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

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 B 1, B 2, …, B N  Each advertiser provides bids for keywords he is interested in. Search Engine

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

The Adwords Problem: N advertisers;  Daily Budgets B 1, B 2, …, B N  Each advertiser provides bids for keywords he is interested in. Search Engine Select one Ad Advertiser pays his bid queries (online)

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

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

Example: $1$0.99 $1 $0 Book CD Bidder1Bidder 2 B 1 = B 2 = $100 Queries: 100 Books then 100 CDs Bidder 1 Bidder 2 Algorithm Greedy LOST Revenue 100$

Example: $1$0.99 $1 $0 Book CD Bidder1Bidder 2 B 1 = B 2 = $100 Queries: 100 Books then 100 CDs Bidder 1 Bidder 2 Optimal Allocation Revenue 199$

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?

General Equilibrium Theory Occupied center stage in Mathematical Economics for over a century Do markets even have inherently stable operating points?

Leon Walras, 1874 Pioneered general equilibrium theory

Supply-demand curves

Irving Fisher, 1891 Fundamental market model

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

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,

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, 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, Highly non-constructive

Adam Smith The Wealth of Nations 2 volumes, ‘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

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

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 Equilibrium

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) queueing delay (in TCP Vegas) p(e):

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):

primal process: packet rates at sources dual process: packet drop at links AIMD + RED converges to equilibrium in the limit TCP Congestion Control

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 with money m(1), … m(k) Find: equilibrium flows and edge prices

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 Equilibrium

$5

$10 $40 $30

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

m(1) m(2) m(3) m(4 ) cap(e)

min-cut separating from all the sinks 6060

6060

6060

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

6060

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;

nested cuts

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

Implementation

Capacity of edge =

min s-t cut 6060

6060

6060

Capacity of edge =

f(2)=10

Eisenberg-Gale Program, 1959

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

Convex Program for Kelly’s Model

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

Irrational for 2 sources & 3 sinks 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)

Given: Network G = (V, E), directed  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.

Eisenberg-Gale-type program for branching market 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 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, 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, JV, 2005) 3 sources branching: irrational Open: (no max-min theorems):  2 source-sink pairs, directed  2 sources, network coding

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]. Chakrabarty, Devanur & V., 2006:

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

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

Rich classification!

MarketEfficiency Single-source1 3-source branching k source-sink undirected 2 source-sink directedarbitrarily 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?