Essential equilibria of discontinuous games

Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria

“…Yu and Xiang [26] proposed an essential set for Nash equilibria and proved the existence of essential component in the second direction by considering perturbations of payoff functions of players. There is a lot of researches about using the essential component to discuss the stability of Nash equilibria (see, [1,2,8,17,18,24,25,27]). However, there are two drawbacks in the second way.…”

Further results on essential Nash equilibria in normal-form games

“…Carmona (2009, Corollary 2 and Proposition 3)], G q X ⊆ G X [Scalzo (2013, Proposition 2)], and G q X ⊆ G g X ⊇ G p X (Scalzo 2013). …”

“…While we cannot guarantee that the topological closure of G contains the collections of games considered in Carbonell-Nicolau (2010) and Scalzo (2013), we do prove the existence of generic essential games within a strict superset of the sets of games considered in the literature, namely the union of (the closure of) G and the collections of games considered in Carbonell-Nicolau (2010) and Scalzo (2013).…”

On the existence of maximal elements, fixed points and equilibria of generalized games in a fuzzy environment

“…Moreover, we give new sufficient conditions for the existence of fuzzy maximal elements and fuzzy fixed points of fuzzy mappings. The main condition considered in our results is the generalized positive quasi-transfer continuity, introduced in [16] in a non-fuzzy setting (later said in a crisp setting); indeed, any fuzzy mapping can be identified with a realvalued function. The condition implies that the crisp map of the strong α-cuts of a fuzzy mapping has a property that we call generalized transfer open lower sections.…”