J1

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J1

  1. 1. Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Polynomial Time Algorithms For Market Equilibria
  2. 2. 1) History and Basic Notions
  3. 3. Markets
  4. 4. Stock Markets
  5. 6. Internet
  6. 7. <ul><li>Revolution in definition of markets </li></ul>
  7. 8. <ul><li>Revolution in definition of markets </li></ul><ul><li>New markets defined by </li></ul><ul><ul><li>Google </li></ul></ul><ul><ul><li>Amazon </li></ul></ul><ul><ul><li>Yahoo! </li></ul></ul><ul><ul><li>Ebay </li></ul></ul>
  8. 9. <ul><li>Revolution in definition of markets </li></ul><ul><li>Massive computational power available </li></ul>
  9. 10. <ul><li>Revolution in definition of markets </li></ul><ul><li>Massive computational power available </li></ul><ul><li>Important to find good models and </li></ul><ul><li>algorithms for these markets </li></ul>
  10. 11. Adwords Market <ul><li>Created by search engine companies </li></ul><ul><ul><li>Google </li></ul></ul><ul><ul><li>Yahoo! </li></ul></ul><ul><ul><li>MSN </li></ul></ul><ul><li>Multi-billion dollar market </li></ul><ul><li>Totally revolutionized advertising, especially </li></ul><ul><li>by small companies. </li></ul>
  11. 14. New algorithmic and game-theoretic questions <ul><li>Queries are coming on-line. Instantaneously decide which bidder gets it. </li></ul><ul><li>Monika Henzinger, 2004: Find on-line alg. </li></ul><ul><li>to maximize Google’s revenue. </li></ul>
  12. 15. New algorithmic and game-theoretic questions <ul><li>Queries are coming on-line. Instantaneously decide which bidder gets it. </li></ul><ul><li>Monika Henzinger, 2004: Find on-line alg. </li></ul><ul><li>to maximize Google’s revenue. </li></ul><ul><li>Mehta, Saberi, Vazirani & Vazirani, 2005: </li></ul><ul><li>1-1/e algorithm. Optimal. </li></ul>
  13. 16. How will this market evolve??
  14. 17. <ul><li>The study of market equilibria has occupied </li></ul><ul><li>center stage within Mathematical Economics </li></ul><ul><li>for over a century. </li></ul>
  15. 18. <ul><li>The study of market equilibria has occupied </li></ul><ul><li>center stage within Mathematical Economics </li></ul><ul><li>for over a century. </li></ul><ul><li>This talk: Historical perspective </li></ul><ul><li>& key notions from this theory. </li></ul>
  16. 19. 2). Algorithmic Game Theory <ul><li>Combinatorial algorithms for </li></ul><ul><li>traditional market models </li></ul>
  17. 20. 3). New Market Models <ul><li>Resource Allocation Model of Kelly, 1997 </li></ul>
  18. 21. 3). New Market Models <ul><li>Resource Allocation Model of Kelly, 1997 </li></ul><ul><li>For mathematically modeling </li></ul><ul><li>TCP congestion control </li></ul><ul><li>Highly successful theory </li></ul>
  19. 22. A Capitalistic Economy <ul><li>Depends crucially on </li></ul><ul><li>pricing mechanisms to ensure: </li></ul><ul><li>Stability </li></ul><ul><li>Efficiency </li></ul><ul><li>Fairness </li></ul>
  20. 23. Adam Smith <ul><li>The Wealth of Nations </li></ul><ul><li>2 volumes, 1776. </li></ul>
  21. 24. Adam Smith <ul><li>The Wealth of Nations </li></ul><ul><li>2 volumes, 1776. </li></ul><ul><li>‘ invisible hand’ of the market </li></ul>
  22. 25. Supply-demand curves
  23. 26. Leon Walras, 1874 <ul><li>Pioneered general </li></ul><ul><li>equilibrium theory </li></ul>
  24. 27. Irving Fisher, 1891 <ul><li>First fundamental </li></ul><ul><li>market model </li></ul>
  25. 28. Fisher’s Model, 1891 milk ¢ $$$$$$$$$ $ $$$$ <ul><li>People want to maximize happiness – assume </li></ul><ul><li>linear utilities. </li></ul>Find prices s.t. market clears cheese wine bread
  26. 29. Fisher’s Model <ul><li>n buyers, with specified money, m(i) for buyer i </li></ul><ul><li>k goods (unit amount of each good) </li></ul><ul><li>Linear utilities: is utility derived by i </li></ul><ul><li>on obtaining one unit of j </li></ul><ul><li>Total utility of i, </li></ul>
  27. 30. Fisher’s Model <ul><li>n buyers, with specified money, m(i) </li></ul><ul><li>k goods (each unit amount, w.l.o.g.) </li></ul><ul><li>Linear utilities: is utility derived by i </li></ul><ul><li>on obtaining one unit of j </li></ul><ul><li>Total utility of i, </li></ul><ul><li>Find prices s.t. market clears, i.e., </li></ul><ul><li>all goods sold, all money spent. </li></ul>
  28. 32. Arrow-Debreu Model, 1954 Exchange Economy <ul><li>Second fundamental market model </li></ul><ul><li>Celebrated theorem in Mathematical Economics </li></ul>
  29. 33. Kenneth Arrow <ul><li>Nobel Prize, 1972 </li></ul>
  30. 34. Gerard Debreu <ul><li>Nobel Prize, 1983 </li></ul>
  31. 35. Arrow-Debreu Model <ul><li>n agents, k goods </li></ul>
