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

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Presentation on theme: "Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Markets and the Primal-Dual Paradigm."— Presentation transcript:

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

2 The new face of computing

3 A paradigm shift in the notion of a market!

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

8 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

9 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

10 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 Scarf, 1973: Algorithms for approximately computing fixed points

11 New markets defined by Internet companies, e.g., Google Yahoo! Amazon eBay Massive computing power available for running markets in a distributed or centralized manner A deep theory of algorithms with many powerful techniques Todays reality

12 What is needed today? An inherently-algorithmic theory of markets and market equilibria

13 What is needed today? An inherently-algorithmic theory of markets and market equilibria Beginnings of such a theory, within Algorithmic Game Theory

14 What is needed today? An inherently-algorithmic theory of markets and market equilibria Beginnings of such a theory, within Algorithmic Game Theory Natural starting point: algorithms for traditional market models

15 What is needed today? An inherently-algorithmic theory of markets and market equilibria Beginnings of such a theory, within Algorithmic Game Theory Natural starting point: algorithms for traditional market models New market models emerging!

16 Theory of algorithms Interestingly enough, recent study of markets has contributed handsomely to this theory!

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

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

19 Supply-demand curves

20 Irving Fisher, 1891 Defined a fundamental market model

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23 utility Utility function amount of milk

24 utility Utility function amount of bread

25 utility Utility function amount of cheese

26 Total utility of a bundle of goods = Sum of utilities of individual goods

27 For given prices,

28 For given prices, find optimal bundle of goods

29 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

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31 Combinatorial Algorithm for Linear Case of Fishers Model Devanur, Papadimitriou, Saberi & V., 2002 Using the primal-dual schema

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

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

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

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36 No LPs known for capturing equilibrium allocations for Fishers model

37 No LPs known for capturing equilibrium allocations for Fishers model Eisenberg-Gale convex program, 1959

38 No LPs known for capturing equilibrium allocations for Fishers model Eisenberg-Gale convex program, 1959 DPSV: Extended primal-dual schema to solving a nonlinear convex program

39 Fishers 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,

40 Fishers 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

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

42 A convex program whose optimal solution is equilibrium allocations.

43 A convex program whose optimal solution is equilibrium allocations. Constraints: packing constraints on the xijs

44 A convex program whose optimal solution is equilibrium allocations. x Constraints: packing constraints on the xijs Objective fn: max utilities derived.

45 A convex program whose optimal solution is equilibrium allocations. x Constraints: packing constraints on the xijs 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

46 Money-weighed geometric mean of utilities

47 Eisenberg-Gale Program, 1959

48 KKT conditions

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

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

51 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

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

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

54 KKT conditions e(i)

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

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

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

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

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

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

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

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

63 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

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

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

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

67 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

68 Network N m p buyers goods bang-per-buck edges

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

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

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

72 Balanced flow surplus vector: vector of surpluses w.r.t. f. A flow that minimizes l 2 norm of surplus vector. Must be a max-flow.

73 Balanced flow surplus vector: vector of surpluses w.r.t. f. A flow that minimizes l 2 norm of surplus vector. Must be a max-flow. All balanced flows have same surplus vector.

74 Balanced flow surplus vector: vector of surpluses w.r.t. f. A flow that minimizes l 2 norm of surplus vector. Makes surpluses as equal as possible.

75 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.

76 Property 1 i surplus(i) < surplus(j) j R:

77 Property 1 i surplus(i) < surplus(j) j R:

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

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

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

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

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

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

84 KKT conditions

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

87 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.

88 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.

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

90 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

91 utility Concave utility function amount of j

92 Concave utility functions Do not satisfy weak gross substitutability

93 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.

94 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!

95 utility Piecewise linear, concave amount of j

96 utility PTAS for concave function amount of j

97 Piecewise linear concave utility Does not satisfy weak gross substitutability

98 utility Piecewise linear, concave amount of j

99 rate rate = utility/unit amount of j amount of j Differentiate

100 rate amount of j rate = utility/unit amount of j money spent on j

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

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

103 $20$40$100 Extend model: assume buyers have utility for money rate

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105 Theorem: Polynomial time algorithm for computing equilibrium prices and allocations for Fishers model with spending constraint utilities. Furthermore, equilibrium prices are unique.

106 Satisfies weak gross substitutability!

107 Old pieces become more complex + there are new pieces

108 But they still fit just right!

109 Don Patinkin, Considered utility functions that are a function of allocations and prices.

110 An unexpected fallout!!

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

112 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 Googles AdWords Market!

113 A digression

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

115 The view 5 years ago: Relevant Search Results

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117 Business worlds view now : (as Advertisement companies)

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

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

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

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

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

123 Spending constraint utilities AdWords Market

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

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

126 Choice of utility function Expressive enough that advertisers get close to their optimal allocations

127 Choice of utility function Expressive enough that advertisers get close to their optimal allocations Efficiently computable

128 Choice of utility function Expressive enough that advertisers get close to their optimal allocations Efficiently computable Easy to specify utilities

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

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

131 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

132 Easier for a buyer To say what are good allocations, for a range of prices, rather than how happy she is with a given bundle.

133 Online shoe business Interested in two keywords: mens clog womens clog Advertising budget: $100/day Expected profit: mens clog: $2/click womens clog: $4/click

134 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

135 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)

136 $60$100 mens clog rate 2 1 rate = utility/click

137 $60$100 womens clog rate 2 4 rate = utility/click

138 $80$100 money rate 0 1 rate = utility/$

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

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

141 AdWords market Suppose Google stays with auctions but allows advertisers to specify bids in the spending constraint model expressivity! Good online algorithm for maximizing Googles revenues?

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

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

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

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

146 Proof follows from Eisenberg-Gale Convex Program, 1959

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

148 Open Is there an LP whose optimal solutions capture equilibrium allocations for Fishers linear case?

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

150 utility Piece-wise linear, concave amount of j

151 rate rate = utility/unit amount of j amount of j Differentiate

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

153 rate money spent on j rate = utility/unit amount of j

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

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

156 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!

157 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 …

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