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Algorithmic Game Theory and Internet Computing. New Market Models Resource Allocation Markets. Vijay V. Vazirani. 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 - PowerPoint PPT Presentation
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Algorithmic Game Theoryand Internet Computing
Vijay V. Vazirani
New Market Models
Resource Allocation Markets
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
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 ij ijj
U u xiju
xuu ijj iji
Eisenberg-Gale Program, 1959
0:
1:
)(:
..)(log)(max
xx
xu
ij
i ij
ijj ij
i
ij
j
iui
tsiuim
Via KKT Conditions can establish:
Optimal solution gives equilibrium allocations
Lagrange variables give prices of goods
Equilibrium exists (under mild conditions) Equilibrium utilities and prices are unique
Eisenberg-Gale program helps establish:
Equilibrium exists (under mild conditions) Equilibrium utilities and prices are unique Rational!!
Eisenberg-Gale program helps establish:
Kelly’s resource allocation model, 1997
Mathematical framework for understanding
TCP congestion control
Kelly’s model
Given:
network G = (V,E)
(directed or undirected)
capacities on edges
source-sink pairs (agents)
m(i): money/unit time agent i
is willing to pay
)1(m
t1
s2
t2)(ecs1
)2(m
Kelly’s model
Network determines:
f(i): flow rate of agent i
Assume utility u(i) = m(i) log f(i)
Total utility is additive
t1
s2
t2s1
Convex Program for Kelly’s Model
0:,
)()(:
)(:..
)(log)(max
f
f
p
i
p
p
i
i
pi
eceflowe
ifits
ifim
Kelly’s model
t1
s2
t2)(eps1
Lagrange variables:
p(e): price/unit flow
Kelly’s model
t1
s2
t2)(eps1
Optimum flow and edge prices
are in equilibrium:
1). p(e)>0 only if e is saturated
2) flows go on cheapest paths
3) money of each agent is fully used
Let rate(i) = cost of cheapest path for i
m(i) = f(i) rate(i)
Kelly’s model
t1
s2
t2)(eps1
Optimum flow and edge prices
are in equilibrium:
1). p(e)>0 only if e is saturated
2) flows go on cheapest paths
3) money of each agent is fully used
Let rate(i) = cost of cheapest path for i
f(i)’s and rate(i)’s are unique!
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 increasing an agent’s flow by 5% is to decrease other agents’ flow by at least 5%
p(e):
TCP Congestion Control
f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas)
Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time).
p(e):
TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas)
Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time).
AIMD + RED converges to equilibrium primal-dual (source-link) alg.
p(e):
TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas)
Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time).
FAST: for high speed networks with large bandwidth
p(e):
Combinatorial Algorithms
Devanur, Papadimitriou, Saberi & V., 2002: for Fisher’s linear utilities case
Kelly & V., 2002: Kelly’s model is a generalization of Fisher’s model.
Find comb. poly time algs!
Irrational for 2 sources & 3 sinks
s1 t1
1
s2
t2
1
t21 2
$1 $1
$1
Irrational for 2 sources & 3 sinks
s1 t1
1
s2
t2
1
t2
313
3
Equilibrium prices
1 source & multiple sinks 2 source-sink pairs
s
t1
t2
2
2
110$
10$
s
t1
t2
2
2
1 10$
10$
$5
$5
s
t1
t2
2
2
1 10$
10$
120$
s
t1
t2
2
2
1 120$
10$
$10
$40
$30
Jain & V., 2005: strongly poly alg
Primal-dual algorithmUsual: linear programs & LP-dualityThis: convex programs & KKT conditions
Ascending price auctionBuyers: sinks (fixed budgets, maximize flow)Sellers: edges (maximize price)
st1
t2
t3
t4
rate(i): cost of cheapest path ts i
st1
t2
t3
t4
t
st1
t2
t3
t4
t
Capacity of edge =tt i
)()(irate
im
st1
t2
t3
t4
t
min s-t cut
st1
t2
t3
t4
t
p
st1
t2
t3
t4
t
p
st1
t2 t3
t4
t
pp 0
prate0
)2(
st1
t2 t3
t4
t
p0
p prate0
)2(
st1
t2 t3
t4
t
p0
p1
prate0
)2(
ppraterate10
)3()1(
st1
t2 t3
t4
t
p0
p1
p
st1
t2 t3
t4
t
p0
p1 p
2 nested cuts
st1
t2 t3
t4
t
p0
p1 p
2
prate0
)2(
ppraterate10
)3()1(
ppprate210
)4(
Find s-t max flow
Flow and prices will:
Saturate all red cutsUse up sinks’ moneySend flow on cheapest paths
s
t1
t2
2
2
1120$
10$
a
b
s
t1
t2
2
2
1 p120
p10
a
b
t
p
s
t1
t2
2
2
1 10120
11010
a
b
t
10p
s
t1
t2
2
2
1 p10120
1
a
b
t
100p p
s
t1
t2
2
2
1 33010
120
1
a
b
t
100p 30p
s
t1
t2
2
2
1 120$
10$
$10
$40
$30
Rational!!
Max-flow min-cut theorem
Other resource allocation markets
2 source-sink pairs (directed/undirected) Branchings rooted at sources (agents) Spanning trees Network coding
Eisenberg-Gale-Type Convex Program
iiuim )(log)(max
s.t. packing constraints
Eisenberg-Gale Market
A market whose equilibrium is captured as an optimal solution to an Eisenberg-Gale-type program
Megiddo, 1974: Let T = set of sinks (agents)
For define v(S) to be the max-flow possible from s to sinks in S.
Then v is a submodular function, i.e., for
TS
)()()()(,
,
AvtAvBvtBvAt
TBA
Simpler convex program for single-source market
0)(:
)()(:
..)(log)(max
ifi
SvifTS
tsifim
Si
i
Submodular Utility Allocation Market
Any market which has simpler program and v is submodular
Submodular Utility Allocation Market
Any market which has simpler program and v is submodular
Theorem: Strongly polynomial algorithm for SUA markets.
Submodular Utility Allocation Market Any market which has simpler program and v is submodular
Theorem: Strongly polynomial algorithm for SUA markets.
Corollary: Rational!!
Theorem: Following markets are SUA: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: Following markets are SUA: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 thoerems):2 source-sink pairs, directed2 sources, network coding
Theorem: Following markets are SUA:2 source-sink pairs, undirected (Hu, 1963)spanning tree (Nash-William & Tutte, 1961)2 sources branching (Edmonds, 1967 + JV)
3 sources branching: irrational
Open (no max-min thoerems):2 source-sink pairs, directed2 sources, network coding
Chakrabarty, Devanur & V., 2006
EG[2]: Eisenberg-Gale markets with 2 agents
Theorem: EG[2] markets are rational.
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].
EG
Rational
Comb EG[2]
SUA
EG[2]
3-source branching
Fisher
2 s-s undir
2 s-s dir
Single-source
Other properties:
Efficiency Fairness (max-min + min-max fair) Competition monotonicity
Open issues
Strongly poly algs for approximatingnonlinear convex programsequilibria
Insights into congestion control protocols?