on the efficiency of markets with two-sided proportional allocation...
TRANSCRIPT
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On the Efficiency of Markets with Two-Sided Proportional
Allocation Mechanisms
Volodymyr Kuleshov Adrian Vetta
Department of Mathematics and School of Computer ScienceMcGill University
1Thursday, January 6, 2011
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What is the most fair way to share a good between people, given their competing interests?
2Thursday, January 6, 2011
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Electricityproduction
Bandwidthsharing
3Thursday, January 6, 2011
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• The Internet is made up of smaller independent networks.
INTERNET AUTONOMOUS SYSTEMS
• They wish to have connectivity to each other.
4Thursday, January 6, 2011
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• The Internet is made up of smaller independent networks.
INTERNET AUTONOMOUS SYSTEMS
• They wish to have connectivity to each other.
4Thursday, January 6, 2011
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• The Internet is made up of smaller independent networks.
INTERNET AUTONOMOUS SYSTEMS
• They wish to have connectivity to each other.
4Thursday, January 6, 2011
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• The Internet is made up of smaller independent networks.
INTERNET AUTONOMOUS SYSTEMS
• They wish to have connectivity to each other.
• Network owners are willing to sell transit
4Thursday, January 6, 2011
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INTERNET AUTONOMOUS SYSTEMS
• How can we efficiently organize supply and demand?
5Thursday, January 6, 2011
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INTERNET AUTONOMOUS SYSTEMS
• How can we efficiently organize supply and demand?
Economic efficiencyLeave the users well-off.
5Thursday, January 6, 2011
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INTERNET AUTONOMOUS SYSTEMS
• How can we efficiently organize supply and demand?
Economic efficiencyLeave the users well-off.
Computational efficiencyScale to the size of the Internet
5Thursday, January 6, 2011
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INTERNET AUTONOMOUS SYSTEMS
• How can we efficiently organize supply and demand?
Economic efficiencyLeave the users well-off.
Computational efficiencyScale to the size of the Internet
5Thursday, January 6, 2011
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INTERNET AUTONOMOUS SYSTEMS
• How can we efficiently organize supply and demand?
Economic efficiencyLeave the users well-off.
Computational efficiencyScale to the size of the Internet
There is a fundamentaltradeoff between them.
