Abstract
In cooperative games, the coalition structure core is, despite its potential emptiness, one of the most popular solutions. While it is a fundamentally static concept, the consideration of a sequential extension of the underlying dominance correspondence gave rise to a selection of non-empty generalizations. Among these, the payoff-equivalence minimal dominant set and the myopic stable set are defined by a similar set of conditions. We identify some problems with the payoff-equivalence minimal dominant set and propose an appropriate reformulation called the minimal dominant set. We show that replacing asymptotic external stability by sequential weak dominance leaves the myopic stable set unaffected. The myopic stable set is therefore equivalent to the minimal dominant set.
Highlights
The core is one of the most popular solution concepts in cooperative game theory
We argue first that any element of the coalition structure core belongs to a payoff-equivalence minimal dominant set (PEMDS)
An attractive feature of the myopic stable set is that it coincides with the coalition structure core whenever the coalition structure core is non-empty and provides a unique non-empty prediction otherwise
Summary
The core is one of the most popular solution concepts in cooperative game theory. It introduces a coalitional notion of stability not unlike the Nash equilibrium: once such a proposal is made, it is not abandoned. Roth and Vande Vate (1990), Diamantoudi et al (2004), Klaus and Klijn (2007) and Chen et al (2016) show for matching markets and Sengupta and Sengupta (1994, 1996), Kóczy and Lauwers (2004), and Kóczy (2006) show for coalitional games that the core can be reached The latter results prove the existence of an upper bound on the number of steps needed. Sengupta and Sengupta (1994) define the set of viable proposals, Kóczy and Lauwers (2007) the payoff-equivalence minimal dominant set and most recently Demuynck et al (2019) the myopic stable set All these concepts rely on notions of sequential dominance and belong to the family of stochastic solutions (Packel, 1981) with appropriate restrictions on the permitted transitions. For the class of proper simple games, we present a general result of payoff equivalence of the payoff-equivalent minimal dominant set, the minimal dominant set, and the myopic stable set
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