algorithmic game theory and internet computing vijay v. vazirani new market models and algorithms
TRANSCRIPT
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Algorithmic Game Theoryand Internet Computing
Vijay V. Vazirani
New Market Models
and Algorithms
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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 byGoogle AmazonYahoo!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 companiesGoogleYahoo!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 Google’s 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 Enginequeries (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 EngineSelect 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 EngineSelect 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 EngineSelect 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:
$1 $0.99
$1 $0
Book
CD
Bidder1 Bidder 2
B1 = B2 = $100
Queries: 100 Books then 100 CDs
Bidder 1 Bidder 2
Algorithm Greedy
LOST
Revenue100$
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Example:
$1 $0.99
$1 $0
Book
CD
Bidder1 Bidder 2
B1 = B2 = $100
Queries: 100 Books then 100 CDs
Bidder 1 Bidder 2
Optimal Allocation
Revenue199$
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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
advertisers queries
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Online bipartite matching
advertisers queries
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Online bipartite matching
advertisers queries
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Online bipartite matching
advertisers queries
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Online bipartite matching
advertisers queries
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Online bipartite matching
advertisers queries
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Online bipartite matching
advertisers queries
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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, b>>1
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Adwords Problem
Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming budgets>>bids.
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Adwords Problem
Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming budgets>>bids.
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 LP’s
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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 Mathematical
Economics 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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Fisher’s Model, 1891
milkcheese
winebread
¢¢
$$$$$$$$$$$$$$$$$$
$$
$$$$$$$$
People want to maximize happiness – assume
linear utilities.Find prices s.t. market clears
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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,
i ij ijj
U u xiju
]1,0[
x
xuuij
ijj iji
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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.
i ij ijj
U u xiju
xuu ijj iji
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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 Fisher’s 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 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
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2s
1s
2t
1t
A combinatorial market
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2s
1s
2t
1t
A combinatorial market
)(ec
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2s
1s
2t
1t
A combinatorial market
)1(m
)2(m
)(ec
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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
1 1( , ),...( , )k ks t s t
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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
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Kelly’s 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 someone’s
dollar is to decrease 5% flow from
someone else’s dollar.
p(e):
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primal process: packet rates at sources
dual process: packet drop at links
AIMD + RED converges to equilibrium
in the limit
TCP Congestion Control
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Kelly & V., 2002: Kelly’s model is a
generalization of Fisher’s 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: sourceCapacities on edges c(e)Agents: sinks
with money m(1), … m(k)
Find: equilibrium flows and edge prices
1,..., kt t
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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
![Page 74: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/74.jpg)
s
t1
t2
2
2
110$
10$
![Page 75: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/75.jpg)
s
t1
t2
2
2
1 10$
10$
$5
$5
![Page 76: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/76.jpg)
s
t1
t2
2
2
1 10$
10$
120$
![Page 77: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/77.jpg)
s
t1
t2
2
2
1 120$
10$
$10
$40
$30
![Page 78: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/78.jpg)
Jain & V., 2005:
Strongly polynomial combinatorial algorithm
for single-source multiple-sink market
Ascending price auctionBuyers: sinks (fixed budgets, maximize flow)Sellers: edges (maximize price)
![Page 79: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/79.jpg)
Auction of k identical goods
p = 0; while there are >k buyers:
raise p; end; sell to remaining k buyers at price p;
![Page 80: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/80.jpg)
s
t1
t2
t3
t4
Find equilibrium prices and flows
![Page 81: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/81.jpg)
s
t1
t2
t3
t4
Find equilibrium prices and flows
m(1)
m(2)
m(3)m(4)
cap(e)
![Page 82: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/82.jpg)
s
t1
t2
t3
t4
min-cut separating from all the sinkss
60
![Page 83: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/83.jpg)
s
t1
t2
t3
t4
p
60
![Page 84: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/84.jpg)
s
t1
t2
t3
t4
p
60
![Page 85: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/85.jpg)
Throughout the algorithm:
s itc(i): cost of cheapest path from to
sink demands flow ( )
( )( )
m if i
c iit
![Page 86: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/86.jpg)
s
t1
t2
t3
t4
p
: ( )i c i p
60
sink demands flow ( )
( )m i
f ip
it
![Page 87: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/87.jpg)
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;
![Page 88: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/88.jpg)
s
t1
t2 t3
t4
pp 0
0(2)c p
60 50 (2) 10f
![Page 89: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/89.jpg)
s
t1
t2 t3
t4
p0
p
0(2)c p
60 50 0(1) (3) (4)c c c p p
![Page 90: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/90.jpg)
s
t1
t2 t3
t4
p0
p1
60 50 20
0(2)c p
0 1(1) (3)c c p p
(1) (3) 30f f
![Page 91: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/91.jpg)
s
t1
t2 t3
t4
p0
p1
p
60 50 20
![Page 92: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/92.jpg)
s
t1
t2 t3
t4
p0
p1 p
2
60 50 200 1 2(4)c p p p
(4) 20f
![Page 93: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/93.jpg)
s
t1
t2 t3
t4
p0
p1 p
2 nested cuts
60 50 20
![Page 94: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/94.jpg)
Flow and prices will:
Saturate all red cutsUse up sinks’ moneySend flow on cheapest paths
![Page 95: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/95.jpg)
s
t1
t2
t3
t4
Implementation
![Page 96: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/96.jpg)
s
t1
t2
t3
t4
t
![Page 97: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/97.jpg)
s
t1
t2
t3
t4
t
Capacity of edge =tt i
( )( )
( )
m if i
c i
![Page 98: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/98.jpg)
s
t1
t2
t3
t4
t
min s-t cut
60
![Page 99: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/99.jpg)
s
t1
t2
t3
t4
t
p
60
![Page 100: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/100.jpg)
s
t1
t2
t3
t4
t
p
60
![Page 101: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/101.jpg)
s
t1
t2
t3
t4
t
p tt i
Capacity of edge =
( )( )
m if i
p
: ( )i c i p
![Page 102: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/102.jpg)
s
t1
t2 t3
t4
t
pp 0
0(2)c p
60 50
f(2)=10
![Page 103: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/103.jpg)
s
t1
t2 t3
t4
t
p0
p
60 50
![Page 104: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/104.jpg)
s
t1
t2 t3
t4
t
p0
p1
0(2)c p
0 1(1) (3) (4)c c c p p
60 50 20
![Page 105: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/105.jpg)
s
t1
t2 t3
t4
t
p0
p1
p
![Page 106: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/106.jpg)
s
t1
t2 t3
t4
t
p0
p1 p
2
0 1 2(4)c p p p
![Page 107: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/107.jpg)
Eisenberg-Gale Program, 1959
max ( ) log
. .