5Thursday, January 6, 2011
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Link with !xed capacity C>0
THE PROPORTIONAL ALLOCATION MECHANISM
Q users
6Thursday, January 6, 2011
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Link with !xed capacity C>0
1. User submits a payment of
THE PROPORTIONAL ALLOCATION MECHANISM
Q users
$$$
q bq
6Thursday, January 6, 2011
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Link with !xed capacity C>0
1. User submits a payment of
THE PROPORTIONAL ALLOCATION MECHANISM
Q users
2. Capacity is allocated proportionally to the bids. If you pay $50 out of $100, you receive one half.
$$$
q bq
6Thursday, January 6, 2011
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Link with !xed capacity C>0
1. Let be a set of demand functions.
THE PROPORTIONAL ALLOCATION MECHANISM
Q users
D = {D(p, b) = b/p | b > 0}
0.2 0.4 0.6 0.8 1.0
5
10
15
20
25
30
7Thursday, January 6, 2011
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Link with !xed capacity C>0
1. Let be a set of demand functions.
THE PROPORTIONAL ALLOCATION MECHANISM
Q users
2. User q chooses a demand function
0.2 0.4 0.6 0.8 1.0
5
10
15
20
25
30
D = {D(p, b) = b/p | b > 0}
Dq(p) = D(p, bq) ∈ D
7Thursday, January 6, 2011
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Link with !xed capacity C>0
3. The mechanism chooses a price so that
THE PROPORTIONAL ALLOCATION MECHANISM
Q users
4. User q buys at price
0.2 0.4 0.6 0.8 1.0
5
10
15
20
25
30
�
q
Dq(p) = C
Dq(p) p
p
8Thursday, January 6, 2011
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Link with !xed capacity C>0
THE PROPORTIONAL ALLOCATION MECHANISM
Q users
0.2 0.4 0.6 0.8 1.0
5
10
15
20
25
30
�
q
Dq(p) =�
q
bqp
= C =⇒ p =
�q bq
p
9Thursday, January 6, 2011
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Link with !xed capacity C>0
THE PROPORTIONAL ALLOCATION MECHANISM
Q users
0.2 0.4 0.6 0.8 1.0
5
10
15
20
25
30
�
q
Dq(p) =�
q
bqp
= C =⇒ p =
�q bq
p
=⇒ Dq(p) =bqp
=bq�q bq
C
9Thursday, January 6, 2011
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THAT WAS AN EXAMPLE OF A PRICING MECHANISM
• We focus on pricing mechanisms.
• A single price minimizes communication with the users.
• Pricing is standard tool for sharing resources, e.g. road tolls, electricity pricing.
10Thursday, January 6, 2011
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• We measure welfare loss using the price of anarchy.
ECONOMIC EFFICIENCY OF THE PROP. ALLOC. MECH.
• User q has utility:
• Every user makes his best bid given the others’ bids:
bq ∈ argmaxb
Uq(b,b−q)
Uq(dq) = Vq(dq)� �� �value
− pdq����money
11Thursday, January 6, 2011
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ECONOMIC EFFICIENCY OF THE PROP. ALLOC. MECH.
Uq(dq) = Vq(dq)− pdq
12Thursday, January 6, 2011
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ECONOMIC EFFICIENCY OF THE PROP. ALLOC. MECH.
Uq(dq) = Vq(dq)− p dq(bq)� �� �allocation
12Thursday, January 6, 2011
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Uq(dq) = Vq(dq)− p(bq)� �� �price
dq(bq)� �� �allocation
ECONOMIC EFFICIENCY OF THE PROP. ALLOC. MECH.
12Thursday, January 6, 2011
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Uq(dq) = Vq(dq)− p(bq)� �� �price
dq(bq)� �� �allocation
ECONOMIC EFFICIENCY OF THE PROP. ALLOC. MECH.
dq =θq�iθi
C =θq
p
C > 0θq, q ∈ Q
θr, r ∈ R
p =�
iθi
Cµdq
Uq(dq) = Vq(dq)� �� �value
− pdq����money
Theorem. (Kelly, 1997) When users do not exercise their mar-ket power, the Kelly mechanism is optimal. It maximizes
�
q∈Q
Vq(dq)
Ur(sr) = µsr − Cr(sr)
�
q∈Q
Vq(dq)
dq(µ) =θq
µ
sr(µ) = 1− θr
µ�
q∈Q
θq
µ=
�
r∈R
�1− θr
µ
�=⇒ µ =
�q∈Q
θq +�
r∈Rθr
R
Uq(θq; θ−q) = Vq(dq(θq; θ−q))− µ(θq; θ−q)dq(θq; θ−q)θq ∈ arg max
θUq(θ, θ−q)
ϕ :=A(dNE)A(dOPT )
A(d) :=�
q∈Q
Vq(dq)
dq =θq
µ
sr = 1− θr
µ
µ =�
q∈Qθq +
�r∈R
θr
R
1
12Thursday, January 6, 2011
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Uq(dq) = Vq(dq)− p(bq)� �� �price
dq(bq)� �� �allocation
ECONOMIC EFFICIENCY OF THE PROP. ALLOC. MECH.
12Thursday, January 6, 2011
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Uq(dq) = Vq(dq)− p(bq)� �� �price
dq(bq)� �� �allocation
ECONOMIC EFFICIENCY OF THE PROP. ALLOC. MECH.A(d) :=
�
q∈Q
Vq(dq)
dq =θq
µ
sr = 1− θr
µ
µ =�
q∈Qθq +
�r∈R
θr
R
Theorem. (Johari and Tsitsiklis, 2004) Given some natural
assumptions on the utility functions, the price of anarchy in
Kelly’s mechanism is 3/4.
