Pseudocontinuous functions and existence of Nash equilibria

Location of Nash equilibria: A Riemannian geometrical approach

“…Although many extensions and applications of Nash's result can be found (see, for instance, Chang [5], Georgiev [6], Kulpa and Szymanski [9], Morgan and Scalzo [11], Yu and Zhang [18], and the references therein), to the best of our knowledge, only a few works can be found in the literature dealing with the above question. Nessah and Kerstens [14] characterize the existence of Nash equilibrium points on non-convex strategy sets via a modified diagonal transfer convexity notion, Tala and Marchi [16] also treat games with non-convex strategies reducing the problem to the convex case via a certain homeomorphism, and Kassay, Kolumbán and Páles [7] and Ziad [19] considered Nash equilibrium points on convex domains for non-convex payoff functions having suitable regularity instead of their convexity.…”

Berge's maximum theorem to vector-valued functions with some applications

“…The well-known Berge's maximum theorem is as follows: Many results about this theorem have been achieved in the literatures and the literatures therein (see [5,7,8,18,23,24,28]). In the early years, Dutta and Mitra [5] presented a maximum theorem for convex structures with weaker continuity requirements and applied to the problem of optimal intertemporal allocation.…”

“…Tian and Zhou [24] generalized Berge's maximum theorem by introducing the feasible path transfer lower semicontinuity and prove the existence of equilibrium for the abstract economy. Morgan and Scalzo [18] introduced the pseudocontinuity and studied the maximum theorem for pseudocontinuous functions and obtained the existence of Nash equilibria for n persons noncooperative games with pseudocontinuous payoffs. Recently, Yu [28] generalized the maximum theorem to the continuous vectorvalued functions and proved the existence of weakly Pareto-Nash equilibria for multiobjective games and generalized multiobjective games both with continuous vector-valued payoffs.…”

“…For this we propose this paper. In this paper, based on the works of Morgan and Scalzo [18] and Yu [28], we prove the Berge's maximum theorem for vector-valued functions with pseudocontinuity and obtain the set-valued mapping of the solutions is upper semicontinuous with nonempty and compact values. Our result is different from the present references against the pseudocontinuous vector-valued functions.…”

On the existence of Nash equilibrium in discontinuous games

“…McManus (1964), Roberts and Sonnenschein (1977), Nishimura and Friedman (1981), Topkis (1979), Vives (1990), and Milgrom and Roberts (1990)), some seek to weaken the continuity of payoff functions (cf. Dasgupta and Maskin (1986), Simon (1987), Simon and Zame (1990), Tian (1992aTian ( , 1992bTian ( , 1992cTian ( , 2009), Zhou (1992, 1995), Reny (1999Reny ( , 2009), Bagh and Jofre (2006), Morgan and Scalzo (2007), Carmona (2009Carmona ( , 2011, and…”