Computational aspects of cooperative game theory by Georgios Chalkiadakis, Edith Elkind, Visit Amazon's Michael

By Georgios Chalkiadakis, Edith Elkind, Visit Amazon's Michael Wooldridge Page, search results, Learn about Author Central, Michael Wooldridge,

Cooperative video game conception is a department of (micro- )economics that stories the habit of self-interested brokers in strategic settings the place binding agreements between brokers are attainable. Our target during this booklet is to provide a survey of labor at the computational facets of cooperative online game concept. we commence by means of officially defining transferable software video games in attribute functionality shape, and introducing key solution Read more...

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Indeed, it is not hard to see that the optimal solution to the following linear program with variables ε, x1 , . . 2. SOLUTION CONCEPTS 31 Similarly, we can design a linear program to compute the cost of stability. 3); we will see examples of this in chapter 4. 3) with i∈N xi and taking the dual of the resulting LP, we can derive a complete characterization of superadditive games that have a non-empty core. This observation was first made by Bondareva [57] and, independently, by Shapley [232], and the resulting characterization became known as the Bondareva–Shapley theorem.

Ij −1 , ij }) − v({i1 , . . , ij −1 }) ≤ v({1, . . , ij }) − v({1, . . , ij − 1}) = xij . By adding up these inequalities for j = 1, . . , coalition C does not ✷ have an incentive to deviate, a contradiction. 28 2. 27, and the core can be shown to be a convex set. 27 does not, in general, enable us to check whether a given outcome is in the core of a given convex game. 4 Least Core The notion of stability captured in the core requires that no coalition can gain anything by deviating. Conversely, an outcome is unstable if a coalition can benefit even by a tiny amount from deviating.

38 2This means that the stable set does not quite fit the definition of solution concept given in chapter 1, in that the mapping from games to the set of all outcomes that it defines is multi-valued. 2. SOLUTION CONCEPTS 35 • J ∩ Dom(J ) = ∅ (internal stability), and • I (N) \ J ⊆ Dom(J ) (external stability). Even though a game can have many stable sets, there are also games that have no stable sets, though constructing such games is not trivial; the first example of a game with no stable set is due to Lucas [170].

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