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Problem with Beliefs
Since the coalition structure already implies a bid (
g
1
,4),which
is as good as possible for this agent,
a
1
will opt for maintaining the
status quo (Option 1 in the mechanism). The same is true for agent
a
2
.
When it is
a
3
's turn to move, however, that agent will realise there
may be better potential bids. Suppose that it decides to try to form
a new coalition in
r
3
. It does so by sending proposal messages to all
agents that are not already in that room. In this case, that include
a
1
,
a
2
and
a
4
.
Upon receiving the proposal, both
a
1
and
a
2
realise that there is
no possible alternative coalition, whatever the size is, that can make
abidfor
g
3
more attractive than the current bid, so both
a
1
and
a
2
will reject the proposal immediately.
Agent
a
4
, on the other hand, will find out that there are four
potential bids, namely (
g
3
,
1), (
g
3
,
2), (
g
3
,
3), and (
g
3
,
4),thatitprefers
over the current bid (
g
1
,
4). Therefore, agent
a
4
will indicate to
a
3
that it will be interested in switching if the resulting new coalition
has at least 1 member (which is the smallest coalition size among the
preferences listed above).
Upon receiving the other agents' responses, agent
a
3
can construct
a new coalition structure such that both switching agents (
a
3
and
a
4
)
are better off than previously (assuming that this coalition structure
has not been reached before). Agent
a
3
then announces the new coali-
tion structure to all agents and a new round begins. After this, no
other agents can make any more successful proposal. The mechanism
stops after 12 moves, and the coalition structure is not changed any
further. The resulting coalition structure is then used as the solution
of the mechanism. It is not di
cult to show that this solution is in the
core.
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