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

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 B 1, B 2, …, B N  Each advertiser provides bids for keywords he is interested in. Search Engine

17. The Adwords Problem: N advertisers;  Daily Budgets B 1, B 2, …, B N  Each advertiser provides bids for keywords he is interested in. Search Engine queries (online)

18. The Adwords Problem: N advertisers;  Daily Budgets B 1, B 2, …, B N  Each advertiser provides bids for keywords he is interested in. Search Engine Select one Ad Advertiser pays his bid queries (online)

19. The Adwords Problem: N advertisers;  Daily Budgets B 1, B 2, …, B N  Each advertiser provides bids for keywords he is interested in. Search Engine Select one Ad Advertiser pays his bid queries (online) Maximize total revenue Online competitive analysis - compare with best offline allocation

20. The Adwords Problem: N advertisers;  Daily Budgets B 1, B 2, …, B N  Each advertiser provides bids for keywords he is interested in. Search Engine Select one Ad Advertiser pays his bid queries (online) Maximize total revenue Example – Assign to highest bidder: only ½ the offline revenue

21. Example: $1$0.99 $1 $0 Book CD Bidder1Bidder 2 B 1 = B 2 = $100 Queries: 100 Books then 100 CDs Bidder 1 Bidder 2 Algorithm Greedy LOST Revenue 100$

22. Example: $1$0.99 $1 $0 Book CD Bidder1Bidder 2 B 1 = B 2 = $100 Queries: 100 Books then 100 CDs Bidder 1 Bidder 2 Optimal Allocation Revenue 199$

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

24. Online bipartite matching advertisers queries

25. Online bipartite matching advertisers queries

26. Online bipartite matching advertisers queries

27. Online bipartite matching advertisers queries

28. Online bipartite matching advertisers queries

29. Online bipartite matching advertisers queries

30. Online bipartite matching advertisers queries

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

42. Leon Walras, 1874 Pioneered general equilibrium theory

43. Supply-demand curves

44. Irving Fisher, 1891 Fundamental market model

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

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

47. Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i, 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

64. 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

65. 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 Equilibrium

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) queueing delay (in TCP Vegas) p(e):

68. 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’s dollar is to decrease 5% flow from someone else’s dollar. p(e):

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

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)  Agents: sinks with money m(1), … m(k) Find: equilibrium flows and edge prices

73. 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 Equilibrium

75. $5

77. $10 $40 $30

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

81. m(1) m(2) m(3) m(4 ) cap(e)

82. min-cut separating from all the sinks 6060

83. 6060

84. 6060

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

86. 6060

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. 6060 5050

89. 6060 5050

90. 6060 5050 2020

91. 6060 5050 2020

92. 6060 5050 2020

93. nested cuts 6060 5050 2020

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

95. Implementation

97. Capacity of edge =

98. min s-t cut 6060

99. 6060

100. 6060

101. Capacity of edge =

102. 6060 5050 f(2)=10

103. 6060 5050

104. 6060 5050 2020

107. Eisenberg-Gale Program, 1959

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

109. Convex Program for Kelly’s Model

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

113. Irrational for 2 sources & 3 sinks 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)

120. 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.

121. Eisenberg-Gale-type program for branching market 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 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. EG[2]: Eisenberg-Gale markets with 2 agents Theorem: EG[2] markets are rational. Chakrabarty, Devanur & V., 2006:

128. 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]. Chakrabarty, Devanur & V., 2006:

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

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

132. Rich classification!

133. MarketEfficiency Single-source1 3-source branching k source-sink undirected 2 source-sink directedarbitrarily 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?