Page 76 - Innovations in Intelligent Machines

Team, Game, and Negotiation based UAV Task Allocation 65
i
An N-tuple {y i∗ ∈ Y ,i ∈ N} is said to constitute a mixed strategy non-
cooperative (Nash) equilibrium solution for a N-person game in normal form,
j
j
if the following N inequalities are satisfied for all y ∈ Y ,j ∈ N :
1
1∗ 2∗
J 1∗ ... y y ...y N∗ m (P 1 ,P 2 ,...,P N )
P 1 P 2 P N
q q q
P P P
1 2 N
1 1
≥ ... y y 2∗ ...y N∗ m (P 1 ,P 2 ,...,P N )
P 1 P 2 P N
q q q
P P P
1 2
N
2
1∗ 2∗
J 2∗ ... y y ...y N∗ m (P 1 ,P 2 ,...,P N )
P 1 P 2 P N
q q q
P 1 P 2 P N
1∗ 2 2
≥ ... y y y 3∗ ...y N∗ m (P 1 ,P 2 ,...,P N )
P 1 P 2 P 3 P N
q q q
P P P
1 2 N
. . .
. . .
. . .
N
1∗ 2∗
J N∗ ... y y ...y N∗ m (P 1 ,P 2 ,...,P N )
P 1 P 2 P N
q q q
P P P
1 2 N
N−1∗ N N
1∗ 2∗
≥ ... y y ...y y m (P 1 ,P 2 ,...,P N ) (37)
P 1 P 2 P N−1 P N
q q q
P P P
1 2 N
The noncooperative Nash equilibrium outcome of a N-person game in mixed
1∗
strategies is given by the N-tuple {J ,...,J N∗ }. If there exists an inner
˘ i
mixed strategy solution then, such a solution {y i∗ ∈ Y ; i ∈ N} of an N-person
game in normal form satisfies the following set of equations:
1 1 l
... y 2∗ ... y N∗ {m (P 1 ,...,P N ) − m (P ,P 2 ,...,P N )} =0,
1
P 2 P N
q q
P P
2 N
q l
P 1 ∈P ,P 1 = P ,
1
1
2 2 l
1∗ 3∗
... y y ... y N∗ {m (P 1 ,...,P N ) − m (P 1 ,P ,...,P N )} =0,
2
P 1 P 3 P N
q q q
P 1 P 3 P N
q l
P 2 ∈P ,P 2 = P ,
2 2
. .
. .
. .
N N l
... y 1∗ ... y N−1∗ {m (P 1 ,...,P N ) − m (P 1 ,...,P N−1 ,P )} =0,
P 1 P N−1 N
q q
P P
1 N−1
q
P N ∈P ,P N = P l (38)
N N
q
i
l
˘ i
where, P is any one of the search paths in P ,and Y is the interior of Y .If
i i
the inner mixed strategy solution does not exist then the above formulation
may not yield a feasible solution. In that case, we may have to choose some
other algorithm. The domain of the search effectiveness function increases