Algorithmic Ratification of the "Invisible Hand of the Market

1. New Market Models and Algorithms Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

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

3. What is the “right” model??

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

5. 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.

6. Piecewise linear, concave utility amount ofj

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

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

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

10. Theorem (V., 2002): Spending constraint utilities: 1). Satisfy weak gross substitutability 2). Polynomial time algorithm for computing equilibrium 3). Equilibrium is rational.

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

12. $20 $40 $60 Application to Adwords market rate = utility/click rate money spent on keywordj

13. 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.”

14. Devanur’s program for linear Fisher

15. C. P. for spending constraint!

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

18. 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.

19. plc utilities

20. Middleman buys all goods and sells to buyers, charging according to utility accrued. • Given p, each buyer picks rate for accruing utility.

21. Middleman buys all goods and sells to buyers, charging according to utility accrued. • Given p, each buyer picks rate for accruing utility. • Equilibrium is captured by a rational convex program!

22. Generalization of EG program works!

23. V., 2010:Generalize to • Continuously differentiable, quasiconcave (non-separable) utilities, satisfying non-satiation.

24. V., 2010:Generalize to • Continuously differentiable, quasiconcave (non-separable) utilities, satisfying non-satiation. • Compare with Arrow-Debreu utilities!! continuous, quasiconcave, satisfying non-satiation.

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

26. Eisenberg-Gale Markets Jain & V., 2007 (Proportional Fairness) (Kelly, 1997) Price disc. market Spending constraint market Nash Bargaining V., 2008 EG convex program = Devanur’s program

27. A combinatorial market

28. A combinatorial market

29. A combinatorial market

30. A combinatorial market • Given: • Network G = (V,E) (directed or undirected) • Capacities on edges c(e) • Agents: source-sink pairs with money m(1), … m(k) • Find: equilibrium flows and edge prices

31. Equilibrium • Flows and edge prices • f(i): flow of agent i • p(e): price/unit flow of edge e • Satisfying: • p(e)>0 only if e is saturated • flows go on cheapest paths • money of each agent is fully spent

32. Kelly’s resource allocation model, 1997 Mathematical framework for understanding TCP congestion control

33. Van Jacobson, 1988: AIMD protocol (Additive Increase Multiplicative Decrease)

34. Van Jacobson, 1988: AIMD protocol (Additive Increase Multiplicative Decrease) • Why does it work so well?

35. Van Jacobson, 1988: AIMD protocol (Additive Increase Multiplicative Decrease) • Why does it work so well? • Kelly, 1977: Highly successful theory

36. TCP Congestion Control • f(i): source rate • prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) p(e):

37. TCP Congestion Control • f(i): source rate • prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) • Low & Lapsley, 1999: AIMD + RED converges to equilibrium in limit p(e):

38. TCP Congestion Control • f(i): source rate • prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) Kelly: Equilibrium flows are proportionally fair: only way of adding 5% flow to someone is to decrease total of 5% flow from rest. p(e):

39. Kelly & V., 2002: Kelly’s model is a generalization of Fisher’s model.

40. Kelly & V., 2002: Kelly’s model is a generalization of Fisher’s model. • Find combinatorial poly time algorithms!

41. Kelly & V., 2002: Kelly’s model is a generalization of Fisher’s model. • Find combinatorial poly time algorithms! (May lead to new insights for TCP congestion control protocol)

42. Jain & V., 2005: • Strongly polynomial combinatorial algorithm for single-source multiple-sink market

43. Single-source multiple-sink market • Given: • Network G = (V,E), s: source • Capacities on edges c(e) • Agents: sinks with money • Find: equilibrium flows and edge prices

44. Equilibrium • Flows and edge prices • f(i): flow of agent i • p(e): price/unit flow of edge e • Satisfying: • p(e)>0 only if e is saturated • flows go on cheapest paths • money of each agent is fully spent

46. $5 $5

48. $30 $10 $40

49. Jain & V., 2005: • Strongly polynomial combinatorial algorithm for single-source multiple-sink market • Ascending price auction • Buyers: sinks (fixed budgets, maximize flow) • Sellers: edges (maximize price)

50. Auction of k identical goods • p = 0; • while there are >k buyers: raise p; • end; • sell to remaining k buyers at price p;