannakakis, 2007: Equilibrium is rational for Fisher and Arrow-Debreu models under separable, plc utilities. • Given prices p, are they equilibrium prices?

25. Case 1 partially allocated utility fully allocated amount ofj

26. Case 2: no p.a. segment utility fully allocated amount ofj

27. p full & partial segments Network N(p) • Theorem:p equilibrium prices iff max-flow in N(p) = unspent money.

28. Network N(p) p’(1) m’(1) p’(2) m’(2) t s p’(3) m’(3) m’(4) p’(4) partially allocated segments

29. LP for max-flow in N(p); variables = fe’s • Next, let p be variables! • “Guess” full & partial segments – gives N(p) • Write max-flow LP -- it is still linear! • variables = fe’s& pj’s

30. Rationality proof • If “guess” is correct, atoptimality, pj’s are equilibrium prices. • Hence rational!

31. Rationality proof • If “guess” is correct, atoptimality, pj’s are equilibrium prices. • Hence rational! • In P??

32. Markets with piecewise-linear, concave utilities • Chen, Dai, Du, Teng, 2009: • PPAD-hardness for Arrow-Debreu model

33. Markets with piecewise-linear, concave utilities • Chen, Dai, Du, Teng, 2009: • PPAD-hardness for Arrow-Debreu model • Chen & Teng, 2009: • PPAD-hardness for Fisher’s model • V. & Yannakakis, 2009: • PPAD-hardness for Fisher’s model

34. Markets with piecewise-linear, concave utilities V, & Yannakakis, 2009: • Membership in PPAD for both models, Sufficient condition: non-satiation.

35. Markets with piecewise-linear, concave utilities V, & Yannakakis, 2009: • Membership in PPAD for both models, Sufficient condition: non-satiation • Otherwise, for both models, deciding existence of equilibrium is NP-hard!

36. Markets with piecewise-linear, concave utilities – same for approx. eq. • Chen, Dai, Du, Teng, 2009: • PPAD-hardness for Arrow-Debreu model • Chen, Teng, 2009: • PPAD-hardness for Fisher’s model • V, & Yannakakis, 2009: • PPAD-hardness for Fisher’s model • Membership in PPAD for both models, Sufficient condition: non-satiation • Otherwise, deciding existence of eq. is NP-hard

37. Markets with piecewise-linear, concave utilities V, & Yannakakis, 2009: • Membership in PPAD for both models. • Path-following algorithm??

38. NP-hardness does not apply • Megiddo, 1988: • Equilibrium NP-hard => NP = co-NP • Papadimitriou, 1991: PPAD • 2-player Nash equilibrium is PPAD-complete • Rational • Etessami & Yannakakis, 2007: FIXP • 3-player Nash equilibrium is FIXP-complete • Irrational

39. FIXP = Problems whose solutions are fixed points of Brouwer functions. (Given as polynomially computable algebraic circuits over the basis {+,-,*,/,max,min}.) • Etessami & Yannakakis, 2007: PPAD = Restriction of FIXP to piecewise-linear Brouwer functions.

40. M: market instance • Prove: PPAD alg. for finding equilibrium for M • Geanakoplos, 2003: Brouwer function-based proof of existence of equilibrium • Piecewise-linear Brouwer function??

41. Instead … • F: correspondence in Kakutani-basedproof. • G: piecewise-linear Brouwer approximation of F • (p*, x*): fixed point of G. Can be found in PPAD.

42. Instead … • F: correspondence in Kakutani-basedproof. • G: piecewise-linear Brouwer approximation of F • (p*, x*): fixed point of G. Can be found in PPAD Yields “guess” • Solve rationality LP to find equilibrium for M!

43. How do we salvage the situation?? Algorithmic ratification of the “invisible hand of the market”

44. Is PPAD really hard?? What is the “right” model??

45. Linear Fisher Market • DPSV, 2002: First polynomial time algorithm • Extend to separable, plc utilities??

46. 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.

47. Piecewise linear, concave utility amount ofj

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

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

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