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Algorithmic Game Theory and Internet Computing

Nash Bargaining via Flexible Budget Markets. Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani. The new platform for computing. Internet. Massive computational power available Sellers (programs) can negotiate with individual buyers!. Internet.

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Algorithmic Game Theory and Internet Computing

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

  2. The new platform for computing

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

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

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

  6. 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.

  7. 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!

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

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

  10. Utilities derived jointly : convex + compact feasible set

  11. Disagreement point = status quo utilities Disagreement point =

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

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

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

  15. Thm: Unique solution satisfying 4 axioms

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

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

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

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

  20. Q: Compute solution combinatoriallyin polynomial time?

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

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

  23. 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

  24. Convex program giving NB solution

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

  26. 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.

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

  28. Goods = edges Objects = flow paths

  29. Given disagreement point,find NB soln.

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

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

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

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

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

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

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

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

  38. Supply-demand curves

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

  40. 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,

  41. 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.

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

  43. 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.

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