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The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric (quota) games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending on invariants of simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts. [ABSTRACT FROM AUTHOR]

This paper considers the computational complexity of the design of voting rules, which is formulated by simple games. We prove that it is an NP -complete problem to decide whether a given simple game is stable, or not. [ABSTRACT FROM AUTHOR]

We study the stability and the stability index of the meet game form defined on a meet semilattice. Given any active coalition structure, we show that the stability index relative to the equilibrium, to the beta core and to the exact core is a function of the Nakamura number, the depth of the semilattice and its gap function. [ABSTRACT FROM AUTHOR]

We study the structure of unstable power mechanisms. A power mechanism is modeled by an interaction form, the solution of which is called a settlement. By stability, we mean the existence of some settlement for any preference profile. Configurations that produce instability are called cycles. We introduce a stability index that measures the difficulty of emergence of cycles. Structural properties such as exactness, superadditivity, subadditivity and maximality provide indications about the type of instability that may affect the mechanism. We apply our analysis to strategic game forms in the context of Nash-like solutions or core-like solutions. In particular, we establish an upper bound on the stability index of maximal interaction forms. [ABSTRACT FROM AUTHOR]

Abstract: Acyclicity of individual preferences is a minimal assumption in social choice theory. We replace that assumption by the direct assumption that preferences have maximal elements on a fixed agenda. We show that the core of a simple game is nonempty for all profiles of such preferences if and only if the number of alternatives in the agenda is less than the Nakamura number of the game. The same is true if we replace the core by the core without majority dissatisfaction, obtained by deleting from the agenda all the alternatives that are non-maximal for all players in a winning coalition. Unlike the core, the core without majority dissatisfaction depends only on the playersʼ sets of maximal elements and is included in the union of such sets. A result for an extended framework gives another sense in which the core without majority dissatisfaction behaves better than the core. [Copyright &y& Elsevier]