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Nessah, Rabia

Weakly continuous security and Nash equilibrium. (English) Zbl 1497.91013

Theory Decis. 93, No. 4, 725-745 (2022).
Summary: This paper investigates the existence of pure strategy Nash equilibria in discontinuous and nonquasiconcave games. We introduce a new notion of continuity, called weakly continuous security, which is weaker than the most known weak notions of continuity, including the surrogate point secure of SSYM game of O. Carbonell-Nicolau and R. P. McLean [Econ. Theory 68, No. 4, 935–965 (2019; Zbl 1443.91065)], the continuous security of P. Barelli and I. Meneghel [Econometrica 81, No. 2, 813–824 (2013; Zbl 1274.91106)], \(C\)-security of A. McLennan et al. [Econometrica 79, No. 5, 1643–1664 (2011; Zbl 1272.91018)], generalized weakly transfer continuity of R. Nessah [J. Math. Econ. 47, No. 4–5, 659–662 (2011; Zbl 1236.91017)], generalized better-reply security of G. Carmona [Econ. Theory 48, No. 1, 31–45 (2011; Zbl 1238.91040)], P. Barelli and I. Soza [“On the existence of Nash equilibria in discontinuous and qualitative games”, Preprint, Univ. of Rochester (2009)], Barelli and Meneghel [loc. cit.], lower single deviation property of P. J. Reny [“Further results on the existence of Nash equilibria in discontinuous games”, Preprint, University of Chicago (2009)], better-reply security of P. J. Reny [Econometrica 67, No. 5, 1029–1056 (1999; Zbl 1023.91501)] and the results of P. Prokopovych [Econ. Theory 48, No. 1, 5–16 (2011; Zbl 1232.91067); Econ. Theory 53, No. 2, 383–402 (2013; Zbl 1268.91013)] and G. Carmona [J. Econ. Theory 144, No. 3, 1333–1340 (2009; Zbl 1159.91305)]. We show that a compact, convex and weakly continuous secure Hausdorff locally convex topological vector space game has a pure strategy Nash equilibrium. Moreover, it holds in a large class of discontinuous games.

MSC:

91A10 Noncooperative games

Keywords:

discontinuous games; nonquasiconcave games; pure strategy Nash equilibrium; weakly continuous security; pseudo upper semicontinuity; generalized payoff security; weakly reciprocal upper semicontinuity

Citations:

Zbl 1443.91065; Zbl 1274.91106; Zbl 1272.91018; Zbl 1236.91017; Zbl 1254.91004; Zbl 1023.91501; Zbl 1232.91067; Zbl 1268.91013; Zbl 1159.91305; Zbl 1238.91040
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References:

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