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
(online)
Search Engine
The Adwords Problem:
N advertisers;
 Daily Budgets B1, B2, …, BN
 Each advertiser provides bids for keywords he is interested in.
queries
(online)
Search Engine
Select one Ad
Advertiser
pays his bid
The Adwords Problem:
N advertisers;
 Daily Budgets B1, B2, …, 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 B1, B2, …, 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:
Bidder1 Bidder 2
Book
$1
CD
$1
$0.99
Queries: 100 Books then 100 CDs
$0
B1 = B2 = $100
LOST
Revenue
100$
Algorithm Greedy
Bidder 1
Bidder 2
Example:
Bidder1 Bidder 2
Book
$1
CD
$1
$0.99
Queries: 100 Books then 100 CDs
$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
Findutilities.
prices s.t. market clears
linear
Fisher’s Model


n buyers, with specified money, m(i) for buyer i
k goods (unit amount of each good) U  u x

Linear utilities: uij is utility derived by i
on obtaining one unit of j
Total utility of i,
i


u  u x
x [0,1]
i
ij
j
ij
ij
j
ij ij
Fisher’s Model


n buyers, with specified money, m(i)
k goods (each unit amount, w.l.o.g.) U  u x

Linear utilities: uij is utility derived by i
on obtaining one unit of j
Total utility of i,
i


u  u x
i

j
Find prices s.t. market clears, i.e.,
all goods sold, all money spent.
ij
ij
j
ij ij
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
Steiner tree
Steiner network
k-MST
scheduling . . .
facility location
k-median
multicut
feedback vertex set

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
s
m ( 2)
2
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
 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
$5
t
1
$10
t
$10
2
s
2
$5
2
1
t
1
$120
10
2
s
2
t
2
$10
$30
1
$10
t
1
$120
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
s
1
t
2
t
3
t
4
Find equilibrium prices and flows
t
s
cap(e)
1
m(1)
t
2
m(2)
m(3)
t 3 m(4
t4 )
6
0
t
s
min-cut separating
1
t
2
t
3
t
s from all the sinks
4
6
0
t
1
t
s
p
2
t
3
t
4
6
0
t
1
t
s
p

2
t
3
t
4
Throughout the algorithm:
c(i): cost of cheapest path from
sink
t
i
s
to
m(i )
demands flow f (i ) 
c(i)
t
i
i : c(i )  p
sink
t
i
m(i )
demands flow f (i ) 
p
6
0
t
1
t
s
p

2
t
3
t
4
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
0
5
0
f (2)  10
t
s
t
p p
0
2
1
t
3
t
4
c(2)  p0
6
0
c(1)  c(3)  c(4)  p0  p
5
0
t
s
t
p
0
1
t
2
p

3
t
4
c(2)  p0
6
0
5
0
t
s
t
p
0
c(1)  c(3)  p0  p1
2
0
1
t
2
p
1
3
t
4
f (1)  f (3)  30
6
0
5
0
2
0
t
s
t
p
0
1
t
2
p
1
3
t
p
4

6
0
5
0
t
s
t
c(4)  p0  p1  p2
2
0
1
t
2
3
t
4
f (4)  20
p
0
p
1
p
2
6
0
5
0
2
0
t
s
t
p
0
1
t
2
p
1
3
t
p
4
nested cuts
2

Flow and prices will:
 Saturate
all red cuts
 Use up sinks’ money
 Send flow on cheapest paths
Implementation
t
s
1
t
2
t
3
t
4
t
t
s
1
t
2
t
3
t
4
t
t
s
Capacity of
1
t
2
t
3
t i  t edge =
t
4
m(i )
f (i ) 
c(i)
t
6
0
t
1
t
s
min s-t cut
2
t
3
t
4
t
6
0
t
1
t
s
p
2
t
3
t
4
t
6
0
t
1
t
s
p

2
t
3
t
4
i : c(i )  p
t
t
1
t
s
p

2
t
3
t
4
Capacity of t i  t edge =
m(i )
f (i ) 

p
f(2)=10
t
6
0
5
0
t
s
t
p p
2
1
t
3
t
4
c(2)  p0
0
t
6
0
5
0
t
s
t
p
0
1
t
2
p

3
t
4
t
6
0
5
0
2
0
t
s
t
1
t
2
3
t
4
c(2)  p0
p
0
p
1
c(1)  c(3)  c(4)  p0  p1
t
t
s
t
p
0
1
t
2
p
1
3
t
p
4

t
t
s
t
p
0
1
t
2
p
1
3
t
p
4
c(4)  p0  p1  p2
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 f i
p
e : flow(e)  c(e)
i, p : f i  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
1
1
t
s
1
2
t
t
1
1
2
2
$1
Irrational for 2 sources & 3 sinks
s
1
3
1
t
s
1
3
1 3
2
Equilibrium prices
2
t
t
1
2
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 ( 2)
m(1)
s
1
c (e)
s
2
m (3)
s
3
Branching market (for broadcasting)
m ( 2)
m(1)
s
1
c (e)
s
2
m (3)
s
3
Branching market (for broadcasting)
m ( 2)
m(1)
s
1
c (e)
s
2
m (3)
s
3
Branching market (for broadcasting)
m ( 2)
m(1)
s
1
c (e)
s
2
m (3)
s
3
Branching market (for broadcasting)

Given: Network G = (V, E),
directed
 edge
capacities
S V
 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 iS 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
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
 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?