students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf ·...
Transcript of students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf ·...
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The efficiency of resource allocationmechanisms for budget-constrained users
Ioannis Caragiannis Alexandros A. Voudouris
University of Patras
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A resource allocation problem
Question: How can we distribute a single divisible resourceamong a set of self-interested users with budget constraints?
Mechanism(auction)
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A resource allocation problem
Question: How can we distribute a single divisible resourceamong a set of self-interested users with budget constraints?
Mechanism(auction)
![Page 4: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/4.jpg)
A resource allocation problem
Question: How can we distribute a single divisible resourceamong a set of self-interested users with budget constraints?
Mechanism(auction)
user signal
𝑠𝑖
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A resource allocation problem
Question: How can we distribute a single divisible resourceamong a set of self-interested users with budget constraints?
Mechanism(auction)
user signal allocation, payment
𝑠𝑖 𝑔𝑖 𝐬 , 𝑝𝑖 𝐬
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Example: pay-your-signal mechanisms
• Each user pays her own signal: pi(s) = si
• Kelly mechanism [Kelly 1997, Kelly et al. 1998]:
gi(s) = si∑j sj
• SHmechanism [Sanghavi and Hajek 2004]:
gi(s) = si
maxℓ sℓ
∫ 1
0
∏j =i
(1 − sj
maxℓ sℓt
)dt
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Example: pay-your-signal mechanisms
• Each user pays her own signal: pi(s) = si
• Kelly mechanism [Kelly 1997, Kelly et al. 1998]:
gi(s) = si∑j sj
• SHmechanism [Sanghavi and Hajek 2004]:
gi(s) = si
maxℓ sℓ
∫ 1
0
∏j =i
(1 − sj
maxℓ sℓt
)dt
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Example: pay-your-signal mechanisms
• Each user pays her own signal: pi(s) = si
• Kelly mechanism [Kelly 1997, Kelly et al. 1998]:
gi(s) = si∑j sj
• SHmechanism [Sanghavi and Hajek 2004]:
gi(s) = si
maxℓ sℓ
∫ 1
0
∏j =i
(1 − sj
maxℓ sℓt
)dt
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User characteristics and selfishness
Each user i has• a valuation function vi : [0, 1] → R≥0 that represents her value forfractions of the resource (increasing, differentiable, concave)
• a budget ci ∈ R≥0 ∪ {+∞} that restricts the payments she can afford
Strategic behavior• Users are utility-maximizers• Select si in order to maximize utility: ui(s) = vi(gi(s)) − pi(s)
• if pi(s) > ci then ui(s) = −∞
Equilibrium• A signal vector s for which all players simultaneously maximize theirpersonal utilities
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User characteristics and selfishness
Each user i has• a valuation function vi : [0, 1] → R≥0 that represents her value forfractions of the resource (increasing, differentiable, concave)
• a budget ci ∈ R≥0 ∪ {+∞} that restricts the payments she can afford
Strategic behavior• Users are utility-maximizers• Select si in order to maximize utility: ui(s) = vi(gi(s)) − pi(s)
• if pi(s) > ci then ui(s) = −∞
Equilibrium• A signal vector s for which all players simultaneously maximize theirpersonal utilities
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User characteristics and selfishness
Each user i has• a valuation function vi : [0, 1] → R≥0 that represents her value forfractions of the resource (increasing, differentiable, concave)
• a budget ci ∈ R≥0 ∪ {+∞} that restricts the payments she can afford
