1. Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Polynomial Time Algorithms For Market Equilibria

2. Markets

3. Stock Markets

5. Internet

6. Revolution in definition of markets

7. Revolution in definition of markets New markets defined by  Google  Amazon  Yahoo!  Ebay

8. Revolution in definition of markets Massive computational power available

9. Revolution in definition of markets Massive computational power available Important to find good models and algorithms for these markets

10. Adwords Market Created by search engine companies  Google  Yahoo!  MSN Multi-billion dollar market Totally revolutionized advertising, especially by small companies.

13. How will this market evolve??

14. The study of market equilibria has occupied center stage within Mathematical Economics for over a century.

15. The study of market equilibria has occupied center stage within Mathematical Economics for over a century. This talk: Historical perspective & key notions from this theory.

16. 2). Algorithmic Game Theory Combinatorial algorithms for traditional market models

17. 3). New Market Models Resource Allocation Model of Kelly, 1997

18. 3). New Market Models Resource Allocation Model of Kelly, 1997 For mathematically modeling TCP congestion control Highly successful theory

19. A Capitalistic Economy Depends crucially on pricing mechanisms to ensure: Stability Efficiency Fairness

20. Adam Smith The Wealth of Nations 2 volumes, 1776.

21. Adam Smith The Wealth of Nations 2 volumes, 1776. ‘invisible hand’ of the market

22. Supply-demand curves

23. Leon Walras, 1874 Pioneered general equilibrium theory

24. Irving Fisher, 1891 First fundamental market model

25. Fisher’s Model, 1891 milk cheese wine bread ¢ $$$$$$$$$ $ $$$$ People want to maximize happiness – assume linear utilities. Find prices s.t. market clears

26. Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i,

27. Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i, Find prices s.t. market clears, i.e., all goods sold, all money spent.

29. Arrow-Debreu Model, 1954 Exchange Economy Second fundamental market model Celebrated theorem in Mathematical Economics

30. Kenneth Arrow Nobel Prize, 1972

31. Gerard Debreu Nobel Prize, 1983

32. Arrow-Debreu Model n agents, k goods

33. Arrow-Debreu Model n agents, k goods Each agent has: initial endowment of goods, & a utility function

34. Arrow-Debreu Model n agents, k goods Each agent has: initial endowment of goods, & a utility function Find market clearing prices, i.e., prices s.t. if  Each agent sells all her goods  Buys optimal bundle using this money  No surplus or deficiency of any good

35. Utility function of agent i Continuous, monotonic and strictly concave For any given prices and money m, there is a unique utility maximizing bundle for agent i.

36. Agents: Buyers/sellers Arrow-Debreu Model

37. Initial endowment of goods Agents Goods

38. Agents Prices Goods = $25 = $15 = $10

39. Incomes Goods Agents =$25 =$15 =$10 $50 $40 $60 $40 Prices

40. Goods Agents Maximize utility $50 $40 $60 $40 =$25 =$15 =$10 Prices

41. Find prices s.t. market clears Goods Agents $50 $40 $60 $40 =$25 =$15 =$10 Prices Maximize utility

42. Observe: If p is market clearing prices, then so is any scaling of p Assume w.l.o.g. that sum of prices of k goods is 1. k-1 dimensional unit simplex

43. Arrow-Debreu Theorem For continuous, monotonic, strictly concave utility functions, market clearing prices exist.

44. Proof Uses Kakutani’s Fixed Point Theorem.  Deep theorem in topology

45. Proof Uses Kakutani’s Fixed Point Theorem.  Deep theorem in topology Will illustrate main idea via Brouwer’s Fixed Point Theorem (buggy proof!!)

46. Brouwer’s Fixed Point Theorem Let be a non-empty, compact, convex set Continuous function Then

47. Brouwer’s Fixed Point Theorem

48. Idea of proof Will define continuous function If p is not market clearing, f(p) tries to ‘correct’ this. Therefore fixed points of f must be equilibrium prices.

49. Use Brouwer’s Theorem

50. When is p an equilibrium price? s(j): total supply of good j. B(i): unique optimal bundle which agent i wants to buy after selling her initial endowment at prices p. d(j): total demand of good j.

51. When is p an equilibrium price? s(j): total supply of good j. B(i): unique optimal bundle which agent i wants to buy after selling her initial endowment at prices p. d(j): total demand of good j. For each good j: s(j) = d(j).

52. What if p is not an equilibrium price? s(j) p(j) s(j) > d(j) => p(j) Also ensure

53. Let S(j) S(j) > d(j) => N is s.t.

54. is a cts. fn. => is a cts. fn. of p => f is a cts. fn. of p

55. is a cts. fn. => is a cts. fn. of p => f is a cts. fn. of p By Brouwer’s Theorem, equilibrium prices exist.

56. is a cts. fn. => is a cts. fn. of p => f is a cts. fn. of p By Brouwer’s Theorem, equilibrium prices exist. q.e.d.!

57. Kakutani’s Fixed Point Theorem convex, compact set non-empty, convex, upper hemi-continuous correspondence s.t.

58. Fisher reduces to Arrow-Debreu Fisher: n buyers, k goods AD: n+1 agents  first n have money, utility for goods  last agent has all goods, utility for money only.