Algorithmic Game Theory New Market Models and Internet

n Revolution in definition of markets n New markets defined by ¨ Google ¨

n Revolution in definition of markets n Massive computational power available for running these

Theory of Algorithms n Powerful tools and techniques developed over last 4 decades.

Theory of Algorithms n Powerful tools and techniques developed over last 4 decades. n

Adwords Market n Created by search engine companies ¨ Google ¨ Yahoo! ¨ MSN

New algorithmic and game-theoretic questions n Monika Henzinger, 2004: Find an on-line algorithm that

The Adwords Problem: N advertisers; ¨ Daily Budgets B 1, B 2, …, BN

Example: Bidder 1 Bidder 2 Book $1 CD $1 $0. 99 Queries: 100 Books

Generalizes online bipartite matching n Each daily budget is $1, and each bid is

Online bipartite matching advertisers queries

Online bipartite matching n Karp, Vazirani & Vazirani, 1990: 1 -1/e factor randomized algorithm.

Adwords Problem n Mehta, Saberi, Vazirani & Vazirani, 2005: 1 -1/e algorithm, assuming budgets>>bids.

$New Algorithmic Technique n Idea: Use both bid and fraction of left-over budget$

$New Algorithmic Technique n Idea: Use both bid and fraction of left-over budget n$

Historically, the study of markets n has been of central importance, especially in the

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

Do markets even have inherently stable operating points?

Do markets even have inherently stable operating points? General Equilibrium Theory Occupied center stage

Leon Walras, 1874 n Pioneered general equilibrium theory

Irving Fisher, 1891 n Fundamental market model

Fisher’s Model, 1891 $ $$$$$ ¢ wine bread cheese n milk $$$$ People want

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

Fisher’s Model n n n buyers, with specified money, m(i) k goods (each unit

Arrow-Debreu Theorem, 1954 n Celebrated theorem in Mathematical Economics n Established existence of market

Arrow-Debreu Theorem, 1954. n Highly non-constructive

Adam Smith n The Wealth of Nations 2 volumes, 1776. n ‘invisible hand’ of

What is needed today? n An inherently algorithmic theory of market equilibrium n New

n Beginnings of such a theory, within Algorithmic Game Theory n Started with combinatorial

Combinatorial Algorithm for Fisher’s Model n Devanur, Papadimitriou, Saberi & V. , 2002 Using

Primal-Dual Schema n Highly successful algorithm design technique from exact and approximation algorithms

Exact Algorithms for Cornerstone Problems in P: n n n Matching (general graph) Network

Approximation Algorithms set cover Steiner tree Steiner network k-MST scheduling. . . facility location

n No LP’s known for capturing equilibrium allocations for Fisher’s model n Eisenberg-Gale convex

A combinatorial market n Given: ¨ Network G = (V, E) (directed or undirected)

Equilibrium n Flows and edge prices f(i): flow of agent i ¨ p(e): price/unit

Kelly’s resource allocation model, 1997 Mathematical framework for understanding TCP congestion control Highly successful

TCP Congestion Control f(i): source rate n p(e): prob. of packet loss (in TCP

TCP Congestion Control n n f(i): source rate p(e): prob. of packet loss (in

TCP Congestion Control primal process: packet rates at sources dual process: packet drop at

n Kelly & V. , 2002: Kelly’s model is a generalization of Fisher’s model.

Jain & V. , 2005: n Strongly polynomial combinatorial algorithm for single-source multiple-sink market

Single-source multiple-sink market n Given: ¨ Network G = (V, E), s: source ¨

Auction of k identical goods p = 0; n while there are >k buyers:

Find equilibrium prices and flows m(1) cap(e) m(2) m(3) m(4)

60 min-cut separating from all the sinks

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

Auction of edges in cut p = 0; n while the cut is over-saturated:

n Flow and prices will: ¨ Saturate all red cuts ¨ Use up sinks’

n Lagrangian variables: prices of goods n Using KKT conditions: optimal primal and dual

JV Algorithm n primal-dual alg. for nonlinear convex program n “primal” variables: flows n

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

Irrational for 2 sources & 3 sinks Equilibrium prices

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

Branching market (for broadcasting) n Given: Network G = (V, E), directed ¨ edge

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

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

Eisenberg-Gale Market n A market whose equilibrium is captured as an optimal solution to

n Theorem: Strongly polynomial algs for following markets : ¨ 2 source-sink pairs, undirected

