1. Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Markets and the Primal-Dual Paradigm

2. Markets

3. Stock Markets

4. Internet

5. Revolution in definition of markets

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

7. Revolution in definition of markets Massive computational power available for running these markets in a centralized or distributed manner

8. Revolution in definition of markets Massive computational power available for running these markets in a centralized or distributed manner Important to find good models and algorithms for these markets

9. Theory of Algorithms Powerful tools and techniques developed over last 4 decades.

10. Theory of Algorithms Powerful tools and techniques developed over last 4 decades. Recent study of markets has contributed handsomely to this theory as well!

11. AdWords Market Created by search engine companies  Google  Yahoo!  MSN Multi-billion dollar market – and still growing! Totally revolutionized advertising, especially by small companies.

14. Historically, the study of markets has been of central importance, especially in the West

15. Historically, the study of markets has been of central importance, especially in the West General Equilibrium Theory Occupied center stage in Mathematical Economics for over a century

16. Leon Walras, 1874 Pioneered general equilibrium theory

17. Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem.

18. Kenneth Arrow Nobel Prize, 1972

19. Gerard Debreu Nobel Prize, 1983

20. General Equilibrium Theory Also gave us some algorithmic results  Convex programs, whose optimal solutions capture equilibrium allocations, e.g., Eisenberg & Gale, 1959 Nenakov & Primak, 1983  Cottle and Eaves, 1960’s: Linear complimentarity  Scarf, 1973: Algorithms for approximately computing fixed points

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

22. What is needed today? An inherently algorithmic theory of market equilibrium New models that capture new markets and are easier to use than traditional models

23. Beginnings of such a theory, within Algorithmic Game Theory Started with combinatorial algorithms for traditional market models New market models emerging

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

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

26. Supply-demand curves

27. Irving Fisher, 1891 Defined a fundamental market model

30. utility Utility function amount of milk

31. utility Utility function amount of bread

32. utility Utility function amount of cheese

33. Total utility of a bundle of goods = Sum of utilities of individual goods

34. For given prices,

35. For given prices, find optimal bundle of goods

36. Fisher market Several goods, fixed amount of each good Several buyers, with individual money and utilities Find equilibrium prices of goods, i.e., prices s.t.,  Each buyer gets an optimal bundle  No deficiency or surplus of any good

38. Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 Using the primal-dual schema

39. Primal-Dual Schema Highly successful algorithm design technique from exact and approximation algorithms

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

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

43. No LP’s known for capturing equilibrium allocations for Fisher’s model

44. No LP’s known for capturing equilibrium allocations for Fisher’s model Eisenberg-Gale convex program, 1959

45. No LP’s known for capturing equilibrium allocations for Fisher’s model Eisenberg-Gale convex program, 1959 DPSV: Extended primal-dual schema to solving a nonlinear convex program

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

47. Fisher’s Model n buyers, money m(i) for buyer i k goods (unit amount of each good) : utility derived by i on obtaining one unit of j Total utility of i, Find market clearing prices

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

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

50. At prices p, buyer i’s most desirable goods, S = Any goods from S worth m(i) constitute i’s optimal bundle Bang-per-buck

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

52. Max flow m(1) m(2) m(3) m(4) p(1) p(2) p(3) p(4) infinite capacities

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

54. 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

55. Idea of algorithm Iterations: execute primal & dual improvements Allocations Prices

56. Two important considerations The price of a good never exceeds its equilibrium price  Invariant: s is a min-cut

57. Max flow m(1) m(2) m(3) m(4) p(1) p(2) p(3) p(4) p: low prices

58. Two important considerations The price of a good never exceeds its equilibrium price  Invariant: s is a min-cut  Identify tight sets of goods

59. Two important considerations The price of a good never exceeds its equilibrium price  Invariant: s is a min-cut  Identify tight sets of goods Rapid progress is made  Balanced flows

60. Network N m p buyers goods bang-per-buck edges

61. Balanced flow in N m p W.r.t. flow f, surplus(i) = m(i) – f(i,t) i

62. Balanced flow surplus vector: vector of surpluses w.r.t. f.

63. Balanced flow surplus vector: vector of surpluses w.r.t. f. A flow that minimizes l 2 norm of surplus vector.

64. Property 1 f: max flow in N. R: residual graph w.r.t. f. If surplus (i) < surplus(j) then there is no path from i to j in R.

