kernel.pdf - Lecture 5 The Kernel The Kernel is another...

Lecture 5
The Kernel
The Kernel is another stability concept that weakens the stability requirements of the
core. It was first introduced by Davis and Maschler [3]. The definition of the kernel is
based on the excess of a coalition. For the nucleolus, a positive excess was interpreted
as an amount of complaint as by forming a coalition with positive excess, some payoff
was lost. In the kernel, a positive excess is interpreted as a measure of threat: in the
current payoff distribution, if some agents deviate by forming coalition with positive
excess, they are able to increase their payoff by redistributing the excess between them.
When any two agents in a coalition have similar threatening powers, the kernel consid-
ers that the payoff is stable. In the following, we will see two definitions of the kernel
and we will see that it is guaranteed to be non-empty.
5.1 Definition of the Kernel
We recall that the excess related to coalition C for a payoff distribution x is defined as
e ( C , x ) = v ( C ) - x ( C ) . We saw that a positive excess can be interpreted as an amount
of complaint for a coalition. We can also interpret the excess as a potential to generate
more utility. Let us consider that the agents are forming a CS S = {C
1
, . . . , C
k
} , and
let consider that the excess of a coalition C / ∈ S is positive. Agent i ∈ C can view the
positive excess as a measure of his strength: if she leaves its current coalition in S and
forms coalition C ⊆ N , she has the power to generate some surplus e ( C , x ) . When
two agents want to compare their strength, they can compare the maximum excess of
a coalition that contains them and excludes the other agent, and the kernel is based on
this idea.
5.1.1. D
EFINITION
. [Maximum surplus] For a TU game ( N, v ) , the maximum surplus
s
k,l
( x ) of agent k over agent l with respect to a payoff distribution x is
s
k,l
( x ) = max
C⊆ N | k ∈C , l/ ∈C
e ( C , x ) .
53

54 Lecture 5. The Kernel
For two agents k and l , the maximum surplus s
k,l
of agent k over agent l with re-
spect to x is the maximum excess from a coalition that includes k but does exclude l .
This maximum surplus can be used by agent k to show its strength over agent l : as-
suming it is positive and that agent k can claim all of it, she can argue that she will
be better off without agent l ; hence she should be compensated with more utility for
staying in the current coalition. When any two agents in a coalition have the same
maximum surplus (except for a special case), the agents are said to be in equilibrium.
A payoff distribution is in the Kernel when all agents are in equilibrium. The formal
definitions follow:
5.1.2. D
EFINITION
. [kernel] Let ( N, v, S ) be a TU game with coalition structure. The
kernel is the set of imputations x ∈ I mp ( S ) such that for every coalition C ∈ S , if
( k, l ) ∈ C 2 , k = l , then we have either s
kl
( x ) ≥ s
lk
( x ) or x
k
= v ( { k } ) .