Algorithmic Game Theory and Internet Computing

1. Algorithms for the Linear Case, and Beyond … Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

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

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

4. Linear Fisher Market • Assume: • Buyer i’s total utility, • mi: money of buyer i. • One unit of each good j. • Findmarket clearing prices!

5. Eisenberg-Gale Program, 1959

6. Eisenberg-Gale Program, 1959 prices pj

7. Convex programs thatcapture market equilibria • Underly all “efficient” markets (so far). • Rational convex programs for many markets! • Algorithms: • combinatorial, for rational (primal-dual) • continuous (ellipsoid/interior point)

8. Combinatorial algorithms for rational convex programs Natural extension of field of combinatorial optimization!

9. Combinatorial algorithms for rational convex programs Natural extension of field of combinatorial optimization! Central aspect of C.O. – efficient algorithms for solving integral LP’s, e.g. matching, flow.

10. 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

11. Combinatorial algorithms Yield deep structural insights. Preferable for applications.

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

13. Primal-Dual Paradigm • Highly successful algorithm design technique from exact and approximation algorithms

14. Exact Algorithms for Cornerstone Problems in P: • Matching (general graph) • Network flow • Shortest paths • Minimum spanning tree • Minimum branching

15. Approximation Algorithms set cover facility location Steiner tree k-median Steiner network multicut k-MST feedback vertex set scheduling . . .

18. Yin & Yang

19. An easier question • Given prices p, are they equilibrium prices? • If so, find equilibrium allocations.

20. An easier question • Given prices p, are they equilibrium prices? • If so, find equilibrium allocations. • Equilibrium prices are unique!

21. Bang-per-buck • At prices p, buyer i’s most desirable goods, Si = • Any goods from Siworth m(i) constitute i’s optimal bundle

22. For each buyer, most desirable goods, i.e. m(1) p(1) m(2) p(2) m(3) p(3) m(4) p(4)

23. Network N(p) p(1) m(1) p(2) m(2) t s p(3) m(3) m(4) p(4) infinite capacities

24. Max flow in N(p) p(1) m(1) p(2) m(2) p(3) m(3) m(4) p(4) p: equilibrium prices iff both cuts saturated

25. Idea of algorithm • “primal” variables: allocations • “dual” variables: prices of goods • Approach equilibrium prices from below: • start with very low prices; buyers have surplus money • iteratively keep raising prices and decreasing surplus

26. An important consideration • The price of a good never exceeds its equilibrium price • Invariant: s is a min-cut

27. Invariant: s is a min-cut in N(p) p(1) m(1) p(2) m(2) s p(3) m(3) m(4) p(4) p: low prices

28. AllocationsPrices Idea of algorithm • Iterations: execute primal & dual improvements

29. How is primal-dual paradigm adapted to nonlinear setting?

30. Fundamental difference betweenLP’s and convex programs • Complementary slackness conditions: involve primal or dual variables, not both. • KKT conditions: involve primal and dual variables simultaneously.

31. KKT conditions

32. KKT conditions

33. Primal-dual algorithms so far(i.e., LP-based) • Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.)

34. Primal-dual algorithms so far • Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.) • Only exception: Edmonds, 1965: algorithm for max weight matching.

35. Primal-dual algorithms so far • Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.) • Only exception: Edmonds, 1965: algorithm for max weight matching. • Otherwise primal objects go tight and loose. Difficult to account for these reversals -- in the running time.

36. Our algorithm • Dual variables (prices) are raised greedily • Yet, primal objects go tight and loose • Because of enhanced KKT conditions

37. Our algorithm • Dual variables (prices) are raised greedily • Yet, primal objects go tight and loose • Because of enhanced KKT conditions • New algorithmic ideas needed!

38. Key Algorithmic Idea • Dual variables (prices) are raised greedily • Yet, primal objects go tight and loose • Because of enhanced KKT conditions • Balanced Flows: For limiting no. of such events

39. Max-flow in N p m s t i W.r.t. a max-flow f, surplus(i) = m(i) – f(i,t)

40. Max-flow in N p m i surplus vector = vector of surpluses w.r.t. f

41. Obvious potential function • Total surplus money = l1 norm of surplus vector • Reduce l1 norm of surplus vector by inverse polynomial fraction in each iteration

42. Balanced flow • A max-flow that minimizes l2 norm of surplus vector. • Makes surpluses as equal as possible.

43. Balanced flow • A max-flow that minimizes l2 norm of surplus vector. • Makes surpluses as equal as possible. • All balanced flows have same surplus vector.

44. Our algorithm • Reduces l2 norm of surplus vector by inverse polynomial fraction in each iteration.

46. Property 1 • f: max-flow in N. • R: residual graph w.r.t. f. • If surplus (i) < surplus(j) then there is no pathfrom i to j in R.

47. Property 1 R: i j surplus(i) < surplus(j)

48. Property 1 R: i j surplus(i) < surplus(j)

49. Property 1 R: i j Circulation gives a more balanced flow.

50. Property 1 • Theorem: A max-flow is balanced iff it satisfies Property 1.