1. Algorithmic Game Theory and Internet Computing Vijay V. Vazirani 3) New Market Models, Resource Allocation Markets

2. 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,

3. 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

4. Eisenberg-Gale Program, 1959

5. Via KKT Conditions can establish: Optimal solution gives equilibrium allocations Lagrange variables give prices of goods

6. Equilibrium exists (under mild conditions) Equilibrium utilities and prices are unique Eisenberg-Gale program helps establish:

7. Equilibrium exists (under mild conditions) Equilibrium utilities and prices are unique Rational!! Eisenberg-Gale program helps establish:

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

9. Kelly’s model Given: network G = (V,E) (directed or undirected) capacities on edges source-sink pairs (agents) m(i): money agent i is willing to pay

10. Kelly’s model Network determines: f(i): flow of agent i Assume utility u(i) = m(i) log f(i) Total utility is additive

11. Convex Program for Kelly’s Model

12. Kelly’s model Lagrange variables: p(e): price/unit flow

13. Kelly’s model Optimum flow and edge prices are in equilibrium: 1). p(e)>0 only if e is saturated 2) flows go on cheapest paths 3) money of each agent is fully used Let rate(i) = cost of cheapest path for i m(i) = f(i) rate(i)

14. Kelly’s model Optimum flow and edge prices are in equilibrium: 1). p(e)>0 only if e is saturated 2) flows go on cheapest paths 3) money of each agent is fully used Let rate(i) = cost of cheapest path for i f(i)’s and rate(i)’s are unique!

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

16. 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):

17. TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). p(e):

18. TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). AIMD + RED converges to equilibrium primal-dual (source-link) alg. p(e):

19. TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). FAST: for high speed networks with large bandwidth p(e):

20. Combinatorial Algorithms Devanur, Papadimitriou, Saberi & V., 2002: for Fisher’s linear utilities case Kelly & V., 2002: Kelly’s model is a generalization of Fisher’s model. Find combinatorial poly time algorithms!

21. Irrational for 2 sources & 3 sinks $1

22. Irrational for 2 sources & 3 sinks Equilibrium prices

23. 1 source & multiple sinks 2 source-sink pairs

25. $5

27. $10 $40 $30

28. Jain & V., 2005: strongly poly alg Primal-dual algorithm  Usual: linear programs & LP-duality  This: convex programs & KKT conditions Ascending price auction  Buyers: sinks (fixed budgets, maximize flow)  Sellers: edges (maximize price)

29. rate(i): cost of cheapest path

31. Capacity of edge =

32. min s-t cut

39. nested cuts

41. Find s-t max flow Flow and prices will:  Saturate all red cuts  Use up sinks’ money  Send flow on cheapest paths

47. $10 $40 $30

48. Rational!!

49. Max-flow min-cut theorem

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

51. Branching market (for broadcasting) Given: Network G = (V, E)  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.

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

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

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

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

56. Megiddo, 1974: Let T = set of sinks (agents) For define v(S) to be the max-flow possible from s to sinks in S. Then v is a submodular function, i.e., for

57. Simpler convex program for single-source market

58. Submodular Utility Allocation Market Any market which has simpler program and v is submodular

59. Submodular Utility Allocation Market Any market which has simpler program and v is submodular Theorem: Strongly polynomial algorithm for SUA markets.

60. Submodular Utility Allocation Market Any market which has simpler program and v is submodular Theorem: Strongly polynomial algorithm for SUA markets. Corollary: Rational!!

61. Theorem: Following markets are SUA:  2 source-sink pairs, undirected (Hu, 1963)  spanning tree (Nash-William & Tutte, 1961)  2 sources branching (Edmonds, 1967 + JV, 2005) 3 sources branching: irrational

62. Theorem: Following markets are SUA:  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 thoerems):  2 source-sink pairs, directed  2 sources, network coding

63. Theorem: Following markets are SUA:  2 source-sink pairs, undirected (Hu, 1963)  spanning tree (Nash-William & Tutte, 1961)  2 sources branching (Edmonds, 1967 + JV) 3 sources branching: irrational Open (no max-min thoerems):  2 source-sink pairs, directed  2 sources, network coding Chakrabarty, Devanur & V., 2006

64. EG[2]: Eisenberg-Gale markets with 2 agents Theorem: EG[2] markets are rational.

65. 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]. Using Tardos, 1986.

66. 2 source-sink market in directed graphs

67. 2 1

70. Polytope of feasible flows

71. LP’s corresponding to facets

76. $30 $60

77. For any m(1), m(2), need to ‘‘price’’ at most two facets, with prices say

78. $10 $5

79. For any m(1), m(2), need to ‘‘price’’ at most two facets, with prices say Exponentially many facets!  Binary search on

80. For any m(1), m(2), need to ‘‘price’’ at most two facets, with prices say Exponentially many facets!  Binary search on Compute duals

83. For any m(1), m(2), need to ‘‘price’’ at most two facets, with prices say Exponentially many facets!  Binary search on Compute duals Compute

84. $5, each

85. 10/2 = $5, each $10, each

86. $30 $60 $5 $10 $15

87. $30 $60 $5 $10 $15

88. $30=$15x2 $60=$20x3 $5 $10 $15

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

90. Observe: Equilibrium is always an s-t max-flow

91. 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.

92. Efficiency

93. Rich classification!

94. MarketEfficiency Single-source1 3-source branching k source-sink undirected 2 source-sink directedarbitrarily small

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

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

99. The End