Algorithmic Game Theory
   New Market Models
 and Internet Computing
     and Algorithms

    Vijay V. Vazirani
Markets
Stock Markets
Internet
Revolution in definition of markets

Revolution in definition of markets





    New markets defined by

     Google
     Amazon
     Yahoo!
     Ebay
Revolution in definition of markets




    Massive computational power available


    for running these markets in a
 ...
Revolution in definition of markets




    Massive computational power available


    for running these markets in a
 ...
Theory of Algorithms

    Powerful tools and techniques


     developed over last 4 decades.
Theory of Algorithms

    Powerful tools and techniques


     developed over last 4 decades.

    Recent study of market...
Adwords Market
    Created by search engine companies

     Google
     Yahoo!
     MSN


    Multi-billion dollar mar...
New algorithmic and
         game-theoretic questions


    Monika Henzinger, 2004: Find an on-line


    algorithm that ...
The Adwords Problem:

N advertisers;
   Daily Budgets B1, B2, …, BN

       Each advertiser provides bids for keywords he...
The Adwords Problem:

N advertisers;
   Daily Budgets B1, B2, …, BN

         Each advertiser provides bids for keywords ...
The Adwords Problem:

N advertisers;
   Daily Budgets B1, B2, …, BN

         Each advertiser provides bids for keywords ...
The Adwords Problem:

N advertisers;
   Daily Budgets B1, B2, …, BN

         Each advertiser provides bids for keywords ...
The Adwords Problem:

N advertisers;
   Daily Budgets B1, B2, …, BN

         Each advertiser provides bids for keywords ...
Example:

       Bidder1 Bidder 2


Book                       Queries: 100 Books then 100 CDs
          $1     $0.99


 C...
Example:

       Bidder1 Bidder 2


Book                      Queries: 100 Books then 100 CDs
          $1    $0.99


 CD ...
Generalizes online bipartite matching

     Each daily budget is $1, and
 

      each bid is $0/1.
Online bipartite matching

                queries
  advertisers
Online bipartite matching

                queries
  advertisers
Online bipartite matching

                queries
  advertisers
Online bipartite matching

                queries
  advertisers
Online bipartite matching

                queries
  advertisers
Online bipartite matching

                queries
  advertisers
Online bipartite matching

                queries
  advertisers
Online bipartite matching

    Karp, Vazirani & Vazirani, 1990:


    1-1/e factor randomized algorithm.
Online bipartite matching

    Karp, Vazirani & Vazirani, 1990:


    1-1/e factor randomized algorithm. Optimal!
Online bipartite matching

    Karp, Vazirani & Vazirani, 1990:


    1-1/e factor randomized algorithm. Optimal!

    Ka...
Adwords Problem

    Mehta, Saberi, Vazirani & Vazirani, 2005:


    1-1/e algorithm, assuming budgets>>bids.
Adwords Problem

    Mehta, Saberi, Vazirani & Vazirani, 2005:


    1-1/e algorithm, assuming budgets>>bids.
    Optimal!
New Algorithmic Technique

    Idea: Use both bid and


           fraction of left-over budget
New Algorithmic Technique

    Idea: Use both bid and


           fraction of left-over budget

    Correct tradeoff giv...
Historically, the study of markets


    has been of central importance,


       especially in the West
A Capitalistic Economy
depends crucially on pricing mechanisms,
 with very little intervention, to ensure:

  Stability

...
Do markets even have inherently
   stable operating points?
Do markets even have inherently
     stable operating points?


    General Equilibrium Theory
Occupied center stage in Ma...
Leon Walras, 1874


              Pioneered general
          

              equilibrium theory
Supply-demand curves
Irving Fisher, 1891


               Fundamental
           

                market model
Fisher’s Model, 1891

                                           $
                $$$$$$$$$

    ¢

                     ...
Fisher’s Model
      n buyers, with specified money, m(i) for buyer i
  

      k goods (unit amount of each good) U = u ...
Fisher’s Model
     n buyers, with specified money, m(i)
 

     k goods (each unit amount, w.l.o.g.)
 