Strategic behavior• Users are utility-maximizers• Select si in order to maximize utility: ui(s) = vi(gi(s)) − pi(s)
• if pi(s) > ci then ui(s) = −∞
Equilibrium• A signal vector s for which all players simultaneously maximize theirpersonal utilities
![Page 12: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/12.jpg)
User characteristics and selfishness
Each user i has• a valuation function vi : [0, 1] → R≥0 that represents her value forfractions of the resource (increasing, differentiable, concave)
• a budget ci ∈ R≥0 ∪ {+∞} that restricts the payments she can afford
Strategic behavior• Users are utility-maximizers• Select si in order to maximize utility: ui(s) = vi(gi(s)) − pi(s)
• if pi(s) > ci then ui(s) = −∞
Equilibrium• A signal vector s for which all players simultaneously maximize theirpersonal utilities
![Page 13: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/13.jpg)
Efficiency without budgets
• The social welfare of an allocation d in game G is
SW(d, G) =∑
i
vi(di)
• Price of anarchy of mechanism M ,
PoA(M) = supG∈M
sups∈EQ(G)
maxd SW(d, G)SW(g(s), G)
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Efficiency without budgets
• PoA(Kelly) = 4/3 [Johari and Tsitsiklis, 2004]• PoA(SH) ≈ 8/7 [Sanghavi and Hajek, 2004]
• There exist mechanisms that achieve full efficiency: MBmechanisms[Maheswaran and Basar, 2006]
gi(s) = si∑j sj
pi(s) = si
∑j =i
sj
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Efficiency without budgets
• PoA(Kelly) = 4/3 [Johari and Tsitsiklis, 2004]• PoA(SH) ≈ 8/7 [Sanghavi and Hajek, 2004]• There exist mechanisms that achieve full efficiency: MBmechanisms[Maheswaran and Basar, 2006]
gi(s) = si∑j sj
pi(s) = si
∑j =i
sj
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Efficiency with budgets
• The price of anarchy may be arbitrarily large
• Example: a player with high value, but really low budget
0 1
v(x)
fraction
value
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Efficiency with budgets
• The price of anarchy may be arbitrarily large• Example: a player with high value, but really low budget
0 1
v(x)
fraction
value
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Efficiency with budgets
• The price of anarchy may be arbitrarily large• Example: a player with high value, but really low budget
0 EQ 1
v(EQ)
v(x)
fraction
value
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Efficiency with budgets
• The price of anarchy may be arbitrarily large• Example: a player with high value, but really low budget
0 EQ OPT 1
v(EQ)
v(OPT)
v(x)
fraction
value
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Efficiency with budgets
• Alternative: liquid welfare [Dobzinski and Paes Leme, 2014]• The liquid welfare of an allocation d in game G is
LW(d, G) =∑
i
min{vi(di), ci}
• Liquid price of anarchy of mechanism M ,
LPoA(M) = supG∈M
sups∈EQ(G)
maxd LW(d, G)LW(g(s), G)
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LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
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LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
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LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞
• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
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LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)
• wlog d1 ≤ d2 ⇔ d1 ≤ 12 , d2 ≥ 1
2• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
![Page 25: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/25.jpg)
LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
