Example Application of the b-Core: Buyer Coalition Problem with Beliefs - Belief-based Stability in Coalition Formation with Uncertainty - page 129

Problem with Beliefs

Beliefs in NTU-Buyer Games) Given two private beliefs bel 1
and bel 2 ,wesay bel 1 is more accurate than bel 2 , if for all agents j ∈ N ,

and any bid b 1 and b 2 ,wehavethefollowings.
1) If bel 1 ( b 1 j b 2 ) holds but bel 2 ( b 1 j b 2 )doesnotholds,then

bel 1 ( b 1 j b 2 )holds.
2) If bel 1 ( b 1 j

b 2 ) does not hold but be 1 2 ( b 1 j b 2 ) holds, then

bel 1 ( b 1 j b 2 )doesnothold.

The following definition correspond to Definition 4.9, which extends
the is-more-accurate-than relation to belief profiles.

Definition 5.7 (Is-more-accurate-than Relation of Belief Pro-
files in NTU-buyer Games) Given two belief profiles

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

and

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

we say B is more accurate than B if there exists i

N such that bel i is
more accurate than bel i , and either bel j = bel j or bel j is more accurate
than bel j

∈

for all j ∈ N \{ i } .

As a specific example of NTU-PD games, the core and the b-core of
a NTU-buyer game coincide with each other when all the agent beliefs
are accurate. The following theorem states this conclusion, which is
the counterpart of Theorem 4.2.

Theorem 5.2 The core of a NTU-Buyer is the same as the b-core
if all agents' beliefs are accurate. That is, given an NTU-buyer games

g = N,G, ( i ) ,B ,

we have

b-core( g )=core( g )

if bel i ( b 1 j b 2 ) ⇒ b 1 j b 2 for all agents i, j ∈ N ,andanybids

b 1 ,b 2 ∈ B .