1. Algorithmic Game Theory
New Market Models
and Internet Computing
and Algorithms
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

Totally revolutionized advertising, especially

by small companies.

15. New algorithmic and
game-theoretic questions
Monika Henzinger, 2004: Find an on-line

algorithm that maximizes Google’s revenue.

16. The Adwords Problem:
N advertisers;
 Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.

Search Engine

17. The Adwords Problem:
N advertisers;
 Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.

queries Search Engine
(online)

18. The Adwords Problem:
N advertisers;
 Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.

Select one Ad
queries Search Engine
(online) Advertiser
pays his bid

19. The Adwords Problem:
N advertisers;
 Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.

Select one Ad
queries Search Engine
(online) Advertiser
pays his bid
Maximize total revenue
Online competitive analysis - compare with best offline allocation

20. The Adwords Problem:
N advertisers;
 Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.

Select one Ad
queries Search Engine
(online) Advertiser
pays his bid
Maximize total revenue
Example – Assign to highest bidder: only ½ the offline revenue

21. Example:
Bidder1 Bidder 2
Book Queries: 100 Books then 100 CDs
$1 $0.99
CD $1 $0
B1 = B2 = $100
LOST
Revenue
100$
Algorithm Greedy
Bidder 1 Bidder 2

22. Example:
Bidder1 Bidder 2
Book Queries: 100 Books then 100 CDs
$1 $0.99
CD $1 $0
B1 = B2 = $100
Revenue
199$
Optimal Allocation
Bidder 1 Bidder 2

23. Generalizes online bipartite matching
Each daily budget is $1, and

each bid is $0/1.

24. Online bipartite matching
queries
advertisers

25. Online bipartite matching
queries
advertisers

26. Online bipartite matching
queries
advertisers

27. Online bipartite matching
queries
advertisers

28. Online bipartite matching
queries
advertisers

29. Online bipartite matching
queries
advertisers

30. Online bipartite matching
queries
advertisers

31. Online bipartite matching
Karp, Vazirani & Vazirani, 1990:

1-1/e factor randomized algorithm.

32. Online bipartite matching
Karp, Vazirani & Vazirani, 1990:

1-1/e factor randomized algorithm. Optimal!

33. Online bipartite matching
Karp, Vazirani & Vazirani, 1990:

1-1/e factor randomized algorithm. Optimal!
Kalyanasundaram & Pruhs, 1996:

1-1/e factor algorithm for b-matching:
Daily budgets $b, bids $0/1, b>>1

34. Adwords Problem
Mehta, Saberi, Vazirani & Vazirani, 2005:

1-1/e algorithm, assuming budgets>>bids.

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

36. New Algorithmic Technique
Idea: Use both bid and

fraction of left-over budget

37. New Algorithmic Technique
Idea: Use both bid and

fraction of left-over budget
Correct tradeoff given by

tradeoff-revealing family of LP’s

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

39. A Capitalistic Economy
depends crucially on pricing mechanisms,
with very little intervention, to ensure:
Stability

 Efficiency
 Fairness

40. Do markets even have inherently
stable operating points?

41. Do markets even have inherently
stable operating points?
General Equilibrium Theory
Occupied center stage in Mathematical
Economics for over a century

42. Leon Walras, 1874
Pioneered general

equilibrium theory

43. Supply-demand curves

44. Irving Fisher, 1891
Fundamental

market model

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

46. Fisher’s Model
n buyers, with specified money, m(i) for buyer i

k goods (unit amount of each good) U = u x

￥
i ij ij
Linear utilities: uij is utility derived by i
j

on obtaining one unit of j
Total utility of i,

u = u x
i ij ij
j
x  [0,1]
ij

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

49. 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.

50. Kenneth Arrow
 Nobel Prize, 1972

51. Gerard Debreu
 Nobel Prize, 1983

52. Arrow-Debreu Theorem, 1954
.
 Highly non-constructive

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

54. What is needed today?
An inherently algorithmic theory of

market equilibrium
New models that capture new markets


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

56. Combinatorial Algorithm
for Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002

Using primal-dual schema

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

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

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

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

61. A combinatorial market
s2
s1
t1
t2

62. A combinatorial market
s2
c(e)
s1
t1
t2

63. A combinatorial market
m ( 2)
s2
c(e)
m(1)
s1
t1
t2

64. A combinatorial market
Given:

 Network G = (V,E) (directed or undirected)
 Capacities on edges c(e)
( s1 , t1 ),...( sk , tk )
 Agents: source-sink pairs
with money m(1), … m(k)
Find: equilibrium flows and edge prices


65. Equilibrium
Flows and edge prices

f(i): flow of agent i

p(e): price/unit flow of edge e

Satisfying:

p(e)>0 only if e is saturated

flows go on cheapest paths

money of each agent is fully spent


66. Kelly’s resource allocation model, 1997
Mathematical framework for understanding
TCP congestion control
Highly successful theory

67. TCP Congestion Control
f(i): source rate

prob. of packet loss (in TCP Reno)

p(e):
queueing delay (in TCP Vegas)

68. TCP Congestion Control
f(i): source rate

prob. of packet loss (in TCP Reno)

p(e):
queueing delay (in TCP Vegas)
Kelly: Equilibrium flows are proportionally fair:
only way of adding 5% flow to someone’s
dollar is to decrease 5% flow from
someone else’s dollar.