![Page 26: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/26.jpg)
LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
![Page 27: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/27.jpg)
LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
![Page 28: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/28.jpg)
LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)
• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
![Page 29: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/29.jpg)
LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
![Page 30: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/30.jpg)
LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well
• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 32
![Page 31: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/31.jpg)
LPoA(any n-player mechanism M ) ≥ 2 − 1n
• Proof for two players: lower bound of 3/2
• Game G: v1(x) = v2(x) = x and c1 = c2 = +∞• Equilibrium: s = (s1, s2) with allocation d = (d1, d2)• wlog d1 ≤ d2 ⇔ d1 ≤ 1
2 , d2 ≥ 12
• Same valuations, no budget constraints ⇒ LPoA(G) = 1
• Game G: v1(x) = x, c1 = +∞ and v2(x) = x + d2, c2 = d2
• u1(y, s2) = v1(g1(y, s2)) − p1(y, s2) = u1(y, s2)• u2(s1, y) = v2(g2(s1, y)) − p2(s1, y) = u2(s1, y) + d2
• s is an equilibrium of G as well• LW(d, G) = d1 + d2 = 1 vs. OPT ≥ 1 + d2 ⇒ LPoA(G) ≥ 3
2
![Page 32: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/32.jpg)
Characterization of worst-case games
For every signal vector s, define the following game:
• player 1 has c1 = +∞ and linear valuation v1(x) = λ1(s)x, with
λ1(s) =
(∂g1(y, s−1)
∂y
∣∣∣∣y=s1
)−1
· ∂p1(y, s−1)∂y
∣∣∣∣y=s1
• all other players i ≥ 2 have budgets ci = pi(s) and affine valuationvi(x) = λi(s)x + ci
• Player 1 contributes her value to the liquid welfare, and all otherscontribute their budgets
LPoA(M) ≤ sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Concave allocation + convex payment
![Page 33: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/33.jpg)
Characterization of worst-case games
For every signal vector s, define the following game:• player 1 has c1 = +∞ and linear valuation v1(x) = λ1(s)x, with
λ1(s) =
(∂g1(y, s−1)
∂y
∣∣∣∣y=s1
)−1
· ∂p1(y, s−1)∂y
∣∣∣∣y=s1
• all other players i ≥ 2 have budgets ci = pi(s) and affine valuationvi(x) = λi(s)x + ci
• Player 1 contributes her value to the liquid welfare, and all otherscontribute their budgets
LPoA(M) ≤ sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Concave allocation + convex payment
![Page 34: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/34.jpg)
Characterization of worst-case games
For every signal vector s, define the following game:• player 1 has c1 = +∞ and linear valuation v1(x) = λ1(s)x, with
λ1(s) =
(∂g1(y, s−1)
∂y
∣∣∣∣y=s1
)−1
· ∂p1(y, s−1)∂y
∣∣∣∣y=s1
• all other players i ≥ 2 have budgets ci = pi(s) and affine valuationvi(x) = λi(s)x + ci
• Player 1 contributes her value to the liquid welfare, and all otherscontribute their budgets
LPoA(M) ≤ sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Concave allocation + convex payment
![Page 35: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/35.jpg)
Characterization of worst-case games
For every signal vector s, define the following game:• player 1 has c1 = +∞ and linear valuation v1(x) = λ1(s)x, with
λ1(s) =
(∂g1(y, s−1)
∂y
∣∣∣∣y=s1
)−1
· ∂p1(y, s−1)∂y
∣∣∣∣y=s1
• all other players i ≥ 2 have budgets ci = pi(s) and affine valuationvi(x) = λi(s)x + ci
• Player 1 contributes her value to the liquid welfare, and all otherscontribute their budgets
