Necessary and sufficient conditions for l-stability of games in constitutional form
This paper is concerned with the core of a ``game in constitutional form'' (henceforth, simply a ``game''). The notion of game generalizes both the notion of ``simple game in characteristic function from'' and that of ``effectivity function''. The fundamental construct defining a game is its constitution, which determines for each coalition and subset of alternatives whether the coalition can veto a component of the subset in favor of its complement. Three notions of domination of subsets of alternatives, determining whether a coalition will exercise its veto, are defined with respect to a given profile of linear orders on the alternatives. In decreasing order of strictness they are o-domination, i-domination, and j-domination. Let \(\ell\) be either o, i, or j. Then the \(\ell\)-core is the set of \(\ell\)- undominated alternatives. A game is said to be \(\ell\)-stable if the \(\ell\)-core is non-empty for all profiles of preferences. Three notions of acyclicity are defined, denoted \(\ell\)-acyclicity for \(\ell =o,i\), or j. The basic content of the existence of an \(\ell\)-cycle is that there will then be some profile of preferences generating a cycle of \(\ell\)-dominations covering all alternatives. The main result of the paper is: Let G be a game in constitutional form, and \(\ell\) be o, i, or j. If G is \(\ell\)-acyclic, then G is \(\ell\)-stable. Two final sections of the paper use games to generalize Keiding's theorems on the stability of effectivity functions [see \textit{H. Keiding}, Methods Oper. Res. 50, 519-530 (1985; Zbl 0649.90104); and Int. J. Game Theory 14, 93-101 (1985; Zbl 0567.90101)] and Nakamura's theorems on the stability of simple games [see \textit{K. Nakamura}, Int. J. Game Theory 8, 55-61 (1979; Zbl 0415.90087)].
scientific article; zbMATH DE number 4127043