Chakrabarty, Devanur & V. , 2006: n EG[2]: Eisenberg-Gale markets with 2 agents n

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

Efficiency of Markets ‘‘price of capitalism’’ n Agents: n ¨ different abilities to control

Market Efficiency Single-source 1 3 -source branching k source-sink undirected 2 source-sink directed arbitrarily

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

Open issues n Strongly poly algs for approximating ¨ nonlinear convex programs ¨ equilibria

Slides: 138

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

Markets

Stock Markets

Internet

n Revolution in definition of markets

n Revolution in definition of markets n New markets defined by ¨ Google ¨ Amazon ¨ Yahoo! ¨ Ebay

n Revolution in definition of markets n Massive computational power available for running these markets in a centralized or distributed manner

n Revolution in definition of markets n Massive computational power available for running these markets in a centralized or distributed manner n Important to find good models and algorithms for these markets

Theory of Algorithms n Powerful tools and techniques developed over last 4 decades.

Theory of Algorithms n Powerful tools and techniques developed over last 4 decades. n Recent study of markets has contributed handsomely to this theory as well!

Adwords Market n Created by search engine companies ¨ Google ¨ Yahoo! ¨ MSN n Multi-billion dollar market n Totally revolutionized advertising, especially by small companies.

New algorithmic and game-theoretic questions n Monika Henzinger, 2004: Find an on-line algorithm that maximizes Google’s revenue.

The Adwords Problem: N advertisers; ¨ Daily Budgets B 1, B 2, …, BN ¨ Each advertiser provides bids for keywords he is interested in. Search Engine

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

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

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

Example: Bidder 1 Bidder 2 Book $1 CD $1 $0. 99 Queries: 100 Books then 100 CDs $0 B 1 = B 2 = $100 LOST Revenue 100$ Algorithm Greedy Bidder 1 Bidder 2

Example: Bidder 1 Bidder 2 Book $1 CD $1 $0. 99 Queries: 100 Books then 100 CDs $0 B 1 = B 2 = $100 Revenue 199$ Optimal Allocation Bidder 1 Bidder 2

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

Online bipartite matching advertisers queries

Online bipartite matching n Karp, Vazirani & Vazirani, 1990: 1 -1/e factor randomized algorithm.

Online bipartite matching n Karp, Vazirani & Vazirani, 1990: 1 -1/e factor randomized algorithm. Optimal!

Online bipartite matching n Karp, Vazirani & Vazirani, 1990: 1 -1/e factor randomized algorithm. Optimal! n Kalyanasundaram & Pruhs, 1996: 1 -1/e factor algorithm for b-matching: Daily budgets $b, bids $0/1, b>>1

Adwords Problem n Mehta, Saberi, Vazirani & Vazirani, 2005: 1 -1/e algorithm, assuming budgets>>bids.

Adwords Problem n Mehta, Saberi, Vazirani & Vazirani, 2005: 1 -1/e algorithm, assuming budgets>>bids. Optimal!

$New Algorithmic Technique n Idea: Use both bid and fraction of left-over budget$

New Algorithmic Technique n Idea: Use both bid and fraction of left-over budget

$New Algorithmic Technique n Idea: Use both bid and fraction of left-over budget n$

New Algorithmic Technique n Idea: Use both bid and fraction of left-over budget n Correct tradeoff given by tradeoff-revealing family of LP’s

Historically, the study of markets n has been of central importance, especially in the West

A Capitalistic Economy depends crucially on pricing mechanisms, with very little intervention, to ensure: Stability n Efficiency n Fairness n

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 n Pioneered general equilibrium theory

Supply-demand curves

Irving Fisher, 1891 n Fundamental market model

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

Fisher’s Model n n 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,

Fisher’s Model n n 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.

Arrow-Debreu Theorem, 1954 n Celebrated theorem in Mathematical Economics n Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem.