69. TCP Congestion Control
primal process: packet rates at sources
dual process: packet drop at links
AIMD + RED converges to equilibrium
in the limit

70. Kelly & V., 2002: Kelly’s model is a

generalization of Fisher’s model.
Find combinatorial polynomial time

algorithms!

71. Jain & V., 2005:
Strongly polynomial combinatorial algorithm

for single-source multiple-sink market

72. Single-source multiple-sink market
Given:

 Network G = (V,E), s: source
 Capacities on edges c(e)
t1 ,..., tk
 Agents: sinks
with money m(1), … m(k)
Find: equilibrium flows and edge prices


73. Equilibrium
Flows and edge prices

f(i): flow of agent i

p(e): price/unit flow of edge e

Satisfying:

p(e)>0 only if e is saturated

flows go on cheapest paths

money of each agent is fully spent


74. t $10
1
1
2
s
2
t $10
2

75. t $10
1
1
$5
2
s
2
t
$5 $10
2

76. t $10
120
1
1
2
s
2
t $10
2

77. t $120
$30 1
1
$10
2
s
2
t
$40 $10
2

78. Jain & V., 2005:
Strongly polynomial combinatorial algorithm

for single-source multiple-sink market
Ascending price auction

 Buyers: sinks (fixed budgets, maximize flow)
 Sellers: edges (maximize price)

79. Auction of k identical goods
p = 0;

 while there are >k buyers:
raise p;
 end;
 sell to remaining k buyers at price p;

80. Find equilibrium prices and flows
t 1
t
s 2
t 3
t 4

81. Find equilibrium prices and flows
t 1
m(1)
t
s m(2)
2
t m(3)
cap(e) 3
t m(4)
4

82. 60
t 1
t
s 2
t 3
t 4
s from all the sinks
min-cut separating

83. 60
t 1
t
s 2
t 3
t 4
p

84. 60
t 1
t
s 2
t 3
t 4
p ﾭ

85. Throughout the algorithm:
sto t i
c(i): cost of cheapest path from
m(i )
t f (i ) =
i
sink demands flow c(i )

86. m(i )
t
quot;i : c(i ) = p f (i ) =
i
sink demands flow p
60
t 1
t
s 2
t 3
t 4
p ﾭ

87. Auction of edges in cut
p = 0;

 while the cut is over-saturated:
raise p;
 end;
 assign price p to all edges in the cut;

88. c(2) = p0
60 50 f (2) = 10
t 1
s t t
2
3
t 4
p =p 0

89. c(2) = p0
c(1) = c(3) = c(4) = p0 + p
60 50
t 1
s t t
2
3
t 4
p
p 
0

90. c(2) = p0
c(1) = c(3) = p0 + p1
60 50 20
t 1
s t t
2
3
t 4
f (1) + f (3) = 30
p p
0 1

91. 60 50 20
t 1
s t t
2
3
t 4
p p p 
0 1

92. c(4) = p0 + p1 + p2
60 50 20
t 1
s t t
2
3
t 4
f (4) = 20
p p p
0 1 2

93. 60 50 20
t 1
s t t
2
3
t 4
p p p nested cuts
0 1 2

94. Flow and prices will:

 Saturate all red cuts
 Use up sinks’money
 Send flow on cheapest paths

95. Implementation
t 1
t
s 2
t 3
t 4

96. t
t 1
t
s 2
t 3
t 4

97. t
t 1
t
s 2
t 3
t 4
m(i )
f (i ) =
t  t edge
i
Capacity of = c(i )

98. t
60
t 1
t
s 2
t 3
t 4
min s-t cut

99. t
60
t 1
t
s 2
t 3
t 4
p

100. t
60
t 1
t
s 2
t 3
t 4
p 

101. t
quot;i : c(i ) = p
t 1
t
s 2
t 3
t 4
m(i )
p ﾯ
f (i ) =
 t  t edge =
i
p
Capacity of

102. f(2)=10
t
60 50
t 1
s t t
2
3
t 4
p =p c(2) = p0
0

103. t
60 50
t 1
s t t
2
3
t 4
p
p 
0

104. t
60 50 20
t 1
s t t
2
3
t 4
c(2) = p0
p p c(1) = c(3) = c(4) = p0 + p1
0 1

105. t
t 1
s t t
2
3
t 4
p p p 
0 1

106. t
t 1
s t t
2
3
t 4
c(4) = p0 + p1 + p2
p p p
0 1 2

107. Eisenberg-Gale Program, 1959
max ￥ (i ) log ui
m
i
s.t.
quot;i : ui = ￥ u ij x ij
j
quot;j : ￥x ij ﾣ 1
i
quot;ij : x ij ﾳ 0

