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  • Network with us in the USA - We can assist in business contacts in North America.

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    Harvard Harvard Presentation Transcript

    • 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 centralized or distributed manner
    • 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
    • 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 markets has contributed  handsomely to this theory as well!
    • Adwords Market Created by search engine companies   Google  Yahoo!  MSN Multi-billion dollar market  Totally revolutionized advertising, especially  by small companies.
    • New algorithmic and game-theoretic questions Monika Henzinger, 2004: Find an on-line  algorithm that maximizes Google’s revenue.
    • The Adwords Problem: N advertisers;  Daily Budgets B1, B2, …, BN Each advertiser provides bids for keywords he is interested in.  Search Engine
    • The Adwords Problem: N advertisers;  Daily Budgets B1, B2, …, BN Each advertiser provides bids for keywords he is interested in.  queries Search Engine (online)
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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! Kalyanasundaram & Pruhs, 1996:  1-1/e factor algorithm for b-matching: Daily budgets $b, bids $0/1, b>>1
    • 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 given by  tradeoff-revealing family of LP’s
    • 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   Efficiency  Fairness
    • Do markets even have inherently stable operating points?
    • Do markets even have inherently stable operating points? General Equilibrium Theory Occupied center stage in Mathematical Economics for over a century
    • Leon Walras, 1874 Pioneered general  equilibrium theory
    • Supply-demand curves
    • Irving Fisher, 1891 Fundamental  market model
    • Fisher’s Model, 1891 $ $$$$$$$$$ ¢ wine bread $$$$ milk cheese People want to maximize happiness – assume  linear utilities. s.t. market clears Find prices
    • 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
    • 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.
    • 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.
    • 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  the market
    • What is needed today? An inherently algorithmic theory of  market equilibrium New models that capture new markets 
    • Beginnings of such a theory, within  Algorithmic Game Theory Started with combinatorial algorithms  for traditional market models New market models emerging 
    • Combinatorial Algorithm for Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002  Using primal-dual schema
    • 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 paths  Minimum spanning tree  Minimum branching 
    • Approximation Algorithms set cover facility location Steiner tree k-median Steiner network multicut k-MST feedback vertex set scheduling . . .
    • 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
    • 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 t1 t2
    • 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 
    • 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 
    • Kelly’s resource allocation model, 1997 Mathematical framework for understanding TCP congestion control Highly successful theory
    • TCP Congestion Control f(i): source rate  prob. of packet loss (in TCP Reno)  p(e): queueing delay (in TCP Vegas)
    • 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.
    • TCP Congestion Control primal process: packet rates at sources dual process: packet drop at links AIMD + RED converges to equilibrium in the limit
    • Kelly & V., 2002: Kelly’s model is a  generalization of Fisher’s model. Find combinatorial polynomial time  algorithms!
    • 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) t1 ,..., tk  Agents: sinks with money m(1), … m(k) Find: equilibrium flows and edge prices 
    • 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 
    • t $10 1 1 2 s 2 t $10 2
    • t $10 1 1 $5 2 s 2 t $5 $10 2
    • t $10 120 1 1 2 s 2 t $10 2
    • t $120 $30 1 1 $10 2 s 2 t $40 $10 2
    • 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)
    • Auction of k identical goods p = 0;   while there are >k buyers: raise p;  end;  sell to remaining k buyers at price p;
    • Find equilibrium prices and flows t 1 t s 2 t 3 t 4
    • Find equilibrium prices and flows t 1 m(1) t s m(2) 2 t m(3) cap(e) 3 t m(4) 4
    • 60 t 1 t s 2 t 3 t 4 s from all the sinks min-cut separating
    • 60 t 1 t s 2 t 3 t 4 p
    • 60 t 1 t s 2 t 3 t 4 p ᆳ
    • Throughout the algorithm: sto t i c(i): cost of cheapest path from m(i ) t f (i ) = i sink demands flow c(i )
    • 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
    • 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;
    • c(2) = p0 60 50 f (2) = 10 t 1 s t t 2 3 t 4 p =p 0
    • 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
    • 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
    • 60 50 20 t 1 s t t 2 3 t 4 p p p  0 1
    • 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
    • 60 50 20 t 1 s t t 2 3 t 4 p p p nested cuts 0 1 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 4
    • t t 1 t s 2 t 3 t 4
    • t t 1 t s 2 t 3 t 4 m(i ) f (i ) = t  t edge i Capacity of = c(i )
    • t 60 t 1 t s 2 t 3 t 4 min s-t cut
    • t 60 t 1 t s 2 t 3 t 4 p
    • t 60 t 1 t s 2 t 3 t 4 p 
    • 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
    • f(2)=10 t 60 50 t 1 s t t 2 3 t 4 p =p c(2) = p0 0
    • t 60 50 t 1 s t t 2 3 t 4 p p  0
    • 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
    • t t 1 s t t 2 3 t 4 p p p  0 1
    • t t 1 s t t 2 3 t 4 c(4) = p0 + p1 + p2 p p p 0 1 2
    • 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
    • Lagrangian variables: prices of goods  Using KKT conditions:  optimal primal and dual solutions are in equilibrium
    • 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
    • JV Algorithm primal-dual alg. for nonlinear convex program  “primal” variables: flows  “dual” variables: prices of edges  algorithm: primal & dual improvements  Allocations Prices
    • Rational!!
    • Irrational for 2 sources & 3 sinks $1 $1 s 2 t 1 t 1 1 1 s t 1 2 2 2 $1
    • Irrational for 2 sources & 3 sinks 3 s 2 t 1 3 t 1+ 3 1 1 1 s t 2 2 Equilibrium prices
    • Max-flow min-cut theorem!
    • Other resource allocation markets 2 source-sink pairs (directed/undirected)   Branchings rooted at sources (agents)
    • Branching market (for broadcasting) m(1) m(3) m ( 2) s s s c(e) 1 2 3
    • Branching market (for broadcasting) m(1) m(3) m ( 2) s s s c(e) 2 1 3
    • Branching market (for broadcasting) m(1) m(3) m ( 2) s s s c(e) 1 2 3
    • Branching market (for broadcasting) m(1) m(3) m ( 2) s s s c(e) 2 1 3
    • 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. 
    • Eisenberg-Gale-type program for branching market max ¥ S m(i ) log bi iᅫ s.t. packing of branchings
    • Other resource allocation markets 2 source-sink pairs (directed/undirected)   Branchings rooted at sources (agents)  Spanning trees  Network coding
    • 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 Eisenberg-Gale-type program
    • 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 
    • 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
    • Chakrabarty, Devanur & V., 2006: EG[2]: Eisenberg-Gale markets with 2 agents  Theorem: EG[2] markets are rational. 
    • 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]. 
    • 3-source branching Single-source 2 s-s undir SUA Comb EG[2] 2 s-s dir Rational Fisher EG[2] EG
    • 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?
    • Efficiency equilibrium  utility ( I ) eff ( M ) = min I max  utility ( I )
    • Efficiency equilibrium  utility ( I ) eff ( M ) = min I max  utility ( I )  Rich classification!
    • 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
    • Other properties: Fairness (max-min + min-max fair)   Competition monotonicity
    • Open issues Strongly poly algs for approximating   nonlinear convex programs  equilibria Insights into congestion control protocols? 