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Before we proceed to define the new stability concepts, we briefly
discuss the idea by considering how they can be applied in randomized
coalition formation mechanisms. Typically, a randomized coalition for-
mation process consists in multiple rounds, and in each round, a new
coalition structure and the associated new consequence are proposed,
and selected agents are invited to join (or switch to) the new coalition
(see, for examples, [4, 7-10]). The mechanisms terminate when no more
new proposals are made and no more agents choose to switch coalition
for a period of time. Given an NTU-IU game, in such mechanisms, the
task facing each agent thus involves the solving of the following classi-
fication problems: for any pair of possible consequences
x, y
X
,each
agent must classify, based on its opinion and preference rules, each
pair of consequences into one of three cases:
1.
x
is preferred to
y
;or
2.
y
is preferred to
x
;or
3. neither consequence is preferred to the other.
The problem is, of course, how to determine which consequence is
preferred to which ones amidst all those uncertainty and conflicts in
the decision rules a illustrated above. One approach is to group each
ordered pair of consequence (
x, y
) into one of three cases.
First, we say
y
is an
definite objection
to
x
for a given agent if the
agent prefers
y
over
x
under all possible scenarios.
Second, we say
y
is an
potential objection
to
x
for a given agent if
the agent prefers
y
to
x
under some of the possible scenarios. Finally,
we say the agent does not object
x
using
y
if it does not prefer
y
over
x
to any of the possible scenarios, and
vice versa
. More formally:
∈
Definition 3.4 (Agent Definite Objection)
An agent
i
,given
the evidences
E
and a consequence
y
,
definitely objects to
aconse-
quence
x
using
y
,if
y
i,B
i
x
holds but not
x
i,B
i
y
,where
B
i
is the
opinion of agent
i
.
Similarly, we say a consequence
y
is a potential objection to another
consequence
x
by the agent
i
, if that agent prefers
y
over
x
given that
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