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

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

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

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

69. Pieces fit just right! Balanced flows Invariant Bang-per-buck edges Tight sets

70. How primal-dual schema is adapted to nonlinear setting

71. A convex program whose optimal solution is equilibrium allocations.

72. A convex program whose optimal solution is equilibrium allocations. Constraints: packing constraints on the xij’s

73. A convex program whose optimal solution is equilibrium allocations. x Constraints: packing constraints on the xij’s Objective fn: max utilities derived.

74. A convex program whose optimal solution is equilibrium allocations. x Constraints: packing constraints on the xij’s Objective fn: max utilities derived. Must satisfy  If utilities of a buyer are scaled by a constant, optimal allocations remain unchanged  If money of buyer b is split among two new buyers, whose utility fns same as b, then union of optimal allocations to new buyers = optimal allocation for b

75. Money-weighed geometric mean of utilities

76. Eisenberg-Gale Program, 1959

77. prices p j

78. KKT conditions

79. Therefore, buyer i buys from only, i.e., gets an optimal bundle

80. Therefore, buyer i buys from only, i.e., gets an optimal bundle Can prove that equilibrium prices are unique!

81. Will relax KKT conditions e(i): money currently spent by i w.r.t. a special allocation surplus money of i

82. Will relax KKT conditions e(i): money currently spent by i w.r.t. a balanced flow in N surplus money of i

83. KKT conditions e(i)

84. Potential function Will show that potential drops by an inverse polynomial factor in each phase (polynomial time).

85. Potential function Will show that potential drops by an inverse polynomial factor in each phase (polynomial time).

86. Point of departure KKT conditions are satisfied via a continuous process Normally: in discrete steps

87. Point of departure KKT conditions are satisfied via a continuous process Normally: in discrete steps Open question: strongly polynomial algorithm?

88. Another point of departure Complementary slackness conditions: involve primal or dual variables, not both. KKT conditions: involve primal and dual variables simultaneously.

89. KKT conditions

91. Primal-dual algorithms so far Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.)

92. 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 weight matching.

93. 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 weight matching. Otherwise primal objects go tight and loose. Difficult to account for these reversals in the running time.

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

95. Deficiencies of linear utility functions Typically, a buyer spends all her money on a single good Do not model the fact that buyers get satiated with goods

96. utility Concave utility function amount of j

97. Concave utility functions Do not satisfy weak gross substitutability

98. Concave utility functions Do not satisfy weak gross substitutability  w.g.s. = Raising the price of one good cannot lead to a decrease in demand of another good.

99. Concave utility functions Do not satisfy weak gross substitutability  w.g.s. = Raising the price of one good cannot lead to a decrease in demand of another good. Open problem: find polynomial time algorithm!

100. utility Piecewise linear, concave amount of j

101. utility PTAS for concave function amount of j

102. Piecewise linear concave utility Does not satisfy weak gross substitutability

103. utility Piecewise linear, concave amount of j

104. rate rate = utility/unit amount of j amount of j Differentiate

105. rate amount of j rate = utility/unit amount of j money spent on j

106. rate rate = utility/unit amount of j money spent on j Spending constraint utility function $20$40 $60

107. Spending constraint utility function Happiness derived is not a function of allocation only but also of amount of money spent.

108. $20$40$100 Extend model: assume buyers have utility for money rate

110. Theorem: Polynomial time algorithm for computing equilibrium prices and allocations for Fisher’s model with spending constraint utilities. Furthermore, equilibrium prices are unique.

111. Satisfies weak gross substitutability!

112. Old pieces become more complex + there are new pieces

113. But they still fit just right!

114. Don Patinkin, 1956 Money, Interest, and Prices. An Integration of Monetary and Value Theory Pascal Bridel, 2002:  Euro. J. History of Economic Thought, Patinkin, Walras and the ‘money-in-the-utility- function’ tradition

115. An unexpected fallout!!

116. A new kind of utility function  Happiness derived is not a function of allocation only but also of amount of money spent.

117. An unexpected fallout!! A new kind of utility function  Happiness derived is not a function of allocation only but also of amount of money spent. Has applications in Google’s AdWords Market!

118. A digression

119. The view 5 years ago: Relevant Search Results

121. Business world’s view now : (as Advertisement companies)

122. Bids for different keywords Daily Budgets So how does this work?

123. An interesting algorithmic question! Monika Henzinger, 2004: Find an on-line algorithm that maximizes Google’s revenue.