                ...
Arrow-Debreu Theorem, 1954

    Celebrated theorem in Mathematical Economics



    Established existence of market equil...
Kenneth Arrow



         Nobel   Prize, 1972
Gerard Debreu


         Nobel   Prize, 1983
Arrow-Debreu Theorem, 1954
.

      Highly   non-constructive
Adam Smith

         The Wealth of Nations
     

         2 volumes, 1776.


         ‘invisible hand’ of
     
       ...
What is needed today?

    An inherently algorithmic theory of


     market equilibrium


    New models that capture ne...
Beginnings of such a theory, within


       Algorithmic Game Theory

    Started with combinatorial algorithms


     f...
Combinatorial Algorithm
          for Fisher’s Model


    Devanur, Papadimitriou, Saberi & V., 2002




    Using primal...
Primal-Dual Schema

 Highly successful algorithm design
  technique from exact and
  approximation algorithms
Exact Algorithms for Cornerstone
 Problems in P:

     Matching (general graph)
 
     Network flow
 
     Shortest path...
Approximation Algorithms


  set cover           facility location
  Steiner tree        k-median
  Steiner network     mu...
No LP’s known for capturing equilibrium

    allocations for Fisher’s model

    Eisenberg-Gale convex program, 1959



...
A combinatorial market
         s2


 s1

                         t1

              t2
A combinatorial market
         s2
              c(e)

 s1

                         t1

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

                          ...
A combinatorial market
    Given:

     Network   G = (V,E) (directed or undirected)
     Capacities on edges c(e)
    ...
Equilibrium
    Flows and edge prices


        f(i): flow of agent i
    
        p(e): price/unit flow of edge e
    ...
Kelly’s resource allocation model, 1997


  Mathematical framework for understanding

           TCP congestion control


...
TCP Congestion Control
  f(i): source rate


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

           prob. of packet loss (in TCP Reno)

     p(e):
           queu...
TCP Congestion Control

    primal process: packet rates at sources
    dual process:   packet drop at links

    AIMD + R...
Kelly & V., 2002: Kelly’s model is a


      generalization of Fisher’s model.


    Find combinatorial polynomial time
...
Jain & V., 2005:
      Strongly polynomial combinatorial algorithm
  

      for single-source multiple-sink market
Single-source multiple-sink market
    Given:

     Network   G = (V,E), s: source
     Capacities on edges c(e)
      ...
Equilibrium
    Flows and edge prices


        f(i): flow of agent i
    
        p(e): price/unit flow of edge e
    ...
t       $10
                1
        1
    2
s
    2
            t       $10
                2
t       $10
                     1
             1
    $5
         2
s
         2
                 t
    $5                ...
t       $10
                     120
                1
        1
    2
s
    2
            t       $10
                2
t       $120
              $30       1
              1
    $10
          2
s
          2
                    t
    $40    ...
Jain & V., 2005:
      Strongly polynomial combinatorial algorithm
  

      for single-source multiple-sink market

    ...
Auction of k identical goods

  p = 0;

 while there are >k buyers:

       raise p;
 end;
 sell to remaining k buyers...
Find equilibrium prices and flows

             t   1


                     t
   s                     2

               ...
Find equilibrium prices and flows

                  t   1
                          m(1)
                          t
   s...
60

                           t   1


                                   t
         s                             2

    ...
60

             t   1


                     t
s                        2

                             t   3

          ...
60

             t   1


                     t
s                        2

                             t   3

          ...
Throughout the algorithm:


