Abstract
A balanced transferable utility game (N, v) has a stable core if its core is externally stable, that is, if each imputation that is not in the core is dominated by some core element. Given two payoff allocations x and y, we say that x outvotes y via some coalition S of a feasible set if x dominates y via S and x allocates at least v(T) to any feasible T that is not contained in S. It turns out that outvoting is transitive and the set M of maximal elements with respect to outvoting coincides with the core if and only if the game has a stable core. By applying the duality theorem of linear programming twice, it is shown that M coincides with the core if and only if a certain nested balancedness condition holds. Thus, it can be checked in finitely many steps whether a balanced game has a stable core. We say that the game has a super-stable core if each payoff vector that allocates less than v(S) to some coalition S is dominated by some core element and prove that core super-stability is equivalent to vital extendability, requiring that each vital coalition is extendable.
Highlights
In their seminal work on game theory, von Neumann and Morgenstern (1953) introduced the notion of stable sets as the main and most natural solution concept for cooperative games
Stable sets are based on the notion of dominance: A payoff vector x dominates another one y if there is a coalition S for which x is strictly better than y for every member of S, but still remains affordable for them in the game under consideration
It should be noted that the condition “for all ∅ = S ⊆ F” in Corollary 7.2 can be replaced by “for all feasible collections S for v”, which would typically reduce the number of required tests for core stability as often not all nonempty subsets of coalitions in F are feasible for v
Summary
In their seminal work on game theory, von Neumann and Morgenstern (1953) introduced the notion of stable sets as the main and most natural solution concept for cooperative games. Documents de travail du Centre d'Economie de la Sorbonne - 2020.09 unique, there may even be uncountably many of them, and as shown by Lucas (1968), there exist games with no stable sets They are very difficult to find and there is no known algorithm to find them. Other than a small number of elementary truisms (e.g., that the core is contained in every stable set) there is no theory, no tools, certainly no algorithms.” This is why the main solution concept of cooperative game theory turned to be the core. It turns out that vital-exact extendability is a sufficient and necessary condition for core stability for some classes of games, like matching and assignment games, simple flow games, and minimum coloring games, but fails to be necessary in general.
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