Assumption. For all q ∈ Q, the valuation functions Vq(dq) :R+ → R+ are strictly increasing and concave. Over dq > 0,the functions are differentiable. At dq = 0, the right derivative
exists, and is denoted V �q (0).
bleh
Assumption. For all r ∈ R, there exists a continuous, convex,
and strictly increasing function pr(t) : R+ → R+ such that
pr(0) = 0, and for all sr ≥ 0 we have:
Cr(sr) =ˆ sr
0p(t)dt
and for sr ∈ (−∞, 0) we have Cr(sr) = 0.
arrgh
maximizeQ�
q=1
Vq(dq)−R�
r=1
Cr(sr)
such thatQ�
q=1
dq =R�
r=1
sr
0 ≤ sr ≤ 10 ≤ dq
ϕ :=A(dNE , sNE)
A(dOPT , sOPT )
A(d, s) =�
q∈Q
Vq(dq)−�
r∈R
Cr(dr)
2
12Thursday, January 6, 2011
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SUPPLY-SIDE PROPORTIONAL ALLOCATION MECHANISM
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SUPPLY-SIDE PROPORTIONAL ALLOCATION MECHANISM
A(d) :=�
q∈Q
Vq(dq)
dq =θq
µ
sr = 1− θr
µ
µ =�
q∈Qθq +
�r∈R
θr
R
Theorem. (Johari and Tsitsiklis, 2004) Given some natural
assumptions on the utility functions, the price of anarchy in
Kelly’s mechanism is 3/4.
bleh
Theorem. (Johari, 2004) Given some natural assumptions on
the cost functions, the price of anarchy in Kelly’s supply-side
mechanism is 1/2.
Assumption. For all q ∈ Q, the valuation functions Vq(dq) :R+ → R+ are strictly increasing and concave. Over dq > 0,the functions are differentiable. At dq = 0, the right derivative
exists, and is denoted V �q (0).
bleh
Assumption. For all r ∈ R, there exists a continuous, convex,
and strictly increasing function pr(t) : R+ → R+ such that
pr(0) = 0, and for all sr ≥ 0 we have:
Cr(sr) =ˆ sr
0p(t)dt
and for sr ∈ (−∞, 0) we have Cr(sr) = 0.
arrgh
maximizeQ�
q=1
Vq(dq)−R�
r=1
Cr(sr)
such thatQ�
q=1
dq =R�
r=1
sr
0 ≤ sr ≤ 10 ≤ dq
ϕ :=A(dNE , sNE)
A(dOPT , sOPT )
A(d, s) =�
q∈Q
Vq(dq)−�
r∈R
Cr(dr)
2
13Thursday, January 6, 2011
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But in reality, competition occurs on both sides of the market.
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WHY STUDY TWO-SIDED PRICING MECHANISMS?
• Real-world markets are two-sided.
• Current pricing mechanisms apply only to one-sided markets.
• VCG mechanisms cannot be used in the two-sided setting.
