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

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Lecture 3The bargaining setThe 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, theywill be unable to reach an agreement about the payoff distribution. If they still want tobenefit 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 reachan argument and to maintain some notion of stability (though of course, it is a weakerversion of stability).The definition of the bargaining set is due to Davis and Maschler [1]. This notion isabout the stability of a given coalition structure (CS). The agents do not try to changethe nature of the CS, but simply to find a way to distribute the value of the differentcoalitions 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 distributionby 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 agreewith this objection, the payoff distribution should change as proposed. However, someother 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 willdescribe the precise notion of objections and counter-objections.3.1Objections, counter-objections and theprebargaining setLet(N, v,S)be a game with coalition structure andxan imputation.For the bar-gaining set, anobjectionfrom an agentiagainst a payoff distributionxis targeting aparticular agentj, in the hope of obtaining a payment fromj. The goal of agentiis toshow that agentjgets too much payoff as there are some ways in which some agents31
32Lecture 3. The bargaining setbutjcan benefit. The objection can take the following (informal) form:I get too little in the imputationx, and agentjgets too much! I can form acoalition that excludesjin which some members benefit and all membersare at least as well off as inx.We recall that in a game with coalition structure(N, v,S), the agents do not try tochange the CS, but only obtain a better payoff. The setX(N,v,S)offeasiblepayoffvectors for(N, v,S)is defined asX(N,v,S)={xRn| ∀C ∈ S,i∈Cxiv(C)}. Weare now ready to formally define an objection.3.1.1.DEFINITION. [Objection] Let(N, v,S)be a game with coalition structure,xX(N,v,S),C ∈ Sbe a coalition, andiandjtwo distinct members ofC((i, j)∈ C2,i=j). Anobjection ofiagainstjis a pair(P, y)wherePNis a coalition such thatiPandj /P.

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Term
Summer
Professor
NoProfessor
Tags
Men in Black, Cooperative game