The maximum Theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities

On the existence of equilibria in generalized games

“…Thus, our results generalize many of the existence theorems, on equilibria in generalized games by relaxing the compactness of strategy space and the openness of graphs or lower sections of preference correspondences. We also give a result on the existence of Nash equilibria in the conventional games, which extends the results in Nash (1950Nash ( , 1951, Nikaido and Isoda (1955), Dasgupta and Maskin (1986), and Tian and Zhou (1992) to allow nontotal-nontransitive preferences. It may be remarked that, since the competitive mechanism can be regarded as a generalized game, the results given in this paper can be used to prove the existence of competitive equilibria in economies with a non-compact and infinite dimensional feasible set, an uncountably infinite number of agents, and with inter-dependent, price-dependent, and nontotal-nontransitive preference correspondences which may not have an open graph or open lower sections.…”

Existence of Generalised Pareto Equilibria for Constrained Multiobjective Games

“…Aubin (1993), Debreu (1952), Ding and Tan (1993), Harker (1991), Harker and Pang (1990), Ichiishi (1983), Tan and Yuan (1993), Tan and Yuan (1994), and Tian and Zhou (1992)], and both the notions of GPE and weak GPE reduce to the notion of generalised Nash equilibrium. That is, a feasible multistrategy x where each player i, by choosing his strategy x i , minimises his (scalar) payoff over the set Γ i (x −i ).…”

“…With respect to scalar criterion games [see e.g. Aubin (1993), Debreu (1952), Ding and Tan (1993), Harker (1991), Harker and Pang (1990), Ichiishi (1983), Tan and Yuan (1993), Tan and Yuan (1994), and Tian and Zhou (1992)], multicriteria games can better describe reallife situations. We refer to Bergstresser and Yu (1977), Szidarovszky et al (1986), and Zeleny (1976) for motivations for studying multicriteria games.…”

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.…”

“…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.…”