The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 100

Example 4.20 We consider the scenario in Example 4.4. Agent b 's
belief bel b is more accurate than agent a 's belief relation bel a , because
in bel a thereisabeliefofpreference( , movie)

which is not in agent b 's belief bel b , and this preference is false. Sim-
ilarly, agent a 's belief bel a is also more accurate than agent b 's belief
bel b ,becausein bel b there is a false belief of preference

{ a, b, c } , movie) b ( { b, c }

( { a, b, c } , movie) a ( { a, b } , movie)

which is not in agent a 's belief bel a . Thisisanexamplethatshows
that the more-accurate-than relation is generally not transitive and not
asymmetric.

Given two arbitrary beliefs bel 1 and bel 2 , it is possible that we
cannot conclude that bel 1 is more accurate than bel 2 , but we cannot
conclude that bel 2 is more accurate than bel 1 , either. A trivial example
of such a case is when all and only those preferences that are correct
are in both bel 1 and bel 2 ,orwhenboth bel 1 and bel 2 are empty. In
these examples, we are unable to find any preference that is in one
of bel 1 and bel 2 but not the other, whether it is correct or incorrect.
Hence, we cannot say that bel 1 is more accurate than bel 2 ,or bel 2 is
more accurate than bel 1 . In general, the same arguments hold for all
cases when bel 1 = bel 2 .

Definition 4.9 (Accuracy Relation of Agents' Private Belief
Profiles) We define the is more accurate than relation of agents'
private belief profiles as follows. Given two belief profiles

B = {bel 1 ,bel 2 ,...,bel n },
B = bel 1 ,bel 2 ,...,bel n } { ,

we say B is more accurate than B

if there exists i ∈ N such that

is more accurate than bel i , and either bel j = bel j

bel i or bel j is more

accurate than bel j

for all j = i .

In short, given two collections of agent beliefs (either from the same
agent or from different agents), we say that one set of agent beliefs is