Example Application of the b-Core: Buyer Coalition Problem with Beliefs - Belief-based Stability in Coalition Formation with Uncertainty - page 122

members to pay more than the others for the same item as unfair and
will discourage people from joining.
An implication of these assumptions, as first pointed out in [12], is
that we can define the preferences of the agents solely in terms of the
obtained unit price and what product is being bought. We follow the
notation of [13]: since the resulting price of an item depends only on
the coalition size and the item, we can describe each buyer coalition
using a ( item , size ) couple, which we called a bid . Any coalition with
thesame( item , size ) bid is thus indistinguishable from an agent's point
of view.
Based on these assumptions, there are only m

n possible bids
where m is the number of products and n is the number of participating
agents, so the number of preferences to be specified is manageable.
Thus we can further simplify the agents' preference by defining each
agent's preference to be a totally ordered relation on all possible bids
(with ties broken arbitrarily). One of such example is given in Table
5.1, which says that agent a 2 's most preferred bid is ( g 1 , 4), followed
by ( g 2 , 4), and so on.

×

Tab l e 5 . 1 Example of Non-transferable utility preference of buyer agents

a 1 a 2 a 3 a 4

Item Size Item Size Item Size Item Size

g 1 4 g 1 4 g 3 4 g 3 4

g 1 3 g 2 4 g 3 3 g 2 4

g 1 2 g 1 3 g 2 4 g 3 3

g 1 1 g 2 3 g 3 g 2 3

2

g 2 4 g 1 2 g 2 3 g 3 4

g 2 3 g 2 2 g 3 1 g 3 1

g 2 2 g 2 1 g 2 2 g 2 2

g 2 1 g 1 1 g 2 1 g 2 1

g 3 4 g 3 4 g 1 4 g 1 4

g 3 3 g 3 3 g 1 3 g 1 3

g 3 2 g 3 2 g 1 2 g 1 2

g 3 1 g 3 1 g 1 1 g 1 1