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TWO-SIDED PROPORTIONAL ALLOCATION MECHANISM
Q users
Centralauthority at a link
0.2 0.4 0.6 0.8 1.0
5
10
15
20
25
30
200 400 600 800 1000
0.975
0.980
0.985
0.990
0.995
R providerssr = 1− brp
dq =bqp
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A TWO-SIDED MARKET
• Users’ utilities are: • The optimal solution is:
dq =θq
p=
θq�i θi
C
C > 0
θq, q ∈ Q
θr, r ∈ R
p =�
i θi
C
µdq
Uq(dq) = Vq(dq)� �� �value
− pdq����money
Uq(θ) = Vq(dq(θ))� �� �value
− p(θ)dq(θ)� �� �money
Uq(dq) = Vq(dq)� �� �value
− p(dq)dq� �� �money
Ur(sr) = p(sr)sr� �� �money
−Cr(sr)� �� �costs
Theorem. (Kelly, 1997) When users do not exercise their mar-ket power, the Kelly mechanism is optimal. It maximizes
�
q∈Q
Vq(dq)
Ur(sr) = µsr − Cr(sr)
�
q∈Q
Vq(dq)
dq(µ) =θq
µ
sr(µ) = 1− θr
µ
�
q∈Q
θq
µ=
�
r∈R
�1− θr
µ
�=⇒ µ =
�q∈Q θq +
�r∈R θr
R
Uq(θq; θ−q) = Vq(dq(θq; θ−q))− µ(θq; θ−q)dq(θq; θ−q)
θq ∈ arg maxθ
Uq(θ, θ−q)
1
dq =θq
p=
θq�i θi
C
C > 0
θq, q ∈ Q
θr, r ∈ R
p =�
i θi
C
µdq
Uq(dq) = Vq(dq)� �� �value
− pdq����money
Uq(θ) = Vq(dq(θ))� �� �value
− p(θ)dq(θ)� �� �money
Uq(dq) = Vq(dq)� �� �value
− p(dq)dq� �� �money
Ur(sr) = p(sr)sr� �� �money
−Cr(sr)� �� �costs
Theorem. (Kelly, 1997) When users do not exercise their mar-ket power, the Kelly mechanism is optimal. It maximizes
�
q∈Q
Vq(dq)
Ur(sr) = µsr − Cr(sr)
�
q∈Q
Vq(dq)
dq(µ) =θq
µ
sr(µ) = 1− θr
µ
�
q∈Q
θq
µ=
�
r∈R
�1− θr
µ
�=⇒ µ =
�q∈Q θq +
�r∈R θr
R
Uq(θq; θ−q) = Vq(dq(θq; θ−q))− µ(θq; θ−q)dq(θq; θ−q)
θq ∈ arg maxθ
Uq(θ, θ−q)
1
maximizeQ�
q=1
Vq(dq)−R�
r=1
Cr(sr)
such that supply equals demand
minimizeU,d,s
�q Uq(dNE
q ) +�
r Ur(sNEr )
�q Uq(dOPT
q ) +�
r Ur(sOPTr )
such that dNE
q ,sNE
r form a Nash equilibrium allocation
dOPT
q ,sOPT
r form an optimal allocation
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss+ 2s2)
S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
�q Uq(dNE
q ) +�
r Ur(dNEr )
�q Uq(dOPT
q ) +�
r Ur(dOPTr )
minimize�
q Uq(dNEq ) +
�r Ur(sNE
r )�
q Uq(dOPTq ) +
�r Ur(sOPT
r )
such that dNEq and sNE
r form a Nash equilibrium allocation
dNEq and sNE
r form an optimal allocation
U �q(dq)
�1− dq
R
�≥ p if dq > 0
U �q(dq)
�1− dq
R
�≤ p
C �r(sr)
�1 +
srR− 1
�≤ p if 0 < sr ≤ 1
C �r(sr)
�1 +
srR− 1
�≥ p if 0 ≤ sr < 1
3
(Valuations are concave.)
(Marginal costs are convex.)
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MAIN RESULT
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss + 2s2)S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
bleh
Theorem. In the two-sided mechanism extended to networks,the price of anarchy is
infS
s2(S2 + 4Ss + 2s2)S(S + 2s)
where s is the unique positive root of the polynomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
This value can be numerically evaluated to approximately 0.588727.Furthermore, this bound is tight.
bleh
Definition. A smooth two-sided market-clearing mechanismis a tuple of functions (D,S), D : (0,∞) × [0,∞) → R, S :(0,∞) × [0,∞) → R such that for all Q, R, and for all θ ∈RQ+R, θ �= 0, θ ≥ 0, there exists a unique p > 0 that satisfiesthe following equation
Q�
q=1
Dq(p, θq) =R�
r=1
Sr(p, θr)
We denote it as p(θ).