124. AdWords Allocation Problem Search EngineSearch Engine Whose ad to put How to maximize revenue? LawyersRus.com Sue.com TaxHelper.com asbestos Search results Ads

125. AdWords Problem Mehta, Saberi, Vazirani & Vazirani, 2005: 1-1/e algorithm, assuming budgets>>bids

126. AdWords Problem Mehta, Saberi, Vazirani & Vazirani, 2005: 1-1/e algorithm, assuming budgets>>bids Optimal!

127. AdWords Problem Mehta, Saberi, Vazirani & Vazirani, 2005: 1-1/e algorithm, assuming budgets>>bids Optimal!

128. Spending constraint utilities AdWords Market

129. AdWords market Assume that Google will determine equilibrium price/click for keywords

130. AdWords market Assume that Google will determine equilibrium price/click for keywords How should advertisers specify their utility functions?

131. Choice of utility function Expressive enough that advertisers get close to their ‘‘optimal’’ allocations

132. Choice of utility function Expressive enough that advertisers get close to their ‘‘optimal’’ allocations Efficiently computable

133. Choice of utility function Expressive enough that advertisers get close to their ‘‘optimal’’ allocations Efficiently computable Easy to specify utilities

134. linear utility function: a business will typically get only one type of query throughout the day!

135. linear utility function: a business will typically get only one type of query throughout the day! concave utility function: no efficient algorithm known!

136. linear utility function: a business will typically get only one type of query throughout the day! concave utility function: no efficient algorithm known!  Difficult for advertisers to define concave functions

137. Easier for a buyer To say how much money she should spend on each good, for a range of prices, rather than how happy she is with a given bundle.

138. Online shoe business Interested in two keywords:  men’s clog  women’s clog Advertising budget: $100/day Expected profit:  men’s clog: $2/click  women’s clog: $4/click

139. Considerations for long-term profit Try to sell both goods - not just the most profitable good Must have a presence in the market, even if it entails a small loss

140. If both are profitable,  better keyword is at least twice as profitable ($100, $0)  otherwise ($60, $40) If neither is profitable ($20, $0) If only one is profitable,  very profitable (at least $2/$) ($100, $0)  otherwise ($60, $0)

141. $60$100 men’s clog rate 2 1 rate = utility/click

142. $60$100 women’s clog rate 2 4 rate = utility/click

143. $80$100 money rate 0 1 rate = utility/$

144. AdWords market Suppose Google stays with auctions but allows advertisers to specify bids in the spending constraint model

145. AdWords market Suppose Google stays with auctions but allows advertisers to specify bids in the spending constraint model  expressivity!

146. AdWords market Suppose Google stays with auctions but allows advertisers to specify bids in the spending constraint model  expressivity! Good online algorithm for maximizing Google’s revenues?

147. Goel & Mehta, 2006: A small modification to the MSVV algorithm achieves 1 – 1/e competitive ratio!

148. Open Is there a convex program that captures equilibrium allocations for spending constraint utilities?

149. Equilibrium exists (under mild conditions) Equilibrium utilities and prices are unique Rational With small denominators Spending constraint utilities satisfy

150. Equilibrium exists (under mild conditions) Equilibrium utilities and prices are unique Rational With small denominators Linear utilities also satisfy

151. Proof follows from Eisenberg-Gale Convex Program, 1959

152. For spending constraint utilities, proof follows from algorithm, and not a convex program!

153. Open Is there an LP whose optimal solutions capture equilibrium allocations for Fisher’s linear case?

154. Use spending constraint algorithm to solve piecewise linear, concave utilities Open

155. Algorithms & Game Theory common origins von Neumann, 1928: minimax theorem for 2-person zero sum games von Neumann & Morgenstern, 1944: Games and Economic Behavior von Neumann, 1946: Report on EDVAC Dantzig, Gale, Kuhn, Scarf, Tucker …

157. utility Piece-wise linear, concave amount of j

158. rate rate = utility/unit amount of j amount of j Differentiate

159. Start with arbitrary prices, adding up to total money of buyers.

160. rate money spent on j rate = utility/unit amount of j

161. Start with arbitrary prices, adding up to total money of buyers. Run algorithm on these utilities to get new prices.

162. Start with arbitrary prices, adding up to total money of buyers. Run algorithm on these utilities to get new prices.

163. Start with arbitrary prices, adding up to total money of buyers. Run algorithm on these utilities to get new prices. Fixed points of this procedure are equilibrium prices for piecewise linear, concave utilities!