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

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. General Equilibrium Theory
An almost entirely non-algorithmic 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 function
utility
amount of milk

31. Utility function
utility
amount of bread

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

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

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

53. p: equilibrium prices iff both cuts saturated
Max flow
p(1)
m(1)
p(2)
m(2)
p(3)
m(3)
m(4)
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
p(1)
m(1)
p(2)
m(2)
p(3)
m(3)
m(4)
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
buyers p m
bang-per-buck edges
goods

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

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 l2 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. surplus(i) < surplus(j)
Property 1 R: i j
surplus(i) < surplus(j)

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

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

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

69. Invariant Balanced flows Bang-per-buck edges Tight sets
Pieces fit just right!
Invariant
Balanced flows
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.
Constraints: packing constraints on the xij’s
Objective fn: max utilities derived.

74. A convex program whose optimal solution is equilibrium allocations.
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. Eisenberg-Gale Program, 1959
prices pj

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)
e(i)

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

85. Will show that potential drops by an inverse polynomial
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

90. 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. 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. Piecewise linear, concave
utility
amount of j

101. PTAS for concave function
utility
amount of j

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

103. Piecewise linear, concave
utility
amount of j

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

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

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

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

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

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. 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.

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

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

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

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 neither is profitable ($20, $0) If only one is profitable,
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. men’s clog
rate = utility/click
rate 2 1
$60
$100

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

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

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

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

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

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

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

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

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

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. Piece-wise linear, concave
utility
amount of j

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

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

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