The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 103

Proof Since for any two coalitional act profiles S 1 and S 2 we have

( S 1 b-dom α S 2 ) ⇔¬ ( S 1 dom α S 2 )

for some α ,wehave,foranyprofile S 1 , there does not exist an alter-
native profile S 2 so that we have both S 2 b-dom α S 1 and S 2 dom α S 1 for
some α . So by Definition 4.6, we have S 1 in the b-core.

The situation is illustrated in Figure 4.13. Theorem 4.3 is a bit
counter-intuitive, as it says that any solution is stable if all the agents'
beliefs are incorrect. We can interpret this as follows. In a game where
all beliefs are inaccurate, the only objections that can be raised are
invalid ones, so that any alternative proposals will always be turned
down because it will be rejected by at least one of the coalition mem-
bers. On the other hand, the valid proposals will never be raised be-
cause they are in conflict with the agents' beliefs. 2 Another way to
understand this is that in a game where all beliefs are not accurate,
the negotiation would not able to make any progress because all raised
objections are based on incorrect beliefs. This makes any incremental
improvement impossible, and no agent will be able to break away from
their existing coalition to form new ones.

Set of all profiles

b-core

core

All beliefs are inaccurate

Fig. 4.13 Illustration of Theorem 4.3.

2 Since we are dealing with static beliefs in this chapter, no objections can be

accepted by all potential partners of a deviating coalition, and any solution is
therefore stable as a result.