Satisfaction Games in
Graphical Multi-resource
Allocation
Sun Ruijia 5110309622
1
Outline

Motivation

System model

Convergence of Nash Equilibria

Simulation

Conclusion
2
Outline

Motivation

System model

Convergence of Nash Equilibria

Simulation

Conclusion
3
Motivation
Communication system
Resource allocation
limited
users
Maximize the total utility
• Centralized manner
• Distributed manner
4
Motivation
•
•
•
•
…
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QoS
Satisfaction game
Multi-resource allocation games
Object replication
Satisfaction multi-resource allocation games
7
Outline

Motivation

System model

Convergence of Nash Equilibria

Simulation

Conclusion
8
System model

N nodes (entities) want to compete for R resources.

Nodes can be represented by an interference graph according to their
locations and each node has a set of neighbors.

The more nodes in neighbor set of node i choose the same resource,
the less the QoS received by node i is.

Each node has a demand QoS.

Once the QoS is larger than or equal to the demand, the user is
satisfied.

Each node can collect more than one resource.
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System model

N nodes (entities) want to compete for R resources.

Nodes can be represented by an interference graph according to their
locations and each node has a set of neighbors.

The more nodes in neighbor set of node i choose the same resource,
the less the QoS received by node i is.

Each node has a demand QoS.

Once the QoS is larger than or equal to the demand, the user is
satisfied.

Each node can collect more than one resource.
10
System model
ℛ = 𝟎, 1,2, … 𝑅
the set of resources
(𝒩,
ℛ,
𝒩 = {1,2,3, … , 𝑁}
the set of nodes
𝐷𝑛 ≥ 0
The QoS demand of node n
(𝑄𝑛𝑟 )𝑛∈𝒩,𝑟∈ℛ ,
QoS received by user
n who has selected
resource 𝑟
(𝐷𝑛 )𝑛∈𝒩 ,
𝒦 = {𝐾(𝑛)|𝑛 ∈ 𝒩} ,
𝐾(𝑖) is the number of
resources node 𝑛
needs to allocate
𝒢,
𝐾,
𝒢 = 𝑁, 𝐸 :
interference graph
Strategy profile: 𝒜 = {𝐴𝑛 |𝑛 ∈ 𝒩},
where 𝐴𝑛 = {𝑎𝑛𝑟 |𝑟 ∈ ℛ}
Indicator 𝑎𝑛𝑟 =
1, if resource 𝑟 is allocated by node 𝑛
0, otherwise
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𝒜)
System model
(𝒩, ℛ, (𝑄𝑛𝑟 )𝑛∈𝒩,𝑟∈ℛ , (𝐷𝑛 )𝑛∈𝒩 , 𝒢, 𝐾, 𝒜)
(𝑄𝑛𝑟 )𝑛∈𝒩,𝑟∈ℛ
QoS
Congestion level: the number of node in the set of
n’s neighbors choosing the resource r
𝐼𝑛𝑟
13
System model
(𝒩, ℛ, (𝑄𝑛𝑟 )𝑛∈𝒩,𝑟∈ℛ , (𝐷𝑛 )𝑛∈𝒩 , 𝒢, 𝐾, 𝒜)
(𝑄𝑛𝑟 )𝑛∈𝒩,𝑟∈ℛ
Congestion level: the number of node in the set of
n’s neighbors choosing the resource r
QoS
𝐼𝑛𝑟 = 𝑛 ∈ 𝒩 𝑛 : 𝑎𝑛𝑟 = 1
Entity n collects
resource r
(𝐷𝑛 )𝑛∈𝒩
Threshold 𝑇𝑛𝑟
𝐼𝑛𝑟 ≤ 𝑇𝑛𝑟 : entity is satisfied
𝐼𝑛𝑟 > 𝑇𝑛𝑟 : entity is dissatisfied
𝑇𝑛𝑟
𝐼𝑛𝑟
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System model
(𝒩, ℛ, (𝑄𝑛𝑟 )𝑛∈𝒩,𝑟∈ℛ , (𝐷𝑛 )𝑛∈𝒩 , 𝒢, 𝐾, 𝒜)
(𝑄𝑛𝑟 )𝑛∈𝒩,𝑟∈ℛ
𝐼𝑛𝑟 = 𝑛 ∈ 𝒩 𝑛 : 𝑥𝑛 = 𝑟
Threshold 𝑇𝑛𝑟
QoS
if 𝐼𝑛𝑟 ≥ 𝑇𝑛𝑟
if 𝐼𝑛𝑟 < 𝑇𝑛𝑟
1,
𝑟
𝑈𝑛 =
0,
𝑈𝑛𝑟
𝑈𝑛 =
(𝐷𝑛 )𝑛∈𝒩
𝑟
(𝒩, ℛ, 𝑇𝑛𝑟 , 𝒢, 𝐾)
𝑇𝑛𝑟
𝐼𝑛𝑟
15
System model
The evicted set: 𝐸𝑛 𝑡 = {𝑟|𝑎𝑛𝑟 0 = 1 ∧ 𝑎𝑛𝑟 𝑡 = 0}
The inserted set: 𝐼𝑛 𝑡 = {𝑟|𝑎𝑛𝑟 0 = 0 ∧ 𝑎𝑛𝑟 𝑡 = 1}
1,2,3
3,5,6
𝐸𝑛 𝑡 = 1,2
𝐼𝑛 𝑡 = {5,6}
16
Outline

