Example Application of the b-Core: Buyer Coalition Problem with Beliefs - Belief-based Stability in Coalition Formation with Uncertainty - page 123

Problem with Beliefs

5.2.1.2 Bids and Agents' Preferences

We will now formally define the agent preferences. However, before we
do that, we first need to define the idea of valid bids, as follows

Definition 5.1 (Bids and Valid Bids) A bid is a couple ( item,
size ), where item is a product being sold, and size is the number of
units requested by a buyer coalition, which is also equal to the size of
the coalition. A bid b =( g i ,s )isa valid bid if its size s is less than or
equal to both the total number of participating buyer agents, and the
total number of items available for sale from the seller.

)torepresentthebid
of that coalition. Similarly, given an agent a and a coalition structure
CS ,wewrite b a|CS to represent the bid of the coalition C in CS ,of
which a is a member, i.e., b a|CS =( g C , |C| )where a ∈ C ∈ CS .
The preference of an agent i in a buyer coalition problem is denoted
by a relation

For a given coalition C ,wewrite b C =( g C , | C |

i on the set of all valid bids, such that, for any two valid
bids b 1 and b 2 ,wehave b 1 i b 2 if agent i considers b 1 to be no less

preferred than b 2 .

5.2.2 Agent Beliefs

Our next step is to define the agent beliefs. As in many other coalition
formation problems, most existing works in buyer coalition assume
that each individual agent's preferences are known to each other, an
assumption that we do not follow in this work.
Once again, we employ a more realistic belief-based model. In many
situations, the agents can have some beliefs regarding the other agents'
preferences even though they do not have access to the others' prefer-
ence directly.
For example, in a repeated coalition formation game, an agent can
have models of his peers' preferences by observing their past actions