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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 $1 $0.99 Queries: 100 Books then 100 CDs CD $1 $0 B1 = B2 = $100 LOST Revenue 100$ Algorithm Greedy Bidder 1 Bidder 2 Example: Bidder1 Bidder 2 Book $1 $0.99 Queries: 100 Books then 100 CDs 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 advertisers queries Online bipartite matching advertisers queries Online bipartite matching advertisers queries Online bipartite matching advertisers queries Online bipartite matching advertisers queries Online bipartite matching advertisers queries Online bipartite matching advertisers queries 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 $ $$$$$$$$$ ¢ wine bread $$$$ cheese milk People want to maximize happiness – assume Find prices linear utilities.s.t. market clears Fisher‟s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) U u x i j ij ij Linear utilities: uij is utility derived by i on obtaining one unit of j Total utility of i, u u x i j ij ij 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 j ij ij Linear utilities: uij is utility derived by i on obtaining one unit of j Total utility of i, u u x i j ij ij 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) Agents: source-sink pairs ( s1 , t1 ),...( sk , tk ) 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 p(e): prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) TCP Congestion Control f(i): source rate p(e): 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. 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) Agents: sinks t1 ,..., tk 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 1 t 1 $10 2 s 2 t 2 $10 1 t 1 $10 $5 2 s 2 $5 t 2 $10 1 t 1 10 $120 2 s 2 t 2 $10 $30 1 t 1 $120 $10 2 s 2 $40 t 2 $10 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 s t 2 t 3 t 4 Find equilibrium prices and flows t 1 m(1) m(2) s cap(e) t 2 m(3) t 3 m(4 t4 ) 6 0 t 1 s t 2 t 3 t 4 min-cut separating s from all the sinks 6 0 t 1 s t 2 t 3 t 4 p 6 0 t 1 s t 2 t 3 t 4 p Throughout the algorithm: c(i): cost of cheapest path from s to t i m(i) sink t i demands flow f (i) c(i) m(i) i : c(i) p sink t i demands flow f (i) p 6 0 t 1 s t 2 t 3 t 4 p 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 6 5 0 0 f (2) 10 t 1 s t 2 t 3 t 4 p p 0 c(2) p0 6 5 c(1) c(3) c(4) p0 p 0 0 t 1 s t 2 t 3 t 4 p p 0 c(2) p0 c(1) c(3) p0 p1 6 5 2 0 0 0 t 1 s t 2 t 3 t 4 f (1) f (3) 30 p 0 p 1 6 5 2 0 0 0 t 1 s t 2 t 3 t 4 p 0 p 1 p c(4) p0 p1 p2 6 5 2 0 0 0 t 1 s t 2 t 3 t 4 f (4) 20 p 0 p 1 p 2 6 5 2 0 0 0 t 1 s t 2 t 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 s t 2 t 3 t 4 t t 1 s t 2 t 3 t 4 t t 1 s t 2 t 3 t 4 m(i) Capacity of t i t edge = f (i) c(i) t 6 0 t 1 s t 2 t 3 t 4 min s-t cut t 6 0 t 1 s t 2 t 3 t 4 p t 6 0 t 1 s t 2 t 3 t 4 p i : c(i) p t t 1 s t 2 t 3 t 4 m(i) p Capacity of t i t edge = f (i) p f(2)=10 t 6 5 0 0 t 1 s t 2 t 3 t 4 p p 0 c(2) p0 t 6 5 0 0 t 1 s t 2 t 3 t 4 p p 0 t 6 5 2 0 0 0 t 1 s t 2 t 3 t 4 c(2) p0 p 0 p 1 c(1) c(3) c(4) p0 p1 t t 1 s t 2 t 3 t 4 p 0 p 1 p t t 1 s t 2 t 3 t 4 c(4) p0 p1 p2 p 0 p 1 p 2 Eisenberg-Gale Program, 1959 max m(i ) log ui i s.t. i : ui j u ij x ij j : i x ij 1 ij : x ij 0 Lagrangian variables: prices of goods Using KKT conditions: optimal primal and dual solutions are in equilibrium Convex Program for Kelly‟s Model max m(i) log f (i) i s.t. i : f (i ) p fi p e : flow(e) c(e) i, p : fi 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 2 s1 t 1 1 t1 1 s 2 2 t 2 $1 Irrational for 2 sources & 3 sinks 3 2 s1 3 t 1 1 1 3 t 1 s 2 t 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 ( 2) m(3) s 1 c (e) s 2 s 3 Branching market (for broadcasting) m(1) m ( 2) m(3) s 1 c (e) s 2 s 3 Branching market (for broadcasting) m(1) m ( 2) m(3) s 1 c (e) s 2 s 3 Branching market (for broadcasting) m(1) m ( 2) m(3) s 1 c (e) s 2 s 3 Branching market (for broadcasting) Given: Network G = (V, E), directed edge capacities sources, S V 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 iS m(i) log bi 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 i m(i)log ui 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 Fisher Rational EG[2] EG Efficiency of Markets „„price of capitalism‟‟ Agents: differentabilities 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 1/ 2 k source-sink undirected 1/(2k 1) 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?