bleh
Theorem. (D,S) is a smooth two-sided market-clearing mech-anism satisfying the three axioms if and only if there exist dif-ferentiable functions a(p), b(p) : (0,∞) → [0,∞) such that forall θ, p
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
3
18Thursday, January 6, 2011
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MAIN RESULT
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss + 2s2)S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
bleh
Theorem. In the two-sided mechanism extended to networks,the price of anarchy is
infS
s2(S2 + 4Ss + 2s2)S(S + 2s)
where s is the unique positive root of the polynomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
This value can be numerically evaluated to approximately 0.588727.Furthermore, this bound is tight.
bleh
Definition. A smooth two-sided market-clearing mechanismis a tuple of functions (D,S), D : (0,∞) × [0,∞) → R, S :(0,∞) × [0,∞) → R such that for all Q, R, and for all θ ∈RQ+R, θ �= 0, θ ≥ 0, there exists a unique p > 0 that satisfiesthe following equation
Q�
q=1
Dq(p, θq) =R�
r=1
Sr(p, θr)
We denote it as p(θ).
bleh
Theorem. (D,S) is a smooth two-sided market-clearing mech-anism satisfying the three axioms if and only if there exist dif-ferentiable functions a(p), b(p) : (0,∞) → [0,∞) such that forall θ, p
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
3
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss + 2s2)S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
bleh
Theorem. In the two-sided mechanism extended to networks,the price of anarchy is
infS
s2(S2 + 4Ss + 2s2)S(S + 2s)
where s is the unique positive root of the polynomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
This value can be numerically evaluated to approximately 0.588727.Furthermore, this bound is tight.
bleh
Definition. A smooth two-sided market-clearing mechanismis a tuple of functions (D,S), D : (0,∞) × [0,∞) → R, S :(0,∞) × [0,∞) → R such that for all Q, R, and for all θ ∈RQ+R, θ �= 0, θ ≥ 0, there exists a unique p > 0 that satisfiesthe following equation
Q�
q=1
Dq(p, θq) =R�
r=1
Sr(p, θr)
We denote it as p(θ).
bleh
Theorem. (D,S) is a smooth two-sided market-clearing mech-anism satisfying the three axioms if and only if there exist dif-ferentiable functions a(p), b(p) : (0,∞) → [0,∞) such that forall θ, p
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
3
18Thursday, January 6, 2011
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OBSERVATIONS
• Supply-side competition improves the price of anarchy.
• In a fully competitive market, the price of anarchy equals 0.64.
20 40 60 80 100
0.6388
0.6390
0.6392
0.6394
0.6396
0.6398
0.6400
19Thursday, January 6, 2011
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OBSERVATIONS
• Demand-side competition worsens the price of anarchy!
• The best price of anarchy occurs in a monopsony market. It equals 0.72.
20Thursday, January 6, 2011
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PROOF TECHNIQUE
•We formulate the price of anarchy as an optimization problem and analytically compute its solution.