LPoA(M) ≤ sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Concave allocation + convex payment
![Page 36: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/36.jpg)
Characterization of worst-case games
For every signal vector s, define the following game:• player 1 has c1 = +∞ and linear valuation v1(x) = λ1(s)x, with
λ1(s) =
(∂g1(y, s−1)
∂y
∣∣∣∣y=s1
)−1
· ∂p1(y, s−1)∂y
∣∣∣∣y=s1
• all other players i ≥ 2 have budgets ci = pi(s) and affine valuationvi(x) = λi(s)x + ci
• Player 1 contributes her value to the liquid welfare, and all otherscontribute their budgets
LPoA(M) ≤ sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Concave allocation + convex payment
![Page 37: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/37.jpg)
Characterization of worst-case games
For every signal vector s, define the following game:• player 1 has c1 = +∞ and linear valuation v1(x) = λ1(s)x, with
λ1(s) =
(∂g1(y, s−1)
∂y
∣∣∣∣y=s1
)−1
· ∂p1(y, s−1)∂y
∣∣∣∣y=s1
• all other players i ≥ 2 have budgets ci = pi(s) and affine valuationvi(x) = λi(s)x + ci
• Player 1 contributes her value to the liquid welfare, and all otherscontribute their budgets
LPoA(M) ≤ sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Concave allocation + convex payment
![Page 38: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/38.jpg)
Characterization of worst-case games
For every signal vector s, define the following game:• player 1 has c1 = +∞ and linear valuation v1(x) = λ1(s)x, with
λ1(s) =
(∂g1(y, s−1)
∂y
∣∣∣∣y=s1
)−1
· ∂p1(y, s−1)∂y
∣∣∣∣y=s1
• all other players i ≥ 2 have budgets ci = pi(s) and affine valuationvi(x) = λi(s)x + ci
• Player 1 contributes her value to the liquid welfare, and all otherscontribute their budgets
LPoA(M) = sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Concave allocation + convex payment
![Page 39: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/39.jpg)
LPoA bounds (overview)
n players• LPoA(Kelly) = 2 (best possible for large number of players)• LPoA(SH) = 3
Two players• LPoA(E2-PYS) = 1.792 (best possible among all pay-your-signalmechanisms with concave allocation functions)
• LPoA(E2-SR) ≤ 1.529 (almost optimal over all two-playermechanisms)
![Page 40: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/40.jpg)
LPoA bounds (overview)
n players• LPoA(Kelly) = 2 (best possible for large number of players)• LPoA(SH) = 3
Two players• LPoA(E2-PYS) = 1.792 (best possible among all pay-your-signalmechanisms with concave allocation functions)
• LPoA(E2-SR) ≤ 1.529 (almost optimal over all two-playermechanisms)
![Page 41: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/41.jpg)
LPoA bounds (overview)
n players• LPoA(Kelly) = 2 (best possible for large number of players)• LPoA(SH) = 3
Two players• LPoA(E2-PYS) = 1.792 (best possible among all pay-your-signalmechanisms with concave allocation functions)
• LPoA(E2-SR) ≤ 1.529 (almost optimal over all two-playermechanisms)
![Page 42: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/42.jpg)
LPoA(Kelly) = 2
LPoA(Kelly) = sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Pay-your-signal:∑
i≥2 pi(s) =∑
i≥2 si =: C
• For player 1: g1(s) = s1s1+C
g1(y, s−1) = yy+C ⇒ ∂g1(y,s−1)
∂y = C(y+C)2
p1(y, s−1) = y ⇒ ∂p1(y,s−1)∂y = 1
λ1(s) = (s1 + C)2
C
LPoA(Kelly) = sups1,C
C + (s1 + C)2/C
C + (s1 + C)s1/C= 2
![Page 43: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/43.jpg)
LPoA(Kelly) = 2
LPoA(Kelly) = sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Pay-your-signal:∑
i≥2 pi(s) =∑
i≥2 si =: C
• For player 1: g1(s) = s1s1+C
g1(y, s−1) = yy+C ⇒ ∂g1(y,s−1)
∂y = C(y+C)2
p1(y, s−1) = y ⇒ ∂p1(y,s−1)∂y = 1