                                        sto t     i
 c(i): cost of cheapest path from


      ...
m(i )
                                          t
quot;i : c(i ) = p                                                      ...
Auction of edges in cut

  p = 0;

 while the cut is over-saturated:

      raise p;
 end;
 assign price p to all edge...
c(2) = p0

     60        50                           f (2) = 10

                    t   1


s          t               ...
c(2) = p0

                                         c(1) = c(3) = c(4) = p0 + p
    60       50

                     t   ...
c(2) = p0

                                                  c(1) = c(3) = p0 + p1
    60       50                      20...
60       50                      20

                     t   1


s            t               t
                 2
      ...
c(4) = p0 + p1 + p2
    60       50                      20

                     t   1


s            t               t
 ...
60       50                      20

                     t   1


s            t               t
                 2
      ...
Flow and prices will:


     Saturate all red cuts
     Use up sinks’money
     Send flow on cheapest paths
Implementation

        t   1


                t
s                   2

                        t   3

                  ...
t

    t   1


            t
s               2

                    t   3

                            t   4
t

                   t   1


                           t
s                              2

                             ...
t
    60

                  t   1


                          t
s                             2

                         ...
t
    60

             t   1


                     t
s                        2

                             t   3

    ...
t
    60

             t   1


                     t
s                        2

                             t   3

    ...
t
quot;i : c(i ) = p


                         t   1


                                 t
          s                    ...
f(2)=10
                                                       t
     60        50

                    t   1


s         ...
t
    60       50

                     t   1


s            t               t
                 2
                        ...
t
    60       50                      20

                     t   1


s            t               t
                 2
...
t

                    t   1


s           t               t
                2
                                3

        ...
t

                    t   1


s           t               t
                2
                                3

        ...
Eisenberg-Gale Program, 1959

       max ¥ (i ) log ui
            m
              i

       s.t.
       quot;i : ui = ¥ u...
Lagrangian variables: prices of goods




    Using KKT conditions:


     optimal primal and dual solutions
     are in...
Convex Program for Kelly’s Model

         max ¥ (i ) log f (i )
              m
                i

         s.t.
        ...
JV Algorithm
    primal-dual alg. for nonlinear convex program


    “primal” variables: flows


    “dual” variables: p...
Rational!!
Irrational for 2 sources & 3 sinks


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


                                   3
         s
                                     ...
Max-flow min-cut theorem!
Other resource allocation markets


    2 source-sink pairs (directed/undirected)
  
   Branchings rooted at sources (ag...
Branching market (for broadcasting)

   m(1)                        m(3)
                     m ( 2)
                     ...
Branching market (for broadcasting)

   m(1)                        m(3)
                     m ( 2)
                     ...
Branching market (for broadcasting)

   m(1)                        m(3)
                     m ( 2)
                     ...
Branching market (for broadcasting)

   m(1)                        m(3)
                     m ( 2)
                     ...
Branching market (for broadcasting)
     Given: Network G = (V, E), directed
 
      edge capacities
                   ...
Eisenberg-Gale-type program
    for branching market


          max ¥ S m(i ) log bi
               iᅫ




        s.t. p...
Other resource allocation markets


    2 source-sink pairs (directed/undirected)
  
   Branchings rooted at sources (ag...
Eisenberg-Gale-Type Convex Program



         max ¥m(i ) log ui
              i


        s.t. packing constraints
Eisenberg-Gale Market


      A market whose equilibrium is captured
  

       as an optimal solution to an
       Eisen...
Theorem: Strongly polynomial algs for


              following markets :
     2 source-sink pairs, undirected (Hu, 1963...
Theorem: Strongly polynomial algs for


              following markets :
     2 source-sink pairs, undirected (Hu, 1963...
Chakrabarty, Devanur & V., 2006:

     EG[2]: Eisenberg-Gale markets with 2 agents
 



     Theorem: EG[2] markets are r...
Chakrabarty, Devanur & V., 2006:

     EG[2]: Eisenberg-Gale markets with 2 agents
 



     Theorem: EG[2] markets are r...
3-source branching




 Single-source
                     2 s-s undir
                                   SUA
           C...
Efficiency of Markets
  ‘‘price of capitalism’’

