Page 78 - Innovations in Intelligent Machines

Team, Game, and Negotiation based UAV Task Allocation 67
N
q q
|P |×| P | (41)
i j
j=1
j =i
The pure strategy Nash equilibrium may not always exist, in which case we
have to use mixed strategy equilibrium. The main disadvantage of using the
earlier model is that, if there is no inner mixed strategy Nash solution then
we may not be able to find a feasible solution. However, in this model we can
directly use the bilinear programming method to compute the mixed strategies
equilibrium.
A pair {y ,z } constitutes a mixed-strategy Nash Equilibrium solution
∗
∗
to a bimatrix game (M 1,i ,M 2,i ) if, and only if, their exists a pair (f ,g )
∗
∗
∗
∗
∗
∗
such that {y ,z ,f ,g } is a solution of the following bilinear programming
problem:
1,i 2,i
min [−y M z − y M z + f + g] (42)
y,z,f,g
subject to
− M 1,i z ≥−f · 1 q , − M 2,i z ≥−g · 1 N
|P | q
i | j=1 P |
j
j =i

q =1,z · 1
y ≥ 0,z ≥ 0,y · 1 |P | N q = 1 (43)
i | j=1 P |
j
j =i
q
where 1 q and 1 N q are column vectors of dimensions |P | and
1 | j=1 P |
|P (C s 1 )| i
j
j =i
N q
| j=1 P |, with all elements equal to 1.
j
j =i
Security Strategy
In security strategy the individual agents try to secure their minimal profits
assuming adversarial behavior of the other players. For this purpose, the coali-
tional form given above is the ideal framework to obtain security strategies.
Then, agent A i chooses a ’row k ’ whose smallest entry is no smaller than the
∗
smallest entry of any other row, which implies
1,i
1,i
∗
V(M 1,i ) = max min m , k = arg max{min m } (44)
kl
kl
¯ k l k l
q
where, k =1, 2,... , |P |,and l represents a particular combination of strate-
i
gies used by the N − 1 agents and
N
q

l =1, 2,... , | P | (45)
j
j=1
j =i
i
Further, k is the security strategy for A i , and V(M ) is the guaranteed payoff
∗
¯
to A i . Every agent computes the security strategy individually and adopts the
route given by k .
∗