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

• Xenocygne2
• 9

Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. This preview shows page 1 - 3 out of 9 pages.

Lecture 5The KernelThe Kernel is another stability concept that weakens the stability requirements of thecore. It was first introduced by Davis and Maschler [3]. The definition of the kernel isbased on the excess of a coalition. For the nucleolus, a positive excess was interpretedas an amount of complaint as by forming a coalition with positive excess, some payoffwas lost. In the kernel, a positive excess is interpreted as a measure of threat: in thecurrent payoff distribution, if some agents deviate by forming coalition with positiveexcess, 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 kerneland we will see that it is guaranteed to be non-empty.5.1Definition of the KernelWe recall that theexcessrelated to coalitionCfor a payoff distribution x is defined ase(C, x) =v(C)-x(C). We saw that a positive excess can be interpreted as an amountof complaint for a coalition. We can also interpret the excess as a potential to generatemore utility. Let us consider that the agents are forming a CSS={C1, . . . ,Ck}, andlet consider that the excess of a coalitionC/∈ Sis positive. Agenti∈ Ccan view thepositive excess as a measure of his strength: if she leaves its current coalition inSandforms coalitionC ⊆N, she has the power to generate some surpluse(C, x). Whentwo agents want to compare their strength, they can compare the maximum excess ofa coalition that contains them and excludes the other agent, and the kernel is based onthis idea.5.1.1.DEFINITION. [Maximum surplus] For a TU game(N, v), themaximum surplussk,l(x)ofagentkover agentlwith respect to a payoff distributionxissk,l(x) =maxC⊆N|k∈C, l/∈Ce(C, x).53
54Lecture 5. The KernelFor two agentskandl, themaximum surplussk,lof agentkover agentlwith re-spect toxis themaximum excessfrom a coalition thatincludeskbut doesexcludel.This maximum surplus can be used by agentkto show its strength over agentl: as-suming it is positive and that agentkcan claim all of it, she can argue that she willbe better off without agentl; hence she should be compensated with more utility forstaying in the current coalition. When any two agents in a coalition have the samemaximum 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 formaldefinitions follow:5.1.2.DEFINITION. [kernel] Let(N, v,S)be a TU game with coalition structure. Thekernelis the set of imputationsx∈ Imp(S)such that for every coalitionC ∈ S, if(k, l)∈ C2,k=l, then we have eitherskl(x)slk(x)orxk=v({k}).

Course Hero member to access this document

Course Hero member to access this document

End of preview. Want to read all 9 pages?

Course Hero member to access this document

Term
Summer
Professor
NoProfessor
Tags
Trigraph, coalition, Software agent, Men in Black