 Agents:
     different abilities to control prices
     idiosyncrati...
Efficiency

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

                   equilibrium  utility ( I )
 eff ( M ) = min I
                      max  utility ( I )


...
Market               Efficiency
     Single-source                 1
  3-source branching
                               ᄈ...
Other properties:

    Fairness (max-min + min-max fair)
  
   Competition monotonicity
Open issues
      Strongly poly algs for approximating
  
       nonlinear convex programs
       equilibria



      I...
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  1. 1. Algorithmic Game Theory New Market Models and Internet Computing and Algorithms Vijay V. Vazirani
  2. 2. Markets
  3. 3. Stock Markets
  4. 4. Internet
  5. 5. Revolution in definition of markets 
  6. 6. Revolution in definition of markets  New markets defined by   Google  Amazon  Yahoo!  Ebay
  7. 7. Revolution in definition of markets  Massive computational power available  for running these markets in a centralized or distributed manner
  8. 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. 9. Theory of Algorithms Powerful tools and techniques  developed over last 4 decades.
  10. 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. 11. Adwords Market Created by search engine companies   Google  Yahoo!  MSN Multi-billion dollar market  Totally revolutionized advertising, especially  by small companies.
  12. 12. New algorithmic and game-theoretic questions Monika Henzinger, 2004: Find an on-line  algorithm that maximizes Google’s revenue.
  13. 13. The Adwords Problem: N advertisers;  Daily Budgets B1, B2, …, BN Each advertiser provides bids for keywords he is interested in.  Search Engine
  14. 14. The Adwords Problem: N advertisers;  Daily Budgets B1, B2, …, BN Each advertiser provides bids for keywords he is interested in.  queries Search Engine (online)
  15. 15. 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
  16. 16. 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
  17. 17. 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
  18. 18. 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
  19. 19. 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
  20. 20. Generalizes online bipartite matching Each daily budget is $1, and  each bid is $0/1.
  21. 21. Online bipartite matching queries advertisers
  22. 22. Online bipartite matching queries advertisers
  23. 23. Online bipartite matching queries advertisers
  24. 24. Online bipartite matching queries advertisers
  25. 25. Online bipartite matching queries advertisers
  26. 26. Online bipartite matching queries advertisers
  27. 27. Online bipartite matching queries advertisers
  28. 28. Online bipartite matching Karp, Vazirani & Vazirani, 1990:  1-1/e factor randomized algorithm.
  29. 29. Online bipartite matching Karp, Vazirani & Vazirani, 1990:  1-1/e factor randomized algorithm. Optimal!
  30. 30. 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
  31. 31. Adwords Problem Mehta, Saberi, Vazirani & Vazirani, 2005:  1-1/e algorithm, assuming budgets>>bids.
  32. 32. Adwords Problem Mehta, Saberi, Vazirani & Vazirani, 2005:  1-1/e algorithm, assuming budgets>>bids. Optimal!
  33. 33. New Algorithmic Technique Idea: Use both bid and  fraction of left-over budget
  34. 34. New Algorithmic Technique Idea: Use both bid and  fraction of left-over budget Correct tradeoff given by  tradeoff-revealing family of LP’s
  35. 35. Historically, the study of markets has been of central importance,  especially in the West
  36. 36. A Capitalistic Economy depends crucially on pricing mechanisms, with very little intervention, to ensure: Stability   Efficiency  Fairness
  37. 37. Do markets even have inherently stable operating points?