maximizeQ�
q=1
Vq(dq)−R�
r=1
Cr(sr)
such that supply equals demand
minimizeU,d,s
�q Uq(dNE
q ) +�
r Ur(sNEr )
�q Uq(dOPT
q ) +�
r Ur(sOPTr )
such that dNE
q ,sNE
r form a Nash equilibrium allocation
dOPT
q ,sOPT
r form an optimal allocation
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss+ 2s2)
S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
�q Uq(dNE
q ) +�
r Ur(dNEr )
�q Uq(dOPT
q ) +�
r Ur(dOPTr )
minimize�
q Uq(dNEq ) +
�r Ur(sNE
r )�
q Uq(dOPTq ) +
�r Ur(sOPT
r )
such that dNEq and sNE
r form a Nash equilibrium allocation
dNEq and sNE
r form an optimal allocation
U �q(dq)
�1− dq
R
�≥ p if dq > 0
U �q(dq)
�1− dq
R
�≤ p
C �r(sr)
�1 +
srR− 1
�≤ p if 0 < sr ≤ 1
C �r(sr)
�1 +
srR− 1
�≥ p if 0 ≤ sr < 1
3
21Thursday, January 6, 2011
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PROOF TECHNIQUE
1. Derive necessary and sufficient conditions for an allocation to be Nash equilibrium:
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss + 2s2)S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
U �q(dq)
�1− dq
R
�≥ p if dq > 0
U �q(dq)
�1− dq
R
�≤ p
C �r(sr)
�1 +
sr
R− 1
�≤ p if 0 < sr ≤ 1
C �r(sr)
�1 +
sr
R− 1
�≥ p if 0 ≤ sr < 1
bleh
Definition. A smooth two-sided market-clearing mechanismis a tuple of functions (D,S), D : (0,∞) × [0,∞) → R, S :(0,∞) × [0,∞) → R such that for all Q, R, and for all θ ∈RQ+R, θ �= 0, θ ≥ 0, there exists a unique p > 0 that satisfiesthe following equation
Q�
q=1
Dq(p, θq) =R�
r=1
Sr(p, θr)
We denote it as p(θ).
bleh
Theorem. (D,S) is a smooth two-sided market-clearing mech-anism satisfying the three axioms if and only if there exist dif-ferentiable functions a(p), b(p) : (0,∞) → [0,∞) such that forall θ, p
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
3
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss + 2s2)S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
U �q(dq)
�1− dq
R
�≥ p if dq > 0
U �q(dq)
�1− dq
R
�≤ p
C �r(sr)
�1 +
sr
R− 1
�≤ p if 0 < sr ≤ 1
C �r(sr)
�1 +
sr
R− 1
�≥ p if 0 ≤ sr < 1
bleh
Definition. A smooth two-sided market-clearing mechanismis a tuple of functions (D,S), D : (0,∞) × [0,∞) → R, S :(0,∞) × [0,∞) → R such that for all Q, R, and for all θ ∈RQ+R, θ �= 0, θ ≥ 0, there exists a unique p > 0 that satisfiesthe following equation
Q�
q=1
Dq(p, θq) =R�
r=1
Sr(p, θr)
We denote it as p(θ).
bleh
Theorem. (D,S) is a smooth two-sided market-clearing mech-anism satisfying the three axioms if and only if there exist dif-ferentiable functions a(p), b(p) : (0,∞) → [0,∞) such that forall θ, p
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
3
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss + 2s2)S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
U �q(dq)
�1− dq
R
�≥ p if dq > 0
U �q(dq)
�1− dq
R
�≤ p
C �r(sr)
�1 +
sr
R− 1
�≤ p if 0 < sr ≤ 1
C �r(sr)
�1 +
sr
R− 1
�≥ p if 0 ≤ sr < 1
dUq(dNEq )
dθq= 0
dUr(sNEr )
dθr= 0
3
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss + 2s2)S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
U �q(dq)
�1− dq
R
�≥ p if dq > 0
U �q(dq)
�1− dq
R
�≤ p
C �r(sr)
�1 +
sr
R− 1
�≤ p if 0 < sr ≤ 1
C �r(sr)
�1 +
sr
R− 1
�≥ p if 0 ≤ sr < 1
dUq(dNEq )
dθq= 0
dUr(sNEr )
dθr= 0
3
22Thursday, January 6, 2011
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PROOF TECHNIQUE
2. Show that the worst case occurs with linear utilities and marginal costs.
NE is unchanged because derivative is unchanged.
But the utility at OPT may be better.
