Algorithmic Game Theory and Internet Computing

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Algorithmic Game Theoryand Internet Computing
New Market Models and Algorithms

• Vijay V. Vazirani

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Markets
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Stock Markets
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Internet
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• Revolution in definition of markets

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

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

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• 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

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Theory of Algorithms

• Powerful tools and techniques
• developed over last 4 decades.

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Theory of Algorithms

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

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Adwords Market

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

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New algorithmic and game-theoretic questions

• Monika Henzinger, 2004 Find an on-line
• algorithm that maximizes Googles revenue.

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The Adwords Problem

• N advertisers
• Daily Budgets B1, B2, , BN
• Each advertiser provides bids for keywords he is
  interested in.

Search Engine
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The Adwords Problem

• N advertisers
• Daily Budgets B1, B2, , BN
• Each advertiser provides bids for keywords he is
  interested in.

Search Engine
queries (online)
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The Adwords Problem

• N advertisers
• Daily Budgets B1, B2, , BN
• Each advertiser provides bids for keywords he is
  interested in.

Search Engine
Select one Ad Advertiser pays his bid
queries (online)
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The Adwords Problem

• N advertisers
• Daily Budgets B1, B2, , BN
• 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
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The Adwords Problem

• N advertisers
• Daily Budgets B1, B2, , BN
• 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
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Example
Bidder1
Bidder 2
1 0.99
1 0
Book
Queries 100 Books then 100 CDs
CD
B1 B2 100
LOST
Revenue 100
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Example
Bidder1
Bidder 2
1 0.99
1 0
Book
Queries 100 Books then 100 CDs
CD
B1 B2 100
Revenue 199
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Generalizes online bipartite matching

• Each daily budget is 1, and
• each bid is 0/1.

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Online bipartite matching
queries
advertisers
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Online bipartite matching
queries
advertisers
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Online bipartite matching
queries
advertisers
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Online bipartite matching
queries
advertisers
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Online bipartite matching
queries
advertisers
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Online bipartite matching
queries
advertisers
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Online bipartite matching
queries
advertisers
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Online bipartite matching

• Karp, Vazirani Vazirani, 1990
• 1-1/e factor randomized algorithm.

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Online bipartite matching

• Karp, Vazirani Vazirani, 1990
• 1-1/e factor randomized algorithm. Optimal!

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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, bgtgt1

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Adwords Problem

• Mehta, Saberi, Vazirani Vazirani, 2005
• 1-1/e algorithm, assuming budgetsgtgtbids.

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Adwords Problem

• Mehta, Saberi, Vazirani Vazirani, 2005
• 1-1/e algorithm, assuming budgetsgtgtbids.
• Optimal!

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New Algorithmic Technique

• Idea Use both bid and
• fraction of left-over budget

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New Algorithmic Technique

• Idea Use both bid and
• fraction of left-over budget
• Correct tradeoff given by
• tradeoff-revealing family of LPs

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Historically, the study of markets

• has been of central importance,
• especially in the West

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A Capitalistic Economy

• depends crucially on pricing mechanisms,
• with very little intervention, to ensure
• Stability
• Efficiency
• Fairness

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Do markets even have inherentlystable operating
points?
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General Equilibrium TheoryOccupied center stage
in MathematicalEconomics for over a century
Do markets even have inherentlystable operating
points?
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Leon Walras, 1874

• Pioneered general
• equilibrium theory

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Supply-demand curves
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Irving Fisher, 1891

• Fundamental
• market model

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Fishers Model, 1891




milk

• People want to maximize happiness assume
• linear utilities.

Find prices s.t. market clears
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Fishers 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,

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Fishers 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.

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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.

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Kenneth Arrow

• Nobel Prize, 1972

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Gerard Debreu

• Nobel Prize, 1983

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Arrow-Debreu Theorem, 1954

• .
• Highly non-constructive

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Adam Smith

• The Wealth of Nations
• 2 volumes, 1776.
• invisible hand of the market
  

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What is needed today?

