Example Application of the b-Core: Buyer Coalition Problem with Beliefs - Belief-based Stability in Coalition Formation with Uncertainty - page 134

The current round ends.
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After the last agent in the proposal order has proposed, the first
agent then proposes again and the process repeats. A round ends and
the next round begins only when one of the proposing agents is able
to make a successful proposal (Option 3) or chooses to switch to r 0
(Option 2) so that the coalition structure is changed as a result.
Note that the beginning of a new round does not mean we jump
back to the first agent in the proposal order. Rather, the next agent
to propose is still the next one in the proposal order.
The mechanism ends after there is no successful proposal in over

n m moves in a row, where m is the number of products and n is

the number of agents.
Note that this mechanism does not assume the proposing agent
would send the proposal messages to all agents that are not currently
in the target room. Rather, the proposing agent should select the re-
cipients of the messages according to their preferred strategy. There
are two strategies worth considering here: a non-belief based strategy
(nb-strategy) and a belief-based one (b-strategy).
The non-belief-based strategy (nb-strategy) is as follows. When it
is an agent a i 's turn to make a proposal, he should check whether there
is a bid

×

b =( g i ,s )

where g ∈ G ,and s is greater than the current size of r i , such that

b i b a i |CS ,

where CS is the current coalition structure as mentioned above.
1) If such a bid b exists and the agent a i has not already proposed
product g i in the current round, then he should select Option 3 and
send proposal messages to all agents that is not already in room r i .
2) Otherwise, if no such bid exists, the agent a i should check if

there exists

b =( g i , 1) ,