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counter-objections. More formally, let
x
be an imputation in a coalition game with
transferable payoff
(
A,v
)
, we define our argumentation scheme as follows:
•
(
S, y
)
is
an
objection
of
coalition
S
to
x
against
T
if
S
excludes
i
and
e
(
S ∪{i} ,x
)
>e
(
S, y
)
•
coalition
T
counteracts to the objection of coalition
S
against accepting player
i
if
e
(
T ∪{i} ,y
)
/e
(
T,x
)
>
1+
μ
or
e
(
T ∪{i} ,y
)+
e
(
S, y
)
>e
(
T,x
)+
e
(
S ∪{i} ,x
)
− τ
.
To correlate the game-theoretic terms introduced above to our setting, we give the in-
terpretation of these terms for our scenario. Specifically, by imputation we mean the
distribution of utilities over the coalitions' set, whereas the excess
of each coalition
represents the difference between its potential maximum in utility (which corresponds
to meeting the desired energetic profile, see Section 3) and its current utility
e
.
We reason that the excess criteria applied for solution concepts such as the kernel
and the nucleolus appears to be an appropriate measure of the coalition's efficiency,
especially in games where the primary concern lies rather in the performance of the
coalition itself. This basis further advocates for argumentation settings where objections
are raised by coalitions and not by single players, such as the case of the bargaining set
or the kernel. The objection
v
(
S, y
)
may be interpreted as an argument of coalition
S
for
excluding
where its excess is being decreased. Our solution
models situations where such objections cause unstable outcomes only if coalition
i
resulting in imputation
y
T
to which the objection has been addressed fails to counterobject by asserting that
S
's
demand is not justified since
T
's excess under
y
by accepting
i
would be larger than
it was under
. Such a response would have hold if we simply presumed players to
be self-interested and not mind the social welfare of the system. If on the contrary,
players are concerned with the overall efficiency of the system, they would consider
accepting the greater sacrifice of
x
y
in comparison to
x
only if this would account for
an improvement of
S
that exceeds the deterioration of
T
's performance by at least the
margin
τ
. Thus,
τ
is the threshhold gain required in order for justifying the deviation,
whereas
's tolerance to suboptimal gains.
Recalling our collaborative VPP scenario, it becomes imperative, as system design-
ers, endowing the system with the possibility for relaxing standards of their individual
performance in the interest of the social welfare. Our proposed mechanisms, thus aims
at assessing how and to what extent this may be achieved in order to satisfy the desired
system functionalities. We further elaborate on this matter in Section 5 based on the
experiments performed.
For applying this solution concept to our setting, we additionally need to take into
account the underlying topology and thus restrain the inter-coalition argumentation to
the given network structure, representing a particularization of the more generic out-
line presented herein. Thus, each coalition perceives a local solution with respect to its
neighborhood. Accordingly, from coalition
μ
represents
S
S
i
's local view point at iteration
l
the local
solution is:
CS
i
(
l
)=
{S
i
1
,S
i
2
, ..., S
ik
} ,S
ik
∈N
S
i
A potential argument of one coalition would trigger reactions in its vicinity and so,
coalitions need to make local adaptive decisions. Therefore, the system is able to