Kenneth Arrow n Nobel Prize, 1972

Gerard Debreu n Nobel Prize, 1983

Arrow-Debreu Theorem, 1954. n Highly non-constructive

Adam Smith n The Wealth of Nations 2 volumes, 1776. n ‘invisible hand’ of the market

What is needed today? n An inherently algorithmic theory of market equilibrium n New models that capture new markets

n Beginnings of such a theory, within Algorithmic Game Theory n Started with combinatorial algorithms for traditional market models n New market models emerging

Combinatorial Algorithm for Fisher’s Model n Devanur, Papadimitriou, Saberi & V. , 2002 Using primal-dual schema

Primal-Dual Schema n Highly successful algorithm design technique from exact and approximation algorithms

Exact Algorithms for Cornerstone Problems in P: n n n Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching

Approximation Algorithms set cover Steiner tree Steiner network k-MST scheduling. . . facility location k-median multicut feedback vertex set

n No LP’s known for capturing equilibrium allocations for Fisher’s model n Eisenberg-Gale convex program, 1959 n DPSV: Extended primal-dual schema to solving nonlinear convex programs

A combinatorial market

A combinatorial market n Given: ¨ Network G = (V, E) (directed or undirected) ¨ Capacities on edges c(e) ¨ Agents: source-sink pairs with money m(1), … m(k) n Find: equilibrium flows and edge prices

Equilibrium n Flows and edge prices f(i): flow of agent i ¨ p(e): price/unit flow of edge e ¨ n 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 n p(e): prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) n

TCP Congestion Control n n 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

n Kelly & V. , 2002: Kelly’s model is a generalization of Fisher’s model. n Find combinatorial polynomial time algorithms!

Jain & V. , 2005: n Strongly polynomial combinatorial algorithm for single-source multiple-sink market

Single-source multiple-sink market n Given: ¨ Network G = (V, E), s: source ¨ Capacities on edges c(e) ¨ Agents: sinks with money m(1), … m(k) n Find: equilibrium flows and edge prices

$5 $5

$30 $10 $40

Jain & V. , 2005: n Strongly polynomial combinatorial algorithm for single-source multiple-sink market n Ascending price auction ¨ Buyers: sinks (fixed budgets, maximize flow) ¨ Sellers: edges (maximize price)

Auction of k identical goods p = 0; n while there are >k buyers: raise p; n end; n sell to remaining k buyers at price p; n

Find equilibrium prices and flows

Find equilibrium prices and flows m(1) cap(e) m(2) m(3) m(4)

60 min-cut separating from all the sinks

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

sink 60 demands flow

Auction of edges in cut p = 0; n while the cut is over-saturated: raise p; n end; n assign price p to all edges in the cut; n

60 50

60 50 20

60 50 20 nested cuts

n Flow and prices will: ¨ Saturate all red cuts ¨ Use up sinks’ money ¨ Send flow on cheapest paths

Implementation

Capacity of edge =

60 min s-t cut

Capacity of edge =

f(2)=10 60 50

60 50

60 50 20

Eisenberg-Gale Program, 1959

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

Convex Program for Kelly’s Model

JV Algorithm n primal-dual alg. for nonlinear convex program n “primal” variables: flows n “dual” variables: prices of edges n algorithm: primal & dual improvements Allocations Prices

Rational!!

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

Irrational for 2 sources & 3 sinks Equilibrium prices

Max-flow min-cut theorem!

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

Branching market (for broadcasting)

Branching market (for broadcasting) n Given: Network G = (V, E), directed ¨ edge capacities ¨ sources, ¨ money of each source n 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 s. t. packing of branchings

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

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

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

n 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) n 3 sources branching: irrational

Chakrabarty, Devanur & V. , 2006: n EG[2]: Eisenberg-Gale markets with 2 agents n Theorem: EG[2] markets are rational.

Chakrabarty, Devanur & V. , 2006: n EG[2]: Eisenberg-Gale markets with 2 agents n Theorem: EG[2] markets are rational. n Combinatorial EG[2] markets: polytope of feasible utilities can be described via combinatorial LP. n 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 EG[2] EG Rational

Efficiency of Markets ‘‘price of capitalism’’ n Agents: n ¨ different abilities to control prices ¨ idiosyncratic ways of utilizing resources n Q: Overall output of market when forced to operate at equilibrium?

Efficiency

Efficiency n Rich classification!

Market Efficiency Single-source 1 3 -source branching k source-sink undirected 2 source-sink directed arbitrarily small

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

Open issues n Strongly poly algs for approximating ¨ nonlinear convex programs ¨ equilibria n Insights into congestion control protocols?