  38. 38. Do markets even have inherently stable operating points? General Equilibrium Theory Occupied center stage in Mathematical Economics for over a century
  39. 39. Leon Walras, 1874 Pioneered general  equilibrium theory
  40. 40. Supply-demand curves
  41. 41. Irving Fisher, 1891 Fundamental  market model
  42. 42. Fisher’s Model, 1891 $ $$$$$$$$$ ¢ wine bread $$$$ milk cheese People want to maximize happiness – assume  linear utilities. s.t. market clears Find prices
  43. 43. 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
  44. 44. 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.
  45. 45. 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.
  46. 46. Kenneth Arrow  Nobel Prize, 1972
  47. 47. Gerard Debreu  Nobel Prize, 1983
  48. 48. Arrow-Debreu Theorem, 1954 .  Highly non-constructive
  49. 49. Adam Smith The Wealth of Nations  2 volumes, 1776. ‘invisible hand’ of  the market
  50. 50. What is needed today? An inherently algorithmic theory of  market equilibrium New models that capture new markets 
  51. 51. Beginnings of such a theory, within  Algorithmic Game Theory Started with combinatorial algorithms  for traditional market models New market models emerging 
  52. 52. Combinatorial Algorithm for Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002  Using primal-dual schema
  53. 53. Primal-Dual Schema  Highly successful algorithm design technique from exact and approximation algorithms
  54. 54. Exact Algorithms for Cornerstone Problems in P: Matching (general graph)  Network flow  Shortest paths  Minimum spanning tree  Minimum branching 
  55. 55. Approximation Algorithms set cover facility location Steiner tree k-median Steiner network multicut k-MST feedback vertex set scheduling . . .
  56. 56. 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
  57. 57. A combinatorial market s2 s1 t1 t2
  58. 58. A combinatorial market s2 c(e) s1 t1 t2
  59. 59. A combinatorial market m ( 2) s2 c(e) m(1) s1 t1 t2
  60. 60. 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 
  61. 61. 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 
  62. 62. Kelly’s resource allocation model, 1997 Mathematical framework for understanding TCP congestion control Highly successful theory
  63. 63. TCP Congestion Control f(i): source rate  prob. of packet loss (in TCP Reno)  p(e): queueing delay (in TCP Vegas)
  64. 64. 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.
  65. 65. TCP Congestion Control primal process: packet rates at sources dual process: packet drop at links AIMD + RED converges to equilibrium in the limit
  66. 66. Kelly & V., 2002: Kelly’s model is a  generalization of Fisher’s model. Find combinatorial polynomial time  algorithms!
  67. 67. Jain & V., 2005: Strongly polynomial combinatorial algorithm  for single-source multiple-sink market
  68. 68. 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 
  69. 69. 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 
  70. 70. t $10 1 1 2 s 2 t $10 2
  71. 71. t $10 1 1 $5 2 s 2 t $5 $10 2
  72. 72. t $10 120 1 1 2 s 2 t $10 2
  73. 73. t $120 $30 1 1 $10 2 s 2 t $40 $10 2
  74. 74. 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)
  75. 75. Auction of k identical goods p = 0;   while there are >k buyers: raise p;  end;  sell to remaining k buyers at price p;
  76. 76. Find equilibrium prices and flows t 1 t s 2 t 3 t 4
  77. 77. Find equilibrium prices and flows t 1 m(1) t s m(2) 2 t m(3) cap(e) 3 t m(4) 4
  78. 78. 60 t 1 t s 2 t 3 t 4 s from all the sinks min-cut separating
  79. 79. 60 t 1 t s 2 t 3 t 4 p
  80. 80. 60 t 1 t s 2 t 3 t 4 p ᆳ
  81. 81. Throughout the algorithm: sto t i c(i): cost of cheapest path from m(i ) t f (i ) = i sink demands flow c(i )
  82. 82. 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
  83. 83. 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;