• An inherently algorithmic theory of
• market equilibrium
• New models that capture new markets

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

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Combinatorial Algorithm for Fishers Model

• Devanur, Papadimitriou, Saberi V., 2002
• Using primal-dual schema

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Primal-Dual Schema

• Highly successful algorithm design
• technique from exact and
• approximation algorithms

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Exact Algorithms for Cornerstone
Problems in P

• Matching (general graph)
• Network flow
• Shortest paths
• Minimum spanning tree
• Minimum branching

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Approximation Algorithms

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

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• No LPs known for capturing equilibrium
  allocations for Fishers model
• Eisenberg-Gale convex program, 1959
• DPSV Extended primal-dual schema to
• solving nonlinear convex
  programs

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A combinatorial market
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A combinatorial market
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A combinatorial market
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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

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Equilibrium

• Flows and edge prices
• f(i) flow of agent i
• p(e) price/unit flow of edge e
• Satisfying
• p(e)gt0 only if e is saturated
• flows go on cheapest paths
• money of each agent is fully spent

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Kellys resource allocation model, 1997
Mathematical framework for understanding TCP
congestion control
Highly successful theory
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TCP Congestion Control

• f(i) source rate
• prob. of packet loss (in TCP Reno)
• queueing delay (in TCP Vegas)

p(e)
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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
  someones
• dollar is to decrease 5 flow from
• someone elses dollar.

p(e)
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TCP Congestion Control

• primal process packet rates at sources
• dual process packet drop at links
• AIMD RED converges to equilibrium
• in the
  limit

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• Kelly V., 2002 Kellys model is a
• generalization of Fishers model.
• Find combinatorial polynomial time
• algorithms!

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Jain V., 2005

• Strongly polynomial combinatorial algorithm
• for single-source multiple-sink market

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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

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Equilibrium

• Flows and edge prices
• f(i) flow of agent i
• p(e) price/unit flow of edge e
• Satisfying
• p(e)gt0 only if e is saturated
• flows go on cheapest paths
• money of each agent is fully spent

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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)

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Auction of k identical goods

• p 0
• while there are gtk buyers
• raise p
• end
• sell to remaining k buyers at price p

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Find equilibrium prices and flows
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Find equilibrium prices and flows
m(1)
m(2)
m(3)
cap(e)
m(4)
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min-cut separating from all the sinks
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Throughout the algorithm
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sink demands flow
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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

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nested cuts
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• Flow and prices will
• Saturate all red cuts
• Use up sinks money
• Send flow on cheapest paths

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Implementation
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Capacity of edge
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min s-t cut
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Capacity of edge
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f(2)10
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Eisenberg-Gale Program, 1959
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• Lagrangian variables prices of goods
• Using KKT conditions
• optimal primal and dual solutions
• are in equilibrium

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Convex Program for Kellys Model
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JV Algorithm

• primal-dual alg. for nonlinear convex program
• primal variables flows
• dual variables prices of edges
• algorithm primal dual improvements

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Rational!!
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Irrational for 2 sources 3 sinks
1
1
1
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Irrational for 2 sources 3 sinks
Equilibrium prices
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Max-flow min-cut theorem!
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Other resource allocation markets

• 2 source-sink pairs (directed/undirected)
• Branchings rooted at sources (agents)

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Branching market (for broadcasting)
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Branching market (for broadcasting)
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Branching market (for broadcasting)
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Branching market (for broadcasting)
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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) gt 0 gt e is saturated
• each branching is cheapest possible
• money of each source fully used.

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Eisenberg-Gale-type program for branching
market
s.t. packing of branchings
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Other resource allocation markets

• 2 source-sink pairs (directed/undirected)
• Branchings rooted at sources (agents)
• Spanning trees
• Network coding

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

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

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• 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

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• 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

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Chakrabarty, Devanur V., 2006

• EG2 Eisenberg-Gale markets with 2 agents
• Theorem EG2 markets are rational.

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Chakrabarty, Devanur V., 2006

• EG2 Eisenberg-Gale markets with 2 agents
• Theorem EG2 markets are rational.
• Combinatorial EG2 markets polytope
• of feasible utilities can be described via
• combinatorial LP.
• Theorem Strongly poly alg for Comb EG2.

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3-source branching
Single-source
SUA
2 s-s undir
Comb EG2
2 s-s dir
Rational
Fisher
EG2
EG
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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?

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Efficiency
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Efficiency

• Rich classification!

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

• Fairness (max-min min-max fair)
• Competition monotonicity

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Open issues

• Strongly poly algs for approximating
• nonlinear convex programs
• equilibria
• Insights into congestion control protocols?

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