:
: 1
: 0
ii
i ij ijj
iji
ij
m i u
s t
i u
j
ij
u xx
x
![Page 108: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/108.jpg)
Lagrangian variables: prices of goods
Using KKT conditions:
optimal primal and dual solutions
are in equilibrium
![Page 109: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/109.jpg)
Convex Program for Kelly’s Model
max ( ) log ( )
. .
: ( )
: ( ) ( )
, : 0
i
pip
pi
m i f i
s t
i f i f
e flow e c e
i p f
![Page 110: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/110.jpg)
JV Algorithm
primal-dual alg. for nonlinear convex program
“primal” variables: flows
“dual” variables: prices of edges
algorithm: primal & dual improvements
Allocations Prices
![Page 111: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/111.jpg)
Rational!!
![Page 112: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/112.jpg)
Irrational for 2 sources & 3 sinks
s1 t1
1
s2
t2
1
t21 2
$1 $1
$1
![Page 113: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/113.jpg)
Irrational for 2 sources & 3 sinks
s1 t1
1
s2
t2
1
t2
31
3
3
Equilibrium prices
![Page 114: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/114.jpg)
Max-flow min-cut theorem!
![Page 115: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/115.jpg)
Other resource allocation markets
2 source-sink pairs (directed/undirected) Branchings rooted at sources (agents)
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Branching market (for broadcasting)
s1 s2
)1(m3s
)2(m)(ec
(3)m
![Page 117: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/117.jpg)
Branching market (for broadcasting)
s1 s2
)1(m3s
)2(m)(ec
(3)m
![Page 118: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/118.jpg)
Branching market (for broadcasting)
s1 s2
)1(m3s
)2(m)(ec
(3)m
![Page 119: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/119.jpg)
Branching market (for broadcasting)
s1 s2
)1(m3s
)2(m)(ec
(3)m
![Page 120: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/120.jpg)
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.
S V
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Eisenberg-Gale-type program for branching market
max ( ) log ii Sm i b
s.t. packing of branchings
![Page 122: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/122.jpg)
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
max ( ) log iim i u
s.t. packing constraints
![Page 124: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/124.jpg)
Eisenberg-Gale Market
A market whose equilibrium is captured
as an optimal solution to an
Eisenberg-Gale-type program
![Page 125: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/125.jpg)
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
![Page 126: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/126.jpg)
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, directed2 sources, network coding
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EG[2]: Eisenberg-Gale markets with 2 agents
Theorem: EG[2] markets are rational.
Chakrabarty, Devanur & V., 2006:
![Page 128: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/128.jpg)
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:
![Page 129: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/129.jpg)
EG
Rational
Comb EG[2]
SUA
EG[2]
3-source branching
Fisher
2 s-s undir
2 s-s dir
Single-source
![Page 130: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/130.jpg)
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?
![Page 131: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/131.jpg)
Efficiency
( )( ) min
max ( )I
equilibrium utility Ieff M
utility I
![Page 132: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/132.jpg)
Efficiency
Rich classification!
( )( ) min
max ( )I
equilibrium utility Ieff M
utility I
![Page 133: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/133.jpg)
1/(2 1)k
Market EfficiencySingle-source 1
3-source branching
k source-sink undirected
2 source-sink directed arbitrarily
small
1/ 2
. . 1/( 1)l b k
![Page 134: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/134.jpg)
Other properties:
Fairness (max-min + min-max fair) Competition monotonicity
![Page 135: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/135.jpg)
Open issues
Strongly poly algs for approximatingnonlinear convex programsequilibria
Insights into congestion control protocols?
![Page 136: Algorithmic Game Theory and Internet Computing Vijay V. Vazirani New Market Models and Algorithms](https://reader037.vdocuments.net/reader037/viewer/2022110210/56649e865503460f94b88d4a/html5/thumbnails/136.jpg)
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