Distributed Optimization and State Based Ordinal Potential Games - Intelligent Computing for Sustainable Energy and Environment - page 116

and action dependent state transition function f : X

× A → X . For every i ∈ N ,

a −i ∈ j = i A j , a i ∈ A i ,given the old state action pair ( x,a i ,a −i )

and the new state action pair ( x,a i ,a −i ) ,if there exists a differential and convex
potential function ϕ : X

A i , a i ∈

× A → R that satisfy the following property

sgn U i ( x,a i ,a −i ) U i ( x,a i ,a −i ) = sgn ϕ ( x,a i ,a −i ) ϕ ( x,a i ,a −i )

− −

(2)

Then G is called state based ordinal potential game with potential funciton ϕ .

4.2 State Based Ordinal Potential Game Design
The design process of state based ordinal potential game is based on the work
of Li [13]. In contrary to Li [13], the desirable global objective can be acquired
even if the interaction topology is not connected at some time intervals based
on the rule of communication.
Given v =( v 1 ,v 2 ,...,v n ) as the tuple of value profile for n players. In Marden
[14], the average of local objective functions of agent i s neighboring agents is
chosen to act as a new kind of local objective function for agent i ,termedas
equally shared utility, which is U i ( v )= j =1 s ij U j ( v ) / j =1 s ij .
Because an agent may not have the complete knowledge about the true value
of the local agents' action as quickly as possible, especially in systems with
intermittent communication or time delays. So we make use of the estimation
term e =( e 1 ,e 2 ,...,e n ) in the state space to estimate the true value of actions.
Accordingly, the equally shared utility above can be rewritten as U i ( e j | s ij =1 )=

j =1 s ij φ ( e j ,e j ,...,e j ) / j =1 s ij .
Next, we will introduce the equation above as one of the components in the
following design for local objective function in the framework of state based
ordinal potential game. Meanwhile, considering the error caused by the in-
troduction of the estimation items, thus we will have to minimize the errors
so that the global objective can be achieved. Accordingly we define the lo-
cal objective function as U i ( x,a )= U i ( x,a )+ αU i ( x,a ), where U i ( x,a )=

j =1 k s ij ( e i −

e j ) 2 / j =1 s ij , which is the component in order to minimize
theerrorbetweenthetruevalueandtheestimation items in our game model,
and α is a positive tradeoff parameter. By inspired from the notion of equally
shared utility, we define U i ( x,a )= U i ( e j | s ij =1 ).

4.3 Analytical Properties of State Based Ordinal Potential Games
Next, we need to further analyze the analytical properties of the model and
verify whether the model we designed meets the desired goals or not. First and
foremost, a very important issue is to verify whether the designed model results
in the framework of potential game or not.
Theorem 1. Model the distributed optimization problem in (1) as game model
in Section 4.2 with any positive constant α . Given the potential function as

ϕ ( x,a )= ϕ φ ( x,a )+ αϕ e ( x,a ) where ϕ φ ( x,a )= j =1 φ ( e j ,...,e j ) /n and

ϕ e ( x,a )= i j =1 k s ij ( e i −

e j ) 2 / 2 n . Then the game model in Section 4.2

is potential game.