Motivation

System model

Convergence of Nash Equilibria

Simulation

Conclusion
17
Convergence of NE
Key concept in game theory

Definition 1 (Better Reply Update). The event where a player n
changes its choice of strategy from 𝑥𝑛 to r is a better reply update if and
only if 𝑈𝑛 𝑟, 𝒙−𝑛 > 𝑈𝑛 𝑥𝑛 , 𝒙−𝑛 , where we write the argument of the
function as 𝒙 = (𝑥𝑛 , 𝑥−𝑛 ) with 𝒙−𝑛 = (𝑥1 , … , 𝑥𝑛−1 , 𝑥𝑛+1 , … , 𝑥𝑁 )
representing the strategy profile of all users except player n.
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Convergence of NE

Definition 2 (Pure Nash Equilibrium). A strategy profile 𝒙 is a pure
NE if no users at 𝒙 can perform a better reply update, i.e., 𝑈𝑛 𝑟, 𝒙−𝑛 >
𝑈𝑛 𝑥𝑛 , 𝒙−𝑛 for any 𝑟 ∈ ℛ 𝑎𝑛𝑑 𝑛 ∈ 𝒩.

Definition 3 (Finite Improvement Property). A game has the finite
improvement property if any asynchronous better reply update process
terminates at a pure NE within a finite number of updates.
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Convergence of NE

Definition 4 (modified better reply) An modified better reply for
multi-resource allocation problem is that for ∀𝑟 ∈ 𝐸𝑛 (𝑡), 𝑈𝑛𝑟 𝑡 = 0;
and for ∀𝑟 ∈ 𝐼𝑛 (𝑡), 𝑈𝑛𝑟 𝑡 = 1. 𝐸𝑛 𝑡 = 𝐼𝑛 𝑡 ≤ 𝐾(𝑛).
Time slot 1 for
node n1
Update
resource 1
…
Update
resource 2
Time slot 1 for
node n
…
…
…
Time slot 1 for
node nn
Update
resource K(n)
20
Convergence of NE
Theorem 1. In satisfaction game for graphical multi-resource allocation,
FIP can be guaranteed under modified better reply.
Proof:
𝐹𝑛𝑟 𝐴𝑛 , 𝐴−𝑛
Φ 𝑨 =
𝑛
′
𝐹𝑛𝑟 −
𝑛
𝑟
𝐹𝑛𝑟
2𝑇𝑛𝑟 − 𝐼𝑛𝑟 𝑨 , if 𝑎𝑛𝑟 = 1
=
0,
if 𝑎𝑛𝑟 = 0
𝐹𝑛𝑟 ≥ 2
𝑛
For a modified better reply by node n, Φ 𝑨 will increase at least 2
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Convergence of NE
Theorem 1. In satisfaction game for graphical multi-resource allocation,
FIP can be guaranteed under modified better reply.
Proof:
𝐹𝑛𝑟 𝐴𝑛 , 𝐴−𝑛
Φ 𝑨 =
𝑛
𝑟
−𝑁 ≤ −𝐼𝑛𝑟 < 𝐹𝑛𝑟 ≤ 2𝑇𝑛𝑟 < 2𝑁 + 2
Φ 𝑨 < 3𝑁 + 2 𝑁
𝐾(𝑛)
𝐹𝑛𝑟
2𝑇𝑛𝑟 − 𝐼𝑛𝑟 𝑨 , if 𝑎𝑛𝑟 = 1
=
0,
if 𝑎𝑛𝑟 = 0
1
3𝑁 + 2 𝑁
2
𝐾(𝑛)
𝑛
𝑛
25
Outline

Motivation

System model

Convergence of Nash Equilibria

Simulation

Conclusion
26
Simulation
N=6
R=10
Set the threshold 𝑇𝑛𝑟 , interference graph and the initial stage randomly
27
Simulation
N=10
R=15
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Simulation
R=[7,15,25,35,45,53,65,75,83,92];
N=[10,20,30,40,50,60,70,80,90,100];
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Outline

Motivation

System model

Convergence of Nash Equilibria

Simulation

Conclusion
30
Conclusion
• We use the distributed manner to solve the resource
allocation problems
• Instead of single resource allocation, we propose that
every node can collect more than one resource
• Utilize the satisfaction game model to make the allocation
problem more practical
• Using a modified better reply strategy guarantees that NE
can be reached.
31
Conclusion
Future work
• Find a faster way than asynchronous update for
converge
• To collect more resources (so that we can reduce the
demand for each resource) to guarantee that the node
is satisfied with every resource he collects.
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Q&A
33
Thank you!
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