  32. 36. Arrow-Debreu Model <ul><li>n agents, k goods </li></ul><ul><li>Each agent has: initial endowment of goods, </li></ul><ul><li>& a utility function </li></ul>
  33. 37. Arrow-Debreu Model <ul><li>n agents, k goods </li></ul><ul><li>Each agent has: initial endowment of goods, </li></ul><ul><li>& a utility function </li></ul><ul><li>Find market clearing prices, i.e., prices s.t. if </li></ul><ul><ul><li>Each agent sells all her goods </li></ul></ul><ul><ul><li>Buys optimal bundle using this money </li></ul></ul><ul><ul><li>No surplus or deficiency of any good </li></ul></ul>
  34. 38. Utility function of agent i <ul><li>Continuous, monotonic and strictly concave </li></ul><ul><li>For any given prices and money m, </li></ul><ul><li>there is a unique utility maximizing bundle </li></ul><ul><li>for agent i . </li></ul>
  35. 39. Arrow-Debreu Model Agents: Buyers/sellers
  36. 40. Initial endowment of goods Agents Goods
  37. 41. Agents Prices Goods = $25 = $15 = $10
  38. 42. Incomes Goods Agents =$25 =$15 =$10 $50 $40 $60 $40 Prices
  39. 43. Goods Agents Maximize utility $50 $40 $60 $40 =$25 =$15 =$10 Prices
  40. 44. Find prices s.t. market clears Goods Agents $50 $40 $60 $40 =$25 =$15 =$10 Prices Maximize utility
  41. 45. <ul><li>Observe: If p is market clearing </li></ul><ul><li>prices, then so is any scaling of p </li></ul><ul><li>Assume w.l.o.g. that sum of </li></ul><ul><li>prices of k goods is 1. </li></ul><ul><li>k-1 dimensional </li></ul><ul><li>unit simplex </li></ul>
  42. 46. Arrow-Debreu Theorem <ul><li>For continuous, monotonic, strictly concave </li></ul><ul><li>utility functions, market clearing prices </li></ul><ul><li>exist. </li></ul>
  43. 47. Proof <ul><li>Uses Kakutani’s Fixed Point Theorem. </li></ul><ul><ul><li>Deep theorem in topology </li></ul></ul>
  44. 48. Proof <ul><li>Uses Kakutani’s Fixed Point Theorem. </li></ul><ul><ul><li>Deep theorem in topology </li></ul></ul><ul><li>Will illustrate main idea via Brouwer’s Fixed </li></ul><ul><li>Point Theorem (buggy proof!!) </li></ul>
  45. 49. Brouwer’s Fixed Point Theorem <ul><li>Let be a non-empty, compact, convex set </li></ul><ul><li>Continuous function </li></ul><ul><li>Then </li></ul>
  46. 50. Brouwer’s Fixed Point Theorem
  47. 51. Idea of proof <ul><li>Will define continuous function </li></ul><ul><li>If p is not market clearing, f(p) tries to </li></ul><ul><li>‘ correct’ this. </li></ul><ul><li>Therefore fixed points of f must be </li></ul><ul><li>equilibrium prices. </li></ul>
  48. 52. Use Brouwer’s Theorem
  49. 53. When is p an equilibrium price? <ul><li>s(j): total supply of good j. </li></ul><ul><li>B(i): unique optimal bundle which agent i wants to buy after selling her initial </li></ul><ul><li>endowment at prices p. </li></ul><ul><li>d(j): total demand of good j . </li></ul>
  50. 54. When is p an equilibrium price? <ul><li>s(j): total supply of good j. </li></ul><ul><li>B(i): unique optimal bundle which agent i wants to buy after selling her initial </li></ul><ul><li>endowment at prices p. </li></ul><ul><li>d(j): total demand of good j . </li></ul><ul><li>For each good j: s(j) = d(j). </li></ul>
  51. 55. What if p is not an equilibrium price? <ul><li>s(j) < d(j) => p(j) </li></ul><ul><li>s(j) > d(j) => p(j) </li></ul><ul><li>Also ensure </li></ul>
  52. 56. <ul><li>Let </li></ul><ul><li>S(j) < d(j) => </li></ul><ul><li>S(j) > d(j) => </li></ul><ul><li>N is s.t. </li></ul>
  53. 57. <ul><li>is a cts. fn. </li></ul><ul><li>=> is a cts. fn. of p </li></ul><ul><li>=> is a cts. fn. of p </li></ul><ul><li>=> f is a cts. fn. of p </li></ul>
  54. 58. <ul><li>is a cts. fn. </li></ul><ul><li>=> is a cts. fn. of p </li></ul><ul><li>=> is a cts. fn. of p </li></ul><ul><li>=> f is a cts. fn. of p </li></ul><ul><li>By Brouwer’s Theorem, equilibrium prices exist. </li></ul>
  55. 59. <ul><li>is a cts. fn. </li></ul><ul><li>=> is a cts. fn. of p </li></ul><ul><li>=> is a cts. fn. of p </li></ul><ul><li>=> f is a cts. fn. of p </li></ul><ul><li>By Brouwer’s Theorem, equilibrium prices exist. </li></ul><ul><li>q.e.d.! </li></ul>
  56. 60. Bug??
  57. 61. <ul><li>Boundaries of </li></ul>
  58. 62. <ul><li>Boundaries of </li></ul><ul><li>B(i) is not defined at boundaries!! </li></ul>
  59. 63. Kakutani’s Fixed Point Theorem <ul><li>convex, compact set </li></ul><ul><li>non-empty, convex, </li></ul><ul><li>upper hemi-continuous correspondence </li></ul><ul><li>s.t. </li></ul>
  60. 64. Fisher reduces to Arrow-Debreu <ul><li>Fisher: n buyers, k goods </li></ul><ul><li>AD: n+1 agents </li></ul><ul><ul><li>first n have money, utility for goods </li></ul></ul><ul><ul><li>last agent has all goods, utility for money only. </li></ul></ul>
  61. 65. Money

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