7. Revolution in definition of markets
Massive computational power available
for running these markets in a
centralized or distributed manner
8. 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
9. Theory of Algorithms
Powerful tools and techniques
developed over last 4 decades.
10. Theory of Algorithms
Powerful tools and techniques
developed over last 4 decades.
Recent study of markets has contributed
handsomely to this theory as well!
11. Adwords Market
Created by search engine companies
Google
Yahoo!
MSN
Multi-billion dollar market
Totally revolutionized advertising, especially
by small companies.
12.
13.
14.
15. New algorithmic and
game-theoretic questions
Monika Henzinger, 2004: Find an on-line
algorithm that maximizes Google’s revenue.
16. The Adwords Problem:
N advertisers;
Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.
Search Engine
17. The Adwords Problem:
N advertisers;
Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.
queries Search Engine
(online)
18. The Adwords Problem:
N advertisers;
Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.
Select one Ad
queries Search Engine
(online) Advertiser
pays his bid
19. The Adwords Problem:
N advertisers;
Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.
Select one Ad
queries Search Engine
(online) Advertiser
pays his bid
Maximize total revenue
Online competitive analysis - compare with best offline allocation
20. The Adwords Problem:
N advertisers;
Daily Budgets B1, B2, …, BN
Each advertiser provides bids for keywords he is interested in.
Select one Ad
queries Search Engine
(online) Advertiser
pays his bid
Maximize total revenue
Example – Assign to highest bidder: only ½ the offline revenue
21. Example:
Bidder1 Bidder 2
Book Queries: 100 Books then 100 CDs
$1 $0.99
CD $1 $0
B1 = B2 = $100
LOST
Revenue
100$
Algorithm Greedy
Bidder 1 Bidder 2
22. Example:
Bidder1 Bidder 2
Book Queries: 100 Books then 100 CDs
$1 $0.99
CD $1 $0
B1 = B2 = $100
Revenue
199$
Optimal Allocation
Bidder 1 Bidder 2
41. Do markets even have inherently
stable operating points?
General Equilibrium Theory
Occupied center stage in Mathematical
Economics for over a century
45. Fisher’s Model, 1891
$
$$$$$$$$$
¢
wine
bread $$$$
milk
cheese
People want to maximize happiness – assume
linear utilities. s.t. market clears
Find prices
46. Fisher’s Model
n buyers, with specified money, m(i) for buyer i
k goods (unit amount of each good) U = u x
¥
i ij ij
Linear utilities: uij is utility derived by i
j
on obtaining one unit of j
Total utility of i,
u = u x
i ij ij
j
x [0,1]
ij
47. Fisher’s Model
n buyers, with specified money, m(i)
k goods (each unit amount, w.l.o.g.)
U = ¥u x
i ij ij
Linear utilities: uij is utility derived by i
j
on obtaining one unit of j
Total utility of i,
u = u x
i ij ij
j
Find prices s.t. market clears, i.e.,
all goods sold, all money spent.
48.
49. 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.
53. Adam Smith
The Wealth of Nations
2 volumes, 1776.
‘invisible hand’ of
the market
54. What is needed today?
An inherently algorithmic theory of
market equilibrium
New models that capture new markets
55. Beginnings of such a theory, within
Algorithmic Game Theory
Started with combinatorial algorithms
for traditional market models
New market models emerging
56. Combinatorial Algorithm
for Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using primal-dual schema
58. Exact Algorithms for Cornerstone
Problems in P:
Matching (general graph)
Network flow
Shortest paths
Minimum spanning tree
Minimum branching
59. Approximation Algorithms
set cover facility location
Steiner tree k-median
Steiner network multicut
k-MST feedback vertex set
scheduling . . .
60. 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
64. 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
65. 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
66. Kelly’s resource allocation model, 1997
Mathematical framework for understanding
TCP congestion control
Highly successful theory
67. TCP Congestion Control
f(i): source rate
prob. of packet loss (in TCP Reno)
p(e):
queueing delay (in TCP Vegas)
68. TCP Congestion Control
f(i): source rate
prob. of packet loss (in TCP Reno)
p(e):
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.
69. TCP Congestion Control
primal process: packet rates at sources
dual process: packet drop at links
AIMD + RED converges to equilibrium
in the limit
70. Kelly & V., 2002: Kelly’s model is a
generalization of Fisher’s model.
Find combinatorial polynomial time
algorithms!
72. Single-source multiple-sink market
Given:
Network G = (V,E), s: source
Capacities on edges c(e)
t1 ,..., tk
Agents: sinks
with money m(1), … m(k)
Find: equilibrium flows and edge prices
73. 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
120. Branching market (for broadcasting)
Given: Network G = (V, E), directed
edge capacities
SᅪV
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.
130. 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?
131. Efficiency
equilibrium utility ( I )
eff ( M ) = min I
max utility ( I )
132. Efficiency
equilibrium utility ( I )
eff ( M ) = min I
max utility ( I )
Rich classification!