Vold
Vnew
23Thursday, January 6, 2011
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PROOF TECHNIQUE
Nash eq.conditions
Optimalityconditions
Price ofanarchy
Supply = demand
Non-negativity
Nash eq.conditions
Optimalityconditions
Price ofanarchy
Supply = demand
Non-negativity
24Thursday, January 6, 2011
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PROOF TECHNIQUE
Nash eq.conditions
Optimalityconditions
Price ofanarchy
Non-negativity
Nash eq.conditions
Optimalityconditions
Price ofanarchy
Non-negativity
24Thursday, January 6, 2011
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PROOF TECHNIQUE
Nash eq.conditionsOptimalityconditions
Price ofanarchy
Non-negativity
Nash eq.conditionsOptimalityconditions
Price ofanarchy
Non-negativity
24Thursday, January 6, 2011
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PROOF TECHNIQUE
24Thursday, January 6, 2011
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PROOF TECHNIQUEminimize
(1− µ)2R + µ�R
j=1 sNEj − µ/2
�Rj=1
sNEj
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
6
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
6
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PROOF TECHNIQUE
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
6
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
6
24Thursday, January 6, 2011
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PROOF TECHNIQUE
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
6
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
6
24Thursday, January 6, 2011
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PROOF TECHNIQUE
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
6
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
6
24Thursday, January 6, 2011
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PROOF TECHNIQUE
Theorem. The price of anarchy of the two-sided market in-volving R > 1 suppliers equals
s2(S2 + 4Ss + 2s2)S(S + 2s)
where S = R− 1, and s is the unique positive root of the poly-nomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
Furthermore, this bound is tight.
bleh
Corollary. The worst inefficiency occurs when R = 2. Itequals approximately 0.588727.
bleh
Theorem. In the two-sided mechanism extended to networks,the price of anarchy is
infS
s2(S2 + 4Ss + 2s2)S(S + 2s)
where s is the unique positive root of the polynomial
γ(s) = 16s4 + 10S2s(3s− 2) + S3(5s− 4) + Ss2(49s− 24)
This value can be numerically evaluated to approximately 0.588727.Furthermore, this bound is tight.
bleh
Definition. A smooth two-sided market-clearing mechanismis a tuple of functions (D,S), D : (0,∞) × [0,∞) → R, S :(0,∞) × [0,∞) → R such that for all Q, R, and for all θ ∈RQ+R, θ �= 0, θ ≥ 0, there exists a unique p > 0 that satisfiesthe following equation
Q�
q=1
Dq(p, θq) =R�
r=1
Sr(p, θr)
We denote it as p(θ).
bleh
Theorem. (D,S) is a smooth two-sided market-clearing mech-anism satisfying the three axioms if and only if there exist dif-ferentiable functions a(p), b(p) : (0,∞) → [0,∞) such that forall θ, p
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
3
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EXTENSION TO NETWORKS
• At each link, there is an independent instance of the single-link market.
• Consumers buy capacity in order to transmit flow from s to t.
s1
t1
s2
t2θ1
θ2
θ3
Q users
(s1,t1)(s2,t2)
R providers
(b1,b2,...)r
25Thursday, January 6, 2011
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EXTENSION TO NETWORKS
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
bleh
6
26Thursday, January 6, 2011
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EXTENSION TO NETWORKS
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
bleh
6
minimize(1− µ)2R + µ
�Rj=1 sNE
j − µ/2�R
j=1sNE
j
1+sNEj /(R−1)
�Rj=1 min(1/βj , 1)− µ/2
�Rj=1
min(1/βj ,1)2
sNEj (1+sNE
j /(R−1))
such that 0 < sNEj ≤ 1 ∀j
βj =µ
sNEj
�1 + sNE
j /(R− 1)� ∀j
0 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
min( s(1+s/(R−1))µ , 1)− µ
2s(1+s/(R−1)) min( s(1+s/(R−1))µ , 1)2
such that 0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µ/2
s1+s/(R−1)
s(1+s/(R−1))2µ
such that s(1 + s/(R− 1)) ≤ µ
0 < s ≤ 10 ≤ µ < 1
minimize(1− µ)2 + µs− µs
2(1+s/(R−1))
1− µ2s(1+s/(R−1))
such that s(1 + s/(R− 1)) ≥ µ
0 < s ≤ 10 ≤ µ < 1
minimizes2((R− 1)2 + 4(R− 1)s + 2s2)
(R− 1)(R− 1 + 2s)such that 0 < s ≤ 1
0 ≤ µ1,2 < 1s(1 + s/R− 1)) ≥ µ1,2
µ1,2 ∈ R
Theorem. When extended to networks, the mechanism has thesame price of anarchy as in the single link case – approximately0.588727.