  84. 84. c(2) = p0 60 50 f (2) = 10 t 1 s t t 2 3 t 4 p =p 0
  85. 85. 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
  86. 86. 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
  87. 87. 60 50 20 t 1 s t t 2 3 t 4 p p p  0 1
  88. 88. 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
  89. 89. 60 50 20 t 1 s t t 2 3 t 4 p p p nested cuts 0 1 2
  90. 90. Flow and prices will:   Saturate all red cuts  Use up sinks’money  Send flow on cheapest paths
  91. 91. Implementation t 1 t s 2 t 3 t 4
  92. 92. t t 1 t s 2 t 3 t 4
  93. 93. t t 1 t s 2 t 3 t 4 m(i ) f (i ) = t  t edge i Capacity of = c(i )
  94. 94. t 60 t 1 t s 2 t 3 t 4 min s-t cut
  95. 95. t 60 t 1 t s 2 t 3 t 4 p
  96. 96. t 60 t 1 t s 2 t 3 t 4 p 
  97. 97. 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
  98. 98. f(2)=10 t 60 50 t 1 s t t 2 3 t 4 p =p c(2) = p0 0
  99. 99. t 60 50 t 1 s t t 2 3 t 4 p p  0
  100. 100. 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
  101. 101. t t 1 s t t 2 3 t 4 p p p  0 1
  102. 102. t t 1 s t t 2 3 t 4 c(4) = p0 + p1 + p2 p p p 0 1 2
  103. 103. 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
  104. 104. Lagrangian variables: prices of goods  Using KKT conditions:  optimal primal and dual solutions are in equilibrium
  105. 105. 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
  106. 106. JV Algorithm primal-dual alg. for nonlinear convex program  “primal” variables: flows  “dual” variables: prices of edges  algorithm: primal & dual improvements  Allocations Prices
  107. 107. Rational!!
  108. 108. Irrational for 2 sources & 3 sinks $1 $1 s 2 t 1 t 1 1 1 s t 1 2 2 2 $1
  109. 109. Irrational for 2 sources & 3 sinks 3 s 2 t 1 3 t 1+ 3 1 1 1 s t 2 2 Equilibrium prices
  110. 110. Max-flow min-cut theorem!
  111. 111. Other resource allocation markets 2 source-sink pairs (directed/undirected)   Branchings rooted at sources (agents)
  112. 112. Branching market (for broadcasting) m(1) m(3) m ( 2) s s s c(e) 1 2 3
  113. 113. Branching market (for broadcasting) m(1) m(3) m ( 2) s s s c(e) 2 1 3
  114. 114. Branching market (for broadcasting) m(1) m(3) m ( 2) s s s c(e) 1 2 3
  115. 115. Branching market (for broadcasting) m(1) m(3) m ( 2) s s s c(e) 2 1 3
  116. 116. 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. 
  117. 117. Eisenberg-Gale-type program for branching market max ¥ S m(i ) log bi iᅫ s.t. packing of branchings
  118. 118. Other resource allocation markets 2 source-sink pairs (directed/undirected)   Branchings rooted at sources (agents)  Spanning trees  Network coding
  119. 119. Eisenberg-Gale-Type Convex Program max ¥m(i ) log ui i s.t. packing constraints
  120. 120. Eisenberg-Gale Market A market whose equilibrium is captured  as an optimal solution to an Eisenberg-Gale-type program
  121. 121. 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 
  122. 122. 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
  123. 123. Chakrabarty, Devanur & V., 2006: EG[2]: Eisenberg-Gale markets with 2 agents  Theorem: EG[2] markets are rational. 
  124. 124. 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]. 
  125. 125. 3-source branching Single-source 2 s-s undir SUA Comb EG[2] 2 s-s dir Rational Fisher EG[2] EG
  126. 126. 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?
  127. 127. Efficiency equilibrium  utility ( I ) eff ( M ) = min I max  utility ( I )
  128. 128. Efficiency equilibrium  utility ( I ) eff ( M ) = min I max  utility ( I )  Rich classification!
  129. 129. 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
  130. 130. Other properties: Fairness (max-min + min-max fair)   Competition monotonicity
  131. 131. Open issues Strongly poly algs for approximating   nonlinear convex programs  equilibria Insights into congestion control protocols? 

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