λ1(s) = (s1 + C)2
C
LPoA(Kelly) = sups1,C
C + (s1 + C)2/C
C + (s1 + C)s1/C= 2
![Page 44: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/44.jpg)
LPoA(Kelly) = 2
LPoA(Kelly) = sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Pay-your-signal:∑
i≥2 pi(s) =∑
i≥2 si =: C
• For player 1: g1(s) = s1s1+C
g1(y, s−1) = yy+C ⇒ ∂g1(y,s−1)
∂y = C(y+C)2
p1(y, s−1) = y ⇒ ∂p1(y,s−1)∂y = 1
λ1(s) = (s1 + C)2
C
LPoA(Kelly) = sups1,C
C + (s1 + C)2/C
C + (s1 + C)s1/C= 2
![Page 45: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/45.jpg)
LPoA(Kelly) = 2
LPoA(Kelly) = sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Pay-your-signal:∑
i≥2 pi(s) =∑
i≥2 si =: C
• For player 1: g1(s) = s1s1+C
g1(y, s−1) = yy+C ⇒ ∂g1(y,s−1)
∂y = C(y+C)2
p1(y, s−1) = y ⇒ ∂p1(y,s−1)∂y = 1
λ1(s) = (s1 + C)2
C
LPoA(Kelly) = sups1,C
C + (s1 + C)2/C
C + (s1 + C)s1/C= 2
![Page 46: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/46.jpg)
LPoA(Kelly) = 2
LPoA(Kelly) = sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Pay-your-signal:∑
i≥2 pi(s) =∑
i≥2 si =: C
• For player 1: g1(s) = s1s1+C
g1(y, s−1) = yy+C ⇒ ∂g1(y,s−1)
∂y = C(y+C)2
p1(y, s−1) = y ⇒ ∂p1(y,s−1)∂y = 1
λ1(s) = (s1 + C)2
C
LPoA(Kelly) = sups1,C
C + (s1 + C)2/C
C + (s1 + C)s1/C= 2
![Page 47: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/47.jpg)
LPoA(Kelly) = 2
LPoA(Kelly) = sups
∑i≥2 pi(s) + λ1(s)∑
i≥2 pi(s) + λ1(s) g1(s)
• Pay-your-signal:∑
i≥2 pi(s) =∑
i≥2 si =: C
• For player 1: g1(s) = s1s1+C
g1(y, s−1) = yy+C ⇒ ∂g1(y,s−1)
∂y = C(y+C)2
p1(y, s−1) = y ⇒ ∂p1(y,s−1)∂y = 1
λ1(s) = (s1 + C)2
C
LPoA(Kelly) = sups1,C
C + (s1 + C)2/C
C + (s1 + C)s1/C= 2
![Page 48: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/48.jpg)
Two players: the design of E2-PYS
• s1 ≤ s2, z = s1s2
• Pay-your-signal:p1(s) = s1 p2(s) = s2
• Allocation:g1(s) = f(z) g2(s) = 1 − f(z)
• Compute λ1(s):
g1(y, s2) = f(
ys2
)⇒ ∂g1(y,s2)
∂y = 1s2
f ′(
ys2
)p1(y, s2) = y ⇒ ∂p1(y,s2)
∂y = 1
λ1(s) = s2
f ′(z)
• Characterization:
p2(s) + λ1(s)p2(s) + λ1(s)g1(s)
=s2 + s2
f ′(z)
s2 + s2f ′(z) f(z)
= f ′(z) + 1f ′(z) + f(z)
![Page 49: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/49.jpg)
Two players: the design of E2-PYS• s1 ≤ s2, z = s1
s2
• Pay-your-signal:p1(s) = s1 p2(s) = s2
• Allocation:g1(s) = f(z) g2(s) = 1 − f(z)
• Compute λ1(s):
g1(y, s2) = f(
ys2
)⇒ ∂g1(y,s2)
∂y = 1s2
f ′(
ys2
)p1(y, s2) = y ⇒ ∂p1(y,s2)
∂y = 1
λ1(s) = s2
f ′(z)
• Characterization:
p2(s) + λ1(s)p2(s) + λ1(s)g1(s)
=s2 + s2
f ′(z)
s2 + s2f ′(z) f(z)
= f ′(z) + 1f ′(z) + f(z)
![Page 50: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/50.jpg)
Two players: the design of E2-PYS• s1 ≤ s2, z = s1
s2
• Pay-your-signal:p1(s) = s1 p2(s) = s2
• Allocation:g1(s) = f(z) g2(s) = 1 − f(z)
• Compute λ1(s):
g1(y, s2) = f(
ys2
)⇒ ∂g1(y,s2)
∂y = 1s2
f ′(
ys2
)p1(y, s2) = y ⇒ ∂p1(y,s2)
∂y = 1
λ1(s) = s2
f ′(z)
• Characterization:
p2(s) + λ1(s)p2(s) + λ1(s)g1(s)
=s2 + s2
f ′(z)
s2 + s2f ′(z) f(z)
= f ′(z) + 1f ′(z) + f(z)
![Page 51: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/51.jpg)
Two players: the design of E2-PYS• s1 ≤ s2, z = s1
s2
• Pay-your-signal:p1(s) = s1 p2(s) = s2
• Allocation:g1(s) = f(z) g2(s) = 1 − f(z)
• Compute λ1(s):
g1(y, s2) = f(
ys2
)⇒ ∂g1(y,s2)
∂y = 1s2
f ′(
ys2
)p1(y, s2) = y ⇒ ∂p1(y,s2)
∂y = 1
λ1(s) = s2
f ′(z)
• Characterization:
p2(s) + λ1(s)p2(s) + λ1(s)g1(s)
=s2 + s2
f ′(z)
s2 + s2f ′(z) f(z)
= f ′(z) + 1f ′(z) + f(z)
![Page 52: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/52.jpg)
Two players: the design of E2-PYS• s1 ≤ s2, z = s1
s2
• Pay-your-signal:p1(s) = s1 p2(s) = s2
• Allocation:g1(s) = f(z) g2(s) = 1 − f(z)
