Telecom SudParis, Universite Paris-Saclay, CNRS, France. INRIA Sophia Antipolis-Mediterranee, France. Resource Allocation Polytope Games. A. B w. B1 w.
Resource Allocation Polytope Games Uniquess of Equilibrium, PoS, PoA Swapnil Dhamal, Walid Ben-Ameur, Tijani Chahed, and Eitan Altman Telecom SudParis, Universite Paris-Saclay, CNRS, France INRIA Sophia Antipolis-Mediterranee, France
Resource Allocation Polytope Games
A x1 wA1
x2
x3
w wA2 A2
w wA3
wB2
wB3
w B1
y2
y1
y3
Price of Stability and Price of Anarchy
Uniqueness of PSNE
xn wAn A4
Unique PSNE if and only if IOS is nonconflicting and either (a) common contiguous set consists of at most one node, or (b) all nodes in common contiguous set invested on by only one player in the IOS.
1
3
2
4
2
3
1
(wBiyi + αwAixi)
subject to
xi, yi ≥ 0, ∀i ∈ N X X xi ≤ kA, yi ≤ kB
i∈N
πA( jA )
JA
IB
πB( jB )
JB
CA = {1, 2} (x, y) is a PSNE if and only if ∃ jA, jB s.t.
1
3
2
4
CB = {4, 2}
∀i ∈ IA ∪ IB : xi + yi = 1
i∈N
IA
3
i∈N
uB (x, y) =
No universal constant bound for PoA
CB = {4}
(wAixi + βwBiyi),
X
α, β > −1 =⇒ PoS = 1
CAB = {} 4
uA(x, y) =
i∈N
CA = {1, 2}
B
X
For i = πA(jA), πB (jB ) : xi + yi ≤ 1
CAB = {2} 4
3
2
∀i ∈ JA : xi = 0, yi ≤ 1
1
3 CA = {1, 2, 3}
∀i ∈ JB : yi = 0, xi ≤ 1 X X xi = kA, yi = kB , ∀i ∈ N : xi, yi ≥ 0
CB = {2, 3}
Worst PSNE:
i∈N
xi + yi ≤ 1, ∀i ∈ N 1
3
2
4
Polytope game - Linear common coupled constraints - Bilinear utility functions
CAB = {2, 3} 2
4
3
1
3
i∈N
i∈N
For jA ← dkAe to dkA + kB e For jB ← dkB e to dkA + kB e X min xi zAi − max{zBi − zBπB (jB ), 0} x
i∈N
Exact Restricted Potential Game
subject to
CA = {1, 2, 3} ∀x0, x00 ∈ F (y) : uA(x0, y) − uA(x00, y) X X = (wAix0i + wBiyi) − (wAix00i + wBiyi) i∈N 0
00
00
∀y , y ∈ F (x) : uB (x, y ) − uB (x, y ) X X 0 00 = (wAixi + wBiyi) − (wAixi + wBiyi ) i∈N
1
3
2
∀i ∈ N : xi ∈ [0, 1]
4
CB = {2, 3, 1}
∀i ∈ IA ∩ JB : xi = 1 X xi + xπA(jA) = kA
CAB = {1, 2, 3}
i∈N 0
(zAixi + zBiyi)
Characterization of PSNE
wB4 Bn
yn
uA(x, y) + uB (x, y) =
X
2
1
3
4
7
i∈IA
(X πB (jB ) ∈ IA :
xi + xπB (jB ) = jB − kB
i∈IB Socially Optimal Strategy Profile
i∈N
Exact restricted potential game P Potential fn Φ(x, y) = i∈N (wAixi +wBiyi)
zAi = (1 + α)wAi, zBi = (1 + β)wBi
X xi ≥ (jB − 1) − kB i∈IB πB (jB ) ∈ / IA : X xi + xπB (jB ) ≤ jB − kB i∈IB
Non-conflicting Independent Optimal Strategies
IOS of a player is its strategy in absence of other player. IOS (ˆ x, y ˆ) is non-conflicting if and only if xˆi + yˆi ≤ 1, ∀i ∈ N . Conflicting IOS =⇒ Non-unique PSNE Non-conflicting IOS =⇒ 6 Unique PSNE
AAAI 2018
max max x
y≤1−x
X i∈N
= max max x
jB
(zAixi + zBiyi) Paradox
X
zAixi +
i∈N
X
(1 − xi)zBi
i∈IB
+ kB −
X
(1 − xi) zBπB (jB )
i∈IB
h
= max max xi zAi − max{zBi − zBπB (jB ), 0} x jB i∈N i X + max{zBi − zBπB (jB ), 0} + kB zBπB (jB ) i∈N
X
M
w mA2
w mB1
w MB2
1
2
M >> m α=β=0 kA = kB = 1 M +M PoA = m+m
M + M − M If kB = 1 − , PoA = =1 M + M − M