Corollary. When extended to a general economy of N goods,the mechanism has the same price of anarchy of 0.588727,under some mild assumptions on costs and utilities.
bleh
6
26Thursday, January 6, 2011
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TWO-SIDED MARKET-CLEARING MECHANISMS
• A two-sided market-clearing mechanism is a pair of sets of functions: and S = {S(b, p) | b > 0}D = {D(b, p) | b > 0}
27Thursday, January 6, 2011
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TWO-SIDED MARKET-CLEARING MECHANISMS
• A two-sided market-clearing mechanism is a pair of sets of functions: and S = {S(b, p) | b > 0}D = {D(b, p) | b > 0}
Q users R providers
27Thursday, January 6, 2011
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TWO-SIDED MARKET-CLEARING MECHANISMS
• A two-sided market-clearing mechanism is a pair of sets of functions: and S = {S(b, p) | b > 0}D = {D(b, p) | b > 0}
Q users R providers
D(p, bq) ∈ D S(p, br) ∈ S
27Thursday, January 6, 2011
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TWO-SIDED MARKET-CLEARING MECHANISMS
• A two-sided market-clearing mechanism is a pair of sets of functions: and S = {S(b, p) | b > 0}D = {D(b, p) | b > 0}
�
q
D(p, bq) =�
r
D(p, br)
Q users R providers
D(p, bq) ∈ D S(p, br) ∈ S
27Thursday, January 6, 2011
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TWO-SIDED MARKET-CLEARING MECHANISMS
• A two-sided market-clearing mechanism is a pair of sets of functions: and S = {S(b, p) | b > 0}D = {D(b, p) | b > 0}
�
q
D(p, bq) =�
r
D(p, br)
Q users R providers
S(p(�b), br)D(p(�b), bq)
27Thursday, January 6, 2011
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TWO-SIDED MARKET-CLEARING MECHANISMS
• Consider the market-clearing mechanisms for which
• The utility to each user in concave is his bid:
• D is bounded from below and S is bounded from above.
• When users have no market power, the mechanism achieves an optimal allocation.
Uq(dq) = Vq(D(p(�b), bq)− p(�b)D(p(�b), bq)
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OPTIMALITY
Lemma. Under some mild assumptions, every mechanism thataccepts a scalar message θ from each user must allocate demandand supply according to:
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
where a(p), b(p) ≥ 0 are some functions of the price p > 0.
Theorem. Within the class of mechanisms for which a(p) =b(p) for all p > 0, the mechanism presented here achieves thebest possible price of anarchy.
bleh
7
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OPTIMALITY
Lemma. Under some mild assumptions, every mechanism thataccepts a scalar message θ from each user must allocate demandand supply according to:
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
where a(p), b(p) ≥ 0 are some functions of the price p > 0.
Theorem. Within the class of mechanisms for which a(p) =b(p) for all p > 0, the mechanism presented here achieves thebest possible price of anarchy.
bleh
7
Lemma. Under some mild assumptions, every mechanism thataccepts a scalar message θ from each user must allocate demandand supply according to:
D(θ, p) = a(p)θS(θ, p) = 1− b(p)θ
where a(p), b(p) ≥ 0 are some functions of the price p > 0.
Theorem. Among the mechanisms that have a(p) = b(p) forall p > 0, the mechanism presented here is the only one thatachieves the best possible price of anarchy of 0.588727.
bleh
7
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IN CONCLUSION
Our results were to:
• Extend the proportional allocation mechanism to two-sided markets.
• Establish a tight bound on the price of anarchy in both the single and multi-resource settings.
• Establish the optimality of the mechanism within a large class.
30Thursday, January 6, 2011