• Compute λ1(s):
g1(y, s2) = f(
ys2
)⇒ ∂g1(y,s2)
∂y = 1s2
f ′(
ys2
)p1(y, s2) = y ⇒ ∂p1(y,s2)
∂y = 1
λ1(s) = s2
f ′(z)
• Characterization:
p2(s) + λ1(s)p2(s) + λ1(s)g1(s)
=s2 + s2
f ′(z)
s2 + s2f ′(z) f(z)
= f ′(z) + 1f ′(z) + f(z)
![Page 53: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/53.jpg)
Two players: the design of E2-PYS• s1 ≤ s2, z = s1
s2
• Pay-your-signal:p1(s) = s1 p2(s) = s2
• Allocation:g1(s) = f(z) g2(s) = 1 − f(z)
• Compute λ1(s):
g1(y, s2) = f(
ys2
)⇒ ∂g1(y,s2)
∂y = 1s2
f ′(
ys2
)p1(y, s2) = y ⇒ ∂p1(y,s2)
∂y = 1
λ1(s) = s2
f ′(z)
• Characterization:
p2(s) + λ1(s)p2(s) + λ1(s)g1(s)
=s2 + s2
f ′(z)
s2 + s2f ′(z) f(z)
= f ′(z) + 1f ′(z) + f(z)
![Page 54: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/54.jpg)
Two players: the design of E2-PYS
• Require:
f ′(z) + 1f ′(z) + f(z)
= β ⇔ f ′(z) + β
β − 1f(z) = 1
β − 1
• Solve the differential equation:
f(z) = 1β
− 1β
exp(
− β
β − 1z
)• Definition of allocation function:
gi(s) =
1β − 1
β exp(
− ββ−1
si
s3−i
)si ≤ s3−i
β−1β + 1
β exp(
− ββ−1
s3−i
si
)si > s3−i
![Page 55: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/55.jpg)
Two players: the design of E2-PYS
• Require:
f ′(z) + 1f ′(z) + f(z)
= β ⇔ f ′(z) + β
β − 1f(z) = 1
β − 1
• Solve the differential equation:
f(z) = 1β
− 1β
exp(
− β
β − 1z
)
• Definition of allocation function:
gi(s) =
1β − 1
β exp(
− ββ−1
si
s3−i
)si ≤ s3−i
β−1β + 1
β exp(
− ββ−1
s3−i
si
)si > s3−i
![Page 56: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/56.jpg)
Two players: the design of E2-PYS
• Require:
f ′(z) + 1f ′(z) + f(z)
= β ⇔ f ′(z) + β
β − 1f(z) = 1
β − 1
• Solve the differential equation:
f(z) = 1β
− 1β
exp(
− β
β − 1z
)• Definition of allocation function:
gi(s) =
1β − 1
β exp(
− ββ−1
si
s3−i
)si ≤ s3−i
β−1β + 1
β exp(
− ββ−1
s3−i
si
)si > s3−i
![Page 57: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/57.jpg)
LPoA(E2-PYS) = 1.792
• For s1 ≤ s2, by definition, LPoA(E2-PYS) = β
• For s1 > s2, we can show that LPoA(E2-PYS) ≤ β
• By anonymity
f(1) = 12
⇔ 1β
− 1β
exp(
− β
β − 1
)= 1
2⇔ β ≈ 1.792
![Page 58: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/58.jpg)
LPoA(E2-PYS) = 1.792
• For s1 ≤ s2, by definition, LPoA(E2-PYS) = β
• For s1 > s2, we can show that LPoA(E2-PYS) ≤ β
• By anonymity
f(1) = 12
⇔ 1β
− 1β
exp(
− β
β − 1
)= 1
2⇔ β ≈ 1.792
![Page 59: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/59.jpg)
LPoA(E2-PYS) = 1.792
• For s1 ≤ s2, by definition, LPoA(E2-PYS) = β
• For s1 > s2, we can show that LPoA(E2-PYS) ≤ β
• By anonymity
f(1) = 12
⇔ 1β
− 1β
exp(
− β
β − 1
)= 1
2⇔ β ≈ 1.792
![Page 60: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/60.jpg)
The design of E2-SR
• Change the payment function: p1(s) = s1s2
and p2(s) = s2s1
• Follow the same reasoning as with E2-PYS
![Page 61: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/61.jpg)
Open problems
• Is there a simple mechanism that achieves the 2 − 1n bound?
• LPoA bounds over more general equilibrium concepts?• Other resource allocation problems with budget constraints?• Other efficiency benchmarks?
Thank you!
![Page 62: students.ceid.upatras.grstudents.ceid.upatras.gr/~voudouris/liquid.voudouris.pdf · Characterizationofworst-casegames For every signal vector s, define the following game: • player](https://reader035.fdocuments.in/reader035/viewer/2022063011/5fc77bb3f6cce417d20b3d40/html5/thumbnails/62.jpg)
Open problems
• Is there a simple mechanism that achieves the 2 − 1n bound?
• LPoA bounds over more general equilibrium concepts?• Other resource allocation problems with budget constraints?• Other efficiency benchmarks?
Thank you!