bs.pdf - Lecture 3 The bargaining set The notion of core...

Lecture 3
The bargaining set
The notion of core stability is maybe the most natural way to describe stability. How-
ever, some games have an empty core. If agents adopts the notion of the core, they
will be unable to reach an agreement about the payoff distribution. If they still want to
benefit from the cooperation with other agents, they need to relax the stability require-
ments. In this lecture, we will see that the notion of bargaining set is one way to reach
an argument and to maintain some notion of stability (though of course, it is a weaker
version of stability).
The definition of the bargaining set is due to Davis and Maschler [1]. This notion is
about the stability of a given coalition structure (CS). The agents do not try to change
the nature of the CS, but simply to find a way to distribute the value of the different
coalitions between the members of each coalition. Let us assume that a payoff distri-
bution is proposed. Some agents may form an objection against this payoff distribution
by pointing out a problem of that distribution and by offering a different payoff distri-
bution that eliminates this issue (or improves the situation). If all other agents agree
with this objection, the payoff distribution should change as proposed. However, some
other agents may form a counter-objection showing some shortcomings of the objec-
tion. The idea of stability in this context is to ensure that, for each possible objection,
there exists a counter-objection. When this is the case, there is no ground for chang-
ing the payoff distribution, which provides some stability. In the following, we will
describe the precise notion of objections and counter-objections.
3.1 Objections, counter-objections and the
prebargaining set
Let ( N, v, S ) be a game with coalition structure and x an imputation. For the bar-
gaining set, an objection from an agent i against a payoff distribution x is targeting a
particular agent j , in the hope of obtaining a payment from j . The goal of agent i is to
show that agent j gets too much payoff as there are some ways in which some agents
31

32 Lecture 3. The bargaining set
but j can benefit. The objection can take the following (informal) form:
I get too little in the imputation x , and agent j gets too much! I can form a
coalition that excludes j in which some members benefit and all members
are at least as well off as in x .
We recall that in a game with coalition structure ( N, v, S ) , the agents do not try to
change the CS, but only obtain a better payoff. The set X
( N,v, S )
of feasible payoff
vectors for ( N, v, S ) is defined as X
( N,v, S )
= { x ∈ R n | ∀C ∈ S ,
∑
i ∈C
x
i
≤ v ( C ) } . We
are now ready to formally define an objection.
3.1.1. D
EFINITION
. [Objection] Let ( N, v, S ) be a game with coalition structure, x ∈
X
( N,v, S )
, C ∈ S be a coalition, and i and j two distinct members of C ( ( i, j ) ∈ C 2 ,
i = j ). An objection of i against j is a pair ( P, y ) where
• P ⊆ N is a coalition such that i ∈ P and j / ∈ P .