Algorithmic Game Theory and Internet Computing

1. Nash Bargaining via Flexible Budget Markets Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani

4. The new platform for computing

6. Internet • Massive computational power available • Sellers (programs) can negotiate with individual buyers!

7. Internet • Massive computational power available • Sellers (programs) can negotiate with individual buyers! Back to bargaining!

8. Internet • Massive computational power available • Sellers (programs) can negotiate with individual buyers! Algorithmic Game Theory

9. Bargaining and Game Theory • Nash (1950): First formalization of bargaining. • von Neumann & Morgenstern (1947): Theory of Games and Economic Behavior • Game Theory: Studies solution concepts for negotiating in situations of conflict of interest.

10. Bargaining and Game Theory • Nash (1950): First formalization of bargaining. • von Neumann & Morgenstern (1947): Theory of Games and Economic Behavior • Game Theory: Studies solution concepts for negotiating in situations of conflict of interest. • Theory of Bargaining: Central!

11. Nash bargaining • Captures the main idea that both players gain if they agree on a solution. Else, they go back to status quo.

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

13. Utilities derived jointly : convex + compact feasible set

14. Disagreement point = status quo utilities Disagreement point =

15. Nash bargaining problem = (S, c) Disagreement point =

16. Nash bargaining Q: Which solution is the “right” one?

17. Solution must satisfy 4 axioms: • Paretto optimality • Invariance under affine transforms • Symmetry • Independence of irrelevant alternatives

20. Thm: Unique solution satisfying 4 axioms

21. Generalizes to n-players • Theorem: Unique solution

22. Generalizes to n-players • Theorem: Unique solution (S, c) is feasible ifS contains a point that makes each i strictly happier than ci

23. Bargaining theory studies promise problem • Restrict to instances (S, c) which are feasible.

24. Linear Nash Bargaining (LNB) • Feasible set is a polytope defined by linear packing constraints • Nash bargaining solution is optimal solution to convex program:

25. Q: Compute solution combinatoriallyin polynomial time?

26. Study promise problem? • Decision problem reduces to promise problem • Therefore, study decision and search problems.

27. Linear utilities • B: n players with disagreement points, ci • G: g goods, unit amount each • S = utility vectors obtained by distributing goods among players

28. e.g., ci = i’s utility for initial endowment • B: n players with disagreement points, ci • G: g goods, unit amount each • S = utility vectors obtained by distributing goods among players

29. Convex program giving NB solution

30. Theorem • If instance is feasible, Nash bargaining solution is rational! • Polynomially many bits in size of instance

31. Theorem • If instance is feasible, Nash bargaining solution is rational! • Polynomially many bits in size of instance • Decision and search problems can be solved in polynomial time.

32. Resource Allocation Nash Bargaining Problems • Players use “goods” to build “objects” • Player’s utility = number of objects • Bound on amount of goods available

33. Goods = edges Objects = flow paths

34. Given disagreement point,find NB soln.

35. Theorem: Strongly polynomial, combinatorial algorithm for single-source multiple-sink case. • Solution is again rational.

36. Insights into game-theoretic properties of Nash bargaining problems • Chakrabarty, Goel, V. , Wang & Yu: • Efficiency (Price of bargaining) • Fairness • Full competitiveness

37. Linear utilities • B: n players with disagreement points, ci • G: g goods, unit amount each • S = utility vectors obtained by distributing goods among players

38. Game plan • Use KKT conditions to transform Nash bargaining problem to computing the equilibrium in a certain market. • Find equilibrium using primal-dual paradigm.

39. Game plan • Use KKT conditions to transform Nash bargaining problem to computing the equilibrium in a certain market. • Find equilibrium using primal-dual paradigm.

41. General Equilibrium Theory Crown jewel of mathematicaleconomics for over a century!

42. A central tenet • Prices are such that demand equals supply, i.e., equilibrium prices.

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

44. Supply-demand curves

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

46. Fisher’s Model • 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,

47. Fisher’s Model • 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.

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

50. Flexible budget market,only difference: • Buyers don’t spend a fixed amount of money. • Instead, they know how much utility they desire. • At any given prices, they spend just enough money to accrue utility desired.