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Proof
: It is straightforward to verify the state based ordinal game designed in
Section 4.2 is a potential game with the potential function
ϕ
(
x,a
).
The potential function
ϕ
(
x,a
) captures the global objective of distributed
optimization problem in multi-agent system. As we know in [4], the potential
game will guarantee the existence of an equilibrium while at the same time allow
the existing learning algorithms to be directly used in our game model. However,
whether the equilibriums of our designed game are solutions to the optimization
problem in (1) becomes our main concern in the following theorems.
Theorem 2.
Model the optimization problem in (1) as the state based ordinal
potential game proposed in section 4.2 with any positive constant
α
.Suppose
the interaction topology is undirected, time-varying, and the sequence of sens-
ing/communication matrixes is sequentially complete, then
N
,
e
i
=
v
k
.
∀
i,k
∈
: Suppose state action pair (
x,a
)=
(
v,e
)
,
(
v,e
)
is the equilibrium of the
game we design.
Proof
N
, and for any action profile
a
=
(
v
i
, v
−i
)(
e
i
, e
−i
)
,we
∀
i
∈
U
i
(
x,a
) according to the notion of Nash equilibrium.
can get
U
i
(
x,a
)
≤
∀
i,k
∈
N
, the adjacent agents of the agent
i
is denoted by
L
i
=
{
l
:
s
il
(
t
)=1
}
,
where
s
il
is the element of the sensing/communication matrix in
i
th row and
l
th
column at time instant
t
.
L
i
, the new action for the agent
i
, denoted
by
a
i
=(
v
i
, e
i
), is defined as
v
i
=
v
i
and
∀
j
1
,j
2
∈
⎧
⎨
e
i→j
+
δ, j
=
j
1
e
i→j
−
k
i→j
=
e
δ, j
=
j
2
(3)
⎩
e
i→j
,
j
∈
L
i
\{
j
1
,j
2
}
R
and
e
i→j
is the estimation that agent
i
passes to agent
j
regarding
the value of agent
k
. Accordingly the change in the local objective function for
where
∀
δ
∈
agent
i
can be expressed as
j
=1
s
ij
ΔU
i
=
j
=1
s
ij
U
i
(
x,a
)
−
j
=1
s
ij
U
i
(
x,a
).
It is noticed that the estimate items regarding to the value of player
k
haven't
changed for all the agents except the agent
j
1
and
j
2
, therefore, the change in
the local objective function for agent
i
simplifies to
n
s
ij
ΔU
i
=
φ
(
e
j
1
,...,e
j
1
+
δ,...,e
j
1
)+
φ
(
e
j
2
,...,e
j
2
−δ,...,e
j
2
)
j
=1
(4)
−φ
(
e
j
1
,...,e
j
1
,...,e
j
1
)
−φ
(
e
j
2
,...,e
j
2
,...,e
j
2
)
+
α
k
(2
δe
j
1
−
2
δe
j
2
+
δ
2
)
∈
N
When
δ
→
0, we can express the equation in (4) as
n
+
α
k
s
ij
ΔU
i
=
∂φ
2
e
j
2
)
δ
+
o
(
δ
2
)
∂φ
∂e
j
2
(2
e
j
1
−
∂e
j
1
−
(5)
j
=1
∈
N
As we suppose the state action pair (
x,a
)=
(
v,e,
)(
v,e
)
is the equilibrium
of the game model, we know that
∀
δ
∈
R
,
U
i
≥
0. Furthermore, as
δ
can
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