108. Lagrangian variables: prices of goods

Using KKT conditions:

optimal primal and dual solutions
are in equilibrium

109. Convex Program for Kelly’s Model
max ￥ (i ) log f (i )
m
i
s.t.
quot;i : f (i ) = ￥ f i p
p
quot;e : flow(e) ﾣ c(e)
quot;i, p : f i ﾳ 0
p

110. JV Algorithm
primal-dual alg. for nonlinear convex program

“primal” variables: flows

“dual” variables: prices of edges

algorithm: primal & dual improvements

Allocations Prices

111. Rational!!

112. Irrational for 2 sources & 3 sinks
$1
$1
s
2
t
1
t
1 1
1
s t
1 2
2 2
$1

113. Irrational for 2 sources & 3 sinks
3
s
2
t
1
3 t 1+ 3
1 1
1
s t
2 2
Equilibrium prices

114. Max-flow min-cut theorem!

115. Other resource allocation markets
2 source-sink pairs (directed/undirected)

 Branchings rooted at sources (agents)

116. Branching market (for broadcasting)
m(1) m(3)
m ( 2)
s
s s
c(e)
1 2 3

117. Branching market (for broadcasting)
m(1) m(3)
m ( 2)
s
s s
c(e) 2
1 3

118. Branching market (for broadcasting)
m(1) m(3)
m ( 2)
s
s s
c(e)
1 2 3

119. Branching market (for broadcasting)
m(1) m(3)
m ( 2)
s
s s
c(e) 2
1 3

120. Branching market (for broadcasting)
Given: Network G = (V, E), directed

 edge capacities
SￍV
 sources,
 money of each source
Find: edge prices and a packing

of branchings rooted at sources s.t.
p(e) > 0 => e is saturated

each branching is cheapest possible

money of each source fully used.


121. Eisenberg-Gale-type program
for branching market
max ￥ S m(i ) log bi
iￎ
s.t. packing of branchings

122. Other resource allocation markets
2 source-sink pairs (directed/undirected)

 Branchings rooted at sources (agents)
 Spanning trees
 Network coding

123. Eisenberg-Gale-Type Convex Program
max ￥m(i ) log ui
i
s.t. packing constraints

124. Eisenberg-Gale Market
A market whose equilibrium is captured

as an optimal solution to an
Eisenberg-Gale-type program

125. Theorem: Strongly polynomial algs for

following markets :
 2 source-sink pairs, undirected (Hu, 1963)
 spanning tree (Nash-William & Tutte, 1961)
 2 sources branching (Edmonds, 1967 + JV, 2005)
3 sources branching: irrational


126. Theorem: Strongly polynomial algs for

following markets :
 2 source-sink pairs, undirected (Hu, 1963)
 spanning tree (Nash-William & Tutte, 1961)
 2 sources branching (Edmonds, 1967 + JV, 2005)
3 sources branching: irrational

Open: (no max-min theorems):

 2 source-sink pairs, directed
 2 sources, network coding

127. Chakrabarty, Devanur & V., 2006:
EG[2]: Eisenberg-Gale markets with 2 agents

Theorem: EG[2] markets are rational.


128. Chakrabarty, Devanur & V., 2006:
EG[2]: Eisenberg-Gale markets with 2 agents

Theorem: EG[2] markets are rational.

Combinatorial EG[2] markets: polytope

of feasible utilities can be described via
combinatorial LP.
Theorem: Strongly poly alg for Comb EG[2].


129. 3-source branching
Single-source
2 s-s undir
SUA
Comb EG[2]
2 s-s dir
Rational
Fisher
EG[2]
EG

130. Efficiency of Markets
‘‘price of capitalism’’

 Agents:
 different abilities to control prices
 idiosyncratic ways of utilizing resources
Q: Overall output of market when forced

to operate at equilibrium?

131. Efficiency
equilibrium  utility ( I )
eff ( M ) = min I
max  utility ( I )

132. Efficiency
equilibrium  utility ( I )
eff ( M ) = min I
max  utility ( I )
 Rich classification!

133. Market Efficiency
Single-source 1
3-source branching
ﾳ 1/ 2
ﾳ 1/(2k  1)
k source-sink undirected
l.b. = 1/(k  1)
2 source-sink directed arbitrarily
small

134. Other properties:
Fairness (max-min + min-max fair)

 Competition monotonicity

135. Open issues
Strongly poly algs for approximating

 nonlinear convex programs
 equilibria
Insights into congestion control protocols?
