The Myopic Stable Set for Social Environments

Abstract

We introduce a new solution concept for models of coalition formation, called the myopic stable set. The myopic stable set is defined for a very general class of social environments and allows for an infinite state space. We show that the myopic stable set exists and is non-empty. Under minor continuity conditions, we also demonstrate uniqueness. Furthermore, the myopic stable set is a superset of the core and of the set of pure strategy Nash equilibria in noncooperative games.Additionally, the myopic stable set generalizes and unifies various results from more specific environments. In particular, the myopic stable set coincides with the coalition structure core in coalition function form games if the coalition structure core is nonempty; with the set of stable matchings in the standard one-to-one matching model; with the set of pairwise stable networks and closed cycles in models of network formation; and with the set of pure strategy Nash equilibria in finite supermodular games, finite potential games, and aggregative games. We illustrate the versatility of our concept by characterizing the myopic stable set in a model of Bertrand competition with asymmetric costs, for which the literature so far has not been able to fully characterize the set of all (mixed) Nash equilibria.

Bertrand competition with asymmetric costs: a solution in pure strategies

“…Observe that an empty set would necessarily violate asymptotic external stability, so it follows that M is non-empty. While we focus on pure strategies, the MSS can be defined analogously on the set of mixed strategies; for details, see Online Supplement A.6 of Demuynck et al (2019). In any (mixed) Nash equilibrium, we have f (s) = f ∞ (s) = s. Thus, by asymptotic external stability, all mixed-strategy Nash equilibria are part of the MSS in mixed strategies.…”

“…We follow Blume (2003) and Kartik (2011) in that we focus on strategy profiles in which no firm chooses a predatory price, i.e., a price below marginal cost. Instead of Nash equilibrium, however, we employ a solution concept recently introduced by Demuynck et al (2019), the Myopic Stable Set. A set of strategy profiles is myopically stable if it satisfies three conditions, deterrence of external deviations, asymptotic external stability, and minimality.…”

“…In Demuynck et al (2019), we defined the Myopic Stable Set for a very general class of social environments (Chwe 1994) that allows for infinite state spaces and includes normal-form games as a special case. We proved that if the state space is compact, then the Myopic Stable Set exists, and under some mild continuity assumptions, it is also unique.…”

“…In light of these results, the Bertrand model with asymmetric costs is interesting for several reasons: it does not satisfy the compactness and continuity assumptions of Demuynck et al (2019), it does not belong to any of the aforementioned classes of games, and the set of pure strategy Nash equilibria of this game is empty. Moreover, the set of mixed-strategy Nash equilibria is large.…”

The Equivalence of the Minimal Dominant Set and the Myopic Stable Set for Coalition Function Form Games

“…It is natural to ask where do these coalitional improvements lead in case the core is (possibly) empty. Sengupta and Sengupta (1994) define the set of viable proposals, Kóczy and Lauwers (2007) the payoff-equivalence minimal dominant set 1 and most recently Demuynck, Herings, Saulle, and Seel (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.…”