1. Algorithmic Game Theory and Internet Computing
New Market Models
and Algorithms
Algorithmic Game Theory and Internet Computing
Vijay V. Vazirani
Georgia Tech

2. How do we salvage the situation??
Algorithmic ratification of the
“invisible hand of the market”
How do we salvage the situation??

3. What is the “right” model??

4. Linear Fisher Market DPSV, 2002: First polynomial time algorithm
Extend to separable, plc utilities??

5. What makes linear utilities easy?
Weak gross substitutability:
Increasing price of one good cannot
decrease demand of another.
Piecewise-linear, concave utilities do not
satisfy this.

6. Piecewise linear, concave
utility
amount of j

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

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

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

10. Theorem (V., 2002):
Spending constraint utilities:
1). Satisfy weak gross substitutability
2). Polynomial time algorithm for
computing equilibrium
3). Equilibrium is rational.

11. An unexpected fallout!!
Has applications to
Google’s AdWords Market!

12. Application to Adwords market
$20
$40
$60
rate = utility/click
rate
money spent on keyword j

13. Is there a convex program for this model?
“We believe the answer to this question should be ‘yes’. In our experience, non-trivial polynomial time algorithms for problems are rare and happen for a good reason – a deep mathematical structure intimately connected to the problem.”

14. Devanur’s program for linear Fisher

15. C. P. for spending constraint!

16. EG convex program = Devanur’s program
Fisher market
with plc utilities
Spending constraint
market
EG convex program = Devanur’s program

18. Price discrimination markets
Business charges different prices from different
customers for essentially same goods or services.
Goel & V., 2009:
Perfect price discrimination market.
Business charges each consumer what
they are willing and able to pay.

19. plc utilities

20. Middleman buys all goods and sells to buyers,
charging according to utility accrued.
Given p, each buyer picks rate for accruing
utility.

21. Middleman buys all goods and sells to buyers,
charging according to utility accrued.
Given p, each buyer picks rate for accruing
utility.
Equilibrium is captured by a
rational convex program!

22. Generalization of EG program works!

23. V., 2010: Generalize to Continuously differentiable, quasiconcave
(non-separable) utilities, satisfying non-satiation.

24. V., 2010: Generalize to Continuously differentiable, quasiconcave
(non-separable) utilities, satisfying non-satiation.
Compare with Arrow-Debreu utilities!!
continuous, quasiconcave, satisfying non-satiation.

25. Price discrimination market (plc utilities) Spending constraint market
EG convex program = Devanur’s program

26. EG convex program = Devanur’s program
Eisenberg-Gale Markets
Jain & V., 2007
(Proportional Fairness)
(Kelly, 1997)
Price disc.
market
Spending constraint
market
Nash Bargaining
V., 2008
EG convex program = Devanur’s program

27. A combinatorial market

28. A combinatorial market

29. A combinatorial market

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

31. Equilibrium Flows and edge prices Satisfying: 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

32. Kelly’s resource allocation model, 1997
Mathematical framework for understanding
TCP congestion control

33. Van Jacobson, 1988: AIMD protocol
(Additive Increase Multiplicative Decrease)

34. Van Jacobson, 1988: AIMD protocol
(Additive Increase Multiplicative Decrease)
Why does it work so well?

35. Van Jacobson, 1988: AIMD protocol
(Additive Increase Multiplicative Decrease)
Why does it work so well?
Kelly, 1977: Highly successful theory

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

37. TCP Congestion Control
f(i): source rate
prob. of packet loss (in TCP Reno)
queueing delay (in TCP Vegas)
Low & Lapsley, 1999:
AIMD + RED converges to equilibrium in limit
p(e):

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

39. Kelly & V., 2002: Kelly’s model is a
generalization of Fisher’s model.

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

41. Kelly & V., 2002: Kelly’s model is a
generalization of Fisher’s model.
Find combinatorial poly time algorithms!
(May lead to new insights for
TCP congestion control protocol)

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

43. Single-source multiple-sink market
Given:
Network G = (V,E), s: source
Capacities on edges c(e)
Agents: sinks
with money
Find: equilibrium flows and edge prices

44. Equilibrium Flows and edge prices Satisfying: 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

46. $5 $5

48. $30
$10
$40

49. 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)

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

51. Find equilibrium prices and flows

52. Find equilibrium prices and flows
cap(e)

53. min-cut separating from all the sinks60
min-cut separating from all the sinks

54. 60

55. 60

56. Throughout the algorithm:
c(i): cost of cheapest path from to
sink demands flow

57. sink demands flow 60

58. 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;

59. 60 50

60. 60 50

61. 60 50 20

62. 60 50 20

63. 60 50 20

64. 60 50 20
nested cuts

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

66. Exercise: Find the red cuts efficiently!

67. Convex Program for Kelly’s Model

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

69. Rational!!

70. Other resource allocation markets
k source-sink pairs (directed/undirected)

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

72. Branching market (for broadcasting)

73. Branching market (for broadcasting)

74. Branching market (for broadcasting)

75. Branching market (for broadcasting)

76. Branching market (for broadcasting)
Given: Network G = (V, E), directed
edge capacities
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.

77. Eisenberg-Gale-type program for branching market
s.t. packing of branchings

78. Eisenberg-Gale-Type Convex Program
s.t. packing constraints

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

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

81. Irrational for 2 sources & 3 sinks$1 $1 $1

82. Irrational for 2 sources & 3 sinks
Equilibrium prices

83. Max-flow min-cut theorem!

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

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

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

87. 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].

88. 2 source-sink market in directed graphs

89. 2 1

92. Polytope of feasible flows

93. LP’s corresponding to facets

98. $30
$60

99. Polytope of feasible flows
(1, 2) 2 1
(0, 1)

100. Find the two (one) facets
Exponentially many facets!
Binary search on

101. $5
$10

102. Find relative ‘‘prices’’ of
two facets, say
Compute duals

105. Find relative ‘‘prices’’ of
two facets, say
Compute duals
Compute

106. $5, each

107. 10/2 = $5, each
$10, each

108. $10 $5
$30
$15
$60

109. $10 $5
$30
$15
$60

110. $10 $5
$30=$15x